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Automated Design of Steel Open Web Joist Floor
Framing Systems Using a Genetic Algorithm
by
Christopher Erwin, B.S.C.E.
A Thesis submitted to the Faculty of the Graduate School, Marquette University, in Partial Fufullment of
the Requirements for the Degree of Masters of Science
Milwaukee, Wisconsin August 2004
i
Preface
Engineers consider straightforward calculations of capacity during the design process,
and then subject the design to serviceability checks. For a steel framed floor system,
these checks often include deflection and vibration. Floor systems have become lighter
due to advances in material science and construction methods. While these advances
help to reduce cost, they can result in structures that are susceptible to vibraiton problems
that may cause discomfort for the building’s occupants. The vibration response can be
determined through numerous calculations that are heavily dependent on the entire design
making design for vibration difficult.
This thesis proposes a methodology for designing optimized systems utilizing a
genetic algorithm (GA), which utilizes a search strategy that is modeled on the same
mechanisms found in genetic evolution. The GA operates with a selected population of
solutions. For each of these solutions, the capacity, cost and vibration response can be
determined and used to evaluate each system’s performance. Using a “survival of the
fittest” methodology, the GA combines components of the individual systems to create a
new generation (new population). This new population is evaluated, and the process
cycles again. Through repetition, the optimal solution will present itself.
Steel systems utilize a finite number of steel wide-flange sections, joists, decks
and concrete thicknesses. These discrete design options are perfectly suited to the genetic
algorithm. The GA has been used successfully in the past to obtain optimum or near
optimum solutions to many design problems. The successful record of the GA with steel
framed structures shows that the GA will be well suited to the problem of designing steel
floor framing systems.
ii
Acknowledgements
I would like to express my earnest appreciation to my thesis director, Dr. Christopher
Foley for his guidance and infinite patience. I count myself lucky to have a mentor who
never stops working, is always available, and is able to deal with my idiosyncrasies.
I would like to thank Professor Sriramulu Vinnakota for being on my thesis
committee, and teaching me the steel design procedures. The time spent in his steel
design classes have been put to good use.
I would also like to thank Professor Stephen M. Heinrich, as a member of my
thesis committee, as well as for making my final year at Marquette possible.
I would also like to thank Benjamin T. Shock, who through his work and
discussions presented new ways of looking at the algorithm and the processes, which
make the program work.
I would also like to thank the rest of the Department of Civil Engineering at
Marquette University. They are the reason I came to Marquette, and why I continued at
Marquette to complete my M.S. Degree.
Lastly, I’d like to thank my parents, who provided financial assistance and
unwavering support in my academic endeavors.
iii
Table of Contents
Chapter 1 – Introduction Section 1.1 – Design Problem Statement 1 Section 1.2 – Introduction to the Genetic Algorithm 2 Section 1.3 – Components of the Genetic Algorithm 3 Section 1.4 – Genetic Algorithm vs. Other Optimization Procedures 5 Section 1.5 – Prior Uses of the Genetic Algorithm 6
Chapter 2 – Design Considerations for a Steel Floor System Section 2.1 – Basic Components of Floor System 8 Section 2.2 – General Design Procedure 9 Section 2.3 – Evaluation of Serviceability Criteria 13
Chapter 3 – Formulation of the Optimization Problem Section 3.1 – Development of Objective(s) and Constraints 25 Section 3.2 – Definition of Fitness for the Evolutionary Algorithm 29
Chapter 4 – Programming the Genetic Algorithm in MATLAB Section 4.1 – Introduction 45 Section 4.2 – Essential Components o the Genetic Algorithm 45 Section 4.3 – Additional Components in the Genetic Algorithm 56
Chapter 5 – Design Examples & Analysis Section 5.1 – Introduction 64 Section 5.2 – Problem Statement 64 Section 5.3 – Running the Evolutionary Algorithm 65 Section 5.4 – Joist Span Variation 69 Section 5.5 – Superimposed Loading Variation 75 Section 5.6 – Analysis of Results 78
Chapter 6 – Conclusion and Recommendation Section 6.1 – Introduction 80 Section 6.2 – Improvements to the Algorithm 80 Section 6.3 – Future Research 81
Appendix A – Genetic Algorithm Code 84
Appendix B – Algorithm Design Results 122
Appendix C – Design Solution Checks 183
Reference 190
iv
List of Figures
Figure 1.1 – Design Problem 1 Figure 2.1 – Typical Floor System 8 Figure 2.2 – Typical Beam Model 12 Figure 2.3 – Shear/Moment Diagrams due to Uniform Load 12 Figure 2.4 – Deflection Cases 14 Figure 2.5 – Composite Joist Cross-Section 18 Figure 2.6 – Girder and Deck Cross-Section 21 Figure 2.7 – Recommended acceleration for human comfort for vibrations due to human activity. (Allen 1997)
24
Figure 3.1 – Girder Model 33 Figure 3.2 – Composite Joist Cross-Section 38 Figure 3.3 – Girder Composite Cross-Section 41 Figure 4.1 – MATLAB Code for Chromosome Generation 48 Figure 4.2 – Chromosomes for Population of Ten Individuals Systems 48 Figure 4.3 – MATLAB Code for Chromosome Decoding 49 Figure 4.4 – MATLAB Code for Roulette Wheel Selection 52 Figure 4.5 – MATLAB Code for Mutation Implementation 58 Figure 4.6 – Population Plotted in Objective Space (Pre-Domination) 61 Figure 4.7 – Domination Code 62 Figure 4.8 – Population Graphed in Objective Space (Post-Domination). 63 Figure 5.1 – Typical Floor System 64 Figure 5.2 – User Input Section of Master.m File 65 Figure 5.3 – Algorithm Output to Screen 66 Figure 5.4 – Graph Illustrating Generational Variation in Fitness 67 Figure 5.5 – Fitness-Acceleration Plot at End of Evolutionary Algorithm 68 Figure 5.6 – Objective Space Plot at End of Evolutionary Algorithm 69 Figure 5.7 – Input for Algorithm Run 70
v
List of Tables
Table 2.1 – K-Series Joist Selection Table 10 Table 2.2 – Girder Loads to Joists 12 Table 3.1 – Beta and Threshold Values (Allen 1997) 44 Table 5.1 – Configuration 1 (L = 40 ft.) Least Expensive Systems 71 Table 5.2 – System Comparison 71 Table 5.3 – Configuration 2 (L = 30 ft.) Lease Expensive Systems 72 Table 5.4 – Pareto Set of Run 7 for Configuration 3 73 Table 5.5 – Configuration 3 (L = 50 ft.) Least Expensive Systems 74 Table 5.6 – Part of Pareto Set from Run 1 of Configuration 3 75 Table 5.7 – Configuration 4 (L = 30 ft.) Output of Pareto Set 76 Table 5.8 – Configuration 5 (L = 40 ft.) Output of Pareto Set 77 Table 5.9 – Configuration 6 (L = 50 ft.) Output of Pareto Set 78 Table 5.1 – Configuration 2 (L = 40 ft.) Least Expensive Systems 184 Table C.2.1 – Configuration 2 (L = 40 ft.) Least Expensive Systems 185 Table C.2.2 – Joist Checks 185 Table C.2.3 – Girder Information 185 Table C.2.4 – Applied Loads and Deflection 186 Table 5.7 – Configuration 3 (L = 50 ft.) Least Expensive Systems 186 Table C.3.1 – Configuration 3 (L=50) Concrete Decks 187 Table C.3.2 – Joist Checks 187 Table C.3.3 – Girder Information 187 Table C.3.4 – Applied Loads and Deflection 187 Table C.4.1 – Least Expensive Systems 188 Table C.4.2 – Concrete Deck Specifications 188 Table C.4.3 – Joist Checks 188 Table C.4.4 – Girder Information 189 Table C.4.5 – Applied Loads and Deflection 189
1
Chapter 1 – Introduction
1.1 Design Problem Statement
The setup for the design problem considered in this thesis is a standard steel joist floor
system shown in Figure 1.1. The bay is assumed to be an interior part of a larger
continuous system. This simplifies the design of the beams and girders, because both
will be assumed to be loaded symmetrically.
Inputs:
W = Bay Width L = Bay Length SDL = Superimposed Dead Load LL = Live Load
vLL = Vibration Live Load
Outputs: Girder Specification Joist Specification s = Joist Spacing ct = Concrete Topping
rt = Steel Deck Rib Height g = Steel Deck Gauge
s
W
L
Girder
K-Series Joist
Figure 1.1: Design Problem
tc
tr g
Concrete Deck
2
The design consists of four components: the girder, the joist, and the composite
steel deck specifications. The engineering design problem consists of objectives and
constraints. The objective is cost, which the designer is trying to minimize. The
constraints apply to each of the components, and provide the limits for the objectives.
The design problem can be stated as
Minimize: Floor Panel Cost
Subject To: Component Flexural Strength
Component Deflection Limits
Four independent components, which are evaluated as a whole, can serve to be an
imposing problem for any solution technique. However, this is definitely not impossible.
At Stanford, a program was created to design entire floor systems for a building. The
floor system for this building was a beam and girder system using wide flange sections
and composite concrete deck (Jain 1991). More recently, optimization algorithms have
provided a means to design an efficient concrete floor system (Lucas 2001). These
problems did not utilize joists in their designs, nor did they consider vibration
performance. This is the first known attempt to develop an algorithm to automatically
design an open web joist steel floor framing system.
1.2 Introduction to the Genetic Algorithm
A Genetic Algorithm (GA) is a general search and optimization methodology inspired by
the process of natural selection (Holland 1975). The algorithm is based on Darwin’s
theory of evolution, with the central concept being that one could start with a primordial
mess and end up with the incredibly diverse set of biological solutions seen today.
Generation by generation the more fit life forms survived while others perished and
3
became extinct. This idea is applicable to any system with a defined fitness. Therefore,
the GA is a general method applicable to an extremely wide range of problems.
Continuing with the parallels to the natural world, each system is defined by a
genome. For the ease in implementation on the computer, the genome of the GA is a
series of 1’s and 0’s (Goldberg 1989; Haupt 1998). The assembly of these binary digits
(i.e. a chromosome) represents a solution to the design problem that the GA is attempting
to optimize. The basic operating components of the GA are an initial population
generator, a fitness calculator, selection/crossover methods, and a mutation operator.
These components, working on the genome, create a Darwinian system of the greatest
simplicity. Additional components may be added to assist the algorithm in efficiently
finding a solution (e.g. elitism). To use a genetic algorithm, the design problem must be
established as an unconstrained optimization problem. This is discussed in Chapter 3.
1.3 Components of the Genetic Algorithm
The population generator creates an initial population for evaluation. This process at its
simplest can be as basic as a random number generator creating 1’s and 0’s (Coley 1999).
This would most accurately use the idea of the primordial mess. However, more complex
methods of population creation can be utilized. Some complex design problems require
populations, which are suitable for satisfying fitness. Therefore, when they generate
their initial populations, the process is not random, but calculated.
Fitness can be any method of system evaluation. In other words, it is a
mechanism by which the “quality” of one solution can be weighed against another
solution in satisfying the design objective(s). Suppose a GA was set up to automatically
design a simply supported beam. Fitness could be assigned using deflection, weight, or
4
cost. The fitness could also be defined as a weighted function of some or all of these
former items. The method used to carry out fitness calculation is flexible. However,
these aspects that are used in defining fitness when using a GA need to be considered
carefully.
Once fitness is defined and evaluated, the next items to consider become selection
and crossover. This is the equivalent of mating and reproduction. There are many
methods of selection, all based on fitness, ranging from roulette wheel selection to
tournament selection to rank-based selection. These methods provide various ways for
individuals to be selected with the preference for selection going to the fitter individuals
with or without excluding the un-fit. Once two parents are chosen, an offspring will be
created using the genetic data from the parents. This can be accomplished through single
point, multiple point (Coley 1999), or component crossover. Each takes a portion of each
parent’s genome and combines it to create an offspring that is unique from the parents.
This process of selection/mating is repeated until a new population is created for fitness
evaluation.
After an initial population is generated, the cycle of fitness evaluation, selection,
and mating continues for many generations until only the fittest individuals inhabit the
population. Mutation components simulate the random changes found in nature. Others,
like elitism, ensure that each generation is no worse than the previous. Fitness scaling
provides a method of preventing a single individual from dominating a population. With
these additions, the GA can be adjusted and refined to create a set of optimized solutions.
5
1.4 Genetic Algorithm vs. Other Optimization Procedures
Now that the fundamentals of a GA have been briefly discussed, the question arises as to
why a GA should be used over another method of automating or optimizing the design
process. This section will consider a few alternate methods for automated and optimal
design to determine the advantages of using a GA for the design problem tackled in this
thesis.
One alternative method would be to evaluate each alternate solution of the
problem (i.e. conduct an exhaustive search of all solutions in the design space). For small
search spaces this might be feasible and efficient. Unfortunately, the search space
associated with the steel floor systems considered in this thesis is approximately
810 excluding all the spacing options. Evaluating 10 systems per second would take over
3 years. Knowing this demands another technique to solve the problem.
Mathematical optimization procedures rely on gradient-based techniques to guide
the algorithm through the solution space. Such a process requires continuous functions to
represent the design variables. This is not well suited to the discrete design problem of a
steel framed floor design. This is not to say that it is not possible to obtain a design, but it
would be a difficult process to define the discrete set of shapes into a continuous function
(Haupt 1998). Another problem with such algorithms is when they encounter local
minima. Because the search process is based upon an “initial guess” at the solution, the
solution can become very dependent on that guess (Burns 2001). If a system was to
make ten initial guesses, it is possible to obtain ten different answers. This can create the
illusion of the optimal solution, but the if the search space is full of local minima, then
solution could be described as falling in a “fox-hole” that is only a local minimum.
6
Another alternative is simulated annealing, which has been applied to
optimization of reinforced concrete floor design (Lucas 2001). This algorithmic
procedure is based on a model of crystallization of metals during cooling from the liquid
state. The equivalent representation of temperature would be a variable such as cost.
The process heats up the system until the pseudo temperature is raised. As the system
cools, components change to match the reducing cost. If the system is cooled too
quickly, it will not crystallize and result in an amorphous mass. Therefore, protective
limits are applied to ensure that the system is slowly cooled (Haupt 1998). This is a very
efficient way of finding an optimized solution. However it does have the shortcoming of
being limited by how the temperature decreases are interpreted in a discrete and
complicated problem.
The GA has several distinct advantages, which make it very suitable for
automating this type of design problem. The GA can analyze problems with discrete
design variables. It simultaneously searches from a wide sampling of the cost surface,
and deals with a large number of parameters (Haupt 1998). These abilities make the GA
a very robust solution technique suitable for steel floor design.
1.5 Prior Uses of Genetic Algorithms
Genetic algorithms have been used to solve a wide range of problems. GAs have
successfully been used to design everything from communication systems to structural
frames. Solutions can even be found for systems with dynamic search environments.
The applicability of this process is closer to the frame design problems. Examples of
these applications can be found in Schinler (2001) and Pezeshk and Camp (2000).
7
Design of structural frames has been performed using Genetic Algorithms or
Evolutionary Algorithms. Detailed procedures are provided in numerous design
examples. One such example is an object oriented evolutionary algorithm created by
Schinler (2001). This evolutionary algorithm, which is similar to a genetic algorithm,
designed systems by minimizing the weight subject to several constraints. In general,
these examples look at minimizing cost while trying to satisfy capacity and deflection
requirements under different loading conditions.
To the author’s knowledge, the Genetic Algorithm has never before been used to
design steel floor systems composed of open web joist and wide flange members with
consideration of vibration. Without any prior works to compare, the algorithm
developed using first principles of steel floor system design as well as the general rules
and theory governing the Genetic Algorithm.
8
Chapter 2 – Traditional Design Considerations for Steel Floor Systems
2.1 Basic Components of Floor System
The basic steel joist floor system, which will be utilized for this problem, is shown in
Figure 2.1. The bay is assumed to be part of a larger continuous system, and is assumed
to be an interior bay in a building. This simplifies the design of the beams and girders.
Both will be assumed to be loaded symmetrically.
A VLI composite concrete deck is set atop steel K-series joists, which are seated
on a girder. The girders are attached to the columns via a simple shear connection, and
therefore they can be designed as if they were simply supported.
30’
40’
s
Girder K-Series Joist
Figure 2.1: Typical Floor System
9
The AISC shape table from the LRFD Manual (AISC 2001) provides over 260
sections for the beams and girders. The SJI Joist Catalogue provides 64 joists, and the
Vulcraft Steel deck catalogue provides about 250 concrete deck options. Given all the
options, it is understandable why the optimum design might not be reached using hand
design procedures.
2.2 General Design Procedure
The design example in this chapter will utilize the following input, which will be used as
input for the computer program developed. These consist of bay dimensions,
superimposed dead load, live load, and vibration loading values.
Dimensions: L = 40 ft W = 30 ft
Loads: SDL = 15 psf LL = 50 psf LLv = 11 psf
All other required values will be determined as the design is completed.
2.2.1 Selection of Joist
The Steel Joist Institute Joist Catalogue provides the Standard Load Table (SJI 2002),
which is used for selection of the joists. The possible spacing values are determined
based on the bay width, W. Because the maximum loading on any joist is 550 plf, no
spacing values greater than 6-ft will work for a superimposed load greater than 90 psf.
Therefore the spacing will begin at the first number below six feet, which yields a round
number of joists. The spacing values will stop at two feet, which serves a lower
economic limit.
The loading is determined by adding the live load (LL), superimposed dead load
(SDL), and the self-weight of the composite deck to be placed on the joists. The value
for the concrete/steel composite deck system will be assumed (for the present design
10
illustration only) to be 26 psf, which is the lightest available. The total load is then
multiplied by the spacing to determine the required joist capacity in pounds per linear
foot.
The Joist is selected from the standard load table using the span of 40 ft and the
loading corresponding to the spacing. It should be noted that the self-weight of the joist
is accounted for in the joist table, so it can be neglected at this point. The weight of the
joist will be recorded for an efficiency calculation.
The equivalent joist load is calculated and the lightest joist specification will be
checked for adequacy. Table 2.1 illustrates that 28 K 10 spaced at 4.29ft are the lightest
joist system to support the superimposed loading.
Spacing, s (ft)
Loading (plf)
Number of Joists
Joist Selection
Weight of Joist (plf)
Joist Weight (psf)
5.00 455.00 6 N/A N/A N/A
4.29 390.39 7 30 K 10 28 K 10 24 K 12
15.0 14.3 16.0
3.49 3.33 3.73
3.75 341.25 8 30 K 8 28 K 9 24 K 10
13.2 13.0 13.1
3.52 3.47 3.49
3.33 303.03 9
30 K 7 28 K 8 26 K 10 26 K 9 24 K 9
12.3 12.7 12.1 12.2 12.0
3.69 3.81 3.64 3.66 3.60
3.00 273.00 10 28 K 7 26 K 7 24 K 8
11.8 10.9 11.5
3.93 3.96 3.83
2.70 245.70 11 28 K 6 26 K 6
11.4 10.6
4.22 3.93
Table 2.1: K-Series Joist Selection Table
11
2.2.2 Concrete Deck Selection
The concrete deck selection is a check of the assumption used in the joist selection
procedure. The Vulcraft Steel Deck Catalog (Vulcraft 2002) will be used as a source of
all the deck capacities. The smallest span in the manual is 5 ft (Vulcraft 2002). For all
spans smaller than 5 ft, the 5 ft span data will be used to evaluate the capacity. From the
previous procedure, a lightweight concrete deck weighing only 26 psf was assumed. In
the Vulcraft Catalog, six sections exist with the weight of 26 psf. The required capacity
is 91 psf. A 2 inch lightweight concrete cover atop a 1.5VL22 deck provides the required
capacity for a continuous three span condition without shoring.
2.2.3 Girder Design Procedure
The girder will be designed according to the LRFD Manual (AISC 2001) beam design
requirements. The girder is assumed to be simply supported as well as laterally braced
where joists are located. This analytical model is shown in Figure 2.2. The concentrated
loads imposed by the joists will be changed to a simpler problem of uniform load using a
tributary width.
L
weff
Figure 2.2 : Typical Girder Model
12
Superimposed Dead Loads weff
Concrete Deck 26.0 psf * 40 ft 1040.0 plf Joist Self Weight (14.3 plf / 4.29 ft ) * 40 ft 133.3 plf Superimposed Dead Load 15.0 psf * 40 ft 600.0 plf Total Dead Load Reaction 1773.3 plf
Live Loads Live Load 50 psf * 40 ft 2000.0 plf Total Live Load 2000.0 plf Total Factored Load (1.2 * SDL + 1.6 * LL) 5328.0 plf Total Service Load 3773.3 plf
Table 2.2: Girder Loads Due to Joists
The uniform superimposed factored load is 5.328 klf. The uniformly distributed
load is modified to account for girder weight. The factored girder self weight is assumed
to be 120plf. The final uniform load used for the shear and moment diagrams shown in
Figure 2.3 is 5.448 klf.
2
max 8wLM = (2.1)
max 2wL
V = (2.2)
Figure 2.3: Shear/Moment Diagrams due to Uniform Load
V M Mmax = 612.9 k-ft
Vmax = 81.7 k
13
These values will be combined to create the design values. The girder must have
sufficient capacity for φVn > 81.7 k and φMn > 612.9 k-ft (AISC 2001).
Using Table 5-3 of the LRFD Manual (AISC 2001), and value of Lb = 4.29 ft.
Select a W 24 x 68, which provides capacities of φMpx = 664 k-ft and φVn = 266 k.
Because Lb < Lp, the moment capacity of the section is the same as the plastic moment
capacity, or φMn = φMpx. The section is compact, and provides adequate capacity.
Therefore, specify a W24 x 68.
2.3 Evaluation of Serviceability Criteria
With all the components selected, they must now be checked for serviceability criteria.
Each component will be subjected to one or more deflection checks, and then the system
as a whole will be checked for vibration using an AISC procedure (Allen 1997).
2.3.1 Deflection Requirements
Three components of the system will be checked for service load, but in addition, the
construction deflection will be checked. This second check is to ensure that the design
being optimized does not fall prey to ponding associated with beam or joist deflections
during construction.
2.3.1.1 Joist Deflection Considerations
Two situations will be considered for joist deflection. The first is a construction
deflection check. During construction, as the concrete deck is placed, the joist will
deflect. It is important to limit the joist deflection, or the concrete will begin to pond.
The second deflection check is under the total service live load of 50 psf. The SJI
Standard Load Table (SJI 2002) lists loads, which correspond to an L/360 deflection. For
14
a 28 K 10 and a span of 40 ft, this value is 284 plf. This is the value, which will be
checked against both loading cases to verify that the joist is adequate.
Taking the weight of the deck and multiplying by the spacing calculates the
construction loading. Performing this calculation yields a value of 111.5 plf. This is less
than the 284 plf capacity corresponding to the deflection limit, so the joist will perform
adequately during construction.
When the concrete deck cures, it will be a new level surface, so there will be no
noticeable deflection in the joist to the building occupants. Because of this, the service
load deflection check will only include the superimposed dead load and the live load.
The calculation for this check is simply to add the live load and superimposed dead loads,
then take that total and multiply it by the spacing. The resulting load is 279-plf, which is
also less than the limit set forth by the SJI Tables (SJI 2002). Therefore, the joist satisfies
the second deflection criteria.
2.3.1.2 Girder Deflection Considerations
The girder will also be subjected to the same two load conditions as the joist. The
deflection formula is provided in equation (2.3), which was taken from the LRFD Manual
Table 5-17 (AISC 2001) is for the simple span model shown in Figure 2.4. The equation
is used to calculate center of span deflection for the two loading cases.
45
384wL
EIδ = (2.3)
Figure 2.4: Deflection Case
w
L
δ
15
The values for w can be calculated based on the loads to which the system will be
subjected. The first is the construction loading, which would take place after the topping
has been placed on the steel deck. This limit is to control ponding. The second case
corresponds to the live loading once the system is in use.
4
4
5(0.068 1.773)30 1728 0.632384(29000)(1830)
5(2.0)30 1728 0.687384(29000)(1830)
δ
δ
+ ⋅= =
⋅= =
The allowable deflection for this problem is L/360. For a span of 30 ft, the
allowable deflection is 1 in. The current specification of a W24 x 68 is sufficient for both
deflection criteria.
2.3.2 Vibration Requirements
With a preliminary design finished and deemed adequate for all strength and deflection
requirements, a vibration check will now be run. This is very important to the
functionality of the building. However, due to the complex nature of the problem, it is
usually left to simplified procedures, which become checks at the end of a design. This
thesis will not be an exception. The process catalogued in Steel Design Guide Series 11
(Allen 1997) will be used to evaluate the system vibration response.
2.3.2.1 Joist Mode Properties
Joist section properties are not readily available, and must be calculated based on the
values provided by the SJI Standard Load Table (SJI 2002). The moment of inertia can
be calculated from the loading corresponding to allowable deflection of L/360. The
processes for calculating moment of inertia and cross sectional area were taken from the
16
Steel Joist Technical Digest (Galambos 1988). Editing equation (2.3), we can obtain
equation (2.4).
45
384jall
wLIEδ
= (2.4)
Substituting in w = 284plf, L = 40ft, E = 29,000ksi, and δall = L/360 = 1.33in,
424jI = in4
Next, the area of the joist cross section must be calculated, which for this case is the area
of the chords. To obtain this area, we must begin by calculating the ultimate moment
capacity of the joist based on the ultimate loading from the SJI Standard Load Table (SJI
2002). Equation (2.1) can be used to determine the ultimate moment capacity.
Substituting in the values of w = 424 klf and L = 40 ft into Equation (2.1), we can
calculate Msji.
84.8sjiM = k-ft
From this we can calculate the chord area based on the allowable tensile strength of the
chords and the reduction of Msji into a moment couple. Equation (2.5) performs the
operation and is adapted from Steel joist Technical Digest 5 (Galambos 1988).
(0.6 )
sjibot
ej y
MA
D F= (2.5)
Assuming effective joist depth, Dej = 27in and Fy = 50ksi and solving Equation (2.5).
1.26botA = in2
For simplicity, the top chord value is assumed to be the same as the bottom chord value.
Therefore to determine the joist cross sectional area, the bottom chord must be doubled.
2 2.52j botA A= = in2
17
With the joist data calculated, the data required for the vibration analysis can be
assembled. The following data is provided for quick reference with later calculations.
Joist Properties (28 K 10) wsji = 14.3 plf
Aj = 2.52 in2
Ij = 424 in4
Dej = 27 in
ybar = 14 in
Concrete Deck Properties
wc = 110 pcf
f’c = 3000 psi
wcd = 26 psf
tr = 1.5 in
ts = 2.0 in
tcd = 3.5 in
The weight and loading of the system cause friction between the joist and the
concrete deck. This friction causes the system to act compositely when undergoing
vibration response. Because of this assumption, composite cross sectional data must be
calculated. Figure 2.5 illustrates the composite cross-section assumed.
Equation (2.6), which is taken from ACI 318-99 (ACI 1999), calculates the
assumed Modulus of Elasticity for concrete.
1.533( ) 'c c cE w f= (2.6)
1.533 110 3000 2085270cE = × = psi
To simplify future calculations, we can determine a modular ratio, n. The dynamic
nature of the loading requires the cE value to be increased by a factor of 1.35.
29000
10.31.35 1.35(2085.3)
s
c
En
E= = =
18
The centroid location of the composite cross section as measured from the top of
the concrete deck is computed as
51.48 22.52(3.5 14) ( 2)( )10.3 2 4.3251.48
2.52 ( 2)10.3
compy+ +
= =+
i
iin.
With the centroid location known, the composite moment of inertia can now be
calculated using (see Figure 2.7)
3
2 2
51.48( ) 2 51.48 210.3424 2.52(14 3.5 4.32) ( ) (( ) 2)(4.32 )12 10.3 2compI = + + − + + −
ii
975.3compI = in4.
The effective moment of inertia for vibration is therefore (Allen 1997)
( 0.28 / ) 2.80.90(1 ) :6 ( / ) 24jL Dt jC e for L D−= − ≤ ≤ (2.7)
( 0.28(40 12)/28) 2.80.90(1 ) 0.88tC e − ×= − =
1 1
1 1 0.1360.88tC
γ = − = − = (2.8)
Figure 2.5: Composite Joist Cross-Section
ycomp
ybar
Dej
tcd
ts tav
19
1 1742.91 0.136 1
424 975.3
eff
j comp
I
I Iγ= = =
+ + in4. (2.9)
The uniformly distributed load is computed as
4.29(11 26 15) 14.3 237.38jw = + + + = plf.
Going back to Equation (2.3), using the effective moment of inertia and the load,
wj, the corresponding deflection of the joists can be calculated.
45(0.2374)(40) (1728)
0.635384(29000)(742.9)jδ = = in
The joist fundamental frequency can now be calculated (Allen 1997).
386.4
0.18 0.18 4.440.635j
j
gf
δ= = = Hz (2.10)
For future calculations, the effective weight of the beam (joists) in the panel must
be calculated. The following calculations illustrate this:
3 3
( )12( ) 12(2.75)2.02
12 12(10.3)s eff
s
tD
n= = = in4/ft
742.9 173.174.29
effj
ID
s= = = in4/ft
For an effective beam panel width, Cj = 2.0 (Allen 1997)
1 14 42.02( ) 2.0( ) (40) 26.29
173.17s
j j jj
DB C LD
= = = ft.
The effective width is the lesser of Bj and 2/3 of the actual floor width. For a typical
interior bay, the actual floor width is usually more than 3 times the girder span, or 90 ft.
For the present system, 2/3 (900) = 60 ft, which is greater than 26.29 ft, therefore, Bj will
control. The effective weight of the joists (beams) in the panel is therefore
20
237.38 (26.29 40) 581894.29
jj j j
wW B L
s= = × = lbs (2.11)
2.3.2.2 Girder Mode Properties
Girder calculations are slightly simpler because all the girder properties are known.
Girder Properties (W24 x 68) wg = 68 plf
Aj = 20.1 in2 Ig = 1830 in4
Deg = 23.7 in ybar = 11.85 in
Concrete Deck Properties wc = 110 pcf
f’c = 3000 psi wcd = 26 psf
tr = 1.5 in ts = 2.0 in tcd = 3.5 in
The girder is initially assumed to act compositely with the concrete deck. This is not
actually possible due to the flexibility of the joist seats; however, a later calculation will
take this into account. Because of the assumed composite action, the first task is to
determine the effective slab width, followed by the composite section properties. The
effective slab width is the minimum of 0.4Lg or Lj.
min[0.4 , ] min[0.4(30),40] 12eff g jb L L= = = ft
With the effective slab width, we can now compute the composite section properties,
using Figure 2.6.
2.7520.1(11.85 2.0 3.5) (144/10.3)(2.75)( )2 6.86
20.1 (144/10.3)(2.75)gy+ + +
= =+
in
2
32
1830 20.1(11.85 2.0 3.5 6.86)
(144/10.3)(2.75) 2.75(144/10.3)(2.75)(6.86 )12 2
gcompI = + + + −
+ + −
5222.7gcompI = in4
21
Because of flexibility of the joist seats, the composite moment of inertia will be reduced
(Allen 1997).
( )5222.7 18301830 2678.2
4 4gcomp g
eff g g
I II I
− −= + = + = in4
For each girder, an equivalent uniform loading is developed for deflection
calculations.
237.38( ) 40( ) 68 2281.34.29
jeq j g
ww L w
s= + = + = plf
The corresponding deflection can now be calculated for using the new values in equation
(2.3).
4 4
( )
5 5(2.2813)(30) (1728) 0.535384 384(29000)(2678.2)
eq gg
eff g
w LEI
δ = = = in
An estimate for the girder system fundamental frequency can now be calculated (Allen
1997).
Figure 2.6: Girder and Deck Cross Section
yg
3.5
2.0
11.85
22
386.4
0.18 0.18 4.840.535g
g
gf
δ= = = Hz (2.12)
The effective weight of the girder in the panel must now be calculated for use in the
combined mode properties (Allen, 1997),
( ) 2678.2 66.9640
eff gg
j
ID
L= = = in4/ft
Dj is known from prior calculations, and Cg = 1.6 for a girder supporting a joist seat
(Allen 1997).
1 14 4173.17( ) 1.6( ) (30) 60.87
66.96j
g g gg
DB C L
D= = = ft
Once again, Bg will control over 2/3 of the total floor width. Therefore, the effective
weight of girders in the panel can now be calculated for use in the combined mode
properties. (Allen 1997)
2281.3( ) ( )(60.87)(30) 10414740
eqg g g
j
wW B L
L= = = lbs (2.13)
2.3.2.3 Combined Mode Properties When the floor framing system is considered in estimating vibration response, both the
joist and the girder contribute. The following steps constitute the process for combining
the response of the joists and girders (Allen 1997). First, is calculation of the combined
fundamental frequency, (Allen 1997)
386.4
0.18 0.18 3.270.635 0.535n
j g
gf
δ δ= = =
+ +Hz (2.14)
23
Second, the combined panel weight is calculated as (Allen 1997),
j gj g
g j g j
W W Wδ δ
δ δ δ δ= +
+ + (2.15)
0.635 0.535
55189 104147 77575.80.635 0.535 0.635 0.535
= + =+ +
lbs
Finally, the peak non-dimensional acceleration value is computed. For this calculation,
an effective damping ratio of β = 0.05 (building with full height partitions and hanging
ceiling) is used (Allen 1997),
( 0.35 )nf
o oa P eg Wβ
−
= (2.16)
( 0.35 3.27)65
0.00534 0.534%0.05(77575.8)
oa eg
− ×⋅= = =
2.3.2.4 Vibration Results and Comments
For the general rule of 0.5%, this system is borderline adequate for an office application.
However, given the low frequency of 3Hz, it is quite possible that this will be acceptable.
The Design Guide 11 tables show a higher tolerance for structures with frequencies lower
than 4Hz (Allen 1997). Figure 2.7 illustrates frequency thresh holds.
24
Figure 2.7: Recommended acceleration for human comfort for vibrations due to human activity.
(Allen 1997)
25
Chapter 3 – Formulation of the Optimization Problem
3.1 Development of Objective(s) and Constraints
The design example in Chapter 2 was created with a focus of minimal weight. This
served to provide an example of the process required to create a good design, which
satisfies all the capacity and serviceability criteria. The challenge of floor design occurs
when one of the criteria is not satisfied. This situation can also occur when dealing with
special design constraints. Special use buildings such as libraries or hospitals are
subjected to more strenuous vibration criteria. If the system did not satisfy the vibration
criteria, how would the design be improved? What should be stiffened? What is more
economical? These are questions, which can be difficult to answer without years of
experience designing floor systems. This knowledge can also be obtained through use of
a design optimization program, which will search through the many design options for
the most efficient designs.
3.1.1 Objective Functions
At the heart of every optimization problem is the objective function. For most structural
design problems, the objective is to minimize either weight or cost. For steel, cost is
often a function of market price and total weight. This relation closely ties the two
prospective objective functions (Schinler 2001). Because of market fluctuations in steel
prices, many design programs will design for minimum weight. However, if the user can
adjust the cost values, then there is no reason to exclude it. This allows for a more
complex look at the cost of the design, such as the cost of shear stud installation or
26
connection expense. For a design problem posed in this manner, the objective
corresponds to cost minimization:
1 min( )F Cost= (3.1)
where:
1 2 3Cost C C C= + + (3.2)
1 ( )girder girder iC c wt= ⋅ (3.3)
2 / /( )deck sf conc cy conc bayC c c t A= + ⋅ (3.4)
3 ( )( )raw fab frt joist iC c c c wt= + + ⋅ (3.5)
and girderc is the cost per ton of the girder, /deck sfc cost of steel decks per square foot,
/conc cyc is the cost of the concrete per cubic yard, rawc is the cost of the raw joist steel,
fabc is the cost of fabrication, frtc is the cost of shipping the joist to the site, ( )girder iwt is the
weight of the girder in tons, ( )joist iwt is the weight of the joist in tons, and bayA is the area
of the bay.
This thesis will not consider the nuances of cost that can be analyzed using this
objective function. All connection costs are assumed to be the same for every system
analyzed and, therefore, are factored into the problem as a total cost.
A second criterion, or objective could be floor vibration sensitivity to walking.
This is not an objective of the GA in the present formulation. If evaluated for the floor
system, it could be established as a second objective as,
2oa
Fg
= (3.6)
27
The evaluation of equation (3.6) is provided to assist the end user in selection of a
design from the set of possible designs generated by the evolutionary algorithm written.
This is because the most fit or least expensive design is not always the best. Some
excessively lightweight designs can cause extreme vibration sensitivity when subjected to
walking loads. The end result is a system, which can make its occupants uncomfortable.
While methods exist for developing and implementing multiple objective GA’s (Deb
2002), this simpler single objective and secondary evaluation allows the user to see how
subtle changes in a system can greatly affect the cost and vibration response.
3.1.2 Design Problem Constraints
The lightest possible (minimum cost) system may not be able to support the loads applied
to the system. For this reason, constraints are applied in the optimization problem. Each
component and the entire system will be subjected to capacity and serviceability
constraints. This section will provide and discuss all the constraints present in the
optimization problem considered. The generalized symbol for a constraint is q .
An inactive constraint is reflected in the optimization problem with 1.0q ≤ . An
evolutionary algorithm requires that the problem be unconstrained and, therefore, a
penalty will be assessed for violations of the constraints. The complete listing of
constraints in the optimization problem is provided below:
shear ugirder
cap
Vq
V= (3.7)
moment ugirder
cap
Mq
M= (3.8)
deflgirder
allow
qδ
δ= (3.9)
load udeck
cap
pq
p= (3.10)
shoringdeck un shor
cap
sq
L −= (3.11)
imposedcapjoist cap
Load
pq
p= (3.12)
28
( )DLimposeddefl DL
joist deflLoad
pq
p= (3.13)
( )LLimposeddefl LL
joist deflLoad
pq
p= (3.14)
bracingjoist
bracing
Lq
L= (3.15)
where:
,cap capM V - beam flexural and shear capacities (AISC 2001)
allowδ - deflection limit corresponding to L/360
capp - composite deck three span superimposed loading capacity (SDI 2002)
un shorcapL − - max un-shored span for composite decking (SDI 2002)
,cap deflLoad Loadp p - superimposed joist load limits for capacity and deflection (SJI 2002)
bracingL - span corresponding to special bracing condition at midspan (SJI 2002)
The calculation of the required values to accompany these capacities are discussed in
following sections.
3.1.3 Penalties for Constraint Violations
As mentioned previously, an evolutionary algorithm requires that the optimization
problem be posed as unconstrained. Therefore, constraint violations will result in a
penalty being assessed to the objective function. The penalty values for any constraint
are shown as φ and are generally represented by the following formula
, 1.0
1.0, 1.0i i i
ii
pen q for qfor q
φ⋅ >
= ≤ (3.16)
where: ipen is a penalty multiplier for constraint i that scales the impact of the constraint
violation on the fitness. The individual φ values for each constraint are combined to
29
create a combined penalty, Φ , for each component of the design. The combination is
multiplicative and is mathematically stated as
1 2 3( )girder girderφ φ φΦ = ⋅ ⋅ (3.17)
1 2 3( )joist joistφ φ φΦ = ⋅ ⋅ (3.18)
1 2( )deck deckφ φΦ = ⋅ (3.19)
Each iφ value will be defined later in this chapter. Systems, which satisfy all the
constraints, will not be penalized. (i.e. 1.0Φ = , for each component)
3.2 Definition of Fitness for the Evolutionary Algorithm
With all the objectives and constraints set, methods must be devised for determining all
the values required for evaluating the constraints and defining the fitness of an individual
in the population. This section will detail how the constraints are evaluated and penalties
assessed for each component of the system. The fitness is determined using the penalties
and cost calculation.
3.2.1 Joist Capacity and Deflection Constraints
The constraints for the K-series joists are simplified compared to the girders because
loads corresponding to capacity and the deflection limit of L/360 are provided in the
design tables of the Steel Joist Institute K-Series Joist Catalog (SJI 2002). These values
are available for spans ranging from 8ft to 60ft in 1 ft increments. For use in MATLAB,
the joist table was entered into an Excel spreadsheet with each row representing a
possible joist selection. The columns provide all the data required for evaluation of a
joist. Each joist specification has two load values for the aforementioned spans. The
tabular load values for the given system are referred to as WSL for the uniform
30
distributed load corresponding to flexural and/or shear capacity, and WDL for the load
corresponding to the deflection limit.
The first criterion examined is the joist flexural and/or shear capacity. The joist
capacity, capLoadp , is obtained directly from the SJI Joist table (SJI 2002). The total load
imposed on the joist, imposedp , is calculated by the following relation,
( )imposed deckp SDL LL w s= + + ⋅ (3.20)
where deckw is the weight per square foot of the deck and all other terms are defined in
Figure 1.1. The penalty is assessed in the following manner,
1
1 1
( 1.0)
1.0
imposedcapjoist cap
Load
capjoist
capjoist
pq
p
if q
else
pen q
φ
φ
=
<
=
= ⋅
where 1 1.0pen = for joist capacity.
Next to be evaluated is the deflection limits. Deflection will be looked at for two
cases. The first case corresponds to deflection during construction. The loading will
represent the load of the newly poured concrete and steel deck to ensure that deflection is
limited to prevent ponding. The second case corresponds to the post construction live
loading. The loads for each case are calculated using the following equations.
( )DLimposed deckp SDL w s= + ⋅ (3.21)
LLimposedP LL s= ⋅ (3.22)
The penalties for deflection can be calculated using
31
( )
( )
2
( )2 2
( 1.0)
1.0
DLimposeddefl DL
joist deflLoad
defl DLjoist
a
defl DLa joist
pq
p
if q
else
pen q
φ
φ
=
<
=
= ⋅
( )
( )
2
( )2 2
( 1.0)
1.0
LLimposeddefl LL
joist deflLoad
defl LLjoist
b
defl LLb joist
pq
p
if q
else
pen q
φ
φ
=
<
=
= ⋅
2 2 2max( , )a bφ φ φ=
where 2 1.0pen = for deflection constraints.
Finally there is a penalty for a special bridging requirement. This penalty
corresponds to the shaded region in the SJI joist table (SJI 2001). The penalty is on
sections requiring bolted bridging at the mid span. It does not impact the adequacy of the
section. It is included to provide a price increase equivalent to the cost associated with
the special bridging. This criterion corresponds to Equation (3.15).
3
3
( 1.0)
1.0
1.02
bracingbracing
bracing
LqL
if q
elseφ
φ
=
<
=
=
3.2.2 Concrete Deck Capacity and Un-Shored Span Constraints
The composite deck has the easiest set of constraints to evaluate. The data used to
evaluate the deck comes from the Vulcraft Steel Deck Catalog (Vulcraft 2002), which
provides capacities for spans ranging from 5-ft to 15-ft in 0.5-ft increments. Like the
joists, the steel deck catalogue tables were coded into an Excel spreadsheet. The
constraints correspond to the ultimate capacity, up , as well as the maximum span without
32
shoring, un shorcapL − , which assumes the deck system is continuous over three spans. Due to
the maximum allowable joist loading of 550 plf, the maximum spacing is assumed to be 6
ft. Therefore, the deck will never have to span a distance longer than 6 ft. Since the joist
spacing varies between 2 and 6 ft, the joist spacing may be less than the minimum span
contained in the composite deck table. For this case, the deck capacity is taken to be the
capacity, capp , of the minimum span in the table.
This check and the maximum un-shored span check constitute the composite deck
checks and the penalties are computed as shown below,
1
1 1
( 1.0)1.0
load udeck
cap
loaddeck
loaddeck
pqp
if q
else
pen q
φ
φ
=
≤=
= ⋅
2
2 2
( 1.0)1.0
shoringdeck un shor
cap
shoringdeck
sqL
if q
elsepen
φ
φ
−=
≤=
=
where 1 2.5pen = and 2 2.0pen = for the deck penalties.
3.2.3 Girder Capacity and Deflection Constraints
The girder is a slightly more difficult analysis problem because it is subjected to a series
of point loads as well as the uniformly distributed loading that results from its self-
weight. While the overall format for measuring performance of the girders has not
changed from the metrics used for the beams, the methods used to compute applied shear
and moment, and deflection have changed.
The girder must have sufficient shear and moment capacity and satisfy the
maximum deflection criteria of L/360. The girder is also assumed to be simply
supported, and laterally braced along its entire length by the joist seats which support the
33
joists framing into the girder at the spacing specified by the genetic algorithm. These
joist seats transmit the loads from the joist to the girder, so they create a set of point loads
along the girder’s length. Modeling a girder with point loads needlessly complicates the
analysis. The joist loads are transformed into a uniform distributed load. This value,
combined with the dead load, is used to determine the shear and moment values. The
analytical model for the girders is shown in Figure 3.1.
Figure 3.1: Girder Model
As with the beams, the process begins with calculating the loads applied to the
girder. The same load combinations applied for the beams are applicable with the girders
to generate Service Loads (SL) and Factored Loads (FL). Girder weight will be ignored
and taken into account in the last equations, and the load factors will be applied in the
following equations for effective unit loads,
1.2( ) 1.6( )FL DL SDL LL= + + (3.23)
SL DL SDL LL= + + (3.24)
The loading magnitudes computed using equations (3.23) and (3.24) can be used
to determine the moment and shear forces required to determine the adequacy of the
girder. They are used to calculate the maximum values, which are a combination of the
values that that correspond to the multiple point loads and self-weight of the girder.
L
FL, SL
34
The maximum factored moment in the girder, the maximum factored transverse
shear force in the girder and the deflection of the girder are computed using the following
equations,
2( )
8uFL LM = (3.25)
( )2u
FL LV = (3.26)
45( )( )
384( )( )x
SL LE I
δ = (3.27)
The above values are compared to the capacity and deflection values provided by
the girder to determine adequacy of the girder under the applied loads. The moment
capacity, the shear capacity and the allowable deflection for the girders are,
n b x yM Z Fφ φ= (3.28)
n v y wV F dtφ φ= (3.29)
360all
Lδ = (3.30)
For all systems, the beam and girder yield strength is assumed to be 50ksi. All
non-compact sections for Fy = 50 ksi have been removed from the database of shapes
utilized by the program. Because the girder is considered to have lateral bracing over its
entire length provided by the joists, and all sections are compact, the section plastic
moment capacity is an adequate measure of the beams capacity for this analysis.
With both required and provided capacity, the values are compared to determine
the penalties (if any), which should be assessed for the individual systems. The penalties
are calculated in the following manner,
35
1
1 1
( 1.0)
1.0
moment ugirder
cap
momentgirder
momentgirder
Mq
M
if q
else
pen q
φ
φ
=
≤
=
= ⋅
2
2 2
( 1.0)
1.0
shear ugirder
cap
sheargirder
sheargirder
Vq
V
if q
else
pen q
φ
φ
=
≥
=
= ⋅
3
3 3
( 1.0)
1.0
deflgirder
allow
deflgirder
deflgirder
q
if q
else
pen q
δδ
φ
φ
=
≤
=
= ⋅
where: 1 1.0pen = , 2 1.0pen = , and 3 1.0pen =
3.2.4 Definition of Individual Fitness
Once the constraints have been evaluated and the penalties assessed, the GA/EA needs a
way to distinguish between the qualities of the competing systems. This is usually
achieved through the creation of a fitness function, which (for the present study) is based
on a combination of the objective function and the multiplicative penalties from the
constraints.
A fitness function is essential to running a genetic algorithm. The function
facilitates evaluation of the system and determines how likely it is to be selected for
reproduction. This makes it a crucial component to obtaining a solution.
The process of creating a fitness function began with the objective to minimize
the total weight of the system. The fitness of the system in this case can be represented
as,
i i iFit W= Φ ⋅ (3.31)
where,
iW = total system weight iΦ = penalties for constraint violations
36
The Darwinian analogy to survival of the fittest used in the fittest used in the GA implies
that fitness should be maximized. Because of this, the fitness in Equation (3.31) is
inverted, so the minimum weight system will yield maximum fitness. This new equation
is shown below,
1
ii i
FitW
=Φ ⋅
(3.32)
The penalty scalar values ( ipen ), defined in prior sections, were constant. The
component penalty system, seen previously, was created to ensure that infeasible systems
would participate in the evolutionary process, but not become the fittest in the population.
Weight was used initially to determine fitness. The fitness was subsequently
changed to reflect the cost of the system. This new fitness function is seen below
1
1,000,000ci N
i ii
FitnessCost
=
=Φ ⋅∑
(3.33)
where, iCost is defined in Equation (3.2), and shall now be referred to as iC and cN is
the number of designed components in the system.
The cost aspect was added to attempt to take the additional concerns of a designer
into account. This change allows the algorithm to consider the many differing costs of
materials and sections. Trends in the impact of component costs on the design can now
be observed through the algorithm fitness calculation.
While testing the algorithm, a problem was discovered. The designs generated by
the algorithm were heavily favoring small joist spacings. Larger spacing values were
disappearing from the population, because according to tabulated capacities, fewer joists
work with larger spacings. The result was a complete lack of any spacing value greater
37
than 3 ft in the fittest designs in the final population. The fitness was modified to fix this
problem. The modified fitness equation is shown below
1
1,000,000
( ) 100c
i N
j jj i
FitnessWCs=
=
Φ ⋅ + ⋅
∑ (3.34)
The new fitness includes a 100 dollar cost is assessed for each joist in the system.
The Bay width, W, divided by the joist spacing, s, yields the number of joists in a system.
This value represents a handling cost, which is based on the fact that it costs more in
terms of crane time and manpower to place a larger number of joists. The $100 value
encourages larger joist spacings, and allows the joist spacing to directly influence the
fitness in a similar manner as the other components of chromosome. There was no
analysis as to how changing the coefficient value (100) affected the overall performance
of the algorithm. The value was chosen because it can added a cost without making the
spacing dominant over the system components (ie. joists, girders).
3.2.5 Vibration Evaluation
The walking floor vibration analysis procedure is taken directly from AISC Design Guide
11 (Allen 1997). It provides a series of equations, which are coded into the floor
vibration portion of the automated design algorithm.
Joist section properties must be calculated using the same procedures from
Chapter 2. Once again, the joist data is provided in the SJI Standard Load Table (SJI
2001). The cross-section model for the composite joist section is shown in Figure 3.2.
38
The moment of inertia of the K-series joist (by itself) and the cross sectional area of the
open web joist chords can be back calculated using the values contained in the Vulcraft
load tables (Vulcraft 2002). These computations are illustrated below,
45
384
deflLoad
jall
p LIEδ
= (3.35)
2
8 (0.6 )
capLoad
botej y
p LAD F
=⋅
(3.36)
2j botA A= (3.37)
With the joist data calculated, the K-series joist selection tables (Vulcraft 2002)
and the user input for the automated algorithm provide the following
Joist Properties - , , , ,j j j jw I A D y
Concrete Deck Properties - cov, , , , ,c cd rib er total effw w t t t b
Figure 3.2: Composite Joist Cross-Section
ycomp
ybar
Dj
ttotal
tcover tav
39
The material properties are established using traditional methods from ACI 318 (ACI
2002) and the modular ratio includes a factor, 1.35, which accounts for the dynamic
effects of the loading. (Allen 1997)
' 3000cf =
1.533( ) 'c c cE w f= (3.38)
29000000sE =
1.35
s
c
En
E= (3.39)
The evaluation of the joist contribution to vibration sensitivity involves
computing the composite moment of inertia for the open web joist acting in conjunction
with the concrete deck. This information is more difficult to compute, but procedures are
available (Galambos 1988). The composite moment of inertia of the K-series joist is
computed as (Allen 1997)
cov
cov
cov
( ) ( )( )2
( )
eff erj total er
compeff
j er
b tA t y tny b
A tn
+ +=
+
i
i (3.40)
3cov
2 2covcov
( )( ) ( ) (( ) )( )
12 2
effer
eff ercomp j j total comp er comp
bt b tnI I A y t y t y
n= + + − + + −
ii (3.41)
The effective moment of inertia for vibration consideration is also computed
using previously established procedures (Allen 1997)
( 0.28 / ) 2.80.90(1 ) :6 ( / ) 24j jL Dt j jC e for L D−= − ≤ ≤ (3.42)
1
1tC
γ = − (3.43)
11eff
j comp
I
I Iγ=
+ (3.44)
40
The uniformly distributed loading to be used in determining the effective loading
for vibration consideration is computed as (Allen 1997)
(1) ( )eff v cd jw LL w SDL s w= + + ⋅ + (3.45)
The deflection of the joist when subjected to the effective uniform distributed loading is
computed using traditional mechanics procedures
4
(1)5( )( )384( )( )
eff jj
s eff
w LE I
δ = (3.46)
The joist system fundamental frequency can now be calculated using the
procedures in design guide 11 (Allen 1997). The frequency is computed using
0.18jj
gf
δ= (3.47)
The weight of the beam panel must be calculated for future vibration calculations.
The following facilitates calculation of the effective panel depth (Allen 1997)
312( )
12av
stDn
= (3.48)
effj
ID
s= (3.49)
The effective beam panel width is computed using (Allen 1997),
14 2min( ( ) , (3 ))
3s
j j j gj
DB C L LD
= (3.50)
This study assumes Cj = 2.0. The panel weight is used to determine the combined mode
properties as shown below (Allen 1997)
(1)effj j j
wW B L
s= (3.51)
41
The girder mode (vibration) calculations also follow the procedure established by
Allen et al (1997). The procedure has significant differences and warrants a separate
explanation. The cross-sectional model for the girder is shown in Figure 3.3.
The AISC shape table (AISC 2002) provides the girder properties.
Concrete Deck Propeties - cov, , , , ,c cd rib er av totalw w t t t t
Girder Properties - , , , ,g g g g gw A I D y
The composite section properties for the girder need to be defined first. The first
task is to define an effective width of slab (Allen 1997).
min[0.4 , ]eff g jb L L= (3.52)
The computation of the composite section properties follows the usual mechanics of
materials procedures
2
( ) ( / )( )( )2 2
( / )( )
g avg s total eff av
compg eff av
D tA t t b n ty
A b n t
+ + +=
+ (3.53)
ttotal
Figure 3.3: Girder Composite Cross-Section
ycomp2 ts
Dg/2
42
32 2
2 2
( / )( )( ) ( / )( )( )
2 12 2g eff av av
gcomp g g s total comp eff av comp
D b n t tI I A t t y b n t y= + + + − + + − (3.54)
The composite moment of inertia will be reduced due to the flexibility of the joist seats.
This reduction is approximated as (Allen 1997)
( ) 4gcomp g
eff g g
I II I
−= + (3.55)
As with the beams, an equivalent uniformly distributed loading needs to be
defined. This equivalent loading is used to compute a reference deflection for defining
the natural frequency of the girder. The equivalent uniform loading represents the dead
and live load present on the girder when the dynamic load occurs. (Allen 1997)
(1)(2) ( )eff
eff j g
ww L w
s= + (3.56)
The deflection of the composite girder (with adjustment for joist seat flexibility) is
computed as
4
(2)
( )
5384
eff gg
s eff g
w LE I
δ = (3.57)
The girder system fundamental frequency can now be calculated (Allen 1997).
0.18gg
gf
δ= (3.58)
The combined mode properties can now be computed for the floor framing
system. The effective depth of the composite girder is computed as
( )eff gg
j
ID
L= (3.59)
With Dj known from prior calculations, and Cg = 1.6 for a girder supporting a joist seat,
the effective girder panel width is computed using (Allen 1997)
43
14 2min( ( ) , (3 ))
3j
g g g jg
DB C L L
D= (3.60)
The effective girder panel weight can now be calculated for use in the combined mode
properties (Allen 1997).
(2)( )effg g g
j
wW B L
L= (3.61)
The natural frequency of the joist-girder system is computed as a weighted
combination of two parts (Allen 1997).
0.18nj g
gf
δ δ=
+ (3.62)
j gj g
g j g j
W W Wδ δ
δ δ δ δ= +
+ + (3.63)
Non-dimensional panel acceleration can finally be computed. This acceleration
can be compared to established vales that define the acceptability thresholds for human
perception. Damping present in the floor framing system is incredibly important when
attempting to define acceptable levels of human perception. The study of damping is
outside the scope of this thesis. Assuming a common damping level (β = 0.05 for a
building with full height partitions and hanging ceiling), the non-dimensional panel
acceleration can be computed using (Allen 1997)
( 0.35 )nf
o oa P eg Wβ
−
= (3.64)
Other non-dimensional acceleration thresholds for different building uses are
shown in Table 3.1.
44
Table 3.1: Beta and Threshold Values (Allen 1997)
The procedure outlined in the previous discussion has been coded into the
MATLAB routine: floorvib2.m, which can be found in Appendix A. With the objectives,
constraints of the problem set up, it is now possible to create the genetic algorithm code
that can be used for automated design of floor framing systems.
45
Chapter 4 – Programming the Genetic Algorithm in MATLAB
4.1 Introduction
The purpose of the genetic algorithm written in this thesis is to create the most
economical floor system using seven design variables. The girder variable is the
simplest, because it is only wide flange member size. The joist design variable is
specification includes spacing and joist size, and the steel deck design variables include
the gauge thickness, rib height, topping thickness, and type of concrete. All of the design
variables have to be sifted by the program to generate feasible systems with least cost.
The GA must represent all these variables and efficiently work through them to obtain the
best solution. The following sections show the tools and processes the algorithm utilizes
to solve the design problem.
4.2 Essential Components of the Genetic Algorithm
The basic genetic algorithm is composed of several modules, which perform various
operations to facilitate the evolutionary process. The modules and concepts most
essential to the algorithm are discussed in the following sections.
4.2.1 Creating the Chromosome
At the center of the genetic algorithm is the chromosome, which contains all the
information needed to identify the components of each system. The chromosome is a
series of “1’s” and “0’s” which (when combined to form a binary string) represent the
possible solutions to the problem. For the case of automated floor design, there are five
components: the steel deck with concrete topping, the K-series joists, and the girders
supporting the joists.
46
Each component (design variable in an optimization problem) must be
represented by a portion of the binary string. This can add to the complexity of the
problem. Binary strings with 2 digits can represent 22 or 4 options. Strings of 6 can
provide for 64 options. This trend makes the power of two very important when setting
up the genetic algorithm.
Through observation of the LRFD manual’s wide-flange shapes table (AISC
2001), it was noted that there were about 260 shapes. This proved to be very important,
because 260 is very close to 256, which is equal to 28. To obtain a useable table, some
uncommon (beam) shapes were removed, such as the W14x90 and W14x99. Additional
sections, such as the lightest and heaviest sections were also cut to reduce the list to 256
shapes. With a table of 256 shapes, a binary string (gene) of 8 digits would be sufficient
to represent the girders.
The next component considered was the composite concrete deck. The Vulcraft
catalog (Vulcraft 2002) provides six tables of 42 combinations of decks and toppings.
All of these tables were assembled in similar fashion with rows corresponding to a steel
deck and topping selection and columns corresponding to selection properties.
Combining them into one large table proved to be a simple operation, which yielded one
database with 252 deck combinations. Because this value is so close to 256, four decks
from the top of the table were appended to the bottom of the table, thereby creating 256
options. With the same number of options, the genome for the steel deck and concrete
cover will consist of 8 digits.
Next to be assembled was the K-series joists. All the data used came from the SJI
tables (SJI 2002). Once again, all the tables were merged into one database providing
47
exactly 64 options. This just happens to equal 26, and, therefore no editing was required
for the database to be used by the genetic algorithm.
The final design variable in the system is the joist spacing. The portion of the
chromosome used to define this variable is defined by the programmer. Eight joists
between each column line is the default maximum. Limits are set for the system at a
minimum of 2-ft and a maximum determined using the user-defined loading. The lower
limit of 2-ft was based on an assumption that any spacing less than 2-ft is not considered
economical (too many joists). The upper limit corresponds to 550-plf divided by the
imposed loading. That establishes the minimum number of joists capable of supporting
the super-imposed load. Spacings between these limits correspond to a number of joists
that are created in a separate table. For the cases when there are not eight possible
spacings, the spacings may repeat. See the spacing.m program file, located in the
Appendix A, for the details of the procedure.
4.2.2 Initial Population
The first step in any genetic algorithm is the creation of an initial population. The
method utilized for this is based upon the algorithm developed by Coley (1999). For the
problem of floor design, a simple random number generator is used to create the initial
population. The process can be schematically described as follows: a random number
between zero and one is generated for each allele in the chromosome. Based on whether
that value is greater than or less than one-half, the allele in the chromosome is set to zero
or one, respectively. This is looped over the entire chromosome length to create the
genetic representation (chromosome) for the individual. This process is displayed in
code provided in Figure 4.1:
48
for i = 1 to Chromlength
allele = rand(0 to 1); if allele > 0.5 chromosome( i ) = 1; else chromosome( i ) = 0; END
END
Figure 4.1: MATLAB Code for Chromosome Generation
The for loop shown in Figure 4.1 would be placed within another loop over the
entire population, which would then yield an initial population of chromosomes. A
sample population of 10 is shown in Figure 4.2.
1 - 001010010011001011111011000011110 2 - 010011101111001011001100000101010 3 - 100000111010011100111011000111000 4 - 011000100000110110010000101001000 5 - 101001101111011110010001001001111 6 - 000111110100011100001110000010111 7 - 000100100000111101101000101110001 8 - 000111001010000100011111000000110 9 - 111000100000010001101101101111010 10- 111011011010010111100000111010010
Figure 4.2: Chromosomes for Population of Ten Individuals Systems
4.2.3 Chromosome Decoding
Once the initial population is created, the question becomes how to turn binary strings
into useful data. The determination that the number provided by the segment of the
chromosome would correspond to a row in the data table corresponding to that segment.
In order to use this, the binary string has to be decoded to a standard decimal number.
The decoding process utilizes a simple formula for decoding the chromosome, shown
below
49
1
2N
ii
i
X D=
= ∑ (4.1)
where: X is the unknown value; N is the number of Binary Digits in the segment; and D
is the value of the binary digit.
The chromosome is stored in an array, and therefore, it must be decoded in loops.
The ordering of the string in the decoding process is important. It was determined that
the direction in which the string is decoded was not important as long as the string is
decoded consistently. The process utilized to decode the chromosome is displayed in
Figure 4.3.
j = 1; seg = 1; X = 0; D = unknoRange(seg); for i = 1 to ChromLength if i <= D X(seg) = Chromosome( i ) * 2j + X(seg); j = j + 1; else seg = seg + 1; D = D + unknoRange(seg); j = 1; X(seg) = Chromosome( i ) * 2j + X(seg);
j = j + 1; END END
Figure 4.3: MATLAB Code for Chromosome Decoding.
With the initial values set, the code simply implements the decoding function for
each segment of the chromosome. This process must be repeated over the entire
population to generate all the numbers to be used with the future processes. If there are 5
segments, then the X array will result in a (5 by population) size array. To demonstrate
the process, the second chromosome from Figure 4.2 will be decoded.
0100111011001100000101010
50
The chromosome can be broken up into segments based on the “unknoRange” array.
unknoRange = [ 8 , 8 , 6 , 3 ];
If the first value is 8, then the first 8 digits constitute segment (design variable) one. The
second number represents the number of digits in the array representing the second
segment (design variable). This process is repeated until all the segments are accounted
for as shown below.
01001110 - 11001100 - 000101 - 010
With the chromosomal array broken up, the binary strings are now decoded into
integer numbers as shown below
Segment 1 = 01001110 = 114
Segment 2 = 11001100 = 51
Segment 3 = 000101 = 40
Segment 4 = 010 = 2
With the values now decoded, the chromosomes can be referenced to the tables (e.g.
LRFD manual, Vulcraft tables, SJI tables, etc...) for evaluation.
4.2.4 Fitness Evaluation
Fitness is a measure of how well a system performs under the constraints set by the
engineer in the definition of the optimization problem. Since the fitness was defined and
developed in Chapter 3, this section will only cover the set up of the MATLAB module,
which performs the fitness calculation.
The first step in the process is to evaluate the adequacy of each component of the
system (e.g. steel girder, k-series joist, steel deck, concrete-steel composite deck). This is
51
done via separate modules for each component. Each module generates the Φ values
(penalties) as shown in Chapter 3.
Next, the cost of each section is calculated. The cost of one girder, several joists,
the steel deck and the concrete topping is calculated. These values can be summed
according to equation (3.2) to create an approximate cost per bay.
Using the Φ values already calculated, the cost of each component is penalized
by its own Φ factor. These penalties create a penalized cost value, which serves as the
denominator in equation (3.34). The fitness values, cost, and Φ values are output, and
the fitness evaluation is finished.
4.2.5 Selection & Crossover
Once each system has a defined fitness, the next step in the GA is to create the next
generation by selecting and mating the individuals, which were evaluated in the last
generation. This process can be performed in one program or broken down into multiple
modules. The selection and crossover process are separated in this genetic algorithm.
This ensures that each evolutionary operator can be applied independently from the other.
The traditional roulette-wheel selection mechanism is implemented in this thesis.
The selection process is modeled using an analogy to the probabilistic basis of the
roulette wheel. The standard American roulette wheel is broken up into 36 equal spaces.
The selection is determined by a ball, which skips and rolls around until it stops in one of
the spaces. The GA roulette wheel is also divided, but the number of spaces is equal to
the population size. For a population of 100, there are 100 spaces. Also, rather than an
equal 100 spaces, each space is sized based on the ratio of individual fitness to the total
fitness of all the systems in a population.
52
The simulated action of the roulette wheel is used to determine the individual to
be selected for mating during the evolution of the system. The process begins with a
random number between 0 and 1 being generated and multiplied by the sum of the fitness
for the generation. This value becomes the limit on a loop over the number of individuals
in the population. Each individual’s fitness is tallied until the value is greater than the
limiting value. Then the individual selected for mating is the individual that breaches the
limit. The code shown in Figure 4.4 illustrates the roulette-wheel selection process.
limit = sum(fit); while sum < limit
i = i + 1; sum = sum + fit(i);
end mate = i;
Figure 4.4: MATLAB Code for Roulette Wheel Selection
Roulette wheel selection is not the only method used in genetic algorithm
implementations, nor can one say it is the best selection method. There are other
approaches that may be used. Other selection methods can be applied if the roulette
wheel selection appears to result in convergence difficulties (e.g. premature
convergence). In the end, the roulette wheel is used for simplicity and because it works.
Once the individuals are selected for mating, there are three crossover methods
available for creating the offspring. The methods exchange portions of the chromosome
between parents. The three methods utilized by the genetic algorithm in the thesis are
explained in the following paragraphs. The MATLAB code, which uses each crossover
method is available in Appendix A.
53
The first method, single point crossover, is a very common method in the GA
literature. This method was adapted to the present work from the code provided by Coley
(1999). Single point crossover is very simple to set up and implement. Using a random
number generator, a point in the chromosome is chosen to be an exchange point.
Suppose the two chromosomes below are systems, which have been selected for
crossover (mating).
Parent 1 - 0000000000000000000000000
Parent 2 - 1111111111111111111111111
For this example, the random number generator has chosen the number eleven. The point
just after the 11th digit will be the crossover point.
Parent 1 - 00000000000 – 00000000000000
Parent 2 - 11111111111 – 11111111111111
The offspring is created by selecting the portion of the chromosome of system 1 up to the
crossover point and the portion of the chromosome after the crossover point for system 2.
The resulting system is then a combination of the two original systems. This new
individual then goes on to become part of the new generation in the evolution.
00000000000 – 11111111111111
This is a highly efficient system of creating offspring, but it has a small (yet
significant) inherent side effect. If the crossover point is located within a gene for a
specific component of the systems, new genetic material will be created in the offspring.
This may be desirable, or undesirable depending upon the users needs in the application.
The previous crossover example will now be recreated, with the catch that you will now
see the result on the system identity (offspring chromosome).
54
Parent 1:
00000000-000 – 00000-000000-000 (1,1,1,1)
Parent 2:
11111111-111 – 11111-111111-111 (256,256,64,8)
The example demonstrates the initial tabular values as well as the initial
chromosome. If the same (11th digit) crossover point is assumed, the offspring that
results from mating of the two individuals above is;
00000000-00011111-111111-111 (1,249,64,8)
The offspring will retain most of the genetic material from the parent systems. There is
new genetic material introduced into the offspring at the gene corresponding to the
crossover location. Neither parent had beam number 249, which is now present in the
offspring. This is a direct result of the crossover. While this can be seen as advantageous
(constantly introducing new material to the process), it can also be seen as disruptive
(material that may be very suitable for creating better systems can be destroyed).
Because of this concern, other methods for performing crossover were developed.
All crossover methods developed utilize component crossover. This is when components
of the systems are exchanged, but genetic material present in the initial population is
preserved. Given the same situation as before, the chromosomes are separated into
system components.
00000000-00000000-000000-000
11111111-11111111-111111-111
When performing crossover, suppose components 2 and 5 are selected for exchange. The
resulting offspring would be:
00000000-11111111-000000-111
55
It can be seen that the existing genetic material present in the parents is preserved, but the
two components of the design system are interchanged.
This crossover implementation is similar to that employed in Foley and Voss
(1999) and Foley, et al (2001). The lack of new material generated in the process can be
offset by an increased rate of mutation.
Elite component crossover is the third method of crossover available for use with
the genetic algorithm. This method is very similar to dual component crossover, except
that it selects the components based on performance. This method was inspired by the
evolutionary algorithm developed in Voss and Foley (1999), with the idea of simplifying
it for use with the already existing code. The Φ factors are utilized for each system to
determine the better component as shown below
Chromosomes Φ Values
00000000-00000000-000000-000 [1.0,2.5,1.2,1.2]
11111111-11111111-111111-111 [4.0,1.0,1.0,1.2]
The process compares the Φ values from each parent to determine which component will
go onto the next generation. Remembering that the Φ values are penalties, the lower
value is preferred. For the sample chromosomes and Φ values, the offspring will be:
00000000-11111111-111111-000
For systems with equal Φ values, the first parent will take priority. Therefore, in the
example the last portion of the chromosome will be “000.”
When employing a mechanism like this, it is possible for systems (individuals) to
become stagnant and begin repeating as the evolution progresses. This phenomenon is
called epitasis in genetic terminology. To avoid this problem, intelligent mutation is
56
suggested as a solution by Foley and Voss (1999). At present, no such process is
employed. However, it could be introduced to the mutation module.
With so many crossover options, which is the best? The overall trend seems to
support that all three yield similar results, if used properly for this algorithm. Because of
this, all three are provided for use in the algorithm and it is left up to the user to utilize
that which best solves the problem at hand.
4.3 Additional Components in the Genetic Algorithm
Having explained all the critical parts of the genetic algorithm, several other components
are used to attain the desired results. These additional parts, while often included in
genetic algorithms, are not considered essential because they can be turned off, or have
their influence on the algorithm as a whole reduced. The purpose of these additional
modules is to assist the process in attaining the most efficient solution. Application of
these components within the MATLAB GA can be inhibited by commenting out m-file
function calls within the source coding.
4.3.1 Elitism
Elitism is a process, which allows one (or more) of the fittest individuals to pass onto the
next generation. The coding for this process is very simple. By design, the algorithm
retains every chromosome created. This makes the crossover and elitism operations
especially easy. The chromosomes of the elite individuals need only to be copied to the
new generation. Because of an apparent redundancy of keeping individuals around for
many generations, only one elite individual is allowed to remain for transfer to the next
generation. The code, which performs this operation, can be viewed in Appendix A.
57
4.3.2 Mutation
As the chromosomes are selected and mated, dominant systems will develop. These
systems will eventually maximize the fitness of the population given the genetic material
present in the population. Once the fitness of the population is maximized, there is no
point in continuing the algorithm. Mutation allows for the introduction of new genetic
material into the population irrespective of the crossover method implemented. While
the single component crossover method inherently performs this task, it is still necessary
to provide mutation. Mutation creates additional material for the algorithm’s continuing
improvement of the population. This role is even more critical in the more specialized
crossover methods (e.g. component crossover, elite component crossover) available in the
algorithm.
The general purpose of the mutation function is to slightly modify a portion of a
chromosome (flip a binary digit) to create a new individual for analysis. This can be
done either to the components or to the entire chromosome. For this algorithm, the entire
chromosome is altered.
The process begins at the user end, when a mutation probability is selected. This
value will become the standard for mutation for the entire algorithm. Every individual in
the population will be subjected to mutation. The process begins with a random number
generator. If the value is lower than the mutation percent, then the individual will be
mutated.
Once an individual is selected for mutation, a loop takes over. For each allele in
the chromosome, a random number is generated, if it is less than 0.125, then the allele is
58
swapped (0 becomes 1 and vice versa). Coding the mutation process is shown in Figure
4.5.
if rand() <= MutationPercent for allele = 1:chromLength
if rand() > 0.125 if chromosome(i,allele,Gen) == 0 chromosome(i,allele,Gen) = 1; else chromosome(i,allele,Gen) = 0; end end end
end Figure 4.5: MATLAB Code for Mutation Implementation.
The 0.125 value is used to limit how many allele changes are performed. If all
alleles were changed, the process would cause a mutation cycle. There would only be
one result possible each time any individual is mutated. If that individual is mutated
again, then the original individual will be recreated. This would defeat the purpose of
mutation. The 0.125 value provides that only one allele in a gene of 8 alleles will be
changed. This would alter the chromosome, but still provide something suitable for
crossover to the next generation.
Mutation is an integral part of the solution process. Mutation parameters should
be experimented with to see what yields the best results. Over the course of working
with the problem to which the thesis is devoted, low mutation rates (below 5%) have
worked well with single point crossover. Higher rates (near and above 10%) work well
with the other crossover methods discussed previously. The mutation begins to degrade
the performance of the algorithm when the mutation probability is above 25%. The rates
of mutation that are successful can also be altered as the population size changes. The
59
process of tailoring the GA parameters to the problem being solved is an integral part of a
GA application.
4.3.3 Fitness Scaling
If one recalls the discussion related to selection, individuals are selected based upon
fitness. If one individual’s fitness is disproportionately large relative to others in the
population, this individual’s slice in the roulette wheel will get very large relative to the
other individuals in the population. As a result, one individual (the fittest) will be
repeatedly selected to the detriment of the genetic algorithm’s ability to explore the
search space. One way to prevent this from happening is to scale the fitness.
Fitness scaling is process of adjusting fitness is to provide a more equitable
distribution of fitness among the individuals in the population. Its goal is to ensure that
no one individual has a fitness that is large enough to dominate the remaining individuals.
There are other approaches available (e.g. rank-based selection -Voss and Foley 1999).
This thesis defines the fitness in a manner that ensures that no individual can end up with
a fitness that is disproportionately large relative to the rest of the population. The fitness
calculation, discussed in Chapter 3, was adjusted to balance fitness values across the
population. The resulting fitness calculation provides evenly distributed values of fitness.
The difference in fitness between the most fit and the least fit (barring those with fitness
approaching zero) is only a factor of four. This assures that slices of the roulette wheel
remain equitable among individuals in the population. A separate scaling module
borrowed from Coley (1999) is provided in the code. However, it is not utilized by the
runs in succeeding chapters.
60
4.3.4 Statistics
The selection method utilizes statistics on the population. A module was written to
calculate the statistical information regarding fitness. These calculations include the
mean, maximum, and sum of the fitness values for the population at any generation. The
fittest individual in the population is also determined. The calculations are performed
using existing MATLAB functions, so there was a minimum of programming required.
The code can be viewed in Appendix A.
4.3.5 Skim and Eliminate
During the testing process, the genetic algorithm was generating a limited number of
feasible solutions for the more demanding systems. Because of this problem, it was
decided that the program would retain every individual created. To avoid keeping track
of all fitness, cost, and vibration data, a skimming function was created to keep track of
every individual that satisfies all the constraints. These individuals are considered
feasible and are maintained in a list.
The skimming m-file (Skim.m) keeps track of the generation and individual
number, so that the feasible individuals can be found when the GA has finished its
evolutionary run. It stores this information in an array. This allows for a stockpile of
feasible individuals to be retained for comparison purposes. This allows the algorithm to
run with a small population size and still generate a large number of suitable solutions.
One side effect of keeping track of all the individuals is the buildup of duplicates.
To prevent the cluttering of the end processes with repeating systems, duplicates are
removed by elimination.m. When dealing with larger populations, this can bog
down the algorithm, but the advantage of dealing with fewer systems at the end results in
61
a faster total run time, provided the population is less than 500. A population larger than
100 is considered “large” in engineering applications.
4.3.6 Domination
This portion of the algorithm is to evaluate the “skimmed” population. It utilizes the
fitness and floor vibration modules to evaluate each system. The data calculated is
plotted in objective space as shown in Figure 4.6.
Figure 4.6: Population Plotted in Objective Space (Pre-Domination)
Figure 4.6 illustrates that the problem has moved to a multiple objective problem where
the user must weigh the importance of non-dimensional acceleration in the system and
the cost of the system. Plots in objective space (such as that in figure 4.6) provide the
user with the ability to see how systems satisfy multiple objectives.
In general, floor framing systems are subject to the need to reduce the cost of
construction as well as the need to satisfy the acceleration limit(s). The best systems are
62
the systems which are lowest in cost and meet thresholds of vibration response (one could
also consider minimizing vibration response in certain instances). Many times, there is
no system lowest in both objectives. In these cases, it becomes necessary to determine
which systems are the best. The Pareto optimum set is the non-dominated solution on the
inner edge of the scatter (Coley 1999, Deb 2001). While Pareto optimality can be used in
a ranking selection process (Coley 1999) and in a maxi-min fitness statement (Balling
and Wilson 2001), it is only used in the final evaluation of the population that results
from the evolution.
The domination code works by comparing every individual in the set of feasible
solutions to every other such individual. This is accomplished by creating a loop within a
loop. If a system is dominated by another, then it receives a zero the third column of the
archive array. Any system without a zero is on the Pareto front. The code used can be
seen below.
for i = 1:popSize for j = 1:popSize if wt(5,i) > wt(5,j) if A_p(i) > A_p(j) archive(i,3) = 0; break; elseif A_p(i) < A_p(j) archive(i,3) = 1; end elseif wt(5,i) < wt(5,j) archive(i,3) = 1; end end end
Figure 4.7: Domination Code
63
Figure 4.8: Population Graphed in Objective Space (Post-Domination).
The above plot is the same plot with the addition of the Pareto set. The Pareto set (or
front) of solutions is identified in Figure 4.8 by the solid line connecting solutions. The
domination module tracks the Pareto set so that it can be output by another module.
4.3.7 Print Generation
Every program requires output if it is to be useful. Print Generation provides a method
for the output of the information generated during a run of the genetic algorithm. The
domination module creates the Pareto set, which is output to a delimited (ASCII) text file.
The least expensive system, as well as the system with lowest non-dimensional
acceleration in the Pareto set, is output to the MATLAB main screen for the user’s
benefit.
64
Chapter 5 – Design Examples & Analysis
5.1 Introduction
Having explained the systems within the algorithm in extensive detail, the algorithm must
be put to the test. It will be used to design three floor panel configurations for two
different loadings. One combination of panel configuration and loading was used in
Chapter 2 to serve as a design example. It will also be used as a check of the
corresponding design from this chapter.
5.2 Problem Statement
The general problem setup is very similar to the one used in Chapter 2. The only
difference is the joist span. Three bay lengths are used: L = 30, 40, 50 feet (Figure 5.1).
s
30’
L
Girder
K-Series Joist
Figure 5.1: Typical floor System
Concrete Deck
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In addition to these three configurations, there will be two loading cases. Both
load cases utilize a superimposed dead load of 15 psf to account for hanging ceiling,
partitions and mechanical equipment. The difference in two loading cases is the live
load. The first loading case uses a live load of 50-psf corresponding to a standard office.
The second loading case uses 80-psf, which corresponds to the upper level (second floor
or higher) office building corridor requirements set by ASCE (1998).
A table of systems from the Pareto set will be provided for each design example.
The computer output and objective space plots will then be provided and discussed.
Lastly, an explanation as to how the results can be judged as acceptable will be given.
5.3 Running the Genetic Algorithm
The algorithm is run within the MATLAB (MathWorks 2003) environment. To set up
the desired design problem, the beginning of the master file can be edited. This portion
of the master file is shown in Figure 5.2, which contains the user specified bay
dimensions and superimposed loadings.
% *** Obtain Additional User Options *** W = 30; % Bay Width (ft) L = 40; % Bay Length (ft) Q(1) = 15; % Superimposed Dead Load (psf) Q(2) = 50; % Live Load (psf) Q(3) = 11; % Waling Vibration Load (psf) % *** Set GA Parameters *** n = 20; % number of generations pop = 100; % population size cm = 1; % crossover method: 0 = single point, % 1 = component, % 2 = elite component mp = 0.05; % mutation probability (0.01 = 1%) fit = 0; % fitness method: 0 = Cost Based, % 1 = Component Weight, % 2 = Total Weight fs = 0; % fitness scaling factor ef = 1; % elistism flag: 0 = elitism off, 1 = elitism on % *** *** *** *** *** DO NOT EDIT ANYTHING BELOW THIS LINE *** *** *** *** **
Figure 5.2: User Input Section of Master.m File
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The options available to the user are defined by comments. Once the desired problem is
set up, the file should be saved. The user can then go to the command line in MATLAB
and type “master”. It should be noted that commands within MATLAB are case
sensitive. An example of the program output is given in Figure 5.3.
Problem Parameters Bay Dimensions: L = 30 W = 30 Loading Information: SDL = 15 LL = 50 LLv = 11.000000 GA Parameters Mutation Rate: mp = 0.050000 Population Size: pop = 200.000000 Generations: gen = 20.000000 Elitism: Elitism Scaling: No Scaling Crossover Method: Dual Componant Crosover 20 7 0:20:14.7610 19 32 0:20:16.9040 18 82 0:20:20.5790
... 1 1349 0:36:26.4280 Current plot released Current plot released Current plot released Domination 0:38:1.6426e+001 Printing 0:38:2.1894e+001 Most Economical Floor System Girder: W 27 x 102 Joist: 24 K 10 at 3.333e+000 ft Deck: 2 in Lwt on 2 VL 22 Total Cost: 1.1e+003 Acceleration Limit: 4.9441026e-003 Stiffest Floor System Girder W 33 x 318 Joist: 30 K 12 at 003 ft Deck: 4.5e+000 in Nwt on 2 VL 19 Total Cost: 1.8e+003 Acceleration Limit: 1.8514477e-003 Number of Dominators on Front: 34 Dominators are output to Dominators.txt
Figure 5.3: Algorithm Output to Screen
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While the algorithm runs, it periodically outputs information to the MATLAB
workspace (screen). The first output is the system parameters set by the user. Next, the
evolutionary countdown tracks the number of generations remaining, how many
individuals in the population are feasible, and the computer clock time when the
calculations for each generation was started. When the algorithm finishes, it outputs the
least expensive system and the stiffest system to the screen (see Figure 5.3). All other
floor systems lying on the Pareto set are output to the ASCII file “dominators.txt.” The
file can be opened and viewed in tabular form, or imported into spreadsheet software.
The algorithm also creates a set of three plots. These plots were originally created
as a diagnostic tool. However, they still serve a purpose in determining if the algorithm is
performing as expected. The first plot is in the fitness vs. generation domain (Figure
5.4). It displays the fitness of the fittest individual in the population and the mean fitness
of the population for each generation.
Figure 5.4: Graph Illustrating Generational Variation in Fitness
Fittest Individual
Mean Population Fitness
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This increasing mean fitness and fitness of the “fittest” individual indicates that
the algorithm is accomplishing its goal (i.e. moving the population in the direction of
increasing fitness). Almost every succeeding generation performs better than the
previous. Despite the (slight) drop offs, the overall average fitness improves over the
run. The slight decreases in mean fitness with generation is a result of crossover and
mutation operations exploring the search space (these drop offs are good).
The second plot generated is fitness versus the non-dimensional acceleration.
This plot was generated to ensure that the range of fitness was reasonable for the roulette
wheel selection mechanism. It is provided to ensure that the individuals are appropriately
distributed in fitness-acceleration space. In the sample plot below, feasible systems
(plotted in green) cloud together with the infeasible (plotted in red). The plot then
indicates that it is unlikely that a single system is taking over the population. This then
indicates good exploration of the search space.
Figure 5.5: Fitness-Acceleration Plot at End of Evolutionary Algorithm
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The third plot is cost versus non-dimensional acceleration (see Figure 5.6). The
algorithm identifies all feasible systems and only these systems are plotted. When the
algorithm has completed its run, the domination module determines the Pareto set. The
Pareto Set is re-plotted in a darker color (blue) in Figure 5.6.
Figure 5.6: Objective Space Plot at End of Evolutionary Algorithm
These three plots generated for each run of the evolutionary algorithm and can be
used to determine if the algorithm is performing an adequate search. The next step is to
run the algorithm for several combinations of load and panel configuration. This will
help to ensure that the algorithm can design reasonable systems.
5.4 Joist Span Variation (Sensitivity)
The first load case uses a live load of 50 psf, which comes from ASCE 7-98 Table 4.1 for
standard office use (ASCE 1998). This loading will be used for the design of three floor
panel configurations in which only the joist length varies. These configurations will
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serve as a diagnostic tool to help determine if the algorithm is working as well as identify
any trends that could be further analyzed. It will also give the reader an indication of the
sensitivity of panel designs to the span of the open web joists. In addition, each
configuration will be run ten times. These additional runs for each configuration will
serve as a diagnostic tool to show repeatability, as well as attempt to show the nuances of
floor system design, which can only be examined through many years of design, or
through an algorithm such as that proposed.
5.4.1 Joist Span 1 (L = 40 ft)
The first system run is the same system designed in Chapter 2. Before discussing the
results it should be noted that the odds of the algorithm arriving at the same solution as
the hand example is extremely small. The hand solution made several assumptions at the
start of the design, which the genetic algorithm does not make. The assumptions made in
the hand computations greatly reduce the search space when compared to the
evolutionary algorithm. It is expected that the evolutionary algorithm designs will be
comparable in cost, weight, and performance, but not identical.
The problem design information shown in Figure 5.7 was entered into the
algorithm and run ten times to completion.
Problem Parameters Bay Dimensions: L = 40 W = 30 Loading Information: SDL = 15 LL = 50 LLv = 11.000000 GA Parameters Mutation Rate: mp = 0.050000 Population Size: pop = 200.000000 Generations: gen = 20.000000 Elitism: Elitism Scaling: No Scaling Crossover Method: Dual Componant Crosover
Figure: 5.7: Input for Algorithm Run
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The ten complete pareto sets output by the algorithm, can be found in Appendix B. Hand
calculations verifying the adequacy of the sets are available in Appendix C. The 10 best
performing systems, subject to an allowable vibration response are provided in Table 5.1.
The algorithm runs produced thousands of feasible systems. Each run produced a
Pareto set of solutions. These systems were output to 10 separate text files, and imported
to a spreadsheet. The least expensive systems from each run were collected, as well as
the least expensive system to satisfy the vibration response requirement. The second of
the sets is shown in Table 5.1.
Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 18 x 106 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1744.10 0.004652 W 16 x 89 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1739.20 0.004803 W 24 x 76 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1735.70 0.004484 W 30 x 99 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1742.00 0.004615 W 24 x 94 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1741.50 0.004466 W 24 x 68 26 K 12 4.3 2.0" Nwt 1.5 VL 22 1733.40 0.004727 W 18 x 86 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1738.10 0.004808 W 30 x 90 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1739.90 0.004469 W 18 x 86 24 K 12 4.3 2.0" Nwt 1.5 VL 22 1757.30 0.00554
10 W 27 x 84 26 K 12 4.3 2.0" Nwt 1.5 VL 22 1738.20 0.00469 Table 5.1: Configuration 1 (L = 40 ft.) Least Expensive Systems
The algorithm produced the least expensive system (sixth line in Table 5.1). This
system can be compared to the design from Chapter 2. A side-by-side comparison is
shown in Table 5.2.
Ch. 2 Example Design Variable GA Solution W24 x 68 Girder W24 x 68
2” LWT 1.5VL22 Concrete Deck 2” NWT 1.5VL22 28 K 10 Joist 26 K 12
4.3 ft Spacing 4.3 ft $1751.40 Cost $1733.40 0.00534 Ap 0.00472
Table 5.2: System Comparison
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Overall, the designs are similar. Both use the same steel deck, girder and spacing.
The only differences are in the joist selection and the concrete type. This indicates that
the algorithm does try to arrive at a light system; however, the lightest system is not
always the least expensive or the best performing.
Looking back at Table 5.1 for further comparisons of the designs show that the
systems seem to favor normal weight concrete because of the added cost associated with
lightweight concrete. This is may not be intuitive to a new designer, because the heavier
decks require heavier support systems. This would make a light deck system very
attractive. Another advantage is with time. While a hand design may take an hour, in
only a half hour, ten algorithm runs yielded 9 best systems, which out performed the hand
design.
5.4.2 Joist Span 2 (L = 30 ft)
The algorithm was run 10 times for a new configuration. All the Pareto Sets are available
in Appendix B.
Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 21 x 93 30 K 7 6 2.0" Nwt 1.5 VL 22 1301.60 0.005342 W 21 x 62 30 K 9 5 2.0" Nwt 2.0 VL 22 1421.30 0.004623 W 21 x 122 24 K 10 5 2.0" Nwt 2.0 VL 22 1422.30 0.004314 W 18 x 71 30 K 11 5 2.0" Nwt 1.5 VL 22 1382.20 0.004495 W 21 x 57 24 K 9 5 2.5" Nwt 1.5 VL 22 1418.70 0.004876 W 24 x 131 30 K 8 5 2.0" Nwt 1.5 VL 22 1399.30 0.004297 W 21 x 68 26 K 7 5 2.0" Nwt 2.0 VL 22 1422.30 0.005068 W 18 x 65 26 K 9 5 2.0" Nwt 1.5 VL 22 1379.20 0.004689 W 40 x 167 26 K 9 6 2.0" Nwt 1.5 VL 20 1413.80 0.00539
Table 5.3: Configuration 2 (L = 30 ft.) Lease Expensive Systems
The designs in Table 5.3 illustrate the ability of the algorithm to provide a wide array of
adequate solutions for the configuration and loading prescribed by the design constraints.
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Comparing the least expensive systems of multiple runs shows that the results of
the algorithm consistently arrive at good solutions, but repeatability can be problematic
from run to run. This is because of the random nature of the algorithm. Since the
algorithm begins with a set of randomly chosen systems, each run chugs along with
different joists, decks, and beams. The best of performing of these tend to provide the
bulk of the genetic material for the next generation. As the generations progress and
exchange parts, lighter systems are developed from the best performing of the previous
generations. Mutation does add to the pool of components, however, the bulk of the
components are from the previous generation. With this setup, the odds of getting the
same components in a system from run to run decrease with the size of the search space.
This is not to say that valuable knowledge cannot be gained from these runs, it is that one
run may not provide the best solution. Nuances, which escape the recognition of an
inexperienced designer, are better noted within a single Pareto Set.
Looking at Table 5.4, the trend of a suitable components becoming dominant is
evident in the first four systems. The only difference is the joist size.
Num Girder Joist Spacing Concrete Deck Cost Ap 1 W 21 x 68 26 K 7 5 2.0" Nwt 2.0 VL 22 1,422.30 0.00506 2 W 21 x 68 22 K 10 5 2.0" Nwt 2.0 VL 22 1,422.80 0.00503 3 W 21 x 68 30 K 11 5 2.0" Nwt 2.0 VL 22 1,424.00 0.00459 4 W 21 x 68 30 K 12 5 2.0" Nwt 2.0 VL 22 1,424.30 0.00458 5 W 27 x 84 30 K 12 5 2.0" Nwt 2.0 VL 22 1,429.10 0.00449 6 W 21 x 68 30 K 12 5 2.0" Nwt 1.5 VL 20 1,471.70 0.00446 7 W 27 x 84 30 K 12 5 2.0" Nwt 1.5 VL 20 1,476.50 0.00436 8 W 21 x 68 30 K 10 5 2.5" Nwt 1.5 VL 20 1,496.10 0.00414 9 W 21 x 68 30 K 11 5 2.5" Nwt 1.5 VL 20 1,496.50 0.00414 10 W 27 x 84 30 K 12 5 2.5" Nwt 1.5 VL 20 1,501.70 0.00399
Table 5.4: Pareto Set of Run 7 for Configuration 3
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These systems probably came from a combination of one dominant system and four
different mates. The presence of other systems in the Pareto set proves that one system
did not dominate the entire process. These designs can be considered as suggestions for
efficient designs. A designer could easily check to see if there are more efficient
systems. However, could the designer arrive at a suitable solution complete with
vibration performance analysis in less than an hour? It is not probable without assistance
from software.
5.4.3 Joist Span 3 (L = 50 ft)
This final problem in the span sensitivity study was much more difficult to solve, because
fewer systems are feasible with the longer span of 50 ft (i.e. there are less open web joists
available to span at the design loading). However, the algorithm was able to run and
generate 10 Pareto Sets, which are available in Appendix B. The least expensive systems
from the 10 Runs are available in Table 5.5.
Num Girder Joist Spacing Concrete Deck Cost Ap 1 W 27 x 84 30 K 12 3.8 2.0" Nwt 1.5 VL 22 2113.80 0.00522 2 W 30 x 90 30 K 11 3.3 2.0" Nwt 2.0 VL 22 2263.00 0.00461 3 W 27 x 84 30 K 11 3.3 3.0" Nwt 1.5 VL 22 2306.80 0.00402 4 W 33 x 241 28 K 12 2.7 2.0" Nwt 1.5 VL 22 2437.40 0.00379 5 W 24 x 117 30 K 11 3.3 2.0" Nwt 2.0 VL 22 2271.10 0.00463 6 W 21 x 111 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2198.60 0.00442 7 W 21 x 101 30 K 12 3.3 2.0" Nwt 2.0 VL 22 2238.60 0.00408 8 W 30 x 90 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2192.30 0.00426 9 W 27 x 94 28 K 12 3.3 2.5" Nwt 1.5 VL 22 2235.30 0.00380 10 W 21 x 101 28 K 12 3 2.5" Nwt 2.0 VL 22 2379.40 0.00358
Table 5.5: Configuration 3 (L = 50 ft.) Least Expensive Systems
The fact that fewer feasible systems are available for the problem is shown by the
fact that all the spacing values are small and close to one another. Small spacing values
have an inherent advantage, because more systems work with smaller spacing values (e.g.
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the joists are able to span the 50-ft distance more effectively). Therefore, larger spacing
values will be less likely to be selected generation after generation. The end result is the
effective phasing out of larger spacing values. It should also be noted that one solution
did not satisfy the vibration criteria. This run has a complete Pareto set, which when
observed, further displays the nuances discussed in the previous section.
Num Girder Joist Spacing Concrete Deck Cost Ap 1 W 27 x 84 30 K 12 3.8 2.0" Nwt 1.5 VL 22 2113.80 0.00522 2 W 30 x 235 30 K 12 3.8 2.0" Nwt 1.5 VL 22 2159.10 0.00492 3 W 33 x 118 30 K 11 3.3 2.0" Nwt 1.5 VL 22 2223.70 0.00491 4 W 33 x 118 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2224.00 0.00483 5 W 40 x 211 28 K 12 3.3 2.0" Nwt 1.5 VL 22 2251.80 0.00479 6 W 40 x 211 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2251.90 0.00469 7 W 24 x 103 30 K 12 3 3.0" Nwt 1.5 VL 22 2380.20 0.00330 8 W 24 x 131 30 K 11 3 3.0" Nwt 1.5 VL 22 2388.20 0.00319
Table 5.6: Part of Pareto Set from Run 1 of Configuration 3
The first two options are systems, which vary only in girder specification.
However, the effect of the different girder makes the system adequate in the vibration
performance category if 0.005 is a “firm” limit. To the designer, this shows that
adjusting the girder can allow a system to satisfy vibration criteria. An increase in girder
size could yield better vibration results with a smaller cost increase than adding a joist
and reducing the spacing. The remainder of this Pareto set as well as the other nine runs
are provided in Appendix B.
5.5 Superimposed Loading Variation
A second parametric study was considered in the design of the floor systems using the
GA. The second case uses a larger live load of 80 psf, which comes from ASCE 7-98
Table 4.1 for corridors in standard offices. This loading would be applicable to a
standard office use with partitions that create corridors over the bay. This loading will be
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used for the design of three bay configurations in which only the joist length specification
varies. These runs will add to the information used to identify any trends that could be
analyzed in future studies.
5.5.1 Joist Span 1 (L = 30 ft)
Just as the increase in span reduces the number of adequate systems, so does increasing
the live load. This will result in the same trend of smaller spacing values controlling.
Despite this, the algorithm will still create feasible systems. Ten runs all yielded Pareto
Sets, which are available in Appendix B. The least expensive of the Pareto solutions are
shown in Table 5.7.
Run Girder Joist Spacing Concrete Deck Cost Ap 1 W 30 x 124 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2502.50 0.00376 2 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.00 0.00389 3 W 27 x 102 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.90 0.00394 4 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.00 0.00389 5 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.00 0.00389 6 W 30 x 108 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2497.70 0.00383 7 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.00 0.00389 8 W 27 x 102 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2519.20 0.00453 9 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.00 0.00389 10 W 30 x 108 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2497.70 0.00383
Table 5.7: Configuration 4 (L = 30 ft.) Output of Pareto Set
Looking at these ten systems, repeatability can be seen with respect to most
components. Designs 2 and 10 are virtually identical, with only one joist size separating
them. Looking at the individual Pareto Sets in appendix B, more repeating systems can
be observed. These show that the algorithm can reproduce similar, if not the same,
results without a large number of runs.
5.5.2 Joist Span 2 (L = 40 ft)
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The increasing span length proved to be more difficult for the algorithm. Increasing the
joist span further depleted the number of possible solutions, which increased the
difficulty to obtain the best solution. The algorithm was able to create 10 Pareto Sets.
The lease expensive of each are available in Table 5.8.
Num Girder Joist Spacing Concrete Deck Cost Ap 1 W 27 x 94 28 K 10 3.3 2.0" Nwt 1.5 VL 22 1959.20 0.00463 2 W 24 x 94 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1941.10 0.00395 3 W 30 x 116 28 K 12 3.3 2.0" Nwt 1.5 VL 22 1966.60 0.00455 4 W 30 x 90 26 K 12 3.3 2.0" Nwt 1.5 VL 22 1940.00 0.00417 5 W 30 x 116 28 K 10 3.3 2.0" Nwt 1.5 VL 22 1965.80 0.00463 6 W 30 x 116 28 K 10 3.3 2.0" Nwt 1.5 VL 22 1965.80 0.00463 7 W 24 x 104 28 K 10 3 2.5" Nwt 2.0 VL 22 2110.70 0.00328 8 W 24 x 104 30 K 11 3.3 2.0" Nwt 1.5 VL 20 2064.10 0.00394 9 W 27 x 94 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1941.50 0.00394 10 W 27 x 84 28 K 10 3.3 2.0" Nwt 1.5 VL 22 1956.20 0.00463
Table 5.8: Configuration 5 (L = 40 ft.) Output of Pareto Set
The difficulty in obtaining feasible solutions was the result of a combination of
longer joist span and a heavier applied loading. The difficulty is exhibited by the fact
that the only spacing values to show up in the Pareto set are 3.3 ft and less. There were
not enough feasible systems to sustain any other spacing values.
5.5.3 Joist Span 3 (L = 50 ft)
The final run proved to be the most difficult run for the algorithm. When running the
algorithm for difficult systems (such as this), it can crash if there are no feasible systems
in the first generation (most likely inadequate open web joists in this case). The skim
module will cause the system to crash if no feasible systems exist in the first generation.
This problem can be overcome by increasing the population size to about 1000. With this
correction, the difficulty was overcome, and the algorithm eventually created 10 Pareto
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Sets, which can be viewed in Appendix B. As in the case of the 40-ft span, the only joist
spacings specified were 2.5-ft. Table 5.9 displays the least expensive systems for each
run.
Num Girder Joist Spacing Concrete Deck Cost Ap 1 W 30 x 124 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2502.50 0.00376 2 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.00 0.00389 3 W 27 x 102 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.90 0.00394 4 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.00 0.00389 5 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.00 0.00389 6 W 30 x 108 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2497.70 0.00383 7 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.00 0.00389 8 W 27 x 102 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2519.20 0.00453 9 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2515.60 0.00453 10 W 30 x 108 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2497.70 0.00383
Table 5.9: Configuration 6 (L = 50 ft.) Output of Pareto Set
5.6 Analysis of Results
This chapter presented six designs, which can be used to evaluate the performance of the
evolutionary algorithm under differing configurations and loadings. Unfortunately, there
has been no published research to provide a comparison. The evaluation of the designs
must come down to a brief analysis of the results and engineering judgment (at least at
this point).
Upon completion of the six runs, several trends can be observed. One trend
corresponds to the span of the joist. When dealing with longer spans, the algorithm has a
difficult time obtaining solutions. This occurs, because for spans of 50-ft and greater,
regardless of loading, there are only 19 joists out of 64, which can be considered (using
the pre-defined SJI tabular data). All the other joists do not have capacities listed in the
SJI catalog (SJI 2001). This retards the GA’s ability to determine the best design.
Therefore, this algorithm is better suited for smaller spans. If a theoretical methodology
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for computing joist capacity were utilized (rather than tabular data) this algorithmic
difficulty may be removed. A similar argument could be made for steel-concrete
composite deck capacities. If first principles were used to assign limiting spans for
deflection, shear strength, and moment strength, there would be no concerns regarding
algorithm crashes for long (or short) deck spans.
Another trend deals with a combination of spacing value and loading impact on
the joists. The joists are not intended to support large loads. So, when dealing with
larger loads, the spacing of the joists will often drop to the minimum, and the joists in the
systems will lighten to match the spacing. A correction was added to the fitness to
encourage larger spacing values. This correction was intended to simulate the added cost
incurred during the construction process. In other words, picking up 10 joists at small
spacing (with a crane) will cost more than picking up 4 heavier joists at a larger spacing.
The “pick time” costs money. The crane does not “feel” the weight of each joist.
Unfortunately the correction factor did not provide enough of an impact to overcome this
trend. This suggests that the joists also perform better for lighter loads.
Other trends were also visible when looking at the span study for the lighter loads.
Overall costs increase as the joist span increases. This seems reasonable because the
longer the span the more material is needed to bridge it. Also, as the span increases, the
spacing decreases. This is because fewer joists will create feasible systems with big
spacings and large spans. This lack of adequate joists is also apparent with the composite
steel deck. Lighter composite decks are more predominant because more joists will be
capable of supporting them. This increases the probability that a lighter deck will be
selected.
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Chapter 6 – Conclusions & Recommendations
6.1 Introduction
The genetic algorithm written in this thesis is capable of automatically designing steel
open web joist floor systems. This does not mean that the algorithm is the best it can be.
With alterations, it can be improved as a design tool, or used as a road map for future
research. This chapter explores possible improvements and future uses of the algorithm
developed in this thesis.
6.2 Improvements to the Algorithm
Before entertaining additional uses of the algorithm, suggestions for improvements to its
performance should be examined. Knowing how the algorithm performs allows the user
to make minor alterations to further the design process and expand its application base.
The first item, which could improve the algorithms perfomrance would be to
reduce the search space. One approach could be to shorten the joist portion of the
chromosome and editing the joist database to reduce the joists to those more suited to
large loads or long spans. This same principle could be applied to the girders or concrete
deck. Reducing the shape database to girders most suited to the design problem will
further reduce the search space, increasing the likely hood of reaching the best solution.
Other similar changes in the chromosome can also be enacted when dealing with
the concrete decks. When long spans or large loads are encountered, the short spacing
values required for the joists, results in decks that do not span long distances. This makes
the composite deck capacity less important to the design. Therefore, the emphasis of the
deck design could be reduced, or the database to the composite decks could be reduced.
This can be accomplished by providing fewer deck options, reducing the available decks
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from 256 to 64. This would provide a good design, while reducing the search space to
assist the algorithm.
The practice of search space reduction is inherent to the engineer. Designers
often automatically exclude components deemed (a-priori) to be uneconomical or
insufficient. The GA does not have this luxury, so it may need the user to intervene and
assist by reducing the search space.
6.3 Future Research
The cost calculations utilized in this algorithm are admittedly simplistic but reflective of
reality. The values were established to represent an approximate cost of the system
components. There are many expenses in creating a floor system, which are not
considered. Additional expense associated with the joists could be included in the cost
functions. One example is the cost associated with fireproofing. The expense of meeting
a fire rating could also greatly impact the total cost of the floor panel. These added costs
could affect the tendency for the algorithm to favor smaller spacing values.
Another option for exploration deals with the concrete deck. The composite VLI
decking utilized by this algorithm is intended to provide capacity over longer spans than
suggested by the spacing of the joists. The VLI deck may not be the best decking for the
problem. More steel deck options can be added to the database, and perhaps other
composite concrete slab systems would prove to be more economical.
Once the algorithm is running to satisfaction, it can be used to perform several
tasks. The best use for this algorithm would be for a comparative study of several panel
configurations using different framing systems. A “sister” algorithm has been written
(Shock 2003), which designs steel wide flange beam floor framing systems. It would be
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interesting to perform a side-by-side comparison of the optimum designs for a range of
panel configurations. From such an analysis, one could determine when joist systems are
more economical than wide-flange beam systems. A similar analysis was performed on
different types of concrete floor framing systems (Lucas and Havey 2001). For a
designer, it could be very valuable to know what system proves to be the most
economical for a given panel configuration.
Other analyses have started with architectural plans and designed entire floor
systems (Jain et al. 1991). This former effort did not consider vibration sensitivity in the
design. The algorithm developed in this former effort could be combined with the
algorithm developed in this thesis to automatically design an entire floor system.
In addition to the floor faming studies, several algorithm specific studies could be
attempted. These range from GA specific topics to floor system evaluation. The first
suggested study could be to make the vibration response another objective function or a
constraint. This would allow the algorithm to design systems, which have very stringent
vibration requirements. This would be suitable for designing the type of floor systems
required for operating rooms and electron microscopes (Allen 1997).
Another enhancement involves a more detailed look at the sources of vibration in
floor framing systems. For example, how does the most economical floor panel design
change if dance activity is the source of vibration rather than walking excitation? The
effect of vibration source on the design can be easily studied using the algorithm
developed.
The algorithm in this thesis assembles a Pareto set of solutions. Perhaps it might
be beneficial to run the algorithm to a single best performing system subject to a
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vibration constraint. This would provide a suitable design, rather than a set of feasible
systems from which one can be selected. That would allow the algorithm to become
more of a design tool.
A final aspect is to study the effect of the different crossover methods developed.
How does each crossover method affect the final design? Multiple options were provided
so that the end user could determine which method worked best. This is an interesting
question, which was scarcely considered during the creation of this algorithm. When
dealing with chromosomes with lengthy components, the impact of a crossover method,
like single point crossover, might be significant. Such a method may have the same
effect as a high mutation rate. In general, there is fertile ground to study the effect that all
genetic/evolutionary operators developed in this thesis have on the resulting designs.
In summary, the problem of designing open web steel joist floor systems is not
solved. This thesis is only a first look into automated and optimized design of these
complex floor framing systems.
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Appendix A – Genetic Algorithm Code
Interface and Algorithm 1 Master.m 85 2 ExecuteGA.m 87 All Algorithm Modules 1 Crossover3.m 91 2 DecodeChromo.m 93 3 Domination.m 95 4 Elimination.m 96 5 Elites.m 97 6 FindFIT.m 98 7 Floorvib2.m 101 8 GCheck.m 105 9 InitialPopulation.m 108
10 KJCheck.m 109 11 Mutate.m 112 12 PrintGeneration.m 113 13 Scaling.m 115 14 Scheck.m 116 15 Selection.m 118 16 Skim.m 119 17 Spacings.m 120 18 Statistics.m 121
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function master %---------------------------------------------------------------------------% % % % Purpose: % % This is the master program in a process to utilize a genetic algorithm % % to optimize the design of a floor system. This floor system consists of % % Beams, Girders, K-series Joists and Composite Metal Deck. % % % % Initial Programmer: % % Christopher W. Erwin (CWE) % % Marquette University % % Milwaukee, Wisconsin % % % % Secondary Programmers: % % % % Code Version: % % V 1.0 By:CWE % % % %---------------------------------------------------------------------------% % *** Obtain Additional User Options *** W = 30; % Bay Width (ft) L = 30; % Bay Length (ft) Q(1) = 15; % Superimposed Dead Load (psf) Q(2) = 50; % Live Load (psf) Q(3) = 11; % Waling Vibration Load (psf) % *** Set GA Parameters *** n = 20; % number of generations pop = 150; % population size cm = 1; % crossover method: 0 = single point, % 1 = component, % 2 = elite component mp = 0.05; % mutation probability (0.01 = 1%) fit = 0; % fitness method: 0 = Cost Based, % 1 = Component Weight, % 2 = Total Weight fs = 0; % fitness scaling factor ef = 1; % elitism flag: 0 = elitism off, 1 = elitism on % *** *** *** *** *** DO NOT EDIT ANYTHING BELOW THIS LINE *** *** *** *** ** % *** Set Genetic Code Properties *** % These properties are set to coincide with the Available Databases used % for Components to the floor system. The must be changed to match the % files provided to the G.A. w1 = 8; % Girder Code y1 = 8; % Slab Code z1 = 6; % Joist Code a1 = 3; % Spacing Code ls = w1+y1+z1+a1; % length of substring
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% *** Output GA Inputs to Screen *** fprintf('Problem Parameters\n\n'); fprintf('Bay Dimensions: L = %2.0f W = %2.0f \n',L,W); fprintf('Loading Information: SDL = %2.0f LL = %2.0f LLv = %2f \n\n',... Q(1),Q(2),Q(3)); fprintf('GA Parameters\n\n'); fprintf('Mutation Rate: mp = %2f \n',mp); fprintf('Population Size: pop = %2f \n',pop); fprintf('Generations: gen = %2f \n\n',n); if ef == 1 fprintf('Elitism: Elitism\n'); else fprintf('Elitism: No Elitism\n'); end if fs == 0 fprintf('Scaling: No Scaling\n'); else fprintf('Scaling: Scaling factor of %3.4f \n',fs); end if cm == 0 fprintf('Crossover Method: Single Point Crossover\n\n'); else if cm == 1 fprintf('Crossover Method: Dual Component Crossover\n\n'); else fprintf('Crossover Method: Elite Component Crossover\n\n'); end end % *** Execute GA *** executeGA([w1,x1,y1,z1,a1],[ls,n,cm,mp,fs,ef,pop],... [W,L,0,Q(1),Q(2),Q(3),fit]); % *** END PROGRAM ***
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function executeGA(unknoRange,GAparameters,SystemInputs) %---------------------------------------------------------------------------% % % % executeGA(unknoRange,GAparameters) % % % % Purpose: % % The mainline coding that orchestrates the GA application. % % % % Program Basis: % % LGADOS.FOR - A FORTRAN version of the LGA Genetic Algorithm. For % % distribution with the book "An Introduction to Genetic Algorithms for % % Scientists and Engineers", World Scientific 1998. % % % % David A. Coley % % Complex Systems Group % % Physics Building % % University of Exeter % % Exeter, EX4 4QL UK % % email: [email protected] % % % % Input Arguments: % % unknoRange = array containing the unknowns and the upper and lower % % bounds for the unknowns. % % Example: % % If 4 design variables 8,8,8,and 6 digits repsectively % % per variable in the chromosome are shown as follows: % % unknoRange[8,8,8,6] % % GAparameters = 1-D array containing Genetic Algorithm parameters % % = (1) - length of chromosome(all must have the same length)% % = (2) - maximum number of generations in the evolution % % = (3) - crossover method: 0 = single point crossover; % % 1 = elite componant crossover; % % = (4) - mutation probability (0.01 = 1%) % % = (5) - fitness scaling factor % % = (6) - elistism flag: 0 = elitism is off % % 1 = elitism on % % = (7) - population size % % Example: % % GAparameters(4,100,0.60,0.01,1.0,0,100) % % SystemInputs = 1-D Array containing Floor System Parameters % % = (1) - Bay Width % % = (2) - Bay Length % % = (3) - Joist Spacing % % = (4) - Dead Load % % = (5) - Live Load % % = (6) - Fitness Type, 0=No, 1=Yes % % Example: % % SystemInputs(20,15,5,10,80,0) % % Coding Audit: % % Sept-27-2002: Version 1.1 (modified to MATLAB functions) % % C.M. Foley - Marquette University % % All output to screen has been removed. % % Nov -25-2002: version 1.2 (further modified for MATLAB) % % C.W. Erwin - Marquette University % % Functions Converted to from Fortran to Matlab Format % % SystemInputs added to allow for GA system properties % %---------------------------------------------------------------------------%
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% Set initial Generation Gen = 1; % Create the initial population [chromosome] = initialPopulation(GAparameters(7), GAparameters(1)); % *** Decode chromosome to unknown real values *** [unknowns] = decodeChromo (chromosome,unknoRange,1); [s] = spacing(SystemInputs); % *** Find the fitness of each member of the initial population *** [fit,wt,phi,Adq] = FindFIT(unknowns,SystemInputs); [archive] = skim(Adq,[0],GAparameters(7),Gen); [archive] = elimination(archive,chromosome,1); % *** Find the mean fitness and the fittest individual.*** [meanfit,sumfit,maxfit,fitind] = Statistics(GAparameters(7),fit); % *** If scaling is on, then scale population prior to selection *** [fit,meanfit,sumfit] = Scaling(fit,GAparameters(5),meanfit,sumfit,... fitind,GAparameters(7)); % *** Plot fitness values to figure figure(1); hold on; x = 1; y = maxfit; scatter(x,y,'k'); y = meanfit; scatter(x,y,'r'); %----- Begin looping over generations after initial population is evaluated while Gen <= GAparameters(2) for i = 1:1:GAparameters(7) [mate1] = Selection(fit,sumfit,GAparameters(7)); [mate2] = Selection(fit,sumfit,GAparameters(7)); [chromosome] = CrossOver3(mate1,mate2,phi,chromosome,... unknoRange,i,(Gen+1),GAparameters(3)); end temp = GAparameters(2) - Gen + 1; temp2 = length(archive); Time = clock; fprintf('%6g %6g %2g:%02d:%06.4f\n',temp,temp2,Time(4),... Time(5),Time(6)); Gen = Gen + 1; [chromosome] = Mutate(chromosome,GAparameters(4),Gen); if GAparameters(6) == 1 [chromosome] = Elites(chromosome,fitind,Gen); end
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[unknowns] = decodeChromo (chromosome,unknoRange,Gen); [fit,wt,phi,Adq] = FindFIT(unknowns,SystemInputs); [archive] = skim(Adq,archive,GAparameters(7),Gen); [archive] = elimination(archive,chromosome,temp2); [f_n,Weqv,A_p] = floorvib2(unknowns,SystemInputs,GAparameters(7)); [meanfit,sumfit,maxfit,fitind] = Statistics(GAparameters(7),fit); [fit,meanfit,sumfit] = Scaling(fit,GAparameters(5),meanfit,... sumfit,fitind,GAparameters(7)); % * OUTPUT Fitness vs. Generation * figure(1); hold on; x = Gen; y = maxfit; scatter(x,y,'k'); y = meanfit; scatter(x,y,'r'); % * OUTPUT Fitness vs. Frequency figure(2); hold on; for i = 1:GAparameters(7) if A_p(i) > 0 x = A_p(i); y = fit(i); if Adq(i) == 1 scatter(x,y,'g'); else scatter(x,y,'r'); end end end % * OUTPUT Weight vs. A_o/g for Adequate Systems figure(3); hold on; for i = 1:GAparameters(7) if A_p(i) > 0 if Adq(i) == 1 x = A_p(i); y = wt(5,i); scatter(x,y,'g'); end end end end % END GENERATIONAL LOOP % * Figure Properties * figure(1); hold; title('Evolution of Fittest Individual'); xlabel('Generation'); ylabel('Fitness'); figure(2); hold; title('Indiviual Fitness and Vibration Performance'); xlabel('Accelleration Limit'); ylabel('Fitness');
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figure(3); hold; title('Indiviual Weight and Vibration Performance'); xlabel('Acceleration Limit'); ylabel('Total System Cost'); % Domination Calculations & Output Data [archive] = Domination(chromosome,archive,SystemInputs,unknoRange); PrintGeneration(archive,chromosome,SystemInputs,unknoRange); % *** END PROGRAM MAIN ***
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function [chromosome] = CrossOver3(mate1,mate2,Phi,chromosome,unknoRange,m,Gen,CM) %---------------------------------------------------------------------------% % % % Subroutine: CrossOver3(mate1,mate2,Phi,chromosome,unknoRange,m,Gen,CM) % % % % Purpose: % % Performs various crossover procedures % % % % Coding Audit: % % Aug-30-2002: Version 2.0 (original source - July-17-1998) % % Obtained at http://www.ex.ac.uk/cee/ga % % D.A. Coley - University of Exeter % % Sept-01-2002: Version 2.1 (modified original source to F90) % % Compaq Fortran Version 6.6 % % C.M. Foley - Marquette University % % All output to the screen has been removed. % % Jan-10-2003: Version 2.2 (modified F90 to Malab 6.0) % % C.W. Erwin - Marquette University % %---------------------------------------------------------------------------% % *** Carry out the crossover between MATE1 and MATE2 *** chromLength = length(chromosome(1,:,1)); popSize = length(chromosome(:,1,1)); numunkno = length(unknoRange(:)); if CM == 0; % SINGLE POINT CROSSOVER temp = 0; for i = 2:1:(numunkno - 1) temp = temp + unknoRange(i); end cp = temp * rand(1) + unknoRange(1); for i = 1:chromLength if i <= cp chromosome(m,i,Gen) = chromosome(mate1,i,(Gen-1)); else chromosome(m,i,Gen) = chromosome(mate2,i,(Gen-1)); end end else if CM == 1; % DUAL COMPONANT CROSSOVER temp = numunkno * rand(1); csite = ceil(temp); temp2 = numunkno * rand(1); csite2 = ceil(temp2); j = 8; if csite > csite2 temp3 = csite; csite = csite2; csite2 = temp3; end begin = 1; if csite > 1 for k = 1:(csite-1) begin = begin + unknoRange(k); end
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end fini = 8; if csite > 1 for k = 2:csite fini = fini + unknoRange(k); end end begin2 = 1; if csite2 > 1 for k = 1:(csite2 - 1) begin2 = begin + unknoRange(k); end end fini2 = 8; if csite2 > 1 for k = 2:csite2 fini2 = fini2 + unknoRange(k); end end for allele = 1:chromLength if allele < begin chromosome(m,allele,Gen) = chromosome(mate1,allele,(Gen-1)); else if allele > fini j = allele; break; else chromosome(m,allele,Gen) = chromosome(mate2,allele,(Gen-1)); end end end for allele = j:chromLength if allele < begin2 chromosome(m,allele,Gen) = chromosome(mate1,allele,(Gen-1)); else if allele > fini2 chromosome(m,allele,Gen) = chromosome(mate1,allele,(Gen-1)); else chromosome(m,allele,Gen) = chromosome(mate2,allele,(Gen-1)); end end end else % ELITE COMPONANT CROSSOVER k = 1.0; Phi(5,mate1) = Phi(4,mate1); Phi(5,mate2) = Phi(4,mate2); for i = 1:numunkno for j = 1:unknoRange(i) if Phi(i,mate1) <= Phi(i,mate2) chromosome(m,k,Gen) = chromosome(mate1,k,(Gen-1)); else chromosome(m,k,Gen) = chromosome(mate2,k,(Gen-1)); end k = k + 1; end end end end % *** END PROGRAM ***
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function [unknowns] = decodeChromo (chromosome,unknoRange,Gen) %---------------------------------------------------------------------------% % % % [unknowns] = decodeChromo ( popSize, chromLength, numUNKNO,... % % geneLength,chromosome, unknoRange ) % % % % Purpose: % % This subroutine decodes the binary string to integers % % % % Inputs: % % chromosome - Binary chromosome used for GA % % unknoRange - array containing the unknowns and the upper and lower % % bounds for the unknowns. % % Example: % % If 2 design variables with first variable bounds of % % 1.0 and 5.0 and second variable with bounds of 4.0 % % and 10.0, one would establish uknoBounds as: % % unknoRange[1.0,5.0;4.0,10.0] % % Gen - generation number % % % % Outputs: % % unknowns - Decoded chromosome values % % % % Coding Audit: % % Aug-30-2002: Version 2.0 (original source - July-17-1998) % % Obtained at http://www.ex.ac.uk/cee/ga % % D.A. Coley - University of Exeter % % % % Sept-01-2002: Version 2.1 (modified original source to F90) % % Compaq Fortran Version 6.6 % % C.M. Foley - Marquette University % % % % Nov-25-2002: Version 2.2 (modified F90 to Matlab) % % % % Dec-01-2002: Version 2.3 (Modified Decoding process to simplify) % % C.W. Erwin - Marquette University (CWE) % % % %---------------------------------------------------------------------------% % *** Simplified decoding of binary string chromosomes to integers *** chromLength = length(chromosome(1,:,1)); popSize = length(chromosome(:,1,1)); numUNKNO = length(unknoRange(:)); % initialize matrix for first generation unknowns(1:popSize,1:numUNKNO) = 0; for indiv = 1:popSize % Loop through the Population i = 1; % initialize power counter j = 1; % initialize componant number k = unknoRange(j); % initialize first componant limit for allele = 1:chromLength % Loop through all alleles per chromosome if allele <= k % Break up chromosome into componants unknowns(indiv,j) = unknowns(indiv,j) + ... chromosome(indiv,allele,Gen) * 2^(i-1); i = i + 1; % Counter increases power of 2 used for calc else % Resets when loop has reached next componant
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if j < numUNKNO % Check to verify that j should be increased j = j + 1; % Increas j to next componant k = k + unknoRange(j); % increase allelle limit end i = 1; % Reset power counter unknowns(indiv,j) = unknowns(indiv,j) + ... chromosome(indiv,allele,Gen) * 2^(i-1); i = i + 1; % Counter increases power of 2 used for calc end end for L = 1:numUNKNO % Add one for range of 1 - 256 unknowns(indiv,L) = unknowns(indiv,L) + 1; end end % *** END PROGRAM ***
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function [archive] = Domination(chromosome,archive,SystemInputs,unknoRange) %---------------------------------------------------------------------------% % % % [archive] = Domination(chromosome,archive); % % % % Purpose: To decode only the feasable chromosome and determine dominance. % % % % Inputs: % % chromosome = Binary chromosome used for GA % % archive = Array containing Locations of all Adequate Individuals % % in Chromosome % % % % Outputs: % % archive = Array of Feasible Solutions % % % % Coding Audit: % % Mar -05-2003: version 1.0 % % C.W. Erwin - Marquette University % % % %---------------------------------------------------------------------------% % Problem Issues popSize = length(archive(:,1)); Time = clock; fprintf('\nDomination %2d:%02d:%+6.4d\n \n',Time(4),Time(5),Time(6)); for i = 1:popSize [unknown] = decodeChromo(chromosome(archive(i,1),:,archive(i,2)),... unknoRange,1); unknowns(i,:) = unknown(:,:); end [fit,wt,Phi,X] = FindFIT(unknowns,SystemInputs); [f_n,Weqv,A_p] = floorvib2(unknowns,SystemInputs,popSize); for i = 1:popSize for j = 1:popSize if wt(5,i) > wt(5,j) if A_p(i) > A_p(j) archive(i,3) = 0; break; elseif A_p(i) < A_p(j) archive(i,3) = 1; end elseif wt(5,i) < wt(5,j) archive(i,3) = 1; end end end figure(3); hold on; for i = 1:popSize if archive(i,3) == 1 x = A_p(i); y = wt(5,i); scatter(x,y,'b'); end end % *** END PROGRAM ***
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function [archive2] = elimination(archive,chromosome,temp); %---------------------------------------------------------------------------% % % % [archive] = elimination(chromosome,archive); % % % % Purpose: To decode only the feasable chromosome and determine dominance. % % % % Inputs: % % chromosome = Binary chromosome used for GA % % archive = Array containing Locations of all Adequate Individuals % % in Chromosome % % % % Outputs: % % archive = array of feasible individuals % % % % Coding Audit: % % Mar -05-2003: version 1.0 % % C.W. Erwin - Marquette University % % % %---------------------------------------------------------------------------% check1 = length(archive(:,1)); chromlength = length(chromosome(1,:,1)); i = 1; %Time = clock; %fprintf(' Elimination %2d:%02d:%+6.4d\n \n',Time(4),Time(5),Time(6)); while i <= temp archive2(i,:) = archive(i,:); i = i + 1; end for x = (temp+1):check1 for y = 1:(i-1) j = 0; for z = 1:chromlength a = chromosome(archive(x,1),z,archive(x,2)); b = chromosome(archive2(y,1),z,archive2(y,2)); if a == b j = j + 1; end end if j == chromlength break; end end if j < chromlength archive2(i,:) = archive(x,:); i = i + 1; end end % *** END PROGRAM ***
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function [chromosome] = Elites(chromosome,fitind,Gen) %------------------------------------------------------------------------% % % % Subroutine: Elites % % % % Purpose: % % Applies ELITE by replacing a randomly chosen individual by the elite % % member from the previous population if the new maximum fitness is % % less then the previous value. % % % % Coding Audit: % % Aug-30-2002: Version 2.0 (original source - July-17-1998) % % Obtained at http://www.ex.ac.uk/cee/ga % % D.A. Coley - University of Exeter % % Sept-01-2002: Version 2.1 (modified original source to F90) % % Compaq Fortran Version 6.6 % % C.M. Foley - Marquette University % % Dec-15-2002: Version 2.3 (modified original source to Matlab 6.0) % % C.W. Erwin - Marquette University % %------------------------------------------------------------------------% chromLength = length(chromosome(1,:,1)); popSize = length(chromosome(:,1,1)); for allele = 1:chromLength chromosome(1,allele,Gen) = chromosome(fitind,allele,(Gen-1)); end
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function [fit,Cost,Phi,X] = FindFIT(unknowns,SystemInputs) %---------------------------------------------------------------------------% % % % [fit] = FindFIT(unknowns,SystemInputs) % % % % Purpose: % % The problem at hand is used to assign a positive (or zero) fitness to % % each beam, girder, joist and slab combination. % % % % Inputs: % % unknowns = Decoded chromosome values % % SystemInputs = 1-D Array containing Floor System Parameters % % = (1) - Bay Width % % = (2) - Bay Length % % = (3) - Joist Spacing % % = (4) - Dead Load % % = (5) - Live Load % % = (6) - Fitness Type, 0=No, 1=Yes % % Example: % % SystemInputs(20,15,5,10,80,0) % % % % Coding Audit: % % Aug-30-2002: Version 2.0 (original source - July-17-1998) % % Obtained at http://www.ex.ac.uk/cee/ga % % D.A. Coley - University of Exeter % % % % Sept-01-2002: Version 2.1 (modified original source to F90) % % Compaq Fortran Version 6.6 % % C.M. Foley - Marquette University % % % % Dec-01-2002: Version 2.2 (modified original source to MATLAB) % % Modify for System Floor Design % % C.W. Erwin - Marquette University (CWE) % %---------------------------------------------------------------------------% popSize = length(unknowns(:,1)); % *** Set Costs *** % For W Shapes [[$/Ton] % For Concrete [Lightweight, Normalweight] % For Deck [Provided in CompDeck.xls] % For K Joist [$/Ton] Costs = [[600, 0, 0] [ 18, 20, 0] [ 0, 0, 0] [600, 0, 0]]; % *** Set Penalties *** % For Girger [Moment, Shear, Defelction] % For Beam N/A [Moment, Shear, Defelction] % For Slab [Capacity, Shoring, Blank ] % For K Joist [Capacity, Bridging, Deflection] if SystemInputs(6) == 1 % Penalty Scheme for Individual Weight Fitness pen = [[4, 4, 4]
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[4, 4, 4] [2, 2, 0] [2, 1.02, 2]]; else if SystemInputs(6) == 2 % Penalty Scheme for Total Weight Fitness pen = [[4, 4, 3] [4, 4, 3] [4, 4, 0] [4, 1.02, 3]]; else % Penalty Scheme for Cost Based Fitness pen = [[ 1, 1 , 1 ] [ 1, 1 , 1 ] [ 1, 2 , 0 ] [ 1, 1.02, 1 ]]; end end % *** Set Geometric Properties *** W = SystemInputs(1); % Width of Bay L = SystemInputs(2); % Length of Bay [s] = spacing(SystemInputs); % Joist Spacing % Girder Capacity Checks [Phi1,wt] = GCheck(unknowns(:,1),unknowns(:,2),unknowns(:,3),... unknowns(:,4),SystemInputs,popSize,[0],pen(1,:)); % Joist Capacity Checks [Phi4,wt] = KJCheck(unknowns(:,3),unknowns(:,2),unknowns(:,4),... SystemInputs,popSize,wt,pen(4,:)); % Slab Capacity Checks [Phi3,wt] = SCheck(unknowns(:,2),unknowns(:,4),SystemInputs,... popSize,wt,pen(3,:)); Decks = xlsread('CompDeck.xls'); % *** Determine Fitness *** for i = 1:popSize wt1(i) = wt(1,i) * W; % Total Girder Weight wt3(i) = wt(3,i) * L * W; % Total Slab Weight wt4(i) = wt(4,i) * L * W/s(unknowns(i,4)); % Total Joist Weight wt(5,i) = wt1(i) + wt3(i) + wt4(i); % Total System Weight Cost(1,i) = wt(1,i)/2000 * Costs(1,1); % Total Girder Cost ($/ton) DeckCost(i) = Decks(unknowns(i,2),37); % Steel Deck Cost ($/100sf) if Decks(unknowns(i,2),2) == 0; SlabCost(i) = Costs(2,2)/27 * Decks(unknowns(i,2),13); else SlabCost(i) = Costs(2,1)/27 * Decks(unknowns(i,2),13); end Cost(3,i) = (DeckCost(i)/100 + SlabCost(i))*L*W; % Total Slab Cost Cost(4,i) = wt(4,i)/2000 * Costs(4,1) + 100*W/s(unknowns(i,4));
% Total Joist Cost Cost(5,i) = Cost(1,i)+Cost(3,i)+Cost(4,i); % Total System Cost Phi1(4,i) = Phi1(1,i) * Phi1(2,i) * Phi1(3,i); % Girder Phi Factors Phi3(3,i) = Phi3(1,i) * Phi3(2,i); % Slab Phi Factors
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Phi4(5,i) = Phi4(1,i) * Phi4(3,i) * Phi4(4,i); % Joist Phi Factors % System Fitness Calculation if SystemInputs(6) == 1 % Individual Weight Fitness fit(i) = 10000000 / (wt1(i) * Phi1(4,i) + ... wt3(i) * Phi3(3,i) + wt4(i) * Phi4(4,i)); else if SystemInputs(6) == 2 % Total Weight Fitness fit(i) = 10000000 / ((wt1(i) + wt3(i) + wt4(i))* ... Phi4(4,i) * Phi1(4,i) * Phi3(3,i)); else % Cost Fitness fit(i) = 1000000 / (Cost(1,i) * Phi1(4,i) + ... Cost(3,i) * Phi3(3,i) + Cost(4,i) * Phi4(5,i) * Phi4(2,i)); end end % Exported Phi Values Phi(1,i) = Phi1(4,i); Phi(3,i) = Phi3(3,i); Phi(4,i) = Phi4(5,i); Phi(5,i) = Phi1(4,i) * Phi3(3,i) * Phi4(5,i); % Adequacy Label if Phi(5,i) < 1.001 X(i,1) = 1; X(i,2) = unknowns(i,1); X(i,3) = unknowns(i,2); X(i,4) = unknowns(i,3); X(i,5) = unknowns(i,4); X(i,6) = Cost(5,i); else X(i,1) = 0; X(i,2) = 0; X(i,3) = 0; X(i,4) = 0; X(i,5) = 0; X(i,6) = 0; end end % *** END PROGRAM ***
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function [f_n,Weqv,A_p] = floorvib2(unknowns,geomInfo,popSize) %---------------------------------------------------------------------------% % % % [f_n,Weqv] = floorVib(unknowns,geomInfo,popSize) % % % % Purpose: % % This function computes the natural frequency and effective weight for % % the combined mode of vibration for a rectangular, regularly framed floor % % system using the procedure found in Murray, et al (199x) "Floor % % Vibrations Due to Human Activity", AISC Steel Design Guide Series 11, % % American Institute of Steel Construction, Chicago, IL. % % % % Input Arguments: % % Girder = Identifier for the girder member, number corresponding to % % database row. % % Beam/Joist = Identifier for the beam/joist member, number corresponding % % to database row. % % Deck = Identifier for the deck member, number corresponding to % % database row. % % geomInfo = = 1-D Array containing Floor System Parameters % % = (1) - Bay Width % % = (2) - Bay Length % % = (3) - Joist Spacing % % = (4) - Dead Load % % = (5) - Live Load % % = (6) - Composite Action, 0=No, 1=Yes % % Example: % % geomInfo = [20,15,5,10,1] % % % % Output Arguments: % % f_n = fundamental vibration frequency for the floor system (Hz) % % Weqv = equivalent combined girder beam mode panel weight (kips) % % A_p = non-dimentional acceleration value % % % % Initial Programmer: % % C.M. Foley % % Marquette University % % Milwaukee, Wisconsin % % % % Secondary Programmer % % C.W. Erwin % % Marquette University % % % % Coding Audit: % % Version 1.0 - October 10, 2002: C.M. Foley, Marquette University % % Only allowed WF beam members % % Version 1.1 - Modified to allow K-series open-web joist members % % Version 1.2 - Modified to fit GA and determine open web joist system % % vibration. % %---------------------------------------------------------------------------% % Decode Unknowns Girder = unknowns(:,1); Deck = unknowns(:,3); Joist = unknowns(:,4); % Geometric Data Decoding
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L = geomInfo(2); % Length of Joists (Bay Length) (ft) W = geomInfo(1); % Length of Girders (Bay Width) (ft) [spacing] = spacing(geomInfo); for i = 1:popSize; s(i) = spacing(unknowns(i,5)); % Joist Spacing end % Loading Data Decoding DL = geomInfo(4); % Superimposed Dead Load (psf) LL = geomInfo(6); % Superimposed Live Load (psf) % Steel Material Properties Fy = 50000; % Yeild Strength of Steel (psi) Es = 29000000; % Young's Modulus of Steel (psi) % *** Member Properties *** % Import Girder Table Girders = xlsread('AISC_W_Shapes.xls'); % Import Joist Table Joists = xlsread('SJI_K_Series.xls'); % Import Deck Table Decks = xlsread('CompDeck.xls'); for i = 1:popSize % Determine Girder Properties wg = Girders(Girder(i),2); % Weight of Girder (plf) Ag = Girders(Girder(i),3); % Gross Area og Girder (in^2) dg = Girders(Girder(i),4); % Depth of Girder (in) Ig = Girders(Girder(i),30); % Moment of Inertia Girder Major Axis (in^4) % Determine Joist Properties wj = Joists(Joist(i),3); % Weight of Joist (plf) dj = Joists(Joist(i),1); % Depth of Joist (in) % Calculate Joist Equivalent Moment of Inertia (in^4) j = 2 * L - 12; % Span Location in Table wSL = Joists(Joist(i),j); j = 2 * L - 11; % Span Location in Table wLL = Joists(Joist(i),j); % Allowable Live Load for 1/360 Deflection if wLL > 0 Ij = (5*wLL*L^4*1728*360)/(384*Es*L*12); % Joist Moment of Inertia % Calculate Equivalent Flange Area (Bottom) j = 2 * L - 12; % Span Location in Table wL = Joists(Joist(i),j); % Allowable Load Capacity (plf) Msji = (wSL * L^2 * 12)/8; % SJI Joist Moment Capacity (lb-in) d_ej = dj - 1; % Moment Couple Arm (in) Aj = Msji/(d_ej*0.6*Fy); % Area of Steel in Chords (in^2) yj = dj / 2; % Centroid Location (in) % Determine Deck Properties ws = Decks(Deck(i),5); % Total Deck weight (psf) tsc = Decks(Deck(i),3); % Cover Depth above Ribs (in) tsr = Decks(Deck(i),1); % Rib Height (in) tst = tsc + tsr; % Total Slab Depth (in) tav = tsc + (tsr/2); % Average depth of slab
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LWT = Decks(Deck(i),2); % Specification for Lightweight Concrete be1 = 0.4 * L * 12; % bf1 is 1/8 of The Span (in) be2 = s(i) * 12; % bf2 is 1/2 Center to Center Spacing (in) bes = min(be1,be2); % Effective Flange is Minimum of bf1, bf2. (in) % Concrete Material Properties Fc = 3000; % Compressive Strenght of Conctrete (psi) % Young's Modulus for Concrete (psi) Ec = (33*(150)^1.50)*sqrt(Fc)*(LWT)+(33*(110)^1.50)*sqrt(Fc)*(1-LWT); % Dynamic Modular Ratio n = Es/(1.35*Ec); % *** Joist Mode Properties *** % Composite Centroid Loacation (in) y_barj = (2*Aj*((dj/2) + tst) + ((bes/n) * tsc)*(tsc/2))/ ... (2*Aj + ((bes/n)*tsc)); % Composite Moment of Inertia (in^4) Icomp = Ij + 2*Aj*((dj/2) + tst - y_barj)^2 ... + (1/12) * (bes/n) * (tsc)^3 ... + ((bes/n)*tsc) * (y_barj-(tsc/2))^2; if (L*12/dj) <= 24 C_t = 0.90*(1-(expm(-0.28*L*12/dj)))^2.8; else C_t = 0.90; end gamma = 1/C_t - 1; % Calculate Effective Moment of Inertia (in^4) I_effj = 1/((gamma/Ij)+(1/Icomp)); % Effective Weight on Joist (plf) W_effj = (DL + LL + ws)*s(i) + wj; % Deflection Corresponding to Effective Loading Delta_j = (5*W_effj*L^4*1728)/(384*Es*I_effj); f_j = 0.18*sqrt(386.4/Delta_j); % Joist Mode Fundamental Frequency % Transformed Moment of Inertia per unit width in slab direction D_s = (12*tav^3)/(12*n); %(in^4/ft) % Transformed Moment of Inertia per unit width in joist direction D_j = (I_effj / s(i)); %(in^4/ft) % Value for non edge girders (Value is 1.0 for edge girders) C_j = 2.0; B_j1 = C_j*L*(D_s/D_j)^0.25; % (ft) % 2/3 or 3 times teh span for a 3 span condition (ft) B_j2 = 2*L; B_j = min(B_j1,B_j2); % Effective Beam Panel Width (ft) W_j = (W_effj/s(i))*B_j*L; % Weight of Beam Panel (lbs) % *** Girder Mode Properties *** be1 = 0.4 * W * 12; % bf1 is 1/8 of The Span (in) be2 = L * 12; % bf2 is 1/2 Center to Center Spacing (in) beg = min(be1,be2); % Effective Flange is Minimum of bf1, bf2 (in) y_barg = (Ag*((dg/2)+tst+2.0)+((beg/n)*tav*(tav/2)))/ ... (Ag + ((beg/n)*(tav))); % Composite Centroid Loacation (in) % Composite Moment of Inertia (in^4) I_compg = Ig + Ag * ((dg/2) + 2.0 + tst - y_barg)^2 ...
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+ (1/12) * (beg/n) * (tav)^3 ... + (beg/n) * (tav) * (y_barg - (tav/2))^2; % Calculate Effective Moment of Inertia (in^4) I_effg = Ig + (I_compg-Ig)/4; % Equivalent Uniform Loading W_effg = wg + (L/s(i)*W_effj); % Deflection Corresponding to Effective Loading Delta_g = (5*W_effg*W^4*1728)/(384*Es*I_effg); f_g = 0.18*sqrt(386.4/Delta_g); % Girder Mode Fundamental Frequency % Transformed Moment of Inertia per unit width in joist direction D_g = I_effg/L; % (in^4/ft) % Girders Supporting Joists on the Flange C_g = 1.6; % (1.8 for Beams Connected to Web) B_g1 = C_g*W*(D_j/D_g)^0.25; % Effective Girder Panel Width (ft) % 2/3 or 3 times teh span for a 3 span condition (ft) B_g2 = 2*W; B_g = min(B_g1,B_g2); % Effective Girder Panel Width (ft) W_g = (W_effg/L)*B_g*W; % Weight of Girder Panel (lbs) % *** Combined Mode Properties *** % Combined Mode Fundamental Frequency f_n(i) = 0.18*sqrt(386.4/(Delta_j+Delta_g)); % Equivalent Panel Weight (lbs) Weqv(i) = (Delta_j/(Delta_j+Delta_g))*W_j + (Delta_g/(Delta_j+Delta_g))*W_g; % Value Definitions P_o = 65; % Constant Force (lbs) Beta = 0.05; % Damping Ratio A_p(i) = (P_o * exp( -0.35 * f_n(i) ) )/( Beta * Weqv(i) ); else Lmax= Joists(Joist(i),110); if L >= Lmax f_n(i) = 0; Weqv(i) = 0; A_p(i) = 0; else f_n(i) = 0; Weqv(i) = 0; A_p(i) = 0; end end end % *** END PROGRAM ***
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function [Phi,wt]=GCheck(Girder,Deck,Joist,Spacing,geomInfo,popSize,wt,pen) %---------------------------------------------------------------------------% % % % Purpose: % % This function checks the adequacy of the Beam componant of a floor % % system and returns a rating based on the adequacy based on Strength and % % Defelction Criteria % % % % Inputs: % % Beam = Beam Number from Unknowns Variable % % Deck = Deck Number from Unknowns Variable % % geomInfo = 1-D Array containing Floor System Parameters % % = (1) - Bay Width % % = (2) - Bay Length % % = (3) - Joist Spacing % % = (4) - Dead Load % % = (5) - Live Load % % Example: % % geomInfo(20,15,5,10,80) % % popSize = Number of unknowns in problem % % % % Outputs: % % Phi = Weight Adjustment Facor % % Mn = Moment Capacity of Beams (ft-lbs) % % Vn = Shear Capacity of the Beam (lbs) % % Def = Beam deflection under aggigned loads (ft) % % wt = Weight of Section % % % % Initial Programmer: % % Christopher W. Erwin (CWE) % % T.A. % % Marquette University % % Milwaukee, Wisconsin % % % % Secondary Programmers: % % % % Code Version: % % V 1.0 By:CWE % %---------------------------------------------------------------------------% % *** Set Required Quantities *** fy = 50000; % Steel Yield Stress (psi) Es = 29000000; % Steel Modulus of Elasticity (psi) % Shear Stud Properties, Assuming use of a fsc = 60000; % Ultimate Stress of Shear Connector Asc = 1; % Area of Shear Connector % *** Set Geometric Properties *** L = geomInfo(2); % Joist Length W = geomInfo(1); % Girder Length [spacings] = spacing(geomInfo); for i = 1:popSize; s(i) = spacings(Spacing(i)); % Joist Spacing end
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% *** Import Beam Table *** Beams = xlsread('AISC_W_Shapes.xls'); % *** Import Concrete Deck Table *** Decks = xlsread('CompDeck.xls'); % *** Import Concrete Joist Table *** Joists = xlsread('SJI_K_Series.xls'); % *** Separate Out Data Required for Beams and Deck *** for i = 1:popSize % Girder Properties Bwt(i)= Beams(Girder(i),2); % Wt per linear ft As(i) = Beams(Girder(i),3); % Area of W Section db(i) = Beams(Girder(i),4); % Depth of Section bfw(i)= Beams(Girder(i),7); % Beam Flange Width tw(i) = Beams(Girder(i),10); % Web Thickness tf(i) = Beams(Girder(i),11); % Beam Flange Thickness Is(i) = Beams(Girder(i),30); % Beam Moment of Inertia Zx(i) = Beams(Girder(i),31); % Zx % K-Series Joist Properties Jwt(i)= Joists(Joist(i),3); % Wt per linear ft % Deck Properties Swt(i)= Decks(Deck(i),5); % Deck Weight (psf) end % *** Loading Information *** DL = geomInfo(4); % Dead Load (psf) LL = geomInfo(5); % Live Load (psf) for i = 1:popSize % Superimposed Factored Loads on Girder (psf) SFL(i) = 1.2*( DL + Swt(i) + Jwt(i)/s(i) ) + 1.6*LL ; % Superimposed Service Loads on Girder (psf) SSL(i) = max( ( DL + Swt(i) + Jwt(i)/s(i) ), LL ); end for i = 1:popSize % *** Evaluate Weight *** % Load Calculations wt(1,i) = Bwt(i); % Girder Self Weight (plf) % ***** Evaluate Service Deflections ***** % Deflection Due to SSL (in) def(i) = ( 5 * ( Bwt(i) + SSL(i) * L ) * W^4 ) * 1728 / (384 * Es * Is(i)) ; Adef(i) = L*12/360; % Allowable Deflection (same for all cases) temps = def(i) / Adef(i); if temps <= 1; Phi(3,i) = 1;
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else Phi(3,i) = pen(3) * temps; end end % ***** Evaluate Shear Capacity ***** for i = 1:popSize % Shear Capacity Required (lbs) Vu(i) = ( ( 1.2*Bwt(i) + SFL(i)*L ) * W) / 2 ; % Shear Capacity Provided (lbs) Vn(i) = (fy * tw(i) * db(i)); temps = Vu(i) / ( 0.85*Vn(i) ) ; if temps <= 1; Phi(2,i) = 1; else Phi(2,i) = pen(2) * temps; end end % ***** Evaluate Plastic Moment Capacity ***** for i = 1:popSize Mx(i) = fy * Zx(i) / 12; % Plastic Moment Capacity (lb-ft) Mu(i) = ( ( 1.2*Bwt(i) + SFL(i)*L )* W^2 )/8 ; % Moment Capacity Required (lb-ft) temps = Mu(i) / ( 0.9*Mx(i) ); if temps < 1; Phi(1,i) = 1; else Phi(1,i) = pen(1) * temps; end end % *** END PROGRAM ***
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function [chromosome] = initialPopulation(popSize, chromLength); %---------------------------------------------------------------------------% % % % [chromosome,unknowns] = initialPopulation(popSize, chromLength,... % % numUNKNO,geneLength, unknoRange)% % % % Purpose: % % Function to create an initial random population. % % % % Inputs: % % popSize - Population Size % % chromLength - Total String Length % % numUNKNO - Number of unknowns % % geneLength - Length of substring % % unknoRange - array containing the unknowns and the upper and lower % % bounds for the unknowns. % % Example: % % If 2 design variables with first variable bounds of % % 1.0 and 5.0 and second variable with bounds of 4.0 % % and 10.0, one would establish uknoBounds as: % % unknoRange[1.0,5.0;4.0,10.0] % % % % Outputs: % % chromosome - randomly generated chromosomes % % unknowns - decoded chromosome values % % % % Coding Audit: % % Sept-27-2002: Version 2.1 (modified original source to MATLAB) % % Matlab Ver. 6.5.0.175711 Release 13 % % The MathWorks, Inc. % % Programmer: C.M. Foley - Marquette University % % % %---------------------------------------------------------------------------% % *** Create initial population of chromosomes *** for indiv = 1:popSize % Loop over the population randNum = rand(1,chromLength) ; % Generate an array of random numbers for allele = 1:chromLength % Loop over all alleles in chromosome if randNum(allele) > 0.5 chromosome(indiv,allele,1) = 1 ; else chromosome(indiv,allele,1) = 0 ; end end end % *** END PROGRAM ***
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function [Phi,wt] = KJCheck(Joist,Deck,Spacing,geomInfo,popSize,wt,pen) %---------------------------------------------------------------------------% % % % Purpose: % % This function checks the adequacy of the Deck componant of a floor % % system and returns a rating based on the adequacy based on Strength and % % Defelction Criteria % % % % Inputs: % % Joist = Joist Number from Unknowns Variable % % Deck = Deck Number from Unknowns Variable % % geomInfo = 1-D Array containing Floor System Parameters % % = (1) - Bay Width % % = (2) - Bay Length % % = (3) - Joist Spacing % % = (4) - Dead Load % % = (5) - Live Load % % Example: % % geomInfo(20,15,5,10,80) % % popSize = Number of unknowns in problem % % % % Outputs: % % Phi = Weight Factors for Fitness % % Mn = Moment Capacity of Beams (ft-lbs) % % Vn = Shear Capacity of the Beam (lbs) % % Def = Beam deflection under aggigned loads (ft) % % wt = Weight of Section % % % % Initial Programmer: % % Christopher W. Erwin (CWE) % % T.A. % % Marquette University % % Milwaukee, Wisconsin % % % % Secondary Programmers: % % % % Code Version: % % V 1.0 By:CWE % % % %---------------------------------------------------------------------------% % *** Set Geometric Properties *** W = geomInfo(1); % Width of Bay L = geomInfo(2); % Length of Bay [spacings] = spacing(geomInfo); for i = 1:popSize; s(i) = spacings(Spacing(i)); % Joist Spacing end % *** Loading Information *** DL = geomInfo(4); % Dead Load (psf) LL = geomInfo(5); % Live Load (psf)
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% *** Import Joist and Deck Data *** Joists = xlsread('SJI_K_Series.xls'); Decks = xlsread('CompDeck.xls'); % *** Section Properites *** for i = 1:popSize wt(4,i) = Joists(Joist(i),3); % Joist Weight (plf) dwt(i) = Decks(Deck(i),5); % Deck Weight (psf) % Column in Joist Table with Joist Capacity k = 2 * L - 12; WTL(i) = Joists(Joist(i),k); % Joist Capacity (plf) % Column in Joist Table with Max Defelction Load j = 2 * L - 11; WSL(i) = Joists(Joist(i),j); % Joist Max Load for Deflection (plf) Smax(i)= Joists(Joist(i),110); % Location of Shaded Region in Tables (ft) end % *** Penalty Assesment *** for i = 1:popSize if L <= Smax(i) Phi(2,i) = 1.0; Load1(i) = (DL + LL + dwt(i)) * s(i); % Max Capacity Load2(i) = (DL + dwt(i)) * s(i); % Construction Deflection Load3(i) = LL * s(i); % Live Load Deflection if WTL(i) <= Load1(i) % Penalty for Insufficient Capacity Phi(1,i) = pen(1) * (Load1(i)/(WTL(i)+1)); else Phi(1,i) = 1.0; end if WSL(i) <= Load2(i) % Penalty for Deflection Violations Phi(3,i) = pen(3) * (Load2(i)/(WSL(i)+1)); else Phi(3,i) = 1.0; end if WSL(i) <= Load3(i) % Penalty for Deflection Violations Phi(4,i) = pen(3) * (Load3(i) / (WSL(i)+1)); else Phi(4,i) = 1.0; end else Phi(2,i) = pen(2); % Penalty for Bridging Required Load1(i) = (DL + LL + dwt(i)) * s(i); % Max Capacity Load2(i) = (DL + dwt(i)) * s(i); % Construction Deflection Load3(i) = LL * s(i); % Live Load Deflection if WTL(i) <= Load1(i) % Penalty for Insufficient Capacity Phi(1,i) = pen(1) * (Load1(i)/(WTL(i)+1)); else Phi(1,i) = 1.0; end if WSL(i) <= Load2(i) % Penalty for Deflection Violations Phi(3,i) = pen(3) * (Load2(i) / (WSL(i)+1)); else
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Phi(3,i) = 1.0; end if WSL(i) <= Load3(i) % Penalty for Deflection Violations Phi(4,i) = pen(3) * (Load3(i) / (WSL(i)+1)); else Phi(4,i) = 1.0; end end end % *** END PROGRAM ***
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function [chromosome] = Mutate(chromosome,mp,Gen) %---------------------------------------------------------------------------% % % % Subroutine: Mutate(chromosome,mp,Gen) % % % % Purpose: % % Visit each bit of each string very occasionally flipping a "1" % % to a "0" or visa-versa. % % % % Coding Audit: % % Aug-30-2002: Version 2.0 (original source - July-17-1998) % % Obtained at http://www.ex.ac.uk/cee/ga % % D.A. Coley - University of Exeter % % Sept-01-2002: Version 2.1 (modified original source to F90) % % Compaq Fortran Version 6.6 % % C.M. Foley - Marquette University % % All output to screen has been removed. % % Jan-13-2003: Version 2.2 (modified F90 to Matlab 6.0) % % C.W. Erwin - Marquette University % %---------------------------------------------------------------------------% % *** Step through population and apply mutation *** popSize = length(chromosome(:,1,1)); chromLength = length(chromosome(1,:,1)); for i = 1:popSize % Loop through Population if rand <= mp % Mutate if mp percent of alleles for allele = 1:chromLength % Loop through Genelength if rand > 0.125 if chromosome(i,allele,Gen) == 0 % Changes Allele to inverse value chromosome(i,allele,Gen) = 1; else chromosome(i,allele,Gen) = 0; end end end end end % *** END PROGRAM ***
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function PrintGeneration(archive,chromosome,SystemInputs,unknoRange) %---------------------------------------------------------------------------% % % % Subroutine: PrintGeneration(archive,chromosome,SystemInputs,unknoRange) % % % % Purpose: % % Prints results to the screen and to the file(s) % % % % Coding Audit: % % Aug-30-2002: Version 2.0 (original source - July-17-1998) % % Obtained at http://www.ex.ac.uk/cee/ga % % D.A. Coley - University of Exeter % % Sept-01-2002: Version 2.1 (modified original source to F90) % % Compaq Fortran Version 6.6 % % C.M. Foley - Marquette University % % All output to screen has been removed (commented out). % %---------------------------------------------------------------------------% Time = clock; fprintf('Printing %2d:%02d:%06.4d\n \n',Time(4),Time(5),Time(6)); % *** Import Tables *** Beams = xlsread('AISC_W_Shapes.xls'); Decks = xlsread('CompDeck.xls'); Joists = xlsread('SJI_K_Series.xls'); popSize = length(archive(:,1)); unknowns(:,:) = archive(:,4:7); for i = 1:popSize wt(5,i) = archive(i,8); end [f_n,Weqv,A_p] = floorvib2(unknowns,SystemInputs,popSize); [sp] = spacing(SystemInputs); for i = 1:popSize s(i) = sp(unknowns(i,4)); % Joist Spacing if Decks(unknowns(i,3),2) == 1 Conc(i) = 'N' ; else Conc(i) = 'L' ; end end minW = max(wt(5,:)); Light = 0; % Search for Lightest System for i = 1:popSize if archive(i,3) == 1; if wt(5,i) <= minW minW = wt(5,i); Light = i; end end end % Display Lightest System fprintf('Most Economical Floor System \n' ); fprintf('Girder: W %2d x %-3d \n',Beams(unknowns(Light,1),1),...
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Beams(unknowns(Light,1),2)); fprintf('Joist: %2d K %2d at %4.3d ft \n',Joists(unknowns(Light,3),1),... Joists(unknowns(Light,3),2),s(Light)); fprintf('Deck: %3.1d in %1swt on %03.1d VL %2d \n',Decks(... unknowns(Light,2),3), Conc(Light),Decks(unknowns(Light,2),1),... Decks(unknowns(Light,2),6)); fprintf('Total Cost: %8.1d \n',wt(5,Light)); fprintf('Acceleration Limit: %08.7d \n\n',A_p(Light)); minA = max(A_p); Rigid = 0; % Search for Stiffest System for i = 1:popSize if archive(i,3) == 1; if A_p(i) <= minA minA = A_p(i); Rigid = i; end end end % Display Stiffest System fprintf('Stiffest Floor System \n' ); fprintf('Girder W %2d x %-3d \n',Beams(unknowns(Rigid,1),1),... Beams(unknowns(Rigid,1),2)); fprintf('Joist: %2d K %2d at %4.3d ft \n',Joists(unknowns(Rigid,3),1),... Joists(unknowns(Rigid,3),2),s(Rigid)); fprintf('Deck: %3.1d in %1swt on %03.1d VL %2d \n',Decks(unknowns(Rigid,2),3), Conc(Rigid), ... Decks(unknowns(Rigid,2),1),Decks(unknowns(Rigid,2),6)); fprintf('Total Cost: %8.1d \n',wt(5,Rigid)); fprintf('Acceleration Limit: %08.7d \n\n',A_p(Rigid)); % Output Front to Text File file = fopen('Dominators.txt','w+'); j = 1; fprintf(file,... 'Num, Girder , Joist , Spacing, Concrete, Deck, Cost, A_p,\n'); for i = 1:popSize if archive(i,3) == 1; fprintf(file,'%3d, ',j); fprintf(file,'W %2d x %-3d, ',Beams(unknowns(i,1),1),... Beams(unknowns(i,1),2)); fprintf(file,'%2d K %2d, %3.1f, ',Joists(unknowns(i,3),1),... Joists(unknowns(i,3),2),s(i)); fprintf(file,' %3.1f" %1swt, %3.1f VL %2d, ',Decks(unknowns(i,2),3), Conc(i), ... Decks(unknowns(i,2),1),Decks(unknowns(i,2),6)); fprintf(file,'%9.1f, ',wt(5,i)); fprintf(file,'%08.7f\n',A_p(i)); j = j+1; end end fprintf('Number of Dominators on Front: %d \n',(j-1)); fprintf('Dominators are output to Dominators.txt\n\n'); fclose('all'); % *** END PROGRAM ***
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function [fit,meanfit,sumfit] = scaling(fit,fsf,meanfit,sumfit,fitind,popSize) %---------------------------------------------------------------------------% % % % [fit,meanfit,sumfit] = scaling(fit,fsf,meanfit,sumfit,fitind,popSize) % % % % Purpose: % % Apply Linear FIT Scaling in the following form: scaledfitness = a* % % fitness + b subject to the requirement that, "meanscaledfitness = % % meanfitness " and " bestscaledfitness = c* MEANF " where c, the scaling % % constant, is set by the user. % % % % INPUTS: % % fit - Individual System Fitness % % fsf - Fitness Scaling Factor % % meanfit - Average fitness from Statistics % % sumfit - Sum of all Fitnesses % % fitind - The most fit individual in the Generation % % popSize - Number of individuals per generation % % Gen - Generation being evaluated % % OUTPUTS: % % fit - Individual System Fitness % % meanfit - Average fitness from Statistics % % sumfit - Sum of all Fitnesses % % % % Coding Audit: % % Aug-30-2002: Version 2.0 (original source - July-17-1998) % % Obtained at http://www.ex.ac.uk/cee/ga % % D.A. Coley - University of Exeter % % Sept-01-2002: Version 2.1 (modified original source to F90) % % Compaq Fortran Version 6.6 % % C.M. Foley - Marquette University % % All output to screen has been removed (commented out). % % Jan-13-2003: Version 2.2 (modified F90 to Matlab 6.0) % % C.W. Erwin - Marquette University % % % %---------------------------------------------------------------------------% if fsf ~= 0 % Routine will only run if Scaling Factor was inputted % Routine will only run if calc will yield useable values if (fit(fitind) - meanfit) > 0 A = (fsf -1) * meanfit / (fit(fitind) - meanfit); B = ( 1 - A ) * meanfit; sumfit = 0; % *** Scale Fitness *** for i = 1:popSize fit(i) = A * fit(i) + B; if fit(i) < 0 % Prevent negative values at end of cycle fit(i) = 0; end sumfit = sumfit + fit(i); % Re-Calc sumfit end meanfit = sumfit/popSize; % Re-Calc meanfit end end % *** END PROGRAM ***
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function [phi,wt] = SCheck(Deck,Spacing,geomInfo,popSize,wt,pen) %---------------------------------------------------------------------------% % % % Purpose: % % This function checks the adequacy of the Deck componant of a floor % % system and returns a rating based on the adequacy based on Strength and % % Defelction Criteria % % % % Inputs: % % Deck = Deck Number from Unknowns Variable % % geomInfo = 1-D Array containing Floor System Parameters % % = (1) - Bay Width % % = (2) - Bay Length % % = (3) - Joist Spacing % % = (4) - Dead Load % % = (5) - Live Load % % Example: % % geomInfo(20,15,5,10,80) % % popSize = Number of unknowns in problem % % % % Outputs: % % Phi = Weight Factors for Fitness % % Mn = Moment Capacity of Beams (ft-lbs) % % Vn = Shear Capacity of the Beam (lbs) % % Def = Beam deflection under aggigned loads (ft) % % wt = Weight of Section % % % % Initial Programmer: % % Christopher W. Erwin (CWE) % % T.A. % % Marquette University % % Milwaukee, Wisconsin % % % % Secondary Programmers: % % % % Code Version: % % V 1.0 By:CWE % % % %---------------------------------------------------------------------------% % *** Set Geometric Properties *** W = geomInfo(1); % Width of Bay L = geomInfo(2); % Length of Bay [spacings] = spacing(geomInfo); for i = 1:popSize; s(i) = spacings(Spacing(i)); % Joist Spacing if s(i) <= 5 s(i) = 5; else s(i) = ceil(s(i)); end end % *** Loading Information *** DL = geomInfo(4); % Dead Load (psf)
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LL = geomInfo(5); % Live Load (psf) FL = 1.2 * DL + 1.6 * LL; % Factored Load (psf) % *** Import Concrete Deck Table *** Decks = xlsread('CompDeck.xls'); % *** Table Retrieval *** for i = 1:popSize wt(3,i) = Decks(Deck(i),5); % Deck Weight (psf) smax(i) = Decks(Deck(i),14); % Maximum un-shored Length j = 2*s(i) + 5; SLmax(i) = Decks(Deck(i),j); % Maximum Superimposed Loads Allowed end % *** Shoaring and Capacity Checks *** for i = 1:popSize if smax(i) >= s(i) phi(2,i) = 1.0; % If shoring is not required, No penalty Asessed Load = DL + LL; if SLmax(i) < Load % Penalty for Insufficient Capacity phi(1,i) = pen(1) * (Load / (SLmax(i)+1)); else phi(1,i) = 1.0; end else phi(2,i) = pen(2); % Penalty for Shoring Load = DL + LL; if SLmax(i) < Load % Penalty for Insufficient Capacity phi(1,i) = pen(1) * (Load / (SLmax(i)+1)); else phi(1,i) = 1.0; % Penalty for Insufficient Capacity end end end % *** END PROGRAM ***
118
function [mate] = Selection(fit,sumfit,popSize) %---------------------------------------------------------------------------% % % % Subroutine: Selection(fit,sumfit,popSize) % % % % Purpose: % % Select a single individual using fitness proportionate selection. % % % % Coding Audit: % % Aug-30-2002: Version 2.0 (original source - July-17-1998) % % Obtained at http://www.ex.ac.uk/cee/ga % % D.A. Coley - University of Exeter % % Sept-01-2002: Version 2.1 (modified original source to F90) % % Compaq Fortran Version 6.6 % % C.M. Foley - Marquette University % % All output to screen has been removed. % % Jan-13-2002: Version 2.1 (modified from F90 to Matlab 6.0) % % C.W. Erwin - Marquette University % % % %---------------------------------------------------------------------------% % *** Initialize values *** i = 0; sum = 0; rwheel = rand * sumfit; % *** Step through population and find mate *** while sum < rwheel if i >= popSize i = 0; end i = i + 1; sum = sum + fit(i); end mate = i; % *** END PROGRAM ***
119
function [archive] = skim(Adq,archive,popSize,Gen); %---------------------------------------------------------------------------% % % % Purpose: % % Skim the locations of individuals which satisfy all requirements % % % % Inputs: % % Adq = Flag for System Adequacy % % % % archive = Array containing Locations of Adequate Chromosomes % % popSize = Population Size % % Gen = Generation Number % % % % Outputs: % % archive = Updated Archive % % % % Initial Programmer: % % Christopher W. Erwin (CWE) % % Marquette University % % Milwaukee, Wisconsin % %---------------------------------------------------------------------------% if Gen == 1 count = 1; a = 1; else count = length(archive(:,1)); if count > 1 count = count + 1; end a = 2; end for i = a:popSize if Adq(i) == 1 archive(count,1) = i; archive(count,2) = Gen; count = count + 1; end end % *** END PROGRAM ***
120
function [s] = spacing(SystemInputs) %---------------------------------------------------------------------------% % % % Purpose: % % Create a set of spacing values % % % % Inputs: % % SystemInputs = 1-D Array containing Floor System Parameters % % = (1) - Bay Width % % = (2) - Bay Length % % = (3) - Joist Spacing % % = (4) - Dead Load % % = (5) - Live Load % % = (6) - Floor Vibration Live Load % % = (7) - Composite Action, 0=No, 1=Yes % % Example: % % SystemInputs(20,15,5,10,1) % % Outputs: % % [s] = array of spacing values % % % % Initial Programmer: % % Christopher W. Erwin (CWE) % % Marquette University % % Milwaukee, Wisconsin % % % % Code Version: % % V 1.0 By:CWE % % % %---------------------------------------------------------------------------% W = SystemInputs(1); % Bay Width smax = 550/(SystemInputs(4)+SystemInputs(5)+16); % Max Allowable Spacing(ft) smin = 2.5; % Min Allowable Spacing (ft) Min = ceil(W/smax); % Min Number of Spacings Max = fix(W/smin); % Max Number of Spacings % Formulate 8 Spacings for i = 1:8 s(i) = W / (Min + (i-1)); end for i = 2:1:8 k = 1; if s(i) < 2.5 s(i) = s(k); k = k + 1; end end % *** END PROGRAM ***
121
function [meanfit,sumfit,maxfit,fitind] = Statistics(popSize,fit) %---------------------------------------------------------------------------% % % % Subroutine: [meanfit,sumfit,maxfit,fitind] = Statistics(popSize,fit) % % % % Purpose: % % Calculate the sum of fitness across the population and find the best % % individual, then apply ELITE if required. % % % % Coding Audit: % % Aug-30-2002: Version 2.0 (original source - July-17-1998) % % Obtained at http://www.ex.ac.uk/cee/ga % % D.A. Coley - University of Exeter % % Sept-01-2002: Version 2.1 (modified original source to F90) % % Compaq Fortran Version 6.6 % % C.M. Foley - Marquette University % % Dec-15-2002: Version 2.2 (modified original source to MATLAB 6.0) % % C.W. Erwin - Marquette University % %---------------------------------------------------------------------------% % *** Set initial Variables *** maxfit = 0; sumfit = 0; % *** Find Most Fit System *** for i=1:popSize if fit(i) > maxfit maxfit = fit(i); fitind = i; end end % Total of all Sytem Fitnesses for i=1:popSize sumfit = sumfit + fit(i); end % Calculate Mean Fitness meanfit = sumfit/popSize; % *** END PROGRAM ***
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Appendix B – Algorithm Design Result
Section B.1 Joist Span 2 (L = 40 ft) Loading 1 (LL = 50 psf) 123 Section B.2 Joist Span 1 (L = 30 ft) Loading 1 (LL = 50 psf) 133 Section B.3 Joist Span 3 (L = 50 ft) Loading 1 (LL = 50 psf) 143 Section B.4 Joist Span 1 (L = 30 ft) Loading 2 (LL = 80 psf) 153 Section B.5 Joist Span 2 (L = 40 ft) Loading 2 (LL = 80 psf) 163 Section B.6 Joist Span 3 (L = 50 ft) Loading 2 (LL = 80 psf) 173
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B.1 Joist Span 2 (L = 30 ft) Loading 1 (LL = 50 psf) B.1 – Run 1
Num Girder Joist Spacing Concrete Deck Cost Ap 1 W 21 x 93 30 K 7 6 2.0" Nwt 1.5 VL 22 1301.60 0.00534 2 W 18 x 71 26 K 7 5 2.0" Nwt 1.5 VL 22 1380.60 0.00492 3 W 18 x 71 30 K 7 5 2.0" Nwt 1.5 VL 22 1381.00 0.00451 4 W 21 x 93 30 K 7 5 2.0" Nwt 1.5 VL 22 1387.60 0.00441 5 W 30 x 99 30 K 7 5 2.0" Nwt 1.5 VL 22 1389.40 0.00431 6 W 24 x 250 30 K 7 5 2.0" Nwt 1.5 VL 22 1434.70 0.00412 7 W 36 x 256 30 K 7 5 2.0" Nwt 1.5 VL 22 1436.50 0.00400 8 W 30 x 391 30 K 7 5 2.0" Nwt 1.5 VL 22 1477.00 0.00395 9 W 27 x 102 30 K 7 5 3.5" Nwt 2.0 VL 22 1516.30 0.00354 10 W 24 x 250 30 K 7 5 3.5" Nwt 2.0 VL 22 1560.70 0.00326 11 W 30 x 326 30 K 7 5 3.5" Nwt 2.0 VL 22 1583.50 0.00307 12 W 30 x 391 30 K 7 5 3.5" Nwt 2.0 VL 22 1603.00 0.00303 13 W 27 x 336 30 K 7 4.3 3.0" Nwt 3.0 VL 22 1749.50 0.00247 14 W 30 x 391 24 K 10 4.3 4.5" Nwt 1.5 VL 21 1882.00 0.00222 15 W 30 x 326 30 K 7 3 3.5" Nwt 2.0 VL 22 1983.50 0.00219 16 W 27 x 336 22 K 9 3 4.0" Nwt 2.0 VL 22 1985.60 0.00212 17 W 30 x 391 24 K 8 3.3 4.5" Nwt 1.5 VL 20 1991.60 0.00204 18 W 30 x 357 30 K 7 3 3.0" Nwt 3.0 VL 22 2055.80 0.00193 19 W 30 x 211 28 K 7 3 4.5" Nwt 2.0 VL 21 2153.40 0.00189 20 W 27 x 336 30 K 7 2.7 4.5" Nwt 1.5 VL 20 2175.30 0.00171 21 W 27 x 336 30 K 7 2.7 4.5" Nwt 2.0 VL 20 2201.10 0.00165 22 W 30 x 391 30 K 7 2.7 4.5" Nwt 1.5 VL 21 2281.80 0.00163 23 W 40 x 278 24 K 10 2.5 4.5" Nwt 1.5 VL 16 2618.10 0.00162
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B.1 – Run 2 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 21 x 62 30 K 9 5 2.0" Nwt 2.0 VL 22 1421.30 0.00462 2 W 21 x 62 30 K 11 5 3.0" Nwt 1.5 VL 22 1429.90 0.00387 3 W 24 x 76 30 K 11 5 3.0" Nwt 1.5 VL 22 1434.10 0.00376 4 W 27 x 84 30 K 11 5 3.0" Nwt 1.5 VL 22 1436.50 0.00368 5 W 24 x 117 30 K 11 5 3.0" Nwt 1.5 VL 22 1446.40 0.00361 6 W 30 x 173 28 K 12 5 3.0" Nwt 1.5 VL 22 1463.40 0.00340 7 W 36 x 230 26 K 10 5 3.3" Nwt 2.0 VL 22 1541.10 0.00335 8 W 30 x 173 24 K 9 4.3 3.0" Nwt 1.5 VL 22 1561.90 0.00326 9 W 21 x 111 28 K 12 4.3 3.5" Nwt 1.5 VL 22 1569.40 0.00320 10 W 30 x 173 24 K 9 4.3 3.5" Nwt 1.5 VL 22 1586.50 0.00296 11 W 36 x 280 28 K 12 4.3 3.0" Nwt 1.5 VL 22 1595.50 0.00290 12 W 30 x 261 24 K 9 4.3 3.5" Nwt 1.5 VL 22 1612.90 0.00282 13 W 30 x 124 28 K 12 3.8 3.5" Nwt 1.5 VL 22 1673.30 0.00277 14 W 30 x 173 24 K 9 3.8 3.5" Nwt 1.5 VL 22 1686.50 0.00275 15 W 36 x 170 28 K 12 3.8 3.5" Nwt 1.5 VL 22 1687.10 0.00256 16 W 30 x 124 30 K 9 3.3 3.5" Nwt 1.5 VL 22 1772.20 0.00256 17 W 33 x 354 24 K 9 3.3 3.5" Nwt 1.5 VL 22 1840.80 0.00232 18 W 30 x 148 22 K 11 3.8 4.5" Nwt 2.0 VL 20 1845.10 0.00230 19 W 36 x 798 30 K 11 3.8 3.5" Nwt 1.5 VL 22 1875.30 0.00218 20 W 24 x 207 26 K 8 3.8 4.5" Nwt 3.0 VL 22 1885.70 0.00206 21 W 24 x 207 30 K 11 3.8 4.5" Nwt 3.0 VL 22 1887.00 0.00202 22 W 36 x 230 22 K 11 3.8 4.5" Nwt 3.0 VL 22 1893.10 0.00188 23 W 36 x 798 30 K 11 2.7 3.0" Nwt 1.5 VL 22 2150.70 0.00187 24 W 24 x 207 22 K 10 2.5 4.5" Nwt 3.0 VL 22 2285.90 0.00173 25 W 14 x 730 22 K 11 2.5 4.5" Nwt 2.0 VL 20 2419.70 0.00166
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B.1 – Run 3 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 21 x 122 24 K 10 5 2.0" Nwt 2.0 VL 22 1422.30 0.00431 2 W 21 x 147 26 K 8 5 2.0" Nwt 2.0 VL 22 1429.50 0.00420 3 W 21 x 101 24 K 10 5 3.0" Nwt 1.5 VL 22 1440.60 0.00389 4 W 21 x 147 24 K 10 5 2.5" Nwt 2.0 VL 22 1454.40 0.00387 5 W 21 x 147 28 K 10 5 2.5" Nwt 2.0 VL 22 1454.80 0.00374 6 W 30 x 191 24 K 10 5 2.5" Nwt 2.0 VL 22 1467.60 0.00363 7 W 21 x 147 26 K 8 4.3 2.5" Nwt 2.0 VL 22 1554.10 0.00351 8 W 21 x 147 26 K 8 4.3 2.5" Nwt 2.0 VL 22 1554.10 0.00351 9 W 40 x 249 26 K 8 4.3 2.0" Nwt 2.0 VL 22 1560.10 0.00349 10 W 30 x 191 26 K 8 4.3 2.5" Nwt 2.0 VL 22 1567.30 0.00328 11 W 30 x 235 24 K 10 4.3 2.5" Nwt 2.0 VL 22 1580.80 0.00327 12 W 40 x 249 24 K 9 4.3 2.5" Nwt 2.0 VL 22 1584.70 0.00317 13 W 40 x 249 26 K 8 4.3 2.5" Nwt 2.0 VL 22 1584.70 0.00309 14 W 40 x 249 28 K 10 4.3 3.3" Nwt 2.0 VL 22 1647.00 0.00293 15 W 40 x 324 26 K 8 4.3 3.0" Nwt 2.0 VL 20 1722.40 0.00272 16 W 40 x 249 30 K 7 4.3 3.5" Nwt 2.0 VL 20 1725.20 0.00243 17 W 40 x 149 28 K 10 4.3 4.5" Nwt 1.5 VL 21 1809.80 0.00234 18 W 40 x 211 24 K 12 3.3 3.5" Nwt 2.0 VL 20 1914.90 0.00225 19 W 40 x 249 30 K 7 3.3 3.5" Nwt 2.0 VL 20 1925.20 0.00209 20 W 36 x 650 20 K 9 3 4.5" Nwt 1.5 VL 22 2079.00 0.00200 21 W 33 x 318 28 K 10 2.7 3.0" Nwt 3.0 VL 20 2234.70 0.00182 22 W 40 x 249 28 K 10 2.5 4.3" Nwt 2.0 VL 18 2472.30 0.00173 23 W 30 x 261 26 K 8 3 4.5" Nwt 3.0 VL 17 2677.90 0.00168
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B.1 – Run 4 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 18 x 71 30 K 11 5 2.0" Nwt 1.5 VL 22 1382.20 0.00449 2 W 30 x 116 30 K 11 5 2.0" Nwt 1.5 VL 22 1395.70 0.00426 3 W 30 x 116 30 K 11 5 2.0" Nwt 1.5 VL 22 1395.70 0.00426 4 W 30 x 116 30 K 12 5 2.0" Nwt 1.5 VL 22 1396.10 0.00426 5 W 30 x 191 30 K 11 5 2.0" Nwt 1.5 VL 22 1418.20 0.00413 6 W 36 x 232 28 K 12 5 2.0" Nwt 1.5 VL 22 1430.70 0.00410 7 W 30 x 116 26 K 12 4.3 2.0" Nwt 1.5 VL 22 1495.80 0.00402 8 W 30 x 116 28 K 12 4.3 2.0" Nwt 1.5 VL 22 1495.90 0.00392 9 W 30 x 132 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1500.50 0.00381 10 W 30 x 132 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1500.90 0.00381 11 W 40 x 331 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1560.20 0.00352 12 W 30 x 116 30 K 11 3.8 2.0" Nwt 1.5 VL 22 1595.70 0.00350 13 W 30 x 132 30 K 12 3.8 2.0" Nwt 1.5 VL 22 1600.90 0.00347 14 W 40 x 235 30 K 12 4.3 2.5" Nwt 2.0 VL 22 1601.80 0.00336 15 W 40 x 167 30 K 8 4.3 4.0" Nwt 2.0 VL 22 1635.50 0.00235 16 W 40 x 167 30 K 8 3.8 4.0" Nwt 2.0 VL 22 1735.50 0.00219 17 W 24 x 176 30 K 8 2.7 4.0" Nwt 2.0 VL 22 2038.20 0.00203 18 W 36 x 256 30 K 11 2.7 3.0" Nwt 2.0 VL 20 2103.30 0.00200 19 W 30 x 148 30 K 11 2.5 4.0" Nwt 2.0 VL 22 2130.70 0.00188 20 W 36 x 256 24 K 12 2.5 3.5" Nwt 3.0 VL 22 2279.60 0.00187 21 W 36 x 232 30 K 8 2.5 2.5" Nwt 3.0 VL 20 2283.40 0.00187
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B.1 – Run 5 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 21 x 57 24 K 9 5 2.5" Nwt 1.5 VL 22 1418.70 0.00487 2 W 21 x 57 30 K 8 5 2.5" Nwt 1.5 VL 22 1419.10 0.00459 3 W 40 x 199 20 K 10 5 2.0" Nwt 2.0 VL 22 1445.20 0.00430 4 W 36 x 194 30 K 10 5 2.5" Nwt 1.5 VL 22 1460.70 0.00405 5 W 36 x 194 30 K 11 5 2.5" Nwt 1.5 VL 22 1461.10 0.00405 6 W 40 x 199 30 K 10 5 2.5" Nwt 1.5 VL 22 1462.20 0.00401 7 W 36 x 232 30 K 8 5 2.5" Nwt 1.5 VL 22 1471.60 0.00400 8 W 36 x 232 30 K 11 5 2.5" Nwt 1.5 VL 22 1472.50 0.00400 9 W 30 x 326 30 K 8 5 2.5" Nwt 1.5 VL 22 1499.80 0.00396 10 W 30 x 326 30 K 10 5 2.5" Nwt 1.5 VL 22 1500.30 0.00396 11 W 30 x 326 30 K 11 5 2.5" Nwt 1.5 VL 22 1500.70 0.00396 12 W 18 x 97 28 K 8 5 2.5" Nwt 2.0 VL 20 1529.30 0.00392 13 W 18 x 97 30 K 8 5 2.5" Nwt 2.0 VL 20 1529.50 0.00387 14 W 18 x 97 30 K 11 5 2.5" Nwt 2.0 VL 20 1530.40 0.00386 15 W 24 x 104 28 K 8 5 2.5" Nwt 2.0 VL 20 1531.40 0.00380 16 W 30 x 108 30 K 8 5 2.5" Nwt 2.0 VL 20 1532.80 0.00365 17 W 40 x 199 30 K 8 4.3 2.0" Nwt 2.0 VL 22 1545.50 0.00339 18 W 33 x 221 30 K 10 5 2.5" Nwt 2.0 VL 20 1567.20 0.00336 19 W 36 x 232 30 K 8 5 2.5" Nwt 2.0 VL 20 1570.00 0.00333 20 W 36 x 232 30 K 10 5 2.5" Nwt 2.0 VL 20 1570.50 0.00332 21 W 36 x 280 30 K 8 5 2.5" Nwt 2.0 VL 20 1584.40 0.00328 22 W 21 x 182 26 K 10 4.3 3.5" Nwt 2.0 VL 22 1615.50 0.00289 23 W 36 x 245 26 K 10 3.8 3.5" Nwt 2.0 VL 22 1734.40 0.00236 24 W 27 x 217 30 K 10 3.8 3.5" Nwt 3.0 VL 22 1839.80 0.00224 25 W 33 x 221 30 K 8 3.8 3.5" Nwt 3.0 VL 22 1840.50 0.00214 26 W 36 x 245 28 K 8 3.8 3.5" Nwt 3.0 VL 22 1847.50 0.00211 27 W 30 x 292 30 K 8 3 3.5" Nwt 3.0 VL 22 2061.80 0.00184 28 W 30 x 292 30 K 10 3 3.5" Nwt 3.0 VL 22 2062.30 0.00184 29 W 27 x 146 28 K 9 2.5 4.0" Nwt 3.0 VL 22 2242.50 0.00183 30 W 30 x 326 20 K 10 2.5 3.5" Nwt 3.0 VL 22 2271.70 0.00182 31 W 33 x 241 30 K 12 2.5 4.3" Nwt 2.0 VL 20 2290.90 0.00178 32 W 40 x 392 28 K 8 2.5 4.0" Nwt 3.0 VL 22 2316.20 0.00140
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B.1 – Run 6 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 24 x 131 30 K 8 5 2.0" Nwt 1.5 VL 22 1399.30 0.00429 2 W 21 x 201 28 K 9 5 2.0" Nwt 1.5 VL 22 1420.20 0.00429 3 W 21 x 201 30 K 8 5 2.0" Nwt 1.5 VL 22 1420.30 0.00421 4 W 36 x 230 30 K 8 5 2.0" Nwt 2.0 VL 22 1471.60 0.00420 5 W 21 x 182 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1514.90 0.00390 6 W 21 x 201 28 K 12 4.3 2.0" Nwt 1.5 VL 22 1521.40 0.00387 7 W 24 x 229 28 K 12 4.3 2.0" Nwt 1.5 VL 22 1529.80 0.00380 8 W 30 x 124 22 K 10 4.3 3.5" Nwt 1.5 VL 22 1594.30 0.00356 9 W 21 x 182 28 K 12 3.8 2.0" Nwt 1.5 VL 22 1615.70 0.00355 10 W 24 x 162 28 K 12 5 3.0" Nwt 3.0 VL 22 1623.70 0.00342 11 W 33 x 221 30 K 12 5 3.0" Nwt 3.0 VL 22 1641.60 0.00310 12 W 24 x 131 22 K 10 4.3 3.0" Nwt 3.0 VL 22 1688.10 0.00301 13 W 30 x 90 22 K 10 4.3 3.5" Nwt 3.0 VL 22 1701.00 0.00281 14 W 30 x 124 28 K 8 4.3 3.5" Nwt 3.0 VL 22 1711.20 0.00258 15 W 36 x 230 28 K 10 4.3 3.0" Nwt 3.0 VL 22 1718.30 0.00249 16 W 24 x 162 28 K 9 3.8 3.5" Nwt 3.0 VL 22 1822.70 0.00243 17 W 21 x 201 28 K 9 3.8 3.5" Nwt 3.0 VL 22 1834.40 0.00242 18 W 30 x 124 22 K 10 3.3 3.5" Nwt 3.0 VL 22 1911.20 0.00238 19 W 30 x 124 30 K 8 3.3 3.5" Nwt 3.0 VL 22 1911.40 0.00226 20 W 36 x 230 26 K 6 3.3 3.5" Nwt 3.0 VL 22 1942.40 0.00226 21 W 30 x 90 30 K 12 3 3.5" Nwt 3.0 VL 22 2002.50 0.00226 22 W 21 x 201 22 K 10 3 3.5" Nwt 3.0 VL 22 2034.30 0.00225 23 W 27 x 84 30 K 8 3 4.5" Nwt 3.0 VL 20 2139.20 0.00211 24 W 36 x 359 22 K 10 2.7 3.0" Nwt 3.0 VL 22 2156.50 0.00190 25 W 36 x 230 22 K 10 2.5 3.5" Nwt 3.0 VL 22 2243.00 0.00176 26 W 36 x 798 24 K 9 2.5 3.5" Nwt 2.0 VL 20 2389.80 0.00164
129
B.1 – Run 7 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 21 x 68 26 K 7 5 2.0" Nwt 2.0 VL 22 1422.30 0.00506 2 W 21 x 68 22 K 10 5 2.0" Nwt 2.0 VL 22 1422.80 0.00503 3 W 21 x 68 30 K 11 5 2.0" Nwt 2.0 VL 22 1424.00 0.00459 4 W 21 x 68 30 K 12 5 2.0" Nwt 2.0 VL 22 1424.30 0.00458 5 W 27 x 84 30 K 12 5 2.0" Nwt 2.0 VL 22 1429.10 0.00449 6 W 21 x 68 30 K 12 5 2.0" Nwt 1.5 VL 20 1471.70 0.00446 7 W 27 x 84 30 K 12 5 2.0" Nwt 1.5 VL 20 1476.50 0.00436 8 W 21 x 68 30 K 10 5 2.5" Nwt 1.5 VL 20 1496.10 0.00414 9 W 21 x 68 30 K 11 5 2.5" Nwt 1.5 VL 20 1496.50 0.00414 10 W 27 x 84 30 K 12 5 2.5" Nwt 1.5 VL 20 1501.70 0.00399 11 W 24 x 162 30 K 10 5 2.5" Nwt 1.5 VL 20 1524.30 0.00383 12 W 24 x 162 30 K 11 5 2.5" Nwt 1.5 VL 20 1524.70 0.00383 13 W 40 x 235 30 K 11 5 2.5" Nwt 1.5 VL 20 1546.60 0.00352 14 W 40 x 235 30 K 12 5 2.5" Nwt 1.5 VL 20 1547.00 0.00352 15 W 21 x 68 30 K 11 4.3 3.5" Nwt 1.5 VL 22 1556.30 0.00335 16 W 30 x 108 22 K 10 4.3 3.5" Nwt 1.5 VL 22 1567.20 0.00321 17 W 40 x 235 30 K 10 4.3 2.5" Nwt 1.5 VL 20 1646.20 0.00319 18 W 21 x 68 30 K 11 3.8 3.5" Nwt 1.5 VL 22 1656.30 0.00314 19 W 21 x 68 30 K 12 3.8 3.5" Nwt 1.5 VL 22 1656.70 0.00314 20 W 27 x 84 30 K 12 3.8 3.5" Nwt 1.5 VL 22 1661.50 0.00294 21 W 14 x 455 30 K 12 4.3 3.5" Nwt 1.5 VL 22 1672.80 0.00280 22 W 40 x 235 30 K 8 3.8 3.5" Nwt 1.5 VL 22 1705.50 0.00240 23 W 40 x 235 28 K 9 3 3.5" Nwt 1.5 VL 22 1905.40 0.00210 24 W 40 x 297 30 K 12 3 3.0" Nwt 2.0 VL 22 1926.00 0.00208 25 W 40 x 235 22 K 10 3.3 3.5" Nwt 3.0 VL 22 1944.50 0.00204 26 W 40 x 235 22 K 10 3 3.5" Nwt 3.0 VL 22 2044.50 0.00192 27 W 36 x 245 22 K 10 2.7 4.5" Nwt 1.5 VL 22 2058.10 0.00182 28 W 40 x 297 30 K 12 2.5 3.0" Nwt 2.0 VL 22 2126.00 0.00181 29 W 27 x 336 28 K 12 2.5 4.0" Nwt 2.0 VL 22 2187.30 0.00169
130
B.1 – Run 8 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 18 x 65 26 K 9 5 2.0" Nwt 1.5 VL 22 1379.20 0.00468 2 W 24 x 68 26 K 9 5 2.0" Nwt 1.5 VL 22 1380.10 0.00462 3 W 24 x 68 30 K 8 5 2.0" Nwt 1.5 VL 22 1380.40 0.00445 4 W 24 x 68 30 K 12 5 2.0" Nwt 1.5 VL 22 1381.70 0.00443 5 W 33 x 141 26 K 9 5 2.0" Nwt 1.5 VL 22 1402.00 0.00436 6 W 33 x 141 30 K 8 5 2.0" Nwt 1.5 VL 22 1402.30 0.00418 7 W 33 x 141 30 K 11 5 2.0" Nwt 1.5 VL 22 1403.20 0.00418 8 W 33 x 141 30 K 12 5 2.0" Nwt 1.5 VL 22 1403.60 0.00418 9 W 24 x 306 30 K 12 5 2.0" Nwt 1.5 VL 22 1453.10 0.00406 10 W 33 x 141 30 K 12 5 3.0" Nwt 1.5 VL 22 1454.00 0.00338 11 W 40 x 392 30 K 12 5 3.3" Nwt 2.0 VL 22 1590.90 0.00309 12 W 24 x 306 30 K 9 4.3 3.0" Nwt 1.5 VL 22 1602.20 0.00299 13 W 33 x 141 22 K 11 4.3 4.0" Nwt 1.5 VL 20 1692.60 0.00276 14 W 33 x 141 30 K 8 3.8 4.3" Nwt 1.5 VL 22 1741.60 0.00263 15 W 40 x 167 26 K 9 3.8 4.3" Nwt 1.5 VL 22 1749.10 0.00257 16 W 30 x 357 22 K 10 3.8 4.3" Nwt 1.5 VL 22 1806.20 0.00256 17 W 40 x 327 30 K 9 3.3 3.0" Nwt 1.5 VL 22 1808.50 0.00235 18 W 40 x 167 22 K 11 3.3 4.0" Nwt 1.5 VL 20 1900.40 0.00231 19 W 30 x 357 30 K 10 3.3 4.3" Nwt 1.5 VL 22 1906.90 0.00219 20 W 33 x 141 30 K 8 3.8 4.0" Nwt 3.0 VL 20 1931.10 0.00214 21 W 40 x 327 26 K 10 2.7 3.0" Nwt 1.5 VL 22 2008.60 0.00211 22 W 40 x 327 28 K 10 2.7 3.0" Nwt 1.5 VL 22 2008.80 0.00206 23 W 30 x 292 26 K 8 3.3 4.5" Nwt 1.5 VL 18 2124.00 0.00199 24 W 30 x 357 30 K 12 3.3 4.0" Nwt 2.0 VL 18 2163.80 0.00194 25 W 40 x 327 28 K 10 3 4.0" Nwt 2.0 VL 18 2253.80 0.00178
131
B.1 – Run 9 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 40 x 167 26 K 9 6 2.0" Nwt 1.5 VL 20 1413.80 0.00539 2 W 24 x 84 26 K 8 5 2.5" Nwt 1.5 VL 22 1426.80 0.00463 3 W 24 x 84 28 K 12 5 2.5" Nwt 1.5 VL 22 1428.30 0.00450 4 W 30 x 90 28 K 12 5 2.5" Nwt 1.5 VL 22 1430.10 0.00442 5 W 30 x 261 26 K 8 5 2.5" Nwt 1.5 VL 22 1479.90 0.00423 6 W 40 x 297 26 K 8 5 2.5" Nwt 1.5 VL 22 1490.70 0.00411 7 W 24 x 84 28 K 12 4.3 2.5" Nwt 1.5 VL 22 1528.30 0.00410 8 W 24 x 84 28 K 9 4.3 2.5" Nwt 2.0 VL 22 1535.50 0.00356 9 W 24 x 146 26 K 8 4.3 2.5" Nwt 2.0 VL 22 1553.80 0.00346 10 W 40 x 297 26 K 10 4.3 2.5" Nwt 2.0 VL 22 1599.60 0.00304 11 W 40 x 167 24 K 10 3.8 2.5" Nwt 2.0 VL 22 1660.40 0.00302 12 W 30 x 90 28 K 12 3.3 2.5" Nwt 2.0 VL 22 1738.50 0.00297 13 W 33 x 201 26 K 12 3.8 4.0" Nwt 2.0 VL 22 1746.70 0.00225 14 W 30 x 261 22 K 10 3.3 4.0" Nwt 2.0 VL 20 1953.50 0.00216 15 W 30 x 357 22 K 10 3.3 4.0" Nwt 2.0 VL 20 1982.30 0.00207 16 W 33 x 201 28 K 9 2.7 4.0" Nwt 2.0 VL 22 2045.60 0.00185 17 W 40 x 277 24 K 10 2.7 4.0" Nwt 3.0 VL 22 2181.80 0.00160 18 W 36 x 527 28 K 12 2.5 4.0" Nwt 2.0 VL 21 2424.60 0.00150
132
B.1 – Run 10 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 24 x 55 20 K 10 5 2.5" Nwt 1.5 VL 22 1418.20 0.00522 2 W 24 x 55 22 K 10 5 2.5" Nwt 1.5 VL 22 1418.30 0.00496 3 W 24 x 55 28 K 8 5 2.5" Nwt 1.5 VL 22 1418.30 0.00464 4 W 24 x 55 28 K 9 5 2.5" Nwt 1.5 VL 22 1418.40 0.00464 5 W 24 x 55 28 K 10 5 2.5" Nwt 1.5 VL 22 1418.80 0.00463 6 W 24 x 55 30 K 11 5 2.5" Nwt 1.5 VL 22 1419.40 0.00454 7 W 24 x 94 28 K 8 5 2.5" Nwt 1.5 VL 22 1430.00 0.00448 8 W 24 x 94 28 K 10 5 2.5" Nwt 1.5 VL 22 1430.50 0.00447 9 W 24 x 94 30 K 11 5 2.5" Nwt 1.5 VL 22 1431.10 0.00439 10 W 21 x 111 30 K 11 5 2.5" Nwt 1.5 VL 22 1436.20 0.00438 11 W 24 x 131 30 K 11 5 2.5" Nwt 1.5 VL 22 1442.20 0.00429 12 W 40 x 149 30 K 11 5 2.5" Nwt 1.5 VL 22 1447.60 0.00410 13 W 36 x 230 28 K 10 5 2.5" Nwt 1.5 VL 22 1471.30 0.00408 14 W 36 x 230 30 K 11 5 2.5" Nwt 1.5 VL 22 1471.90 0.00400 15 W 40 x 324 28 K 8 5 2.5" Nwt 1.5 VL 22 1499.00 0.00397 16 W 40 x 324 28 K 10 5 2.5" Nwt 1.5 VL 22 1499.50 0.00397 17 W 12 x 305 30 K 11 5 2.5" Nwt 2.0 VL 22 1522.40 0.00396 18 W 40 x 324 30 K 11 5 2.5" Nwt 2.0 VL 22 1528.10 0.00364 19 W 40 x 324 28 K 8 4.3 2.5" Nwt 1.5 VL 22 1599.00 0.00356 20 W 33 x 169 28 K 10 3.8 2.5" Nwt 1.5 VL 22 1653.00 0.00343 21 W 36 x 230 28 K 10 3.8 2.5" Nwt 1.5 VL 22 1671.30 0.00333 22 W 36 x 230 30 K 11 3.8 2.5" Nwt 1.5 VL 22 1671.90 0.00325 23 W 30 x 211 20 K 10 4.3 2.5" Nwt 3.0 VL 22 1686.80 0.00313 24 W 40 x 324 22 K 10 4.3 2.5" Nwt 3.0 VL 22 1720.80 0.00278 25 W 40 x 324 30 K 11 4.3 3.0" Nwt 3.0 VL 20 1862.10 0.00266 26 W 12 x 305 28 K 9 3.8 4.5" Nwt 3.0 VL 22 1915.40 0.00220 27 W 40 x 183 28 K 8 3.3 4.0" Nwt 1.5 VL 21 1994.90 0.00214 28 W 40 x 431 24 K 12 3.3 3.5" Nwt 3.0 VL 22 2032.10 0.00212 29 W 36 x 359 28 K 9 2.7 4.0" Nwt 2.0 VL 22 2093.00 0.00167 30 W 36 x 650 26 K 9 2.5 3.5" Nwt 2.0 VL 19 2597.50 0.00162
133
B.2 Joist Span 1 (L = 40 ft) Loading 1 (LL = 50 psf) B.2 – Run 1 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 18 x 106 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1744.10 0.00465 2 W 18 x 106 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1744.30 0.00454 3 W 18 x 106 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1744.70 0.00453 4 W 27 x 146 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1756.70 0.00439 5 W 40 x 211 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1775.60 0.00436 6 W 40 x 211 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1776.20 0.00420 7 W 36 x 300 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1802.90 0.00415 8 W 40 x 372 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1824.50 0.00409 9 W 36 x 150 30 K 11 4.3 3.0" Nwt 1.5 VL 22 1851.30 0.00384 10 W 40 x 211 30 K 11 4.3 3.0" Nwt 1.5 VL 22 1869.60 0.00372 11 W 21 x 93 30 K 11 3.8 3.0" Nwt 1.5 VL 22 1908.00 0.00352 12 W 27 x 146 28 K 10 3.8 3.0" Nwt 1.5 VL 22 1923.30 0.00335 13 W 30 x 173 30 K 11 3.8 2.5" Nwt 2.0 VL 22 1932.00 0.00328 14 W 27 x 194 30 K 11 3.8 3.0" Nwt 1.5 VL 22 1938.30 0.00316 15 W 36 x 210 28 K 10 3.8 3.0" Nwt 1.5 VL 22 1942.50 0.00314 16 W 40 x 211 30 K 11 3.8 3.0" Nwt 1.5 VL 22 1943.40 0.00300 17 W 33 x 291 30 K 10 3.8 3.0" Nwt 1.5 VL 22 1967.00 0.00298 18 W 27 x 307 30 K 11 3.8 3.5" Nwt 1.5 VL 22 2005.00 0.00271 19 W 36 x 393 30 K 11 3.8 3.5" Nwt 1.5 VL 22 2030.80 0.00258 20 W 36 x 393 26 K 12 3.3 3.5" Nwt 1.5 VL 22 2130.90 0.00255 21 W 40 x 211 30 K 11 3 3.5" Nwt 1.5 VL 22 2176.20 0.00244 22 W 27 x 194 30 K 11 2.7 3.5" Nwt 2.0 VL 22 2305.50 0.00235 23 W 30 x 108 26 K 12 2.7 4.5" Nwt 1.5 VL 22 2311.80 0.00231 24 W 33 x 241 28 K 10 2.7 3.5" Nwt 2.0 VL 22 2319.00 0.00229 25 W 36 x 160 30 K 10 3.3 4.0" Nwt 2.0 VL 21 2367.70 0.00226 26 W 40 x 277 28 K 10 2.7 4.3" Nwt 2.0 VL 22 2418.50 0.00224 27 W 21 x 201 24 K 12 2.5 4.5" Nwt 2.0 VL 22 2473.90 0.00211 28 W 14 x 500 30 K 12 3 4.0" Nwt 3.0 VL 22 2481.70 0.00200 29 W 14 x 500 30 K 12 2.7 4.0" Nwt 3.0 VL 22 2581.70 0.00193 30 W 33 x 354 28 K 10 2.5 3.5" Nwt 3.0 VL 20 2724.10 0.00190
134
B.2 – Run 2 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 16 x 89 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1739.20 0.00480 2 W 30 x 99 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1742.00 0.00461 3 W 30 x 99 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1742.00 0.00461 4 W 30 x 99 28 K 12 4.3 2.0" Nwt 1.5 VL 22 1742.80 0.00454 5 W 18 x 106 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1744.30 0.00454 6 W 27 x 114 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1746.70 0.00444 7 W 33 x 141 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1754.80 0.00435 8 W 30 x 191 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1769.80 0.00431 9 W 40 x 215 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1777.40 0.00419 10 W 40 x 235 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1783.00 0.00418 11 W 40 x 327 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1810.60 0.00412 12 W 40 x 397 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1831.60 0.00409 13 W 40 x 397 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1831.60 0.00409 14 W 33 x 141 30 K 11 3.8 2.0" Nwt 1.5 VL 22 1855.20 0.00408 15 W 33 x 141 30 K 10 4.3 3.3" Nwt 1.5 VL 22 1865.90 0.00370 16 W 30 x 191 30 K 10 4.3 3.3" Nwt 1.5 VL 22 1880.90 0.00365 17 W 40 x 215 26 K 12 4.3 3.3" Nwt 1.5 VL 22 1888.60 0.00364 18 W 30 x 191 28 K 12 3.8 2.5" Nwt 1.5 VL 22 1904.00 0.00360 19 W 40 x 235 30 K 10 3.8 2.5" Nwt 1.5 VL 22 1916.60 0.00340 20 W 40 x 397 30 K 10 4.3 3.3" Nwt 1.5 VL 22 1942.70 0.00339 21 W 30 x 191 30 K 11 3.3 2.5" Nwt 1.5 VL 22 2003.80 0.00334 22 W 40 x 362 28 K 12 3.3 2.5" Nwt 1.5 VL 22 2055.30 0.00320 23 W 24 x 131 26 K 12 3.3 4.0" Nwt 1.5 VL 22 2085.90 0.00272 24 W 24 x 176 26 K 12 3.3 3.5" Nwt 2.0 VL 22 2100.20 0.00271 25 W 24 x 176 28 K 12 3.3 3.5" Nwt 2.0 VL 22 2100.30 0.00266 26 W 40 x 397 28 K 12 3 3.0" Nwt 2.0 VL 22 2233.00 0.00246 27 W 27 x 258 30 K 10 3.8 4.3" Nwt 2.0 VL 20 2233.00 0.00264 28 W 24 x 207 30 K 10 3 4.0" Nwt 1.5 VL 20 2328.20 0.00240 29 W 24 x 207 28 K 12 2.7 4.0" Nwt 1.5 VL 20 2428.80 0.00235 30 W 40 x 397 28 K 12 2.5 3.0" Nwt 2.0 VL 22 2433.00 0.00226 31 W 36 x 393 26 K 12 3 4.0" Nwt 3.0 VL 22 2449.30 0.00184 32 W 36 x 393 30 K 10 2.5 4.0" Nwt 3.0 VL 22 2648.80 0.00165
135
B.2 – Run 3 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 24 x 76 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1735.70 0.00448 2 W 24 x 104 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1744.10 0.00446 3 W 21 x 111 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1746.20 0.00446 4 W 21 x 132 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1752.50 0.00444 5 W 30 x 132 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1752.50 0.00439 6 W 27 x 178 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1766.30 0.00435 7 W 27 x 217 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1778.00 0.00430 8 W 40 x 235 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1783.40 0.00418 9 W 36 x 328 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1811.30 0.00414 10 W 40 x 331 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1811.80 0.00412 11 W 21 x 111 30 K 11 4.3 3.3" Nwt 1.5 VL 22 1857.30 0.00394 12 W 27 x 114 30 K 11 4.3 3.3" Nwt 1.5 VL 22 1858.20 0.00385 13 W 24 x 103 30 K 11 3.8 2.5" Nwt 1.5 VL 22 1877.40 0.00375 14 W 40 x 235 30 K 10 4.3 3.3" Nwt 1.5 VL 22 1894.10 0.00349 15 W 40 x 235 30 K 11 4.3 3.3" Nwt 1.5 VL 22 1894.50 0.00349 16 W 40 x 235 30 K 11 3.8 2.5" Nwt 1.5 VL 22 1917.00 0.00339 17 W 36 x 300 30 K 11 3.8 2.5" Nwt 1.5 VL 22 1936.50 0.00337 18 W 36 x 328 30 K 11 3.8 2.5" Nwt 1.5 VL 22 1944.90 0.00335 19 W 36 x 359 30 K 11 3.8 2.5" Nwt 1.5 VL 22 1954.20 0.00333 20 W 12 x 252 30 K 10 3.8 3.0" Nwt 2.0 VL 22 1988.90 0.00315 21 W 36 x 300 28 K 10 3.8 3.3" Nwt 2.0 VL 22 2051.60 0.00315 22 W 36 x 359 28 K 10 3.8 3.3" Nwt 2.0 VL 22 2069.30 0.00311 23 W 21 x 132 28 K 10 3.3 3.5" Nwt 2.0 VL 22 2086.30 0.00284 24 W 40 x 392 28 K 12 3.8 2.5" Nwt 3.0 VL 22 2149.10 0.00270 25 W 27 x 178 26 K 12 3.3 4.0" Nwt 1.5 VL 20 2220.00 0.00257 26 W 27 x 84 30 K 11 3.3 4.5" Nwt 1.5 VL 20 2224.50 0.00253 27 W 36 x 300 30 K 10 3 3.5" Nwt 2.0 VL 22 2236.90 0.00224 28 W 36 x 527 30 K 11 3 3.5" Nwt 2.0 VL 22 2305.40 0.00214 29 W 36 x 135 30 K 11 2.7 4.5" Nwt 1.5 VL 20 2439.80 0.00209 30 W 40 x 235 26 K 12 2.7 4.5" Nwt 3.0 VL 22 2535.50 0.00169 31 W 36 x 359 26 K 12 2.7 4.5" Nwt 3.0 VL 22 2572.70 0.00164
136
B.2 – Run 4 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 99 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1742.00 0.00461 2 W 30 x 99 28 K 12 4.3 2.0" Nwt 1.5 VL 22 1742.80 0.00454 3 W 18 x 106 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1744.30 0.00454 4 W 18 x 106 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1745.10 0.00452 5 W 24 x 117 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1747.60 0.00446 6 W 40 x 167 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1762.40 0.00441 7 W 40 x 183 28 K 12 4.3 2.0" Nwt 1.5 VL 22 1768.00 0.00432 8 W 40 x 235 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1783.40 0.00418 9 W 40 x 327 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1811.00 0.00412 10 W 30 x 99 30 K 10 4.3 3.3" Nwt 1.5 VL 22 1853.30 0.00386 11 W 30 x 99 28 K 10 3.8 2.5" Nwt 1.5 VL 22 1875.60 0.00384 12 W 30 x 99 30 K 10 3.8 2.5" Nwt 1.5 VL 22 1875.80 0.00371 13 W 30 x 99 28 K 10 3.8 2.5" Nwt 2.0 VL 22 1909.20 0.00355 14 W 30 x 99 30 K 12 3.8 2.5" Nwt 2.0 VL 22 1910.20 0.00343 15 W 40 x 167 28 K 10 3.8 2.5" Nwt 2.0 VL 22 1929.60 0.00331 16 W 30 x 99 30 K 10 3.8 3.5" Nwt 1.5 VL 22 1942.20 0.00302 17 W 27 x 281 30 K 10 3.8 3.5" Nwt 1.5 VL 22 1996.80 0.00274 18 W 40 x 324 30 K 10 3.8 3.5" Nwt 1.5 VL 22 2009.70 0.00259 19 W 40 x 331 30 K 12 3.8 3.5" Nwt 1.5 VL 22 2012.60 0.00258 20 W 40 x 149 30 K 10 3.3 3.5" Nwt 2.0 VL 22 2091.60 0.00248 21 W 40 x 167 30 K 12 3.3 4.0" Nwt 1.5 VL 22 2097.00 0.00236 22 W 33 x 241 30 K 11 3.3 4.0" Nwt 1.5 VL 22 2118.80 0.00232 23 W 40 x 277 28 K 10 3.3 4.0" Nwt 1.5 VL 22 2129.00 0.00230 24 W 40 x 278 30 K 12 3.3 4.0" Nwt 1.5 VL 22 2130.30 0.00224 25 W 33 x 354 30 K 10 3.3 4.0" Nwt 1.5 VL 22 2152.30 0.00224 26 W 36 x 359 30 K 12 3.3 4.0" Nwt 2.0 VL 22 2188.20 0.00208 27 W 40 x 167 30 K 11 2.7 4.0" Nwt 2.0 VL 22 2330.20 0.00205 28 W 40 x 331 30 K 10 2.7 4.0" Nwt 2.0 VL 22 2379.00 0.00191 29 W 40 x 249 28 K 10 3 4.0" Nwt 3.0 VL 22 2405.40 0.00188 30 W 40 x 278 30 K 12 3 4.5" Nwt 2.0 VL 20 2417.50 0.00184 31 W 40 x 331 30 K 10 3 4.0" Nwt 3.0 VL 22 2430.20 0.00180 32 W 40 x 331 30 K 9 2.5 4.5" Nwt 2.0 VL 20 2632.10 0.00177 33 W 40 x 324 28 K 10 2.7 4.5" Nwt 3.0 VL 19 3070.30 0.00163
137
B.2 – Run 5 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 24 x 94 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1741.50 0.00446 2 W 30 x 99 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1742.20 0.00445 3 W 30 x 116 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1747.30 0.00442 4 W 40 x 167 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1762.40 0.00441 5 W 40 x 167 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1762.60 0.00426 6 W 30 x 235 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1783.00 0.00426 7 W 30 x 235 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1783.80 0.00425 8 W 33 x 318 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1808.70 0.00415 9 W 40 x 331 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1811.80 0.00412 10 W 40 x 167 30 K 10 3.8 2.0" Nwt 1.5 VL 22 1862.60 0.00401 11 W 30 x 235 30 K 10 3.8 2.0" Nwt 1.5 VL 22 1883.00 0.00400 12 W 30 x 235 30 K 12 3.8 2.0" Nwt 1.5 VL 22 1883.80 0.00399 13 W 40 x 249 30 K 10 3.8 2.0" Nwt 1.5 VL 22 1887.20 0.00391 14 W 30 x 99 28 K 10 3.8 3.0" Nwt 1.5 VL 22 1909.20 0.00343 15 W 30 x 99 30 K 10 3.8 3.0" Nwt 1.5 VL 22 1909.40 0.00334 16 W 40 x 167 26 K 12 3.8 3.0" Nwt 1.5 VL 22 1930.30 0.00318 17 W 30 x 235 30 K 12 3.8 3.0" Nwt 1.5 VL 22 1951.00 0.00306 18 W 33 x 318 30 K 11 3.8 3.0" Nwt 1.5 VL 22 1975.50 0.00296 19 W 40 x 397 30 K 12 3.8 3.0" Nwt 1.5 VL 22 1999.60 0.00288 20 W 30 x 326 30 K 10 3.3 3.0" Nwt 1.5 VL 22 2077.50 0.00285 21 W 30 x 235 30 K 10 3.8 2.5" Nwt 3.0 VL 22 2101.40 0.00283 22 W 40 x 264 26 K 12 3.8 2.5" Nwt 3.0 VL 22 2110.60 0.00282 23 W 40 x 167 30 K 10 3 3.0" Nwt 1.5 VL 22 2129.80 0.00279 24 W 40 x 324 30 K 10 3.8 3.0" Nwt 2.0 VL 20 2130.50 0.00270 25 W 40 x 397 30 K 12 3 3.0" Nwt 1.5 VL 22 2199.60 0.00259 26 W 30 x 326 30 K 10 2.7 3.0" Nwt 1.5 VL 22 2277.50 0.00259 27 W 40 x 397 30 K 9 3 4.5" Nwt 1.5 VL 22 2297.50 0.00199 28 W 30 x 326 26 K 12 2.7 4.5" Nwt 1.5 VL 22 2377.20 0.00199 29 W 40 x 392 30 K 10 3 4.5" Nwt 1.5 VL 21 2536.50 0.00190 30 W 40 x 331 26 K 12 3 4.0" Nwt 3.0 VL 20 2550.70 0.00185 31 W 33 x 318 30 K 11 2.7 4.5" Nwt 3.0 VL 22 2560.30 0.00162 32 W 40 x 372 30 K 10 2.7 4.5" Nwt 3.0 VL 22 2576.10 0.00156 33 W 40 x 372 30 K 11 2.7 4.5" Nwt 3.0 VL 22 2576.50 0.00156 34 W 40 x 372 30 K 10 2.5 4.5" Nwt 3.0 VL 22 2676.10 0.00151
138
B.2 – Run 6 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 24 x 68 26 K 12 4.3 2.0" Nwt 1.5 VL 22 1733.40 0.00472 2 W 30 x 99 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1743.00 0.00444 3 W 33 x 118 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1748.30 0.00439 4 W 36 x 170 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1763.50 0.00428 5 W 36 x 170 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1763.90 0.00428 6 W 33 x 241 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1785.20 0.00422 7 W 30 x 99 30 K 11 4.3 3.0" Nwt 1.5 VL 22 1836.00 0.00404 8 W 33 x 118 30 K 12 4.3 3.0" Nwt 1.5 VL 22 1842.00 0.00394 9 W 27 x 161 30 K 11 4.3 3.0" Nwt 1.5 VL 22 1854.60 0.00393 10 W 36 x 170 30 K 12 4.3 3.0" Nwt 1.5 VL 22 1857.60 0.00380 11 W 36 x 256 30 K 11 4.3 3.0" Nwt 1.5 VL 22 1883.10 0.00370 12 W 27 x 146 30 K 11 3.8 2.5" Nwt 1.5 VL 22 1890.30 0.00364 13 W 24 x 103 30 K 10 3.8 2.5" Nwt 2.0 VL 22 1910.60 0.00350 14 W 33 x 118 26 K 12 3.8 2.5" Nwt 2.0 VL 22 1915.60 0.00349 15 W 21 x 147 30 K 10 3.8 2.5" Nwt 2.0 VL 22 1923.80 0.00346 16 W 21 x 147 30 K 11 3.8 2.5" Nwt 2.0 VL 22 1924.20 0.00345 17 W 27 x 161 30 K 10 3.8 2.5" Nwt 2.0 VL 22 1928.00 0.00335 18 W 27 x 161 30 K 11 3.8 2.5" Nwt 2.0 VL 22 1928.40 0.00334 19 W 27 x 161 30 K 12 3.8 2.5" Nwt 2.0 VL 22 1928.80 0.00334 20 W 36 x 170 30 K 11 3.8 2.5" Nwt 2.0 VL 22 1931.10 0.00323 21 W 36 x 170 30 K 12 3.8 2.5" Nwt 2.0 VL 22 1931.50 0.00322 22 W 40 x 199 30 K 10 3.8 2.5" Nwt 2.0 VL 22 1939.40 0.00316 23 W 40 x 199 30 K 11 3.8 2.5" Nwt 2.0 VL 22 1939.80 0.00315 24 W 24 x 117 30 K 12 3.8 3.0" Nwt 2.0 VL 22 1949.20 0.00313 25 W 33 x 118 30 K 12 3.8 3.0" Nwt 2.0 VL 22 1949.50 0.00300 26 W 40 x 149 30 K 11 3.8 3.0" Nwt 2.0 VL 22 1958.40 0.00288 27 W 36 x 170 30 K 10 3.8 3.0" Nwt 2.0 VL 22 1964.30 0.00287 28 W 36 x 170 30 K 12 3.8 3.0" Nwt 2.0 VL 22 1965.10 0.00286 29 W 36 x 194 30 K 12 3.8 3.0" Nwt 2.0 VL 22 1972.30 0.00283 30 W 40 x 235 30 K 11 3.8 3.0" Nwt 2.0 VL 22 1984.20 0.00276 31 W 36 x 439 30 K 10 3.8 3.0" Nwt 2.0 VL 22 2045.00 0.00267 32 W 36 x 439 30 K 12 3.8 3.0" Nwt 2.0 VL 22 2045.80 0.00267 33 W 36 x 650 30 K 12 3.8 3.0" Nwt 2.0 VL 22 2109.10 0.00261 34 W 33 x 141 28 K 12 3.3 4.5" Nwt 1.5 VL 22 2121.80 0.00229 35 W 36 x 160 30 K 11 3.3 4.5" Nwt 1.5 VL 22 2127.30 0.00219 36 W 36 x 170 30 K 11 3.3 4.5" Nwt 1.5 VL 22 2130.30 0.00217 37 W 36 x 170 30 K 12 3 4.5" Nwt 1.5 VL 22 2230.70 0.00209 38 W 30 x 357 28 K 10 3 4.5" Nwt 1.5 VL 22 2285.80 0.00203 39 W 30 x 357 28 K 12 3 4.5" Nwt 1.5 VL 22 2286.60 0.00201 40 W 40 x 199 30 K 11 2.7 4.5" Nwt 1.5 VL 22 2339.00 0.00194 41 W 30 x 326 30 K 10 2.7 4.5" Nwt 1.5 VL 22 2376.70 0.00192 42 W 30 x 326 30 K 11 2.7 4.5" Nwt 1.5 VL 22 2377.10 0.00192 43 W 30 x 357 30 K 10 2.7 4.5" Nwt 1.5 VL 22 2386.00 0.00190 44 W 33 x 241 30 K 11 3 4.5" Nwt 3.0 VL 21 2701.20 0.00174
139
B.2 – Run 7 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 18 x 86 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1738.10 0.00480 2 W 18 x 86 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1738.70 0.00468 3 W 18 x 86 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1739.10 0.00467 4 W 27 x 102 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1742.90 0.00462 5 W 27 x 102 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1743.90 0.00445 6 W 33 x 118 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1748.70 0.00438 7 W 24 x 176 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1765.70 0.00438 8 W 24 x 176 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1766.10 0.00437 9 W 36 x 182 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1767.10 0.00427 10 W 36 x 182 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1767.90 0.00426 11 W 40 x 277 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1796.00 0.00414 12 W 40 x 327 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1810.60 0.00412 13 W 40 x 327 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1811.40 0.00411 14 W 40 x 372 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1824.10 0.00410 15 W 40 x 372 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1824.50 0.00409 16 W 40 x 372 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1824.90 0.00409 17 W 36 x 182 28 K 12 4.3 2.5" Nwt 2.0 VL 22 1861.10 0.00405 18 W 36 x 160 30 K 12 3.8 2.0" Nwt 1.5 VL 22 1861.30 0.00403 19 W 36 x 182 30 K 10 3.8 2.0" Nwt 1.5 VL 22 1867.10 0.00401 20 W 36 x 182 30 K 11 3.8 2.0" Nwt 1.5 VL 22 1867.50 0.00401 21 W 36 x 182 30 K 12 3.8 2.0" Nwt 1.5 VL 22 1867.90 0.00400 22 W 30 x 261 30 K 12 4.3 2.5" Nwt 2.0 VL 22 1884.90 0.00393 23 W 30 x 132 30 K 11 3.8 2.5" Nwt 1.5 VL 22 1886.10 0.00363 24 W 36 x 182 24 K 12 3.8 3.5" Nwt 1.5 VL 22 1967.40 0.00289 25 W 40 x 277 30 K 12 3.8 3.5" Nwt 1.5 VL 22 1996.40 0.00261 26 W 24 x 370 30 K 11 3.3 3.5" Nwt 2.0 VL 22 2158.30 0.00242 27 W 33 x 118 28 K 10 3 4.5" Nwt 2.0 VL 22 2248.50 0.00217 28 W 33 x 118 30 K 10 3 4.5" Nwt 2.0 VL 22 2248.70 0.00212 29 W 33 x 118 30 K 12 3 4.5" Nwt 2.0 VL 22 2249.50 0.00211 30 W 36 x 300 30 K 10 3 4.5" Nwt 2.0 VL 22 2303.30 0.00185 31 W 40 x 372 30 K 10 3 4.5" Nwt 2.0 VL 22 2324.90 0.00179
140
B.2 – Run 8 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 90 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1739.90 0.00446 2 W 30 x 124 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1749.70 0.00441 3 W 30 x 124 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1750.10 0.00440 4 W 27 x 161 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1761.20 0.00437 5 W 27 x 194 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1771.10 0.00433 6 W 30 x 211 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1776.20 0.00428 7 W 30 x 261 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1790.80 0.00424 8 W 30 x 261 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1791.20 0.00423 9 W 40 x 327 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1810.60 0.00412
10 W 40 x 327 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1811.00 0.00412 11 W 40 x 331 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1812.20 0.00412 12 W 30 x 90 30 K 11 4.3 3.3" Nwt 1.5 VL 22 1851.00 0.00388 13 W 30 x 90 30 K 11 3.8 2.0" Nwt 2.0 VL 22 1874.30 0.00386 14 W 18 x 119 30 K 11 3.8 2.5" Nwt 1.5 VL 22 1882.20 0.00383 15 W 27 x 161 30 K 11 3.8 2.0" Nwt 2.0 VL 22 1895.60 0.00377 16 W 21 x 182 30 K 11 3.8 2.5" Nwt 1.5 VL 22 1901.10 0.00366 17 W 18 x 119 30 K 11 3.8 3.0" Nwt 1.5 VL 22 1915.80 0.00348 18 W 24 x 162 26 K 12 3.8 3.0" Nwt 1.5 VL 22 1928.80 0.00339 19 W 18 x 119 30 K 11 3.8 3.5" Nwt 1.5 VL 22 1948.60 0.00319 20 W 24 x 162 26 K 12 3.8 3.5" Nwt 1.5 VL 22 1961.60 0.00304 21 W 40 x 278 30 K 11 3.8 3.0" Nwt 1.5 VL 22 1963.50 0.00295 22 W 24 x 207 30 K 11 3.8 3.5" Nwt 1.5 VL 22 1975.00 0.00287 23 W 30 x 211 26 K 12 3.8 3.5" Nwt 1.5 VL 22 1976.30 0.00286 24 W 30 x 261 30 K 11 3.8 3.5" Nwt 1.5 VL 22 1991.20 0.00271 25 W 36 x 280 30 K 12 3.8 3.5" Nwt 1.5 VL 22 1997.30 0.00263 26 W 40 x 327 30 K 11 3.8 3.5" Nwt 1.5 VL 22 2011.00 0.00259 27 W 36 x 439 30 K 11 3.8 3.5" Nwt 1.5 VL 22 2044.60 0.00256 28 W 40 x 327 30 K 11 3.3 3.5" Nwt 1.5 VL 22 2111.00 0.00246 29 W 36 x 245 26 K 12 3.3 3.5" Nwt 2.0 VL 22 2120.90 0.00246 30 W 36 x 439 30 K 11 3.3 3.5" Nwt 1.5 VL 22 2144.60 0.00243 31 W 40 x 327 30 K 11 3.3 3.5" Nwt 2.0 VL 22 2145.40 0.00230 32 W 40 x 278 30 K 11 3 3.5" Nwt 2.0 VL 22 2230.70 0.00223 33 W 36 x 650 30 K 11 3.3 3.5" Nwt 2.0 VL 22 2242.30 0.00222 34 W 36 x 245 26 K 12 2.5 3.5" Nwt 2.0 VL 22 2420.90 0.00219 35 W 40 x 278 30 K 9 2.5 3.5" Nwt 2.0 VL 22 2429.80 0.00218 36 W 36 x 650 26 K 12 2.7 3.5" Nwt 2.0 VL 22 2442.40 0.00211 37 W 40 x 327 26 K 12 2.5 4.5" Nwt 1.5 VL 22 2477.50 0.00185 38 W 40 x 431 26 K 12 2.5 4.5" Nwt 3.0 VL 20 2814.30 0.00154 39 W 36 x 650 26 K 12 2.5 4.5" Nwt 3.0 VL 19 3268.80 0.00150
141
B.2 – Run 9 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 18 x 86 24 K 12 4.3 2.0" Nwt 1.5 VL 22 1757.30 0.00554 2 W 18 x 130 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1770.60 0.00498 3 W 12 x 210 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1794.60 0.00491 4 W 18 x 86 30 K 11 4.3 2.5" Nwt 1.5 VL 22 1794.70 0.00479 5 W 27 x 102 30 K 11 4.3 2.5" Nwt 1.5 VL 22 1799.50 0.00451 6 W 27 x 102 30 K 12 4.3 2.5" Nwt 1.5 VL 22 1799.90 0.00451 7 W 27 x 114 30 K 10 4.3 2.5" Nwt 1.5 VL 22 1802.70 0.00450 8 W 21 x 147 30 K 10 4.3 2.5" Nwt 1.5 VL 22 1812.60 0.00450 9 W 21 x 147 30 K 11 4.3 2.5" Nwt 1.5 VL 22 1813.00 0.00449 10 W 18 x 175 30 K 10 4.3 2.5" Nwt 1.5 VL 22 1821.00 0.00449 11 W 18 x 175 30 K 11 4.3 2.5" Nwt 1.5 VL 22 1821.40 0.00448 12 W 18 x 86 30 K 11 4.3 3.0" Nwt 1.5 VL 22 1832.10 0.00438 13 W 27 x 102 30 K 10 4.3 3.0" Nwt 1.5 VL 22 1836.40 0.00407 14 W 27 x 114 30 K 10 4.3 3.0" Nwt 1.5 VL 22 1840.00 0.00404 15 W 27 x 114 30 K 11 4.3 3.0" Nwt 1.5 VL 22 1840.50 0.00404 16 W 33 x 118 30 K 11 4.3 3.0" Nwt 1.5 VL 22 1841.70 0.00395 17 W 36 x 135 30 K 10 4.3 3.0" Nwt 1.5 VL 22 1846.30 0.00389 18 W 18 x 175 30 K 11 4.3 3.3" Nwt 1.5 VL 22 1876.50 0.00387 19 W 24 x 104 30 K 11 3.8 2.5" Nwt 1.5 VL 22 1877.70 0.00375 20 W 27 x 114 30 K 12 3.8 2.5" Nwt 1.5 VL 22 1881.10 0.00370 21 W 33 x 118 30 K 11 3.8 2.5" Nwt 1.5 VL 22 1881.90 0.00363 22 W 40 x 215 30 K 10 4.3 3.3" Nwt 1.5 VL 22 1888.10 0.00350 23 W 36 x 256 30 K 10 4.3 3.3" Nwt 1.5 VL 22 1900.40 0.00350 24 W 40 x 215 30 K 10 3.8 2.5" Nwt 1.5 VL 22 1910.60 0.00341 25 W 40 x 215 30 K 11 3.8 2.5" Nwt 1.5 VL 22 1911.00 0.00340 26 W 30 x 116 30 K 11 3.8 2.5" Nwt 2.0 VL 22 1914.90 0.00340 27 W 40 x 324 30 K 10 3.8 2.5" Nwt 1.5 VL 22 1943.30 0.00333 28 W 40 x 324 30 K 11 3.8 2.5" Nwt 1.5 VL 22 1943.70 0.00333 29 W 21 x 147 26 K 12 3.8 3.5" Nwt 1.5 VL 22 1957.10 0.00314 30 W 30 x 173 24 K 12 3.8 3.0" Nwt 2.0 VL 22 1965.50 0.00309 31 W 24 x 192 26 K 12 3.8 3.5" Nwt 1.5 VL 22 1970.60 0.00299 32 W 36 x 280 30 K 11 3.8 3.5" Nwt 1.5 VL 22 1996.90 0.00264 33 W 40 x 324 30 K 10 3.8 3.5" Nwt 1.5 VL 20 2129.70 0.00259 34 W 27 x 194 30 K 10 3.3 4.0" Nwt 2.0 VL 22 2137.90 0.00232 35 W 27 x 194 30 K 11 3.3 4.0" Nwt 2.0 VL 22 2138.30 0.00231 36 W 30 x 173 30 K 11 3 4.0" Nwt 2.0 VL 22 2232.00 0.00222 37 W 40 x 324 30 K 11 3 4.0" Nwt 2.0 VL 22 2277.30 0.00198 38 W 30 x 326 30 K 10 3 4.5" Nwt 3.0 VL 20 2582.30 0.00171 39 W 30 x 326 30 K 10 2.7 4.5" Nwt 3.0 VL 20 2682.30 0.00165
142
B.2 – Run 10 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 27 x 84 26 K 12 4.3 2.0" Nwt 1.5 VL 22 1738.20 0.00469 2 W 24 x 94 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1740.50 0.00465 3 W 24 x 94 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1740.70 0.00448 4 W 24 x 94 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1741.10 0.00447 5 W 24 x 94 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1741.50 0.00446 6 W 24 x 104 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1744.10 0.00446 7 W 24 x 104 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1744.50 0.00445 8 W 30 x 124 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1749.70 0.00441 9 W 24 x 162 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1761.10 0.00440 10 W 36 x 160 28 K 12 4.3 2.0" Nwt 1.5 VL 22 1761.10 0.00439 11 W 40 x 167 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1762.60 0.00426 12 W 36 x 256 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1789.70 0.00418 13 W 36 x 256 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1790.10 0.00418 14 W 36 x 280 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1796.50 0.00417 15 W 30 x 357 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1820.40 0.00416 16 W 36 x 359 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1821.00 0.00411 17 W 24 x 94 30 K 12 4.3 3.0" Nwt 1.5 VL 22 1834.80 0.00411 18 W 24 x 104 30 K 11 4.3 3.0" Nwt 1.5 VL 22 1837.50 0.00409 19 W 30 x 124 28 K 12 4.3 3.0" Nwt 1.5 VL 22 1843.70 0.00405 20 W 30 x 124 30 K 12 4.3 3.0" Nwt 1.5 VL 22 1843.80 0.00397 21 W 21 x 201 30 K 12 4.3 3.0" Nwt 1.5 VL 22 1866.90 0.00396 22 W 36 x 280 30 K 10 4.3 3.0" Nwt 1.5 VL 22 1889.80 0.00369 23 W 36 x 280 30 K 12 4.3 3.0" Nwt 1.5 VL 22 1890.60 0.00368 24 W 36 x 359 30 K 10 4.3 3.0" Nwt 1.5 VL 22 1913.50 0.00364 25 W 36 x 393 30 K 10 4.3 3.0" Nwt 1.5 VL 22 1923.70 0.00362 26 W 24 x 94 30 K 10 3.8 3.5" Nwt 1.5 VL 22 1940.70 0.00312 27 W 24 x 162 30 K 10 3.8 3.5" Nwt 1.5 VL 22 1961.10 0.00295 28 W 21 x 201 30 K 10 3.8 3.5" Nwt 1.5 VL 22 1972.80 0.00294 29 W 21 x 201 30 K 12 3.8 3.5" Nwt 1.5 VL 22 1973.60 0.00293 30 W 36 x 256 30 K 10 3.8 3.0" Nwt 2.0 VL 22 1990.10 0.00277 31 W 33 x 318 30 K 10 3.8 3.5" Nwt 1.5 VL 22 2007.90 0.00264 32 W 36 x 280 30 K 10 3.3 3.5" Nwt 1.5 VL 22 2096.50 0.00252 33 W 33 x 291 26 K 12 3.3 4.5" Nwt 1.5 VL 22 2166.70 0.00213 34 W 36 x 210 28 K 10 2.7 4.0" Nwt 2.0 VL 22 2342.50 0.00209 35 W 36 x 527 30 K 12 3 3.5" Nwt 3.0 VL 20 2577.00 0.00191 36 W 36 x 393 28 K 12 2.5 4.0" Nwt 2.0 VL 18 2858.20 0.00186
143
B.3 Joist Span 3 (L = 50 ft) Loading 1 (LL = 50 psf) B.3 – Run 1 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 27 x 84 30 K 12 3.8 2.0" Nwt 1.5 VL 22 2113.80 0.00522 2 W 30 x 235 30 K 12 3.8 2.0" Nwt 1.5 VL 22 2159.10 0.00492 3 W 33 x 118 30 K 11 3.3 2.0" Nwt 1.5 VL 22 2223.70 0.00491 4 W 33 x 118 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2224.00 0.00483 5 W 40 x 211 28 K 12 3.3 2.0" Nwt 1.5 VL 22 2251.80 0.00479 6 W 40 x 211 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2251.90 0.00469 7 W 24 x 103 30 K 12 3 3.0" Nwt 1.5 VL 22 2380.20 0.00330 8 W 24 x 131 30 K 11 3 3.0" Nwt 1.5 VL 22 2388.20 0.00319 9 W 40 x 264 30 K 11 3 3.0" Nwt 1.5 VL 22 2428.10 0.00283 10 W 40 x 211 30 K 11 2.7 3.0" Nwt 1.5 VL 22 2512.20 0.00279 11 W 40 x 264 30 K 12 2.7 3.0" Nwt 1.5 VL 22 2528.50 0.00271 12 W 40 x 211 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2612.60 0.00267 13 W 40 x 278 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2632.70 0.00262 14 W 27 x 368 30 K 11 2.7 2.5" Nwt 3.0 VL 22 2748.30 0.00256 15 W 40 x 211 30 K 11 2.5 2.5" Nwt 3.0 VL 22 2801.20 0.00249 16 W 27 x 146 28 K 12 2.5 4.0" Nwt 1.5 VL 20 2825.90 0.00236 17 W 27 x 146 30 K 12 2.5 4.0" Nwt 1.5 VL 20 2826.10 0.00233 18 W 40 x 211 28 K 12 2.5 4.0" Nwt 1.5 VL 20 2845.40 0.00214 19 W 40 x 211 30 K 12 2.5 3.5" Nwt 3.0 VL 16 3560.60 0.00195 20 W 40 x 264 30 K 12 2.5 3.5" Nwt 3.0 VL 16 3576.50 0.00192
144
B.3 – Run 2 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 90 30 K 11 3.3 2.0" Nwt 2.0 VL 22 2263.00 0.00461 2 W 30 x 90 30 K 12 3.3 2.0" Nwt 2.0 VL 22 2263.40 0.00455 3 W 33 x 118 30 K 11 3.3 2.0" Nwt 2.0 VL 22 2271.40 0.00449 4 W 33 x 118 30 K 12 3.3 2.0" Nwt 2.0 VL 22 2271.80 0.00442 5 W 24 x 176 30 K 12 3.3 2.0" Nwt 2.0 VL 22 2289.20 0.00442 6 W 30 x 191 30 K 12 3.3 2.0" Nwt 2.0 VL 22 2293.70 0.00435 7 W 36 x 194 30 K 12 3.3 2.0" Nwt 2.0 VL 22 2294.60 0.00430 8 W 36 x 230 30 K 12 3.3 2.0" Nwt 2.0 VL 22 2305.40 0.00427 9 W 40 x 297 30 K 12 3.3 2.0" Nwt 2.0 VL 22 2325.50 0.00421 10 W 36 x 359 30 K 12 3.3 2.0" Nwt 2.0 VL 22 2344.10 0.00420 11 W 33 x 118 30 K 12 3.3 2.0" Nwt 1.5 VL 20 2350.70 0.00412 12 W 36 x 194 30 K 12 3.3 2.0" Nwt 1.5 VL 20 2373.50 0.00400 13 W 40 x 297 30 K 12 3.3 2.0" Nwt 1.5 VL 20 2404.40 0.00390 14 W 40 x 327 30 K 12 3.3 2.0" Nwt 1.5 VL 20 2413.40 0.00389 15 W 33 x 118 30 K 12 3.3 3.0" Nwt 1.5 VL 20 2467.30 0.00369 16 W 40 x 149 30 K 12 3.3 3.0" Nwt 1.5 VL 20 2476.60 0.00357 17 W 36 x 359 30 K 12 3.3 3.0" Nwt 1.5 VL 20 2539.60 0.00342 18 W 33 x 118 30 K 12 3 2.5" Nwt 3.0 VL 22 2573.70 0.00278 19 W 24 x 176 30 K 12 3 4.3" Nwt 2.0 VL 22 2597.00 0.00259 20 W 36 x 230 30 K 12 3 4.3" Nwt 2.0 VL 22 2613.20 0.00239 21 W 36 x 280 30 K 12 3 4.3" Nwt 2.0 VL 22 2628.20 0.00235 22 W 36 x 230 30 K 12 2.5 4.3" Nwt 2.0 VL 22 2813.20 0.00227 23 W 36 x 230 30 K 12 2.5 3.0" Nwt 3.0 VL 20 2999.30 0.00218 24 W 40 x 327 30 K 11 2.7 4.3" Nwt 3.0 VL 20 3085.20 0.00207 25 W 40 x 324 30 K 12 2.5 4.5" Nwt 1.5 VL 17 3700.50 0.00184
145
B.3 – Run 3 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 27 x 84 30 K 11 3.3 3.0" Nwt 1.5 VL 22 2306.80 0.00402 2 W 27 x 84 30 K 12 3.3 3.0" Nwt 1.5 VL 22 2307.10 0.00398 3 W 27 x 84 28 K 12 3.3 3.3" Nwt 1.5 VL 22 2329.20 0.00382 4 W 27 x 194 30 K 11 3.3 3.0" Nwt 1.5 VL 22 2339.80 0.00368 5 W 27 x 194 30 K 12 3.3 3.0" Nwt 1.5 VL 22 2340.10 0.00363 6 W 27 x 235 30 K 12 3.3 3.0" Nwt 1.5 VL 22 2352.40 0.00358 7 W 36 x 245 30 K 12 3.3 3.0" Nwt 1.5 VL 22 2355.40 0.00349 8 W 27 x 194 30 K 12 3.3 3.3" Nwt 1.5 VL 22 2362.40 0.00342 9 W 36 x 245 30 K 12 3.3 3.3" Nwt 1.5 VL 22 2377.70 0.00328 10 W 40 x 249 30 K 12 3.3 3.3" Nwt 1.5 VL 22 2378.90 0.00324 11 W 36 x 328 30 K 12 3.3 3.3" Nwt 1.5 VL 22 2402.60 0.00322 12 W 40 x 249 30 K 12 3 3.3" Nwt 1.5 VL 22 2478.90 0.00314 13 W 36 x 328 30 K 12 3 3.5" Nwt 1.5 VL 22 2488.70 0.00244 14 W 36 x 359 30 K 12 3 3.5" Nwt 1.5 VL 22 2498.00 0.00242 15 W 36 x 328 30 K 11 2.7 3.5" Nwt 1.5 VL 22 2588.30 0.00240 16 W 36 x 328 30 K 12 2.7 3.5" Nwt 1.5 VL 22 2588.70 0.00237 17 W 36 x 359 30 K 12 2.7 3.5" Nwt 1.5 VL 22 2598.00 0.00236 18 W 36 x 328 30 K 11 2.5 3.5" Nwt 1.5 VL 22 2688.30 0.00234 19 W 36 x 328 30 K 12 2.5 3.5" Nwt 1.5 VL 22 2688.70 0.00231 20 W 36 x 359 30 K 12 2.5 3.5" Nwt 1.5 VL 22 2698.00 0.00229 21 W 36 x 393 30 K 12 2.5 3.5" Nwt 1.5 VL 22 2708.20 0.00228 22 W 27 x 235 30 K 12 2.7 4.0" Nwt 1.5 VL 18 3022.80 0.00225 23 W 40 x 249 30 K 12 2.7 4.0" Nwt 1.5 VL 18 3027.00 0.00213 24 W 33 x 354 30 K 12 2.7 4.0" Nwt 1.5 VL 18 3058.50 0.00211 25 W 36 x 359 30 K 12 2.7 4.0" Nwt 1.5 VL 18 3060.00 0.00210 26 W 36 x 393 30 K 12 2.7 4.0" Nwt 1.5 VL 18 3070.20 0.00209 27 W 36 x 328 30 K 11 2.7 4.3" Nwt 3.0 VL 20 3085.50 0.00208 28 W 36 x 328 30 K 12 2.7 4.3" Nwt 3.0 VL 20 3085.90 0.00205 29 W 36 x 359 30 K 12 2.7 4.3" Nwt 3.0 VL 20 3095.20 0.00204 30 W 36 x 328 28 K 12 2.5 4.3" Nwt 3.0 VL 20 3185.80 0.00203 31 W 36 x 328 30 K 12 2.5 4.3" Nwt 3.0 VL 20 3185.90 0.00200 32 W 36 x 359 30 K 12 2.5 4.3" Nwt 3.0 VL 20 3195.20 0.00199 33 W 36 x 393 30 K 12 2.5 4.3" Nwt 3.0 VL 20 3205.40 0.00197
146
B.3 – Run 4 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 33 x 241 28 K 12 2.7 2.0" Nwt 1.5 VL 22 2437.40 0.00379 2 W 27 x 114 30 K 12 3 3.3" Nwt 2.0 VL 22 2486.10 0.00327 3 W 40 x 167 28 K 12 3 3.3" Nwt 2.0 VL 22 2501.90 0.00304 4 W 40 x 167 30 K 12 3 3.3" Nwt 2.0 VL 22 2502.00 0.00299 5 W 40 x 211 28 K 12 3 3.3" Nwt 2.0 VL 22 2515.10 0.00299 6 W 40 x 211 30 K 12 3 3.3" Nwt 2.0 VL 22 2515.20 0.00294 7 W 27 x 114 30 K 12 3 4.3" Nwt 1.5 VL 22 2531.70 0.00289 8 W 40 x 211 30 K 11 3 3.5" Nwt 2.0 VL 22 2538.20 0.00282 9 W 40 x 167 30 K 12 3 4.3" Nwt 1.5 VL 22 2547.60 0.00260 10 W 40 x 211 30 K 11 3 4.3" Nwt 1.5 VL 22 2560.40 0.00258 11 W 33 x 241 30 K 12 3 4.3" Nwt 1.5 VL 22 2569.80 0.00257 12 W 40 x 324 30 K 11 3 4.3" Nwt 1.5 VL 22 2594.30 0.00251 13 W 40 x 392 28 K 12 3 4.3" Nwt 1.5 VL 22 2615.00 0.00250 14 W 40 x 392 30 K 12 3 4.3" Nwt 1.5 VL 22 2615.10 0.00245 15 W 40 x 324 30 K 11 2.7 4.3" Nwt 1.5 VL 22 2694.30 0.00244 16 W 40 x 392 28 K 12 2.7 4.3" Nwt 1.5 VL 22 2715.00 0.00242 17 W 40 x 211 30 K 12 2.5 4.3" Nwt 1.5 VL 22 2760.80 0.00241 18 W 40 x 324 28 K 12 2.5 4.3" Nwt 1.5 VL 22 2794.60 0.00238 19 W 40 x 324 30 K 12 2.5 4.3" Nwt 1.5 VL 22 2794.70 0.00233 20 W 40 x 392 30 K 12 2.5 4.3" Nwt 1.5 VL 22 2815.10 0.00231 21 W 40 x 167 30 K 11 2.5 4.0" Nwt 1.5 VL 19 3252.00 0.00219 22 W 33 x 263 30 K 11 2.5 4.0" Nwt 1.5 VL 19 3280.80 0.00214 23 W 40 x 324 30 K 11 2.5 4.0" Nwt 1.5 VL 19 3299.10 0.00207
147
B.3 – Run 5 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 24 x 117 30 K 11 3.3 2.0" Nwt 2.0 VL 22 2271.10 0.00463 2 W 33 x 152 30 K 11 3.3 2.0" Nwt 2.0 VL 22 2281.60 0.00444 3 W 33 x 152 30 K 12 3.3 2.0" Nwt 2.0 VL 22 2282.00 0.00437 4 W 36 x 194 30 K 12 3.3 2.0" Nwt 2.0 VL 22 2294.60 0.00430 5 W 30 x 90 30 K 12 3.3 2.0" Nwt 1.5 VL 20 2342.30 0.00426 6 W 30 x 391 30 K 12 3.3 2.0" Nwt 2.0 VL 22 2353.70 0.00422 7 W 27 x 258 30 K 12 3 2.0" Nwt 2.0 VL 22 2413.80 0.00417 8 W 24 x 306 30 K 12 3 2.0" Nwt 2.0 VL 22 2428.20 0.00417 9 W 30 x 391 30 K 11 3 2.0" Nwt 2.0 VL 22 2453.30 0.00414 10 W 30 x 391 30 K 12 3 2.0" Nwt 2.0 VL 22 2453.70 0.00407 11 W 30 x 90 30 K 12 3.3 2.0" Nwt 3.0 VL 22 2456.70 0.00401 12 W 24 x 131 30 K 12 3.3 2.0" Nwt 3.0 VL 22 2469.00 0.00397 13 W 33 x 152 30 K 11 3.3 2.0" Nwt 3.0 VL 22 2475.00 0.00386 14 W 33 x 152 30 K 12 3.3 2.0" Nwt 3.0 VL 22 2475.30 0.00380 15 W 33 x 152 30 K 11 3.3 3.0" Nwt 1.5 VL 20 2477.20 0.00367 16 W 24 x 250 30 K 11 3.3 3.0" Nwt 1.5 VL 20 2506.60 0.00366 17 W 33 x 152 30 K 12 3.3 3.0" Nwt 2.0 VL 20 2524.20 0.00333 18 W 36 x 194 30 K 12 3.3 2.5" Nwt 3.0 VL 22 2533.50 0.00326 19 W 36 x 194 30 K 12 3.3 3.0" Nwt 2.0 VL 20 2536.80 0.00326 20 W 33 x 152 30 K 11 3 2.0" Nwt 3.0 VL 22 2542.50 0.00313 21 W 33 x 152 30 K 12 3 2.0" Nwt 3.0 VL 22 2542.90 0.00308 22 W 36 x 194 30 K 11 2.7 3.0" Nwt 2.0 VL 22 2549.10 0.00260 23 W 33 x 152 30 K 11 2.7 3.5" Nwt 1.5 VL 20 2685.50 0.00258 24 W 36 x 194 30 K 11 2.7 3.5" Nwt 1.5 VL 20 2698.10 0.00250 25 W 27 x 258 30 K 11 2.7 4.3" Nwt 2.0 VL 22 2721.20 0.00242 26 W 40 x 392 30 K 11 2.5 3.5" Nwt 1.5 VL 20 2857.50 0.00230 27 W 30 x 391 30 K 12 2.5 4.3" Nwt 3.0 VL 20 3204.80 0.00201
148
B.3 – Run 6 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 21 x 111 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2198.60 0.00442 2 W 33 x 130 30 K 11 3.3 2.0" Nwt 1.5 VL 22 2203.90 0.00417 3 W 33 x 130 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2204.30 0.00410 4 W 40 x 211 30 K 11 3.3 2.0" Nwt 1.5 VL 22 2228.20 0.00402 5 W 40 x 211 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2228.60 0.00396 6 W 36 x 280 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2249.30 0.00393 7 W 40 x 297 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2254.40 0.00390 8 W 40 x 211 30 K 11 3 2.0" Nwt 1.5 VL 22 2328.20 0.00388 9 W 40 x 211 30 K 12 3 2.0" Nwt 1.5 VL 22 2328.60 0.00382 10 W 36 x 280 30 K 12 3 2.0" Nwt 1.5 VL 22 2349.30 0.00379 11 W 40 x 297 30 K 12 3 2.0" Nwt 1.5 VL 22 2354.40 0.00376 12 W 33 x 130 30 K 12 3 3.0" Nwt 1.5 VL 22 2388.30 0.00299 13 W 30 x 191 30 K 12 3 3.0" Nwt 1.5 VL 22 2406.60 0.00293 14 W 40 x 211 28 K 12 3 3.0" Nwt 1.5 VL 22 2412.40 0.00287 15 W 40 x 297 28 K 12 3 3.0" Nwt 1.5 VL 22 2438.20 0.00281 16 W 33 x 130 28 K 12 3 4.3" Nwt 1.5 VL 22 2536.40 0.00276 17 W 40 x 297 30 K 12 2.7 3.0" Nwt 1.5 VL 22 2538.40 0.00268 18 W 40 x 331 30 K 12 2.7 3.0" Nwt 1.5 VL 22 2548.60 0.00267 19 W 40 x 211 30 K 12 3 4.3" Nwt 1.5 VL 22 2560.80 0.00254 20 W 40 x 211 30 K 11 2.7 4.3" Nwt 1.5 VL 22 2660.40 0.00251 21 W 36 x 280 30 K 11 2.7 4.3" Nwt 1.5 VL 22 2681.10 0.00248 22 W 36 x 280 30 K 12 2.7 4.3" Nwt 1.5 VL 22 2681.50 0.00245 23 W 40 x 397 28 K 12 2.7 4.3" Nwt 1.5 VL 22 2716.50 0.00242 24 W 40 x 297 28 K 12 2.5 4.3" Nwt 1.5 VL 22 2786.50 0.00239 25 W 36 x 650 30 K 12 3 3.5" Nwt 1.5 VL 19 3155.30 0.00236 26 W 40 x 211 30 K 12 2.5 3.5" Nwt 1.5 VL 19 3223.60 0.00236 27 W 40 x 331 28 K 12 2.5 3.5" Nwt 1.5 VL 19 3259.40 0.00233 28 W 36 x 650 30 K 12 2.7 3.0" Nwt 3.0 VL 18 3400.30 0.00210 29 W 40 x 249 30 K 12 2.5 4.0" Nwt 2.0 VL 17 3769.00 0.00193
149
B.3 – Run 7 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 21 x 101 30 K 12 3.3 2.0" Nwt 2.0 VL 22 2238.60 0.00408 2 W 36 x 232 30 K 12 3.3 2.5" Nwt 1.5 VL 22 2304.90 0.00402 3 W 12 x 336 30 K 12 3.3 2.0" Nwt 2.0 VL 22 2309.10 0.00379 4 W 36 x 439 28 K 12 3.3 2.0" Nwt 2.0 VL 22 2339.80 0.00353 5 W 33 x 221 30 K 11 3 2.0" Nwt 2.0 VL 22 2374.20 0.00353 6 W 33 x 221 30 K 12 3 2.0" Nwt 2.0 VL 22 2374.60 0.00348 7 W 36 x 232 30 K 12 3 2.0" Nwt 2.0 VL 22 2377.90 0.00345 8 W 36 x 260 30 K 12 3 2.0" Nwt 2.0 VL 22 2386.30 0.00343 9 W 27 x 336 30 K 12 3.3 3.3" Nwt 1.5 VL 22 2405.00 0.00330 10 W 36 x 232 30 K 11 3 3.3" Nwt 1.5 VL 22 2473.40 0.00323 11 W 36 x 260 30 K 11 3 3.3" Nwt 1.5 VL 22 2481.80 0.00321 12 W 27 x 336 30 K 12 3 3.3" Nwt 1.5 VL 22 2505.00 0.00319 13 W 21 x 122 30 K 12 2.7 3.5" Nwt 2.0 VL 22 2569.90 0.00273 14 W 24 x 176 30 K 12 3 4.3" Nwt 2.0 VL 22 2597.00 0.00259 15 W 33 x 221 30 K 12 3 4.3" Nwt 2.0 VL 22 2610.50 0.00241 16 W 36 x 260 30 K 12 3 4.3" Nwt 2.0 VL 22 2622.20 0.00236 17 W 24 x 306 30 K 12 2.7 3.5" Nwt 2.0 VL 22 2625.10 0.00231 18 W 36 x 439 30 K 12 2.7 3.5" Nwt 2.0 VL 22 2665.00 0.00216
150
B.3 – Run 8 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 90 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2192.30 0.00426 2 W 36 x 135 30 K 11 3.3 2.0" Nwt 1.5 VL 22 2205.40 0.00414 3 W 36 x 135 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2205.80 0.00407 4 W 36 x 170 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2216.30 0.00402 5 W 30 x 326 30 K 11 3.3 2.0" Nwt 1.5 VL 22 2262.70 0.00401 6 W 30 x 326 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2263.10 0.00395 7 W 33 x 387 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2281.40 0.00389 8 W 40 x 392 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2282.90 0.00386 9 W 36 x 439 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2297.00 0.00386 10 W 36 x 135 30 K 11 3.3 2.5" Nwt 2.0 VL 22 2322.10 0.00385 11 W 36 x 135 30 K 12 3.3 2.5" Nwt 2.0 VL 22 2322.40 0.00379 12 W 30 x 90 28 K 12 3.3 3.3" Nwt 1.5 VL 22 2331.00 0.00369 13 W 36 x 135 28 K 12 3.3 3.3" Nwt 1.5 VL 22 2344.50 0.00347 14 W 40 x 392 28 K 12 3.3 3.3" Nwt 1.5 VL 22 2421.60 0.00324 15 W 40 x 392 30 K 12 3.3 3.3" Nwt 1.5 VL 22 2421.80 0.00318 16 W 30 x 90 28 K 12 2.7 3.0" Nwt 1.5 VL 22 2476.10 0.00317 17 W 14 x 426 28 K 12 3 3.0" Nwt 1.5 VL 22 2476.90 0.00304 18 W 40 x 431 28 K 12 3 3.0" Nwt 1.5 VL 22 2478.40 0.00277 19 W 40 x 392 30 K 11 2.7 3.0" Nwt 1.5 VL 22 2566.50 0.00269 20 W 40 x 392 30 K 12 2.7 3.0" Nwt 1.5 VL 22 2566.90 0.00265 21 W 40 x 392 30 K 11 2.5 3.0" Nwt 1.5 VL 22 2666.50 0.00262 22 W 40 x 431 28 K 12 2.5 3.0" Nwt 1.5 VL 22 2678.40 0.00261 23 W 40 x 392 30 K 12 3 3.0" Nwt 2.0 VL 21 2808.90 0.00251 24 W 33 x 141 30 K 12 2.7 3.0" Nwt 3.0 VL 20 2872.60 0.00236 25 W 36 x 135 28 K 12 2.5 3.0" Nwt 3.0 VL 20 2970.60 0.00233 26 W 40 x 324 28 K 12 2.5 3.0" Nwt 3.0 VL 20 3027.30 0.00214 27 W 40 x 392 30 K 11 2.5 3.0" Nwt 3.0 VL 20 3047.50 0.00212 28 W 40 x 392 30 K 12 2.5 3.0" Nwt 3.0 VL 20 3047.90 0.00209 29 W 36 x 439 30 K 12 2.5 3.0" Nwt 3.0 VL 20 3062.00 0.00208 30 W 36 x 650 30 K 12 2.5 3.0" Nwt 3.0 VL 17 3935.30 0.00204
151
B.3 – Run 9 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 27 x 94 28 K 12 3.3 2.5" Nwt 1.5 VL 22 2235.30 0.00380 2 W 27 x 94 30 K 12 3.3 2.5" Nwt 1.5 VL 22 2235.50 0.00375 3 W 33 x 141 30 K 12 3.3 2.5" Nwt 1.5 VL 22 2249.60 0.00350 4 W 33 x 152 30 K 12 3.3 2.5" Nwt 1.5 VL 22 2252.90 0.00348 5 W 40 x 183 28 K 12 3.3 2.5" Nwt 1.5 VL 22 2262.00 0.00344 6 W 40 x 235 28 K 12 3.3 2.5" Nwt 1.5 VL 22 2277.60 0.00340 7 W 40 x 235 30 K 12 3.3 2.5" Nwt 1.5 VL 22 2277.80 0.00334 8 W 40 x 372 28 K 12 3.3 2.5" Nwt 1.5 VL 22 2318.70 0.00333 9 W 40 x 372 30 K 12 3.3 2.5" Nwt 1.5 VL 22 2318.90 0.00327 10 W 40 x 235 30 K 12 3.3 3.3" Nwt 1.5 VL 22 2374.70 0.00326 11 W 40 x 235 30 K 12 3 2.5" Nwt 1.5 VL 22 2377.80 0.00323 12 W 40 x 372 30 K 11 3 2.5" Nwt 1.5 VL 22 2418.50 0.00321 13 W 36 x 527 28 K 12 3 2.5" Nwt 1.5 VL 22 2465.20 0.00320 14 W 40 x 235 30 K 11 2.7 2.5" Nwt 1.5 VL 22 2477.40 0.00319 15 W 40 x 235 30 K 12 2.7 2.5" Nwt 1.5 VL 22 2477.80 0.00313 16 W 40 x 278 30 K 12 3 3.3" Nwt 1.5 VL 22 2487.60 0.00313 17 W 40 x 372 30 K 12 3 3.3" Nwt 1.5 VL 22 2515.80 0.00308 18 W 40 x 372 30 K 12 2.7 2.5" Nwt 1.5 VL 22 2518.90 0.00306 19 W 27 x 94 30 K 11 3 4.3" Nwt 1.5 VL 22 2525.30 0.00302 20 W 40 x 235 30 K 11 3 4.3" Nwt 1.5 VL 22 2567.60 0.00256 21 W 21 x 182 30 K 12 2.7 3.5" Nwt 2.0 VL 22 2587.90 0.00251 22 W 40 x 235 30 K 12 2.7 3.5" Nwt 2.0 VL 22 2603.80 0.00223
152
B.3 – Run 10 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 21 x 101 28 K 12 3 2.5" Nwt 2.0 VL 22 2379.40 0.00358 2 W 33 x 130 30 K 11 3 2.5" Nwt 2.0 VL 22 2387.90 0.00317 3 W 33 x 141 28 K 12 3 2.5" Nwt 2.0 VL 22 2391.40 0.00315 4 W 33 x 387 30 K 11 3 2.5" Nwt 2.0 VL 22 2465.00 0.00295 5 W 27 x 178 30 K 12 3 3.5" Nwt 2.0 VL 22 2528.70 0.00295 6 W 33 x 130 30 K 11 3 4.3" Nwt 1.5 VL 22 2536.10 0.00275 7 W 33 x 130 30 K 12 3 4.3" Nwt 1.5 VL 22 2536.50 0.00272 8 W 27 x 178 30 K 12 3 4.3" Nwt 1.5 VL 22 2550.90 0.00272 9 W 36 x 245 30 K 12 3 4.3" Nwt 1.5 VL 22 2571.00 0.00254 10 W 30 x 326 30 K 12 3 4.3" Nwt 1.5 VL 22 2595.30 0.00254 11 W 33 x 387 28 K 12 3 4.3" Nwt 1.5 VL 22 2613.50 0.00252 12 W 36 x 393 30 K 11 3 4.3" Nwt 1.5 VL 22 2615.00 0.00250 13 W 36 x 245 30 K 12 2.7 4.3" Nwt 1.5 VL 22 2671.00 0.00247 14 W 33 x 387 28 K 12 2.7 4.3" Nwt 1.5 VL 22 2713.50 0.00245 15 W 27 x 178 30 K 12 2.7 3.5" Nwt 2.0 VL 20 2736.70 0.00240 16 W 33 x 387 30 K 12 2.5 4.3" Nwt 1.5 VL 22 2813.60 0.00234 17 W 27 x 178 28 K 12 2.5 4.0" Nwt 1.5 VL 20 2835.50 0.00230 18 W 33 x 387 30 K 12 2.7 4.0" Nwt 1.5 VL 21 2948.40 0.00210 19 W 36 x 393 30 K 12 2.5 3.0" Nwt 3.0 VL 16 3573.20 0.00210
153
B.4 Joist Span 1 (L = 30 ft) Loading 2 (LL = 80 psf) B.4 – Run 1 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 99 30 K 7 4.3 2.0" Nwt 1.5 VL 22 1489.40 0.00390 2 W 30 x 99 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1490.20 0.00389 3 W 30 x 124 30 K 7 4.3 2.0" Nwt 1.5 VL 22 1496.90 0.00383 4 W 30 x 132 30 K 7 4.3 2.0" Nwt 1.5 VL 22 1499.30 0.00382 5 W 33 x 130 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1499.50 0.00379 6 W 24 x 207 30 K 7 4.3 2.0" Nwt 1.5 VL 22 1521.80 0.00375 7 W 27 x 217 30 K 7 4.3 2.0" Nwt 1.5 VL 22 1524.80 0.00371 8 W 27 x 217 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1525.60 0.00371 9 W 36 x 256 30 K 7 4.3 2.0" Nwt 1.5 VL 22 1536.50 0.00359 10 W 30 x 99 30 K 9 3.8 2.0" Nwt 1.5 VL 22 1589.70 0.00355 11 W 30 x 132 30 K 9 3.8 2.0" Nwt 1.5 VL 22 1599.60 0.00347 12 W 36 x 527 30 K 7 4.3 2.0" Nwt 1.5 VL 22 1617.80 0.00344 13 W 24 x 207 30 K 7 3.8 2.0" Nwt 1.5 VL 22 1621.80 0.00341 14 W 24 x 207 30 K 9 3.8 2.0" Nwt 1.5 VL 22 1622.10 0.00341 15 W 24 x 207 30 K 10 3.8 2.0" Nwt 1.5 VL 22 1622.60 0.00341 16 W 36 x 256 30 K 9 3.8 2.0" Nwt 1.5 VL 22 1636.80 0.00325 17 W 30 x 99 30 K 7 3.8 3.5" Nwt 1.5 VL 22 1664.40 0.00284 18 W 30 x 99 30 K 10 3.8 3.5" Nwt 1.5 VL 22 1665.20 0.00283 19 W 30 x 108 30 K 10 3.8 3.5" Nwt 1.5 VL 22 1667.90 0.00280 20 W 30 x 132 30 K 9 3.8 3.5" Nwt 1.5 VL 22 1674.60 0.00272 21 W 14 x 455 30 K 7 3.8 3.5" Nwt 1.5 VL 22 1771.20 0.00260 22 W 36 x 135 30 K 9 3.3 3.5" Nwt 1.5 VL 22 1775.50 0.00244 23 W 36 x 527 30 K 7 3.8 3.5" Nwt 1.5 VL 22 1792.80 0.00224 24 W 24 x 335 30 K 9 3.3 3.5" Nwt 2.0 VL 22 1861.30 0.00221 25 W 36 x 256 28 K 7 3 3.5" Nwt 1.5 VL 22 1911.30 0.00215 26 W 36 x 527 30 K 9 3.3 3.5" Nwt 2.0 VL 22 1918.90 0.00196 27 W 36 x 527 28 K 7 2.7 3.5" Nwt 2.0 VL 22 2118.40 0.00177 28 W 36 x 527 30 K 9 2.7 3.5" Nwt 2.0 VL 22 2118.90 0.00170 29 W 40 x 331 24 K 9 2.5 4.0" Nwt 2.0 VL 22 2184.30 0.00164 30 W 36 x 650 24 K 10 2.5 4.0" Nwt 3.0 VL 20 2483.70 0.00138
154
B.4 – Run 2 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 27 x 84 26 K 10 4.3 2.0" Nwt 1.5 VL 20 1575.30 0.00414 2 W 21 x 93 28 K 12 4.3 2.0" Nwt 1.5 VL 20 1579.00 0.00407 3 W 33 x 130 26 K 10 4.3 2.0" Nwt 1.5 VL 20 1589.10 0.00397 4 W 33 x 130 28 K 12 4.3 2.0" Nwt 1.5 VL 20 1590.10 0.00387 5 W 36 x 210 26 K 9 4.3 2.0" Nwt 1.5 VL 20 1612.70 0.00382 6 W 36 x 210 26 K 10 4.3 2.0" Nwt 1.5 VL 20 1613.10 0.00382 7 W 21 x 93 26 K 9 3.8 2.0" Nwt 2.0 VL 22 1613.40 0.00365 8 W 21 x 93 30 K 8 3.8 2.0" Nwt 2.0 VL 22 1613.70 0.00351 9 W 33 x 118 28 K 7 3.8 2.0" Nwt 2.0 VL 22 1620.70 0.00342 10 W 33 x 118 28 K 9 3.8 2.0" Nwt 2.0 VL 22 1621.10 0.00335 11 W 33 x 118 28 K 12 3.8 2.0" Nwt 2.0 VL 22 1622.30 0.00335 12 W 33 x 130 28 K 12 3.8 2.0" Nwt 2.0 VL 22 1625.90 0.00332 13 W 36 x 150 30 K 10 3.8 2.0" Nwt 2.0 VL 22 1631.30 0.00319 14 W 36 x 210 30 K 8 3.8 2.0" Nwt 2.0 VL 22 1648.80 0.00310 15 W 36 x 210 28 K 9 3.8 3.0" Nwt 1.5 VL 22 1692.90 0.00305 16 W 33 x 118 28 K 12 3.8 3.5" Nwt 1.5 VL 22 1693.90 0.00304 17 W 33 x 130 30 K 7 3.8 3.5" Nwt 1.5 VL 22 1696.00 0.00295 18 W 40 x 278 24 K 12 3.8 3.5" Nwt 1.5 VL 22 1741.50 0.00282 19 W 36 x 300 26 K 10 3.8 3.5" Nwt 1.5 VL 22 1747.50 0.00276 20 W 24 x 131 26 K 9 3.3 4.5" Nwt 1.5 VL 22 1823.80 0.00239 21 W 40 x 183 24 K 12 3.3 4.5" Nwt 1.5 VL 22 1840.50 0.00204 22 W 36 x 232 26 K 9 3 4.5" Nwt 1.5 VL 22 1954.10 0.00186 23 W 40 x 331 28 K 9 3 4.0" Nwt 1.5 VL 21 2139.40 0.00186 24 W 30 x 235 26 K 10 3 4.0" Nwt 3.0 VL 20 2159.40 0.00182 25 W 33 x 318 20 K 10 2.5 4.5" Nwt 3.0 VL 22 2319.10 0.00153 26 W 36 x 798 28 K 9 2.7 4.0" Nwt 2.0 VL 21 2404.70 0.00153 27 W 40 x 278 26 K 9 2.5 4.5" Nwt 3.0 VL 16 2712.10 0.00140
155
B.4 – Run 3 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 24 x 68 24 K 10 4.3 2.0" Nwt 1.5 VL 22 1480.30 0.00433 2 W 24 x 68 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1480.70 0.00413 3 W 33 x 152 26 K 9 4.3 2.0" Nwt 1.5 VL 22 1505.30 0.00393 4 W 33 x 152 30 K 9 4.3 2.0" Nwt 1.5 VL 22 1505.60 0.00374 5 W 33 x 152 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1506.10 0.00374 6 W 36 x 170 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1511.50 0.00369 7 W 40 x 235 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1530.80 0.00367 8 W 40 x 235 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1531.00 0.00359 9 W 40 x 235 30 K 8 4.3 2.5" Nwt 1.5 VL 22 1572.50 0.00356 10 W 33 x 387 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1576.60 0.00351 11 W 33 x 152 28 K 8 3.8 2.0" Nwt 1.5 VL 22 1605.40 0.00348 12 W 33 x 152 30 K 10 3.8 2.0" Nwt 1.5 VL 22 1606.10 0.00340 13 W 24 x 103 30 K 10 3.8 2.5" Nwt 1.5 VL 22 1616.60 0.00335 14 W 40 x 235 30 K 10 3.8 2.0" Nwt 1.5 VL 22 1631.00 0.00324 15 W 33 x 152 30 K 10 3.8 2.5" Nwt 1.5 VL 22 1631.30 0.00309 16 W 36 x 210 30 K 10 3.8 2.5" Nwt 1.5 VL 22 1648.70 0.00297 17 W 36 x 182 28 K 8 3.8 2.5" Nwt 2.0 VL 22 1664.80 0.00289 18 W 33 x 221 30 K 10 3.8 2.5" Nwt 2.0 VL 22 1677.20 0.00281 19 W 36 x 210 30 K 10 3.8 2.0" Nwt 3.0 VL 22 1762.70 0.00276 20 W 24 x 103 22 K 11 3.3 4.0" Nwt 2.0 VL 22 1816.40 0.00261 21 W 36 x 210 24 K 12 3.3 4.0" Nwt 2.0 VL 22 1849.20 0.00210 22 W 33 x 221 22 K 11 3 4.0" Nwt 2.0 VL 22 1951.80 0.00204 23 W 40 x 235 22 K 11 3 4.0" Nwt 2.0 VL 22 1956.00 0.00196 24 W 33 x 291 28 K 8 3 4.0" Nwt 2.0 VL 22 1972.50 0.00185 25 W 33 x 387 30 K 11 3 4.0" Nwt 1.5 VL 22 1977.20 0.00184 26 W 33 x 221 28 K 7 2.5 4.0" Nwt 2.0 VL 22 2151.20 0.00176 27 W 33 x 387 18 K 10 2.7 4.5" Nwt 3.0 VL 21 2437.60 0.00171 28 W 40 x 372 22 K 9 2.5 4.5" Nwt 2.0 VL 18 2491.60 0.00162
156
B.4 – Run 4 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 24 x 192 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1518.90 0.00377 2 W 30 x 235 30 K 9 4.3 2.0" Nwt 1.5 VL 22 1530.50 0.00366 3 W 30 x 108 24 K 9 3.8 2.5" Nwt 1.5 VL 22 1617.20 0.00349 4 W 30 x 108 26 K 8 3.8 2.5" Nwt 1.5 VL 22 1617.20 0.00340 5 W 30 x 124 30 K 9 3.8 2.5" Nwt 1.5 VL 22 1622.40 0.00320 6 W 30 x 191 30 K 12 3.8 2.5" Nwt 1.5 VL 22 1643.80 0.00306 7 W 36 x 232 30 K 12 3.8 2.5" Nwt 1.5 VL 22 1656.10 0.00294 8 W 27 x 336 30 K 12 3.8 2.5" Nwt 1.5 VL 22 1687.30 0.00293 9 W 30 x 191 30 K 12 3.8 3.5" Nwt 1.5 VL 22 1715.90 0.00286 10 W 30 x 191 30 K 11 3.3 2.5" Nwt 1.5 VL 22 1743.40 0.00281 11 W 30 x 235 30 K 11 3.3 2.5" Nwt 1.5 VL 22 1756.60 0.00275 12 W 33 x 241 28 K 10 3.3 2.5" Nwt 2.0 VL 22 1783.00 0.00262 13 W 27 x 194 28 K 8 3.8 4.3" Nwt 2.0 VL 22 1785.30 0.00254 14 W 40 x 324 28 K 10 3.3 2.5" Nwt 2.0 VL 22 1807.90 0.00249 15 W 24 x 192 28 K 9 3.3 4.0" Nwt 1.5 VL 22 1817.70 0.00237 16 W 27 x 146 30 K 9 3.3 4.5" Nwt 2.0 VL 22 1854.40 0.00213 17 W 30 x 191 28 K 8 3.3 4.5" Nwt 2.0 VL 22 1867.70 0.00200 18 W 27 x 194 30 K 7 3 4.5" Nwt 1.5 VL 22 1942.70 0.00200 19 W 27 x 161 30 K 12 3 4.5" Nwt 2.0 VL 22 1960.20 0.00198 20 W 30 x 191 30 K 12 3 4.5" Nwt 1.5 VL 20 2033.40 0.00195 21 W 30 x 235 30 K 12 3 4.5" Nwt 1.5 VL 20 2046.60 0.00188 22 W 36 x 439 30 K 7 3 4.0" Nwt 1.5 VL 20 2081.60 0.00179 23 W 36 x 650 28 K 9 3.3 4.3" Nwt 3.0 VL 22 2138.20 0.00177 24 W 14 x 605 24 K 10 3 4.5" Nwt 3.0 VL 21 2403.40 0.00173 25 W 36 x 300 26 K 8 2.7 4.5" Nwt 3.0 VL 18 2528.60 0.00149
157
B.4 – Run 5 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 21 x 111 30 K 7 4.3 2.0" Nwt 1.5 VL 22 1493.00 0.00396 2 W 21 x 111 26 K 9 4.3 2.0" Nwt 1.5 VL 22 1493.00 0.00413 3 W 21 x 111 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1494.20 0.00395 4 W 33 x 130 28 K 8 4.3 2.0" Nwt 1.5 VL 22 1498.80 0.00387 5 W 33 x 130 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1499.30 0.00387 6 W 33 x 130 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1499.90 0.00379 7 W 40 x 215 28 K 9 4.3 2.0" Nwt 1.5 VL 22 1524.40 0.00368 8 W 40 x 249 28 K 8 4.3 2.0" Nwt 1.5 VL 22 1534.50 0.00365 9 W 36 x 439 28 K 8 4.3 2.0" Nwt 1.5 VL 22 1591.50 0.00355 10 W 24 x 146 30 K 11 3.8 2.0" Nwt 1.5 VL 22 1604.70 0.00350 11 W 27 x 258 30 K 11 3.8 2.0" Nwt 1.5 VL 22 1638.30 0.00332 12 W 40 x 331 30 K 7 3.8 2.0" Nwt 1.5 VL 22 1659.00 0.00317 13 W 40 x 215 26 K 12 3.3 2.0" Nwt 1.5 VL 22 1725.50 0.00314 14 W 40 x 235 24 K 10 3.8 3.0" Nwt 2.0 VL 22 1728.40 0.00297 15 W 40 x 235 28 K 10 3.8 3.0" Nwt 2.0 VL 22 1728.80 0.00283 16 W 40 x 149 24 K 9 3.8 4.3" Nwt 1.5 VL 22 1743.60 0.00268 17 W 40 x 215 24 K 9 3.8 4.3" Nwt 1.5 VL 22 1763.40 0.00254 18 W 40 x 215 28 K 10 3.8 4.3" Nwt 1.5 VL 22 1764.10 0.00243 19 W 40 x 149 28 K 8 3.3 4.3" Nwt 2.0 VL 22 1871.80 0.00229 20 W 14 x 370 28 K 10 3.3 4.0" Nwt 2.0 VL 22 1896.70 0.00227 21 W 40 x 215 28 K 9 3.3 4.5" Nwt 1.5 VL 20 1939.20 0.00193 22 W 40 x 215 28 K 9 3 4.5" Nwt 1.5 VL 20 2039.20 0.00181 23 W 40 x 215 28 K 10 3 4.5" Nwt 1.5 VL 20 2039.60 0.00181 24 W 40 x 331 26 K 8 3 4.5" Nwt 2.0 VL 19 2351.50 0.00167 25 W 27 x 217 30 K 11 2.7 4.5" Nwt 3.0 VL 16 2595.00 0.00166 26 W 33 x 354 26 K 7 2.7 4.5" Nwt 3.0 VL 16 2634.50 0.00156
158
B.4 – Run 6 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 24 x 103 26 K 9 4.3 2.0" Nwt 2.0 VL 20 1623.20 0.00425 2 W 30 x 116 30 K 7 4.3 2.0" Nwt 2.0 VL 20 1627.20 0.00397 3 W 30 x 116 30 K 7 4.3 2.0" Nwt 1.5 VL 21 1674.50 0.00385 4 W 18 x 130 30 K 9 3.8 3.0" Nwt 2.0 VL 22 1674.60 0.00302 5 W 40 x 183 24 K 10 3.8 3.0" Nwt 2.0 VL 22 1690.40 0.00269 6 W 36 x 170 30 K 7 3.8 3.5" Nwt 2.0 VL 22 1736.70 0.00269 7 W 33 x 201 30 K 11 3.3 3.0" Nwt 1.5 VL 22 1771.60 0.00253 8 W 40 x 183 26 K 8 3.3 3.0" Nwt 2.0 VL 22 1790.10 0.00246 9 W 40 x 327 30 K 9 3.8 4.3" Nwt 1.5 VL 22 1797.50 0.00230 10 W 27 x 336 30 K 7 3.3 4.3" Nwt 1.5 VL 22 1899.80 0.00225 11 W 40 x 199 30 K 9 3.3 3.0" Nwt 3.0 VL 22 1908.70 0.00212 12 W 36 x 170 26 K 8 3.3 3.5" Nwt 3.0 VL 22 1924.80 0.00211 13 W 40 x 183 26 K 8 3.3 3.5" Nwt 3.0 VL 22 1928.70 0.00205 14 W 40 x 327 28 K 8 3.3 4.5" Nwt 1.5 VL 20 1972.70 0.00184 15 W 27 x 539 22 K 10 3.3 4.0" Nwt 3.0 VL 22 2060.30 0.00181 16 W 40 x 327 28 K 8 3 4.5" Nwt 1.5 VL 20 2072.70 0.00172 17 W 27 x 539 28 K 7 2.7 3.5" Nwt 3.0 VL 22 2235.40 0.00167 18 W 40 x 327 24 K 12 2.5 4.5" Nwt 1.5 VL 20 2273.70 0.00159 19 W 40 x 372 24 K 10 2.7 4.5" Nwt 3.0 VL 16 2640.50 0.00143
159
B.4 – Run 7 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 24 x 84 28 K 8 4.3 2.0" Nwt 1.5 VL 22 1485.00 0.00408 2 W 30 x 99 30 K 9 4.3 2.0" Nwt 1.5 VL 22 1489.70 0.00390 3 W 30 x 99 30 K 9 4.3 2.0" Nwt 1.5 VL 22 1489.70 0.00390 4 W 30 x 99 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1490.20 0.00389 5 W 30 x 124 30 K 9 4.3 2.0" Nwt 1.5 VL 22 1497.20 0.00383 6 W 30 x 173 30 K 9 4.3 2.0" Nwt 1.5 VL 22 1511.90 0.00374 7 W 30 x 261 28 K 8 4.3 2.0" Nwt 1.5 VL 22 1538.10 0.00371 8 W 30 x 261 30 K 9 4.3 2.0" Nwt 1.5 VL 22 1538.30 0.00363 9 W 40 x 277 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1543.60 0.00354 10 W 27 x 84 30 K 9 3.8 2.5" Nwt 1.5 VL 22 1610.40 0.00338 11 W 30 x 99 28 K 8 3.8 2.5" Nwt 1.5 VL 22 1614.70 0.00335 12 W 27 x 102 30 K 7 3.8 2.5" Nwt 1.5 VL 22 1615.50 0.00331 13 W 21 x 182 30 K 8 3.8 2.5" Nwt 1.5 VL 22 1639.80 0.00321 14 W 40 x 277 30 K 9 3.8 2.0" Nwt 1.5 VL 22 1643.10 0.00320 15 W 24 x 250 26 K 12 3.8 2.5" Nwt 1.5 VL 22 1661.20 0.00320 16 W 24 x 250 28 K 12 3.8 2.5" Nwt 1.5 VL 22 1661.30 0.00312 17 W 30 x 261 28 K 7 3.8 2.5" Nwt 1.5 VL 22 1663.00 0.00309 18 W 40 x 277 28 K 7 3.8 2.5" Nwt 1.5 VL 22 1667.80 0.00298 19 W 40 x 277 30 K 7 3.8 2.5" Nwt 1.5 VL 22 1668.00 0.00286 20 W 40 x 331 30 K 8 3.8 2.5" Nwt 1.5 VL 22 1684.50 0.00283 21 W 30 x 261 30 K 9 3.8 3.3" Nwt 2.0 VL 22 1750.30 0.00273 22 W 33 x 130 28 K 8 3.8 4.3" Nwt 2.0 VL 22 1766.10 0.00259 23 W 40 x 277 30 K 9 3.8 4.3" Nwt 2.0 VL 22 1810.50 0.00222 24 W 40 x 362 26 K 9 3.3 3.3" Nwt 3.0 VL 22 1996.30 0.00220 25 W 30 x 391 28 K 6 3 4.5" Nwt 1.5 VL 22 2001.50 0.00191 26 W 40 x 277 30 K 8 3.3 3.5" Nwt 3.0 VL 20 2047.30 0.00188 27 W 40 x 277 28 K 7 2.7 3.5" Nwt 3.0 VL 20 2246.80 0.00170 28 W 40 x 277 28 K 7 2.5 3.5" Nwt 3.0 VL 20 2346.80 0.00160 29 W 40 x 277 28 K 7 3 4.5" Nwt 3.0 VL 19 2488.20 0.00156 30 W 36 x 527 30 K 10 2.5 4.0" Nwt 2.0 VL 18 2514.00 0.00147
160
B.4 – Run 8 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 24 x 103 28 K 8 4.3 2.0" Nwt 1.5 VL 22 1490.70 0.00402 2 W 24 x 103 30 K 8 4.3 2.0" Nwt 1.5 VL 22 1490.90 0.00394 3 W 24 x 103 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1492.20 0.00393 4 W 24 x 103 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1492.20 0.00393 5 W 30 x 108 30 K 8 4.3 2.0" Nwt 1.5 VL 22 1492.40 0.00387 6 W 30 x 108 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1493.70 0.00386 7 W 36 x 150 30 K 8 4.3 2.0" Nwt 1.5 VL 22 1505.00 0.00373 8 W 36 x 170 30 K 8 4.3 2.0" Nwt 1.5 VL 22 1511.00 0.00369 9 W 36 x 170 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1511.50 0.00369 10 W 36 x 170 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1512.30 0.00369 11 W 40 x 183 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1516.20 0.00364 12 W 36 x 245 30 K 8 4.3 2.0" Nwt 1.5 VL 22 1533.50 0.00360 13 W 30 x 108 30 K 8 3.8 2.0" Nwt 1.5 VL 22 1592.40 0.00353 14 W 30 x 108 30 K 12 3.8 2.0" Nwt 1.5 VL 22 1593.70 0.00352 15 W 36 x 135 28 K 12 3.8 2.0" Nwt 1.5 VL 22 1601.60 0.00349 16 W 36 x 170 30 K 8 3.8 2.0" Nwt 1.5 VL 22 1611.00 0.00335 17 W 30 x 108 30 K 8 3.8 2.0" Nwt 2.0 VL 22 1618.20 0.00335 18 W 36 x 150 30 K 8 3.8 2.0" Nwt 2.0 VL 22 1630.80 0.00319 19 W 36 x 170 30 K 8 3.8 2.0" Nwt 2.0 VL 22 1636.80 0.00315 20 W 36 x 170 30 K 11 3.8 2.0" Nwt 2.0 VL 22 1637.70 0.00315 21 W 36 x 170 30 K 12 3.8 2.0" Nwt 2.0 VL 22 1638.10 0.00315 22 W 30 x 211 30 K 8 3.8 2.0" Nwt 2.0 VL 22 1649.10 0.00315 23 W 36 x 245 28 K 8 3.8 2.0" Nwt 2.0 VL 22 1659.10 0.00313 24 W 36 x 245 30 K 12 3.8 2.5" Nwt 1.5 VL 22 1660.00 0.00293 25 W 40 x 167 30 K 8 3.8 3.0" Nwt 2.0 VL 22 1708.10 0.00288 26 W 30 x 261 30 K 8 3.8 3.0" Nwt 2.0 VL 22 1736.30 0.00285 27 W 40 x 264 30 K 10 3.3 2.0" Nwt 2.0 VL 22 1765.50 0.00277 28 W 30 x 148 28 K 12 3.8 4.3" Nwt 2.0 VL 22 1772.90 0.00258 29 W 24 x 103 28 K 12 3.3 4.0" Nwt 2.0 VL 22 1817.40 0.00249 30 W 30 x 108 22 K 10 3.3 4.0" Nwt 2.0 VL 22 1817.60 0.00248 31 W 33 x 241 28 K 12 3.3 4.0" Nwt 2.0 VL 22 1858.80 0.00203 32 W 33 x 291 28 K 12 3 4.3" Nwt 3.0 VL 22 2131.80 0.00184 33 W 36 x 170 30 K 11 2.5 4.0" Nwt 2.0 VL 22 2137.30 0.00175 34 W 40 x 264 26 K 12 2.5 4.0" Nwt 2.0 VL 22 2165.60 0.00165 35 W 36 x 798 24 K 12 2.5 4.3" Nwt 2.0 VL 21 2547.50 0.00161
161
B.4 – Run 9 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 27 x 114 30 K 7 4.3 2.0" Nwt 2.0 VL 22 1536.60 0.00400 2 W 27 x 114 30 K 9 4.3 2.0" Nwt 2.0 VL 22 1536.90 0.00400 3 W 27 x 114 30 K 12 4.3 2.0" Nwt 2.0 VL 22 1538.10 0.00399 4 W 36 x 150 30 K 7 4.3 2.0" Nwt 2.0 VL 22 1547.40 0.00386 5 W 36 x 150 30 K 9 4.3 2.0" Nwt 2.0 VL 22 1547.70 0.00386 6 W 27 x 161 30 K 9 4.3 2.5" Nwt 1.5 VL 22 1550.30 0.00378 7 W 33 x 201 30 K 9 4.3 2.5" Nwt 1.5 VL 22 1562.30 0.00365 8 W 21 x 93 30 K 12 3.8 2.5" Nwt 1.5 VL 22 1614.40 0.00344 9 W 24 x 117 30 K 9 3.8 2.5" Nwt 1.5 VL 22 1620.30 0.00331 10 W 27 x 161 30 K 9 3.8 2.5" Nwt 1.5 VL 22 1633.50 0.00315 11 W 27 x 161 30 K 12 3.8 2.5" Nwt 1.5 VL 22 1634.80 0.00315 12 W 27 x 194 30 K 9 3.8 2.5" Nwt 1.5 VL 22 1643.40 0.00309 13 W 40 x 392 30 K 12 3.8 2.5" Nwt 1.5 VL 22 1704.10 0.00280 14 W 24 x 94 22 K 11 3.3 4.5" Nwt 1.5 VL 22 1813.10 0.00260 15 W 24 x 94 26 K 12 3.3 4.5" Nwt 1.5 VL 22 1814.00 0.00252 16 W 24 x 117 26 K 12 3.3 4.5" Nwt 1.5 VL 22 1820.90 0.00243 17 W 36 x 439 30 K 7 3.3 3.3" Nwt 2.0 VL 22 1903.40 0.00233 18 W 24 x 117 28 K 9 3 4.5" Nwt 1.5 VL 22 1919.80 0.00229 19 W 27 x 114 26 K 12 3 4.5" Nwt 1.5 VL 22 1920.00 0.00226 20 W 24 x 117 30 K 12 3 4.5" Nwt 1.5 VL 22 1921.20 0.00226 21 W 40 x 264 28 K 8 3.3 4.5" Nwt 1.5 VL 20 1953.80 0.00189 22 W 40 x 264 28 K 8 3 4.5" Nwt 1.5 VL 20 2053.80 0.00177 23 W 27 x 235 28 K 8 2.7 4.0" Nwt 3.0 VL 20 2259.10 0.00175
162
B.4 – Run 10 Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 27 x 84 26 K 9 4.3 2.0" Nwt 1.5 VL 22 1484.90 0.00414 2 W 24 x 84 26 K 9 4.3 2.0" Nwt 1.5 VL 22 1484.90 0.00417 3 W 27 x 84 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1485.50 0.00404 4 W 24 x 84 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1486.10 0.00399 5 W 27 x 84 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1486.10 0.00396 6 W 27 x 84 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1486.50 0.00395 7 W 27 x 94 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1489.10 0.00393 8 W 27 x 94 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1489.50 0.00392 9 W 24 x 117 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1496.00 0.00390 10 W 24 x 117 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1496.40 0.00390 11 W 24 x 131 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1500.60 0.00387 12 W 40 x 149 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1506.00 0.00371 13 W 33 x 201 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1521.60 0.00367 14 W 36 x 245 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1534.40 0.00360 15 W 40 x 211 30 K 12 4.3 2.5" Nwt 1.5 VL 22 1566.60 0.00358 16 W 36 x 245 30 K 7 4.3 2.5" Nwt 1.5 VL 22 1575.20 0.00357 17 W 36 x 393 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1579.20 0.00350 18 W 27 x 146 30 K 7 3.8 2.0" Nwt 1.5 VL 22 1603.50 0.00347 19 W 40 x 149 30 K 7 3.8 2.0" Nwt 1.5 VL 22 1604.40 0.00336 20 W 36 x 210 30 K 12 3.8 2.0" Nwt 1.5 VL 22 1624.30 0.00330 21 W 27 x 84 30 K 11 3.8 2.5" Nwt 2.0 VL 22 1636.50 0.00321 22 W 36 x 393 30 K 12 3.8 2.0" Nwt 1.5 VL 22 1679.20 0.00316 23 W 40 x 211 30 K 8 3.8 3.3" Nwt 1.5 VL 22 1706.60 0.00283 24 W 40 x 215 28 K 10 3.8 3.5" Nwt 1.5 VL 20 1812.10 0.00274 25 W 40 x 235 30 K 7 3.3 3.3" Nwt 1.5 VL 22 1813.50 0.00256 26 W 27 x 281 28 K 10 3.8 3.5" Nwt 3.0 VL 22 1886.60 0.00246 27 W 27 x 281 30 K 12 3.8 3.5" Nwt 3.0 VL 22 1887.60 0.00243 28 W 36 x 245 30 K 11 3.8 4.3" Nwt 2.0 VL 20 1891.80 0.00230 29 W 36 x 393 30 K 11 3.8 4.3" Nwt 2.0 VL 20 1936.20 0.00218 30 W 40 x 215 22 K 10 3.3 4.3" Nwt 3.0 VL 22 2007.60 0.00208 31 W 36 x 300 30 K 7 2.7 3.0" Nwt 2.0 VL 22 2025.30 0.00196 32 W 33 x 354 26 K 8 2.7 3.5" Nwt 3.0 VL 22 2180.00 0.00170 33 W 33 x 354 26 K 9 2.7 3.5" Nwt 3.0 VL 22 2180.10 0.00169 34 W 36 x 393 30 K 9 2.7 4.0" Nwt 2.0 VL 21 2283.30 0.00163 35 W 40 x 362 28 K 10 2.7 4.0" Nwt 3.0 VL 18 2522.70 0.00150 36 W 40 x 331 24 K 9 2.7 4.5" Nwt 3.0 VL 19 2604.50 0.00147
163
B.5 Joist Span 2 (L = 40 ft) Loading 2 (LL = 80 psf) B.5 – Run 1
Num Girder Joist Spacing Concrete Deck Cost Ap 1 W 27 x 94 28 K 10 3.3 2.0" Nwt 1.5 VL 22 1959.20 0.00463 2 W 27 x 94 30 K 10 3.3 2.0" Nwt 1.5 VL 22 1959.40 0.00442 3 W 27 x 94 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1959.80 0.00441 4 W 27 x 94 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1960.10 0.00440 5 W 36 x 135 30 K 10 3.3 2.0" Nwt 1.5 VL 22 1971.70 0.00438 6 W 36 x 135 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1972.10 0.00438 7 W 36 x 135 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1972.40 0.00437 8 W 33 x 201 30 K 10 3.3 2.0" Nwt 1.5 VL 22 1991.50 0.00433 9 W 33 x 201 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1991.90 0.00432 10 W 27 x 94 30 K 11 3.3 2.0" Nwt 2.0 VL 22 1998.00 0.00414 11 W 33 x 201 30 K 10 3.3 2.0" Nwt 2.0 VL 22 2029.70 0.00401 12 W 33 x 201 30 K 11 3.3 2.0" Nwt 2.0 VL 22 2030.10 0.00400 13 W 27 x 94 28 K 10 3 2.0" Nwt 2.0 VL 22 2074.90 0.00363 14 W 27 x 94 30 K 10 3 2.0" Nwt 2.0 VL 22 2075.10 0.00348 15 W 36 x 135 30 K 10 3 2.0" Nwt 2.0 VL 22 2087.40 0.00336 16 W 33 x 201 30 K 10 3 2.0" Nwt 2.0 VL 22 2107.20 0.00329 17 W 36 x 328 30 K 10 3 2.0" Nwt 2.0 VL 22 2145.30 0.00318 18 W 27 x 194 28 K 12 3 3.5" Nwt 1.5 VL 22 2201.10 0.00307 19 W 24 x 250 28 K 12 3 3.5" Nwt 1.5 VL 22 2217.90 0.00305 20 W 27 x 281 28 K 12 3 3.5" Nwt 1.5 VL 22 2227.20 0.00297 21 W 36 x 328 30 K 10 3 3.5" Nwt 1.5 VL 22 2240.70 0.00277 22 W 27 x 94 28 K 10 2.7 4.0" Nwt 1.5 VL 22 2274.10 0.00258 23 W 27 x 94 30 K 10 2.7 4.0" Nwt 1.5 VL 22 2274.30 0.00252 24 W 27 x 94 30 K 11 2.7 4.0" Nwt 1.5 VL 22 2274.70 0.00251 25 W 36 x 135 28 K 10 2.7 4.0" Nwt 1.5 VL 22 2286.40 0.00235 26 W 36 x 135 30 K 10 2.7 4.0" Nwt 1.5 VL 22 2286.60 0.00228 27 W 36 x 135 30 K 11 2.7 4.0" Nwt 1.5 VL 22 2287.00 0.00227 28 W 33 x 201 28 K 10 2.7 4.0" Nwt 1.5 VL 22 2306.20 0.00225 29 W 33 x 201 28 K 12 2.7 4.0" Nwt 1.5 VL 22 2307.00 0.00222 30 W 36 x 328 28 K 10 2.7 4.0" Nwt 1.5 VL 22 2344.30 0.00212 31 W 36 x 328 30 K 11 2.7 4.0" Nwt 1.5 VL 22 2344.90 0.00205 32 W 33 x 387 30 K 11 2.7 4.0" Nwt 1.5 VL 22 2362.60 0.00204 33 W 36 x 798 28 K 12 2.7 4.0" Nwt 1.5 VL 22 2486.10 0.00196 34 W 40 x 331 24 K 12 2.5 4.0" Nwt 3.0 VL 17 3398.50 0.00176
164
B.5 – Run 2 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 24 x 94 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1941.10 0.00395 2 W 24 x 94 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1941.50 0.00395 3 W 27 x 114 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1947.10 0.00392 4 W 27 x 114 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1947.10 0.00392 5 W 27 x 114 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1947.50 0.00392 6 W 33 x 118 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1948.30 0.00388 7 W 33 x 118 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1948.70 0.00388 8 W 36 x 170 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1963.90 0.00379 9 W 40 x 199 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1973.00 0.00373 10 W 40 x 215 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1977.40 0.00371 11 W 40 x 215 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1977.80 0.00371 12 W 30 x 124 30 K 11 3.3 2.5" Nwt 2.0 VL 22 2043.50 0.00368 13 W 33 x 118 30 K 11 3 2.0" Nwt 1.5 VL 22 2048.30 0.00367 14 W 30 x 173 30 K 11 3.3 2.5" Nwt 2.0 VL 22 2058.20 0.00360 15 W 36 x 170 30 K 11 3 2.0" Nwt 1.5 VL 22 2063.90 0.00359 16 W 40 x 199 28 K 12 3.3 2.5" Nwt 2.0 VL 22 2066.20 0.00357 17 W 36 x 182 30 K 11 3 2.0" Nwt 1.5 VL 22 2067.50 0.00357 18 W 36 x 210 30 K 11 3.3 2.5" Nwt 2.0 VL 22 2069.30 0.00351 19 W 40 x 215 30 K 12 3.3 2.5" Nwt 2.0 VL 22 2071.10 0.00346 20 W 40 x 278 30 K 11 3.3 2.5" Nwt 2.0 VL 22 2089.70 0.00342 21 W 40 x 278 30 K 12 3.3 2.5" Nwt 2.0 VL 22 2090.00 0.00342 22 W 36 x 170 30 K 11 3 2.5" Nwt 2.0 VL 22 2157.30 0.00337 23 W 40 x 167 26 K 12 3 3.5" Nwt 1.5 VL 22 2163.10 0.00260 24 W 40 x 372 26 K 12 3 3.5" Nwt 1.5 VL 22 2224.60 0.00243 25 W 27 x 258 30 K 12 2.7 3.5" Nwt 1.5 VL 22 2290.70 0.00241 26 W 40 x 278 26 K 12 2.7 3.5" Nwt 1.5 VL 22 2296.40 0.00239 27 W 40 x 372 28 K 10 2.7 3.5" Nwt 1.5 VL 22 2323.90 0.00231 28 W 33 x 221 30 K 11 2.7 4.0" Nwt 2.0 VL 22 2346.40 0.00203 29 W 40 x 199 28 K 12 2.7 4.5" Nwt 1.5 VL 21 2579.20 0.00197 30 W 33 x 221 30 K 12 2.7 4.5" Nwt 1.5 VL 21 2586.00 0.00197 31 W 40 x 278 30 K 12 2.7 4.5" Nwt 1.5 VL 21 2603.10 0.00187 32 W 40 x 199 30 K 12 2.5 4.5" Nwt 1.5 VL 21 2679.40 0.00187 33 W 30 x 235 28 K 10 2.5 4.0" Nwt 3.0 VL 19 3110.00 0.00186 34 W 36 x 650 28 K 12 2.7 4.5" Nwt 1.5 VL 17 3218.50 0.00180 35 W 36 x 650 28 K 12 2.5 4.5" Nwt 1.5 VL 17 3318.50 0.00173
165
B.5 – Run 3 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 116 28 K 12 3.3 2.0" Nwt 1.5 VL 22 1966.60 0.00455 2 W 40 x 149 30 K 10 3.3 2.0" Nwt 1.5 VL 22 1975.90 0.00436 3 W 40 x 149 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1976.60 0.00435 4 W 36 x 210 30 K 10 3.3 2.0" Nwt 1.5 VL 22 1994.20 0.00431 5 W 36 x 210 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1994.90 0.00430 6 W 30 x 116 28 K 12 3.3 2.0" Nwt 2.0 VL 22 2004.80 0.00424 7 W 36 x 328 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2030.30 0.00422 8 W 30 x 116 30 K 11 3 2.0" Nwt 1.5 VL 22 2066.40 0.00415 9 W 40 x 149 30 K 10 3 2.0" Nwt 1.5 VL 22 2075.90 0.00411 10 W 40 x 149 30 K 11 3 2.0" Nwt 1.5 VL 22 2076.30 0.00410 11 W 36 x 194 30 K 10 3 2.0" Nwt 1.5 VL 22 2089.40 0.00408 12 W 36 x 194 30 K 11 3 2.0" Nwt 1.5 VL 22 2089.80 0.00407 13 W 14 x 145 30 K 11 3 2.0" Nwt 2.0 VL 22 2090.80 0.00382 14 W 40 x 149 30 K 10 3 2.0" Nwt 2.0 VL 22 2091.60 0.00332 15 W 40 x 149 30 K 12 3 2.0" Nwt 2.0 VL 22 2092.40 0.00331 16 W 36 x 210 30 K 10 3 2.0" Nwt 2.0 VL 22 2109.90 0.00327 17 W 36 x 210 30 K 11 3 2.0" Nwt 2.0 VL 22 2110.30 0.00326 18 W 36 x 210 30 K 12 3 2.0" Nwt 2.0 VL 22 2110.70 0.00326 19 W 27 x 194 24 K 12 3 3.0" Nwt 1.5 VL 22 2138.20 0.00309 20 W 27 x 194 26 K 12 3 3.0" Nwt 1.5 VL 22 2138.40 0.00301 21 W 30 x 211 30 K 12 3 3.0" Nwt 1.5 VL 22 2143.80 0.00281 22 W 36 x 280 24 K 12 2.7 3.0" Nwt 1.5 VL 22 2264.00 0.00278 23 W 30 x 326 28 K 10 2.7 3.0" Nwt 1.5 VL 22 2277.30 0.00269 24 W 40 x 149 28 K 12 3 3.0" Nwt 2.0 VL 20 2278.60 0.00268 25 W 30 x 211 30 K 10 3 3.0" Nwt 2.0 VL 20 2296.60 0.00263 26 W 36 x 210 30 K 11 3 3.0" Nwt 2.0 VL 20 2296.70 0.00257 27 W 40 x 149 30 K 12 2.7 3.0" Nwt 2.0 VL 20 2378.80 0.00252 28 W 36 x 210 28 K 12 2.7 3.0" Nwt 2.0 VL 20 2396.90 0.00251 29 W 36 x 210 30 K 12 2.7 3.0" Nwt 2.0 VL 20 2397.10 0.00245 30 W 36 x 328 26 K 12 3 3.5" Nwt 3.0 VL 22 2434.00 0.00244 31 W 36 x 210 28 K 12 2.5 3.0" Nwt 2.0 VL 20 2496.90 0.00241 32 W 30 x 326 30 K 12 2.7 4.5" Nwt 1.5 VL 20 2497.50 0.00192 33 W 40 x 431 26 K 12 2.7 4.5" Nwt 1.5 VL 20 2528.70 0.00187 34 W 14 x 370 30 K 11 2.5 4.5" Nwt 3.0 VL 22 2675.90 0.00183
166
B.5 – Run 4 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 90 26 K 12 3.3 2.0" Nwt 1.5 VL 22 1940.00 0.00417 2 W 30 x 90 28 K 12 3.3 2.0" Nwt 1.5 VL 22 1940.10 0.00405 3 W 30 x 90 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1940.30 0.00393 4 W 33 x 118 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1948.30 0.00388 5 W 33 x 118 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1948.70 0.00388 6 W 33 x 169 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1963.60 0.00381 7 W 33 x 169 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1964.00 0.00381 8 W 33 x 201 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1973.20 0.00377 9 W 33 x 201 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1973.60 0.00377 10 W 40 x 211 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1976.60 0.00372 11 W 33 x 118 30 K 10 3 2.0" Nwt 1.5 VL 22 2047.90 0.00368 12 W 33 x 118 30 K 12 3 2.0" Nwt 1.5 VL 22 2048.70 0.00367 13 W 30 x 148 30 K 12 3 2.0" Nwt 1.5 VL 22 2057.70 0.00365 14 W 36 x 194 30 K 12 3 2.0" Nwt 1.5 VL 22 2071.50 0.00356 15 W 40 x 211 30 K 10 3 2.0" Nwt 1.5 VL 22 2075.80 0.00352 16 W 40 x 211 30 K 11 3 2.0" Nwt 1.5 VL 22 2076.20 0.00352 17 W 40 x 211 30 K 12 3 2.0" Nwt 1.5 VL 22 2076.60 0.00352 18 W 33 x 354 30 K 12 3 2.0" Nwt 1.5 VL 22 2119.50 0.00346 19 W 33 x 201 30 K 11 2.7 2.0" Nwt 1.5 VL 22 2173.20 0.00338 20 W 33 x 201 30 K 12 2.7 2.0" Nwt 1.5 VL 22 2173.60 0.00338 21 W 33 x 118 28 K 10 3 3.0" Nwt 2.0 VL 22 2178.40 0.00328 22 W 33 x 118 30 K 11 3 3.0" Nwt 2.0 VL 22 2179.00 0.00315 23 W 33 x 118 30 K 12 3 3.0" Nwt 2.0 VL 22 2179.30 0.00315 24 W 33 x 169 30 K 11 3 3.0" Nwt 2.0 VL 22 2194.30 0.00305 25 W 33 x 201 30 K 10 3 3.0" Nwt 2.0 VL 22 2203.50 0.00301 26 W 33 x 201 30 K 11 3 3.0" Nwt 2.0 VL 22 2203.90 0.00301 27 W 33 x 201 30 K 12 3 3.0" Nwt 2.0 VL 22 2204.20 0.00300 28 W 36 x 230 30 K 11 3 3.0" Nwt 2.0 VL 22 2212.60 0.00295 29 W 36 x 230 30 K 12 3 3.0" Nwt 2.0 VL 22 2212.90 0.00295 30 W 30 x 326 30 K 10 3 3.0" Nwt 2.0 VL 22 2241.00 0.00293 31 W 33 x 201 28 K 12 3 2.5" Nwt 3.0 VL 22 2291.80 0.00263 32 W 36 x 230 30 K 10 3 3.5" Nwt 3.0 VL 22 2404.20 0.00242 33 W 36 x 230 30 K 12 3 3.5" Nwt 3.0 VL 22 2404.90 0.00241 35 W 36 x 230 28 K 12 2.7 3.5" Nwt 3.0 VL 20 2587.70 0.00201 36 W 36 x 230 30 K 10 2.5 3.5" Nwt 3.0 VL 20 2687.10 0.00191 37 W 33 x 118 28 K 10 2.5 4.5" Nwt 3.0 VL 18 3019.70 0.00187
167
B.5 – Run 5 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 116 28 K 10 3.3 2.0" Nwt 1.5 VL 22 1965.80 0.00463 2 W 30 x 116 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1966.70 0.00440 3 W 36 x 135 30 K 10 3.3 2.0" Nwt 1.5 VL 22 1971.70 0.00438 4 W 36 x 135 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1972.40 0.00437 5 W 18 x 158 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1979.30 0.00434 6 W 40 x 167 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1982.00 0.00432 7 W 33 x 241 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2004.20 0.00429 8 W 27 x 129 30 K 12 3.3 2.5" Nwt 1.5 VL 22 2008.00 0.00395 9 W 36 x 135 30 K 10 3.3 2.5" Nwt 1.5 VL 22 2009.00 0.00388 10 W 36 x 300 30 K 12 3.3 2.5" Nwt 1.5 VL 22 2059.30 0.00369 11 W 36 x 135 30 K 10 3 2.5" Nwt 1.5 VL 22 2109.00 0.00367 12 W 33 x 387 30 K 11 3.3 2.0" Nwt 1.5 VL 20 2149.00 0.00364 13 W 24 x 162 30 K 12 3 2.5" Nwt 2.0 VL 22 2155.20 0.00348 14 W 30 x 191 30 K 12 3 2.5" Nwt 2.0 VL 22 2163.90 0.00339 15 W 27 x 129 28 K 10 3 3.5" Nwt 1.5 VL 22 2180.80 0.00324 16 W 27 x 129 30 K 10 3 3.5" Nwt 1.5 VL 22 2181.00 0.00313 17 W 27 x 129 30 K 11 3 3.5" Nwt 1.5 VL 22 2181.40 0.00312 18 W 36 x 135 30 K 10 3 3.5" Nwt 1.5 VL 22 2182.80 0.00300 19 W 30 x 191 30 K 12 3 3.5" Nwt 1.5 VL 22 2200.40 0.00296 20 W 36 x 230 30 K 11 3 3.5" Nwt 1.5 VL 22 2211.70 0.00285 21 W 33 x 387 30 K 11 3 3.5" Nwt 1.5 VL 22 2258.80 0.00275 22 W 33 x 387 30 K 12 3 3.5" Nwt 1.5 VL 22 2259.20 0.00275 23 W 36 x 527 30 K 11 3 3.5" Nwt 1.5 VL 22 2300.80 0.00268 24 W 36 x 230 30 K 12 2.5 3.5" Nwt 1.5 VL 22 2412.10 0.00259 25 W 33 x 291 30 K 12 2.5 3.5" Nwt 1.5 VL 22 2430.40 0.00256 26 W 24 x 162 26 K 12 2.7 3.5" Nwt 3.0 VL 22 2447.20 0.00230 27 W 27 x 129 28 K 10 2.5 4.5" Nwt 2.0 VL 22 2451.80 0.00210 28 W 36 x 135 28 K 10 2.5 4.5" Nwt 2.0 VL 22 2453.60 0.00197 29 W 36 x 135 30 K 10 2.5 4.5" Nwt 2.0 VL 22 2453.80 0.00192 30 W 33 x 221 28 K 10 2.5 4.5" Nwt 2.0 VL 22 2479.40 0.00186 31 W 36 x 527 28 K 10 2.5 4.5" Nwt 2.0 VL 22 2571.20 0.00168
168
B.5 – Run 6 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 116 28 K 10 3.3 2.0" Nwt 1.5 VL 22 1965.80 0.00463 2 W 21 x 132 28 K 10 3.3 2.0" Nwt 1.5 VL 22 1970.60 0.00460 3 W 33 x 152 28 K 10 3.3 2.0" Nwt 1.5 VL 22 1976.60 0.00459 4 W 33 x 152 30 K 10 3.3 2.0" Nwt 1.5 VL 22 1976.80 0.00437 5 W 27 x 235 30 K 10 3.3 2.0" Nwt 1.5 VL 22 2001.70 0.00433 6 W 27 x 368 30 K 10 3.3 2.0" Nwt 1.5 VL 22 2041.60 0.00426 7 W 40 x 372 30 K 10 3.3 2.0" Nwt 1.5 VL 22 2042.80 0.00419 8 W 30 x 116 30 K 10 3.3 2.0" Nwt 1.5 VL 20 2067.30 0.00391 9 W 33 x 152 30 K 10 3.3 2.0" Nwt 1.5 VL 20 2078.10 0.00384 10 W 33 x 241 28 K 12 3.3 2.0" Nwt 1.5 VL 20 2105.40 0.00384 11 W 40 x 277 28 K 12 3.3 2.0" Nwt 1.5 VL 20 2116.20 0.00377 12 W 36 x 280 30 K 10 3.3 2.0" Nwt 1.5 VL 20 2116.50 0.00369 13 W 40 x 372 30 K 10 3.3 2.0" Nwt 1.5 VL 20 2144.10 0.00362 14 W 33 x 152 30 K 11 3 3.0" Nwt 2.0 VL 22 2189.20 0.00308 15 W 36 x 160 30 K 11 3 3.0" Nwt 2.0 VL 22 2191.60 0.00304 16 W 33 x 141 26 K 12 3 4.3" Nwt 1.5 VL 22 2241.10 0.00275 17 W 33 x 152 26 K 12 3 4.3" Nwt 1.5 VL 22 2244.40 0.00273 18 W 14 x 455 30 K 10 3 3.5" Nwt 1.5 VL 22 2249.00 0.00260 19 W 36 x 280 26 K 12 3 4.3" Nwt 1.5 VL 22 2282.80 0.00253 20 W 30 x 116 26 K 12 2.7 4.0" Nwt 2.0 VL 22 2315.00 0.00234 21 W 33 x 152 26 K 12 2.7 4.0" Nwt 2.0 VL 22 2325.80 0.00221 22 W 36 x 280 26 K 12 2.7 4.0" Nwt 2.0 VL 22 2364.20 0.00203 23 W 24 x 207 26 K 12 2.5 4.5" Nwt 3.0 VL 21 2891.10 0.00184 24 W 24 x 176 30 K 12 2.5 4.5" Nwt 3.0 VL 17 3386.10 0.00184
169
B.5 – Run 7 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 24 x 104 28 K 10 3 2.5" Nwt 2.0 VL 22 2110.70 0.00328 2 W 24 x 104 30 K 10 3 2.5" Nwt 2.0 VL 22 2110.90 0.00317 3 W 33 x 130 30 K 11 3 2.5" Nwt 2.0 VL 22 2119.10 0.00302 4 W 30 x 191 30 K 10 3 2.5" Nwt 2.0 VL 22 2137.00 0.00296 5 W 24 x 104 28 K 12 3 3.5" Nwt 1.5 VL 22 2144.30 0.00288 6 W 27 x 114 30 K 10 3 3.5" Nwt 1.5 VL 22 2146.70 0.00277 7 W 36 x 135 30 K 10 3 3.5" Nwt 1.5 VL 22 2153.00 0.00260 8 W 33 x 169 30 K 10 3 3.5" Nwt 1.5 VL 22 2163.20 0.00256 9 W 27 x 307 30 K 10 3 3.5" Nwt 1.5 VL 22 2204.60 0.00248 10 W 40 x 327 30 K 10 3 3.5" Nwt 1.5 VL 22 2210.60 0.00235 11 W 40 x 211 30 K 10 2.7 3.5" Nwt 1.5 VL 22 2275.80 0.00234 12 W 30 x 326 30 K 10 2.7 3.5" Nwt 1.5 VL 22 2310.30 0.00232 13 W 40 x 327 30 K 10 2.7 3.5" Nwt 1.5 VL 22 2310.60 0.00225 14 W 36 x 439 30 K 10 2.7 3.5" Nwt 1.5 VL 22 2344.20 0.00222 15 W 40 x 327 30 K 10 2.5 3.5" Nwt 1.5 VL 22 2410.60 0.00215 16 W 40 x 297 28 K 10 2.5 3.5" Nwt 2.0 VL 22 2435.80 0.00211 17 W 40 x 297 30 K 10 2.5 3.5" Nwt 2.0 VL 22 2436.00 0.00203 18 W 40 x 327 30 K 10 2.7 4.0" Nwt 2.0 VL 20 2497.80 0.00191 19 W 40 x 327 30 K 10 2.5 3.5" Nwt 3.0 VL 22 2596.20 0.00182 20 W 36 x 230 30 K 10 2.5 4.5" Nwt 3.0 VL 22 2633.50 0.00162 21 W 30 x 326 30 K 10 2.5 4.5" Nwt 3.0 VL 16 3202.30 0.00160
170
B.5 – Run 8 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 24 x 104 30 K 11 3.3 2.0" Nwt 1.5 VL 20 2064.10 0.00394 2 W 21 x 122 30 K 11 3.3 2.0" Nwt 1.5 VL 20 2069.50 0.00393 3 W 30 x 132 30 K 11 3.3 2.0" Nwt 1.5 VL 20 2072.50 0.00388 4 W 30 x 132 30 K 12 3.3 2.0" Nwt 1.5 VL 20 2072.90 0.00388 5 W 30 x 148 30 K 10 3.3 2.0" Nwt 1.5 VL 20 2076.90 0.00387 6 W 30 x 148 30 K 11 3.3 2.0" Nwt 1.5 VL 20 2077.30 0.00386 7 W 30 x 148 30 K 12 3.3 2.0" Nwt 1.5 VL 20 2077.70 0.00386 8 W 40 x 167 30 K 11 3.3 2.0" Nwt 1.5 VL 20 2083.00 0.00377 9 W 40 x 167 30 K 12 3.3 2.0" Nwt 1.5 VL 20 2083.40 0.00377 10 W 30 x 261 30 K 11 3.3 2.0" Nwt 1.5 VL 20 2111.20 0.00375 11 W 30 x 261 30 K 12 3.3 2.0" Nwt 1.5 VL 20 2111.60 0.00374 12 W 40 x 278 30 K 11 3.3 2.0" Nwt 1.5 VL 20 2116.30 0.00367 13 W 40 x 327 30 K 11 3.3 2.0" Nwt 1.5 VL 20 2131.00 0.00364 14 W 30 x 132 28 K 12 3 3.3" Nwt 1.5 VL 22 2163.80 0.00329 15 W 30 x 148 28 K 12 3 3.3" Nwt 1.5 VL 22 2168.60 0.00325 16 W 40 x 167 28 K 12 3 3.3" Nwt 1.5 VL 22 2174.30 0.00312 17 W 30 x 132 30 K 12 3 3.5" Nwt 1.5 VL 22 2182.70 0.00307 18 W 40 x 167 30 K 12 3 3.5" Nwt 1.5 VL 22 2193.20 0.00290 19 W 40 x 215 30 K 11 3 3.5" Nwt 1.5 VL 22 2207.20 0.00282 20 W 40 x 235 30 K 10 3 3.5" Nwt 1.5 VL 22 2212.80 0.00282 21 W 40 x 264 30 K 11 3 3.5" Nwt 1.5 VL 22 2221.90 0.00279 22 W 40 x 278 30 K 11 3 3.5" Nwt 1.5 VL 22 2226.10 0.00278 23 W 40 x 297 30 K 10 3 3.5" Nwt 1.5 VL 22 2231.40 0.00276 24 W 30 x 261 28 K 12 3 2.5" Nwt 3.0 VL 22 2309.80 0.00260 25 W 40 x 264 30 K 10 3 2.5" Nwt 3.0 VL 22 2310.10 0.00247 26 W 40 x 264 30 K 12 3 2.5" Nwt 3.0 VL 22 2310.90 0.00247 27 W 40 x 278 30 K 11 3 2.5" Nwt 3.0 VL 22 2314.70 0.00246 28 W 21 x 122 30 K 12 2.7 4.5" Nwt 1.5 VL 22 2316.30 0.00238 29 W 40 x 278 30 K 12 2.7 4.5" Nwt 1.5 VL 22 2363.10 0.00187 30 W 40 x 327 28 K 12 2.5 4.5" Nwt 1.5 VL 22 2477.60 0.00181 31 W 40 x 278 30 K 12 2.5 4.5" Nwt 3.0 VL 22 2648.70 0.00155 32 W 36 x 650 26 K 12 2.5 4.5" Nwt 3.0 VL 22 2760.00 0.00150 33 W 36 x 650 30 K 12 2.5 4.5" Nwt 3.0 VL 22 2760.30 0.00145
171
B.5 – Run 9 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 27 x 94 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1941.50 0.00394 2 W 30 x 108 30 K 10 3.3 2.0" Nwt 1.5 VL 22 1944.90 0.00393 3 W 30 x 108 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1945.30 0.00392 4 W 24 x 162 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1961.50 0.00389 5 W 40 x 199 30 K 10 3.3 2.0" Nwt 1.5 VL 22 1972.20 0.00374 6 W 40 x 199 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1972.60 0.00373 7 W 36 x 260 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1991.30 0.00370 8 W 40 x 264 30 K 10 3.3 2.0" Nwt 1.5 VL 22 1991.70 0.00369 9 W 40 x 264 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1992.10 0.00368 10 W 36 x 300 30 K 10 3.3 2.0" Nwt 1.5 VL 22 2002.50 0.00368 11 W 36 x 300 30 K 11 3.3 2.0" Nwt 1.5 VL 22 2002.90 0.00368 12 W 36 x 300 30 K 11 3.3 2.0" Nwt 1.5 VL 22 2002.90 0.00368 13 W 36 x 300 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2003.30 0.00367 14 W 40 x 324 30 K 11 3.3 2.0" Nwt 1.5 VL 22 2010.10 0.00364 15 W 40 x 199 28 K 12 3 2.0" Nwt 1.5 VL 22 2072.80 0.00363 16 W 40 x 199 30 K 12 3 2.0" Nwt 1.5 VL 22 2073.00 0.00352 17 W 40 x 264 30 K 11 3.3 3.0" Nwt 1.5 VL 22 2085.50 0.00327 18 W 36 x 260 30 K 12 3 3.0" Nwt 1.5 VL 22 2184.60 0.00312 19 W 40 x 362 28 K 12 3 3.0" Nwt 1.5 VL 22 2215.10 0.00312 20 W 40 x 362 26 K 12 3 3.0" Nwt 2.0 VL 22 2222.40 0.00254 21 W 36 x 300 26 K 12 2.7 3.5" Nwt 2.0 VL 22 2337.40 0.00223 22 W 36 x 300 28 K 12 2.7 3.5" Nwt 2.0 VL 22 2337.50 0.00218 23 W 36 x 300 30 K 12 2.7 3.5" Nwt 2.0 VL 22 2337.70 0.00214 24 W 36 x 300 30 K 12 2.5 3.5" Nwt 2.0 VL 22 2437.70 0.00205 25 W 40 x 149 30 K 10 2.5 3.5" Nwt 3.0 VL 22 2542.80 0.00200 26 W 33 x 387 30 K 12 2.7 4.3" Nwt 3.0 VL 22 2607.20 0.00195 27 W 40 x 199 24 K 12 2.5 4.5" Nwt 3.0 VL 19 3133.30 0.00170 28 W 40 x 264 24 K 12 2.5 4.5" Nwt 3.0 VL 19 3152.80 0.00165
172
B.5 – Run 10 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 27 x 84 28 K 10 3.3 2.0" Nwt 1.5 VL 22 1956.20 0.00463 2 W 27 x 84 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1957.10 0.00440 3 W 24 x 104 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1962.80 0.00440 4 W 24 x 103 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1962.80 0.00439 5 W 21 x 122 30 K 10 3.3 2.0" Nwt 1.5 VL 22 1967.80 0.00439 6 W 21 x 122 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1968.20 0.00438 7 W 21 x 132 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1971.50 0.00437 8 W 18 x 158 30 K 10 3.3 2.0" Nwt 1.5 VL 22 1978.60 0.00436 9 W 18 x 158 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1979.00 0.00435 10 W 18 x 158 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1979.30 0.00434 11 W 36 x 170 30 K 11 3.3 2.0" Nwt 1.5 VL 22 1982.60 0.00434 12 W 36 x 170 30 K 12 3.3 2.0" Nwt 1.5 VL 22 1982.90 0.00433 13 W 36 x 300 30 K 10 3.3 2.0" Nwt 1.5 VL 22 2021.20 0.00425 14 W 36 x 300 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2021.90 0.00424 15 W 40 x 331 30 K 10 3.3 2.0" Nwt 1.5 VL 22 2030.50 0.00422 16 W 40 x 331 30 K 11 3.3 2.0" Nwt 1.5 VL 22 2030.90 0.00421 17 W 40 x 331 30 K 12 3.3 2.0" Nwt 1.5 VL 22 2031.20 0.00421 18 W 21 x 122 30 K 10 3.3 2.5" Nwt 2.0 VL 22 2042.40 0.00376 19 W 21 x 132 30 K 10 3.3 2.5" Nwt 2.0 VL 22 2045.40 0.00374 20 W 33 x 318 30 K 10 3.3 2.5" Nwt 2.0 VL 22 2101.20 0.00344 21 W 40 x 331 30 K 10 3.3 2.5" Nwt 2.0 VL 22 2105.10 0.00339 22 W 40 x 331 30 K 11 3.3 2.5" Nwt 2.0 VL 22 2105.60 0.00339 23 W 24 x 104 30 K 11 3 3.0" Nwt 2.0 VL 22 2174.80 0.00328 24 W 40 x 331 26 K 12 3 2.5" Nwt 2.0 VL 22 2179.50 0.00290 25 W 24 x 103 30 K 11 3 3.5" Nwt 1.5 VL 20 2263.80 0.00284 26 W 30 x 148 30 K 10 3 3.5" Nwt 1.5 VL 20 2276.90 0.00264 27 W 33 x 354 30 K 11 2.7 4.0" Nwt 1.5 VL 22 2352.70 0.00205 28 W 27 x 368 26 K 12 2.7 4.5" Nwt 1.5 VL 22 2389.80 0.00200 29 W 33 x 318 30 K 12 2.5 4.0" Nwt 1.5 VL 22 2442.30 0.00199 30 W 33 x 354 30 K 12 2.5 4.0" Nwt 1.5 VL 21 2693.10 0.00197 31 W 40 x 331 26 K 12 2.5 3.5" Nwt 3.0 VL 20 2717.90 0.00189 32 W 24 x 335 28 K 12 2.5 4.5" Nwt 3.0 VL 21 2929.60 0.00169
173
B.6 Joist Span 3 (L = 50 ft) Loading 2 (LL = 80 psf) B.6 – Run 1
Num Girder Joist Spacing Concrete Deck Cost Ap 1 W 30 x 124 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2502.50 0.00376 2 W 36 x 160 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2513.30 0.00365 3 W 33 x 221 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2531.60 0.00360 4 W 36 x 232 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2534.90 0.00358 5 W 40 x 235 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2535.80 0.00356 6 W 24 x 131 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2546.60 0.00342 7 W 24 x 176 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2561.10 0.00341 8 W 27 x 194 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2566.50 0.00335 9 W 33 x 221 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2573.60 0.00309 10 W 36 x 256 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2584.10 0.00305 11 W 30 x 124 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2586.50 0.00302 12 W 36 x 150 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2594.30 0.00289 13 W 36 x 160 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2597.30 0.00288 14 W 40 x 183 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2604.20 0.00283 15 W 36 x 256 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2626.10 0.00279 16 W 40 x 362 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2657.90 0.00271 17 W 36 x 182 30 K 12 2.5 3.5" Nwt 2.0 VL 22 2729.90 0.00269 18 W 33 x 221 30 K 12 2.5 3.5" Nwt 2.0 VL 22 2741.60 0.00267 19 W 30 x 261 30 K 12 2.5 3.5" Nwt 2.0 VL 22 2753.60 0.00266 20 W 33 x 263 30 K 12 2.5 3.5" Nwt 2.0 VL 22 2754.20 0.00263 21 W 40 x 362 30 K 12 2.5 3.5" Nwt 2.0 VL 22 2783.90 0.00254 22 W 40 x 362 30 K 12 2.5 3.5" Nwt 2.0 VL 20 2933.90 0.00254 23 W 36 x 439 30 K 12 2.5 3.5" Nwt 2.0 VL 20 2957.00 0.00253 24 W 36 x 439 30 K 12 2.5 3.0" Nwt 3.0 VL 20 3103.60 0.00253
174
B.6 – Run 2 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.00 0.00389 2 W 30 x 108 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2497.70 0.00383 3 W 30 x 116 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2500.10 0.00379 4 W 33 x 118 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2500.70 0.00372 5 W 33 x 130 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2504.30 0.00370 6 W 33 x 141 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2507.60 0.00369 7 W 36 x 150 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2510.30 0.00366 8 W 36 x 160 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2513.30 0.00365 9 W 40 x 199 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2525.00 0.00358 10 W 36 x 230 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2534.30 0.00358 11 W 36 x 232 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2534.90 0.00358 12 W 30 x 99 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2537.00 0.00341 13 W 30 x 108 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2539.70 0.00336 14 W 33 x 130 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2546.30 0.00321 15 W 36 x 150 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2552.30 0.00316 16 W 33 x 201 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2567.60 0.00311 17 W 36 x 232 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2576.90 0.00307 18 W 33 x 263 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2586.20 0.00306 19 W 33 x 291 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2594.60 0.00304 20 W 40 x 372 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2618.90 0.00297 21 W 36 x 439 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2639.00 0.00296 22 W 33 x 141 30 K 12 2.5 3.3" Nwt 2.0 VL 22 2694.20 0.00292 23 W 33 x 221 30 K 12 2.5 3.3" Nwt 2.0 VL 22 2718.20 0.00281 24 W 36 x 439 30 K 12 2.5 3.3" Nwt 2.0 VL 22 2783.60 0.00267 25 W 36 x 300 30 K 12 2.5 3.0" Nwt 3.0 VL 22 2911.90 0.00258 26 W 30 x 391 30 K 12 2.5 3.0" Nwt 3.0 VL 22 2939.20 0.00258 27 W 40 x 327 30 K 12 2.5 3.5" Nwt 2.0 VL 21 3073.40 0.00256 28 W 40 x 397 30 K 12 2.5 3.0" Nwt 3.0 VL 18 3466.00 0.00252
175
B.6 – Run 3 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 27 x 102 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.90 0.00394 2 W 30 x 108 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2497.70 0.00383 3 W 33 x 118 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2500.70 0.00372 4 W 30 x 148 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2509.70 0.00370 5 W 33 x 152 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2510.90 0.00367 6 W 36 x 160 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2513.30 0.00365 7 W 36 x 182 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2519.90 0.00362 8 W 40 x 199 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2525.00 0.00358 9 W 36 x 230 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2534.30 0.00358 10 W 30 x 108 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2540.70 0.00351 11 W 27 x 146 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2552.10 0.00342 12 W 33 x 152 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2553.90 0.00334 13 W 33 x 169 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2559.00 0.00332 14 W 40 x 199 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2568.00 0.00325 15 W 36 x 256 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2585.10 0.00323 16 W 40 x 297 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2597.40 0.00318 17 W 33 x 387 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2624.40 0.00318 18 W 40 x 392 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2625.90 0.00315 19 W 33 x 118 30 K 12 2.5 3.0" Nwt 2.0 VL 22 2664.00 0.00312 20 W 33 x 152 30 K 12 2.5 3.0" Nwt 2.0 VL 22 2674.20 0.00306 21 W 36 x 170 30 K 12 2.5 3.0" Nwt 2.0 VL 22 2679.60 0.00301 22 W 40 x 199 30 K 12 2.5 3.0" Nwt 2.0 VL 22 2688.30 0.00294 23 W 33 x 221 30 K 12 2.5 3.5" Nwt 1.5 VL 22 2693.80 0.00286 24 W 33 x 241 30 K 12 2.5 3.5" Nwt 1.5 VL 22 2699.80 0.00285 25 W 30 x 292 30 K 12 2.5 3.5" Nwt 1.5 VL 22 2715.10 0.00284 26 W 33 x 221 30 K 12 2.5 3.3" Nwt 2.0 VL 22 2718.20 0.00281 27 W 33 x 318 30 K 12 2.5 3.5" Nwt 1.5 VL 22 2722.90 0.00279 28 W 36 x 256 30 K 12 2.5 2.0" Nwt 3.0 VL 22 2774.10 0.00279 29 W 40 x 264 30 K 12 2.5 2.5" Nwt 2.0 VL 20 2778.50 0.00276 30 W 40 x 278 30 K 12 2.5 2.0" Nwt 3.0 VL 22 2780.70 0.00276 31 W 40 x 324 30 K 12 2.5 2.0" Nwt 3.0 VL 22 2794.50 0.00273 32 W 40 x 372 30 K 12 2.5 2.0" Nwt 3.0 VL 22 2808.90 0.00271 33 W 40 x 199 30 K 12 2.5 3.5" Nwt 2.0 VL 20 2885.00 0.00264 34 W 36 x 256 30 K 12 2.5 3.0" Nwt 3.0 VL 22 2898.70 0.00261 35 W 40 x 264 30 K 12 2.5 3.5" Nwt 2.0 VL 20 2904.50 0.00259 36 W 36 x 359 30 K 12 2.5 3.5" Nwt 2.0 VL 20 2933.00 0.00256
176
B.6 – Run 4 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.00 0.00389 2 W 27 x 114 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2499.50 0.00388 3 W 30 x 116 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2500.10 0.00379 4 W 33 x 118 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2500.70 0.00372 5 W 36 x 135 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2505.80 0.00368 6 W 36 x 150 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2510.30 0.00366 7 W 40 x 167 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2515.40 0.00362 8 W 36 x 194 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2523.50 0.00361 9 W 40 x 211 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2528.60 0.00357 10 W 40 x 215 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2529.80 0.00357 11 W 30 x 99 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2538.00 0.00356 12 W 27 x 114 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2541.50 0.00340 13 W 30 x 116 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2542.10 0.00331 14 W 33 x 118 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2542.70 0.00324 15 W 36 x 135 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2547.80 0.00318 16 W 40 x 167 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2557.40 0.00311 17 W 36 x 210 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2570.30 0.00309 18 W 40 x 215 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2571.80 0.00305 19 W 27 x 114 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2583.50 0.00302 20 W 30 x 132 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2588.90 0.00287 21 W 33 x 141 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2591.60 0.00281 22 W 33 x 201 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2609.60 0.00272 23 W 36 x 210 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2612.30 0.00270 24 W 40 x 392 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2666.90 0.00257 25 W 40 x 362 30 K 12 2.5 3.5" Nwt 2.0 VL 22 2783.90 0.00254 26 W 40 x 397 30 K 12 2.5 3.5" Nwt 2.0 VL 20 2944.40 0.00252
177
B.6 – Run 5 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.00 0.00389 2 W 30 x 108 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2497.70 0.00383 3 W 30 x 124 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2502.50 0.00376 4 W 36 x 135 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2505.80 0.00368 5 W 40 x 149 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2510.00 0.00365 6 W 40 x 167 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2515.40 0.00362 7 W 36 x 194 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2523.50 0.00361 8 W 40 x 199 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2525.00 0.00358 9 W 40 x 211 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2528.60 0.00357 10 W 40 x 235 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2535.80 0.00356 11 W 30 x 99 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2538.00 0.00356 12 W 40 x 249 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2540.00 0.00354 13 W 30 x 108 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2540.70 0.00351 14 W 30 x 124 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2545.50 0.00343 15 W 36 x 135 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2548.80 0.00335 16 W 36 x 150 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2553.30 0.00333 17 W 36 x 170 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2559.30 0.00330 18 W 36 x 194 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2566.50 0.00328 19 W 40 x 199 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2568.00 0.00325 20 W 40 x 211 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2571.60 0.00324 21 W 30 x 108 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2581.70 0.00309 22 W 30 x 108 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2581.70 0.00298 23 W 30 x 124 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2586.50 0.00290 24 W 36 x 135 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2589.80 0.00280 25 W 40 x 149 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2594.00 0.00275 26 W 36 x 170 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2600.30 0.00274 27 W 36 x 245 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2622.80 0.00266 28 W 40 x 331 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2648.60 0.00260 29 W 40 x 362 30 K 12 2.5 3.5" Nwt 2.0 VL 22 2783.90 0.00254 30 W 40 x 362 30 K 12 2.5 3.0" Nwt 3.0 VL 16 3605.50 0.00253
178
B.6 – Run 6 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 108 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2497.70 0.00383 2 W 33 x 141 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2507.60 0.00369 3 W 30 x 191 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2522.60 0.00366 4 W 30 x 108 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2539.70 0.00336 5 W 33 x 130 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2546.30 0.00321 6 W 36 x 135 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2547.80 0.00318 7 W 27 x 217 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2572.40 0.00316 8 W 36 x 260 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2585.30 0.00304 9 W 33 x 387 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2623.40 0.00299 10 W 40 x 249 30 K 12 2.5 3.3" Nwt 1.5 VL 22 2678.90 0.00295 11 W 40 x 372 30 K 12 2.5 3.3" Nwt 1.5 VL 22 2715.80 0.00289 12 W 40 x 392 30 K 12 2.5 3.3" Nwt 1.5 VL 22 2721.80 0.00289 13 W 40 x 431 30 K 12 2.5 3.3" Nwt 1.5 VL 22 2733.50 0.00287 14 W 36 x 182 30 K 12 2.5 2.0" Nwt 3.0 VL 22 2751.90 0.00286 15 W 36 x 194 30 K 12 2.5 2.0" Nwt 3.0 VL 22 2755.50 0.00284 16 W 40 x 199 30 K 12 2.5 2.0" Nwt 3.0 VL 22 2757.00 0.00281 17 W 36 x 280 30 K 12 2.5 2.0" Nwt 3.0 VL 22 2781.30 0.00277 18 W 33 x 354 30 K 12 2.5 2.0" Nwt 3.0 VL 22 2803.50 0.00275 19 W 40 x 372 30 K 12 2.5 2.0" Nwt 3.0 VL 22 2808.90 0.00271 20 W 36 x 439 30 K 12 2.5 2.0" Nwt 3.0 VL 22 2829.00 0.00271 21 W 30 x 326 30 K 12 2.5 3.0" Nwt 3.0 VL 22 2919.70 0.00262
179
B.6 – Run 7 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.00 0.00389 2 W 30 x 108 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2497.70 0.00383 3 W 30 x 116 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2500.10 0.00379 4 W 30 x 124 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2502.50 0.00376 5 W 33 x 130 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2504.30 0.00370 6 W 40 x 149 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2510.00 0.00365 7 W 36 x 170 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2516.30 0.00364 8 W 36 x 194 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2523.50 0.00361 9 W 40 x 199 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2525.00 0.00358 10 W 40 x 211 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2528.60 0.00357 11 W 30 x 99 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2537.00 0.00341 12 W 30 x 116 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2542.10 0.00331 13 W 40 x 149 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2552.00 0.00314 14 W 36 x 194 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2565.50 0.00310 15 W 40 x 211 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2570.60 0.00306 16 W 30 x 124 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2586.50 0.00302 17 W 27 x 146 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2593.10 0.00300 18 W 36 x 150 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2594.30 0.00289 19 W 36 x 194 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2607.50 0.00284 20 W 40 x 264 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2628.50 0.00276 21 W 40 x 297 30 K 12 2.5 3.5" Nwt 1.5 VL 22 2716.60 0.00276 22 W 36 x 650 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2744.30 0.00266 23 W 36 x 798 30 K 12 2.5 3.5" Nwt 1.5 VL 22 2866.90 0.00265 24 W 40 x 211 30 K 12 2.5 3.0" Nwt 3.0 VL 22 2885.20 0.00262 25 W 40 x 264 30 K 12 2.5 3.0" Nwt 3.0 VL 22 2901.10 0.00259 26 W 36 x 527 30 K 12 2.5 3.5" Nwt 2.0 VL 20 2983.40 0.00251 27 W 36 x 650 30 K 12 2.5 3.5" Nwt 2.0 VL 19 3440.30 0.00248
180
B.6 – Run 8 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 27 x 102 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2519.20 0.00453 2 W 33 x 118 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2524.00 0.00432 3 W 30 x 211 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2528.60 0.00364 4 W 36 x 245 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2538.80 0.00357 5 W 33 x 118 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2542.70 0.00324 6 W 27 x 161 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2555.60 0.00323 7 W 40 x 167 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2557.40 0.00311 8 W 40 x 215 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2571.80 0.00305 9 W 40 x 235 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2577.80 0.00304 10 W 40 x 249 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2582.00 0.00302 11 W 30 x 124 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2586.50 0.00302 12 W 40 x 297 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2596.40 0.00300 13 W 40 x 167 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2599.40 0.00285 14 W 33 x 221 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2615.60 0.00283 15 W 40 x 235 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2619.80 0.00278 16 W 40 x 327 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2647.40 0.00273 17 W 40 x 331 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2648.60 0.00273 18 W 36 x 439 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2681.00 0.00271 19 W 40 x 327 30 K 12 2.5 3.3" Nwt 2.0 VL 22 2750.00 0.00270 20 W 40 x 392 30 K 12 2.5 3.3" Nwt 2.0 VL 22 2769.50 0.00268 21 W 36 x 798 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2788.70 0.00264 22 W 36 x 245 30 K 12 2.5 3.5" Nwt 2.0 VL 20 2898.80 0.00263 23 W 33 x 263 30 K 12 2.5 3.0" Nwt 3.0 VL 22 2900.80 0.00263 24 W 40 x 297 30 K 12 2.5 3.0" Nwt 3.0 VL 22 2911.00 0.00256 25 W 36 x 359 30 K 12 2.5 3.0" Nwt 3.0 VL 22 2929.60 0.00255 26 W 27 x 539 30 K 12 2.5 3.0" Nwt 3.0 VL 22 2983.60 0.00255 27 W 40 x 327 30 K 12 2.5 3.0" Nwt 3.0 VL 20 3070.00 0.00255 28 W 36 x 798 30 K 12 2.5 3.0" Nwt 1.5 VL 21 3088.70 0.00250 29 W 36 x 798 30 K 12 2.5 3.0" Nwt 1.5 VL 18 3208.70 0.00250
181
B.6 – Run 9 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495 0.003886 2 W 33 x 130 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2504.3 0.003703 3 W 40 x 215 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2529.8 0.003565 4 W 30 x 99 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2537 0.003408 5 W 36 x 135 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2547.8 0.003183 6 W 40 x 167 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2557.4 0.003109 7 W 40 x 183 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2562.2 0.003085 8 W 36 x 230 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2576.3 0.003068 9 W 36 x 256 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2584.1 0.003046 10 W 40 x 277 30 K 12 2.5 2.5" Nwt 1.5 VL 22 2590.4 0.003007 11 W 30 x 148 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2593.7 0.002953 12 W 33 x 152 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2594.9 0.002916 13 W 24 x 250 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2624.3 0.002908 14 W 27 x 258 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2626.7 0.002866 15 W 40 x 277 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2632.4 0.002749 16 W 40 x 392 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2666.9 0.002707 17 W 36 x 650 30 K 12 2.5 3.0" Nwt 1.5 VL 22 2744.3 0.002524 18 W 40 x 397 30 K 12 2.5 3.5" Nwt 2.0 VL 17 3814.4 0.002524
182
B.6 – Run 10 Num Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 108 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2497.70 0.00383 2 W 30 x 148 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2509.70 0.00370 3 W 40 x 149 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2510.00 0.00365 4 W 40 x 199 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2525.00 0.00358 5 W 40 x 211 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2528.60 0.00357 6 W 30 x 108 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2540.70 0.00351 7 W 33 x 118 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2543.70 0.00340 8 W 30 x 148 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2552.70 0.00338 9 W 40 x 149 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2553.00 0.00331 10 W 40 x 199 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2568.00 0.00325 11 W 40 x 211 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2571.60 0.00324 12 W 40 x 235 30 K 12 2.5 2.0" Nwt 2.0 VL 22 2578.80 0.00322 13 W 27 x 114 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2583.50 0.00314 14 W 33 x 118 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2584.70 0.00298 15 W 30 x 148 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2593.70 0.00295 16 W 40 x 149 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2594.00 0.00288 17 W 40 x 199 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2609.00 0.00281 18 W 40 x 211 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2612.60 0.00280 19 W 40 x 278 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2632.70 0.00276 20 W 40 x 297 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2638.40 0.00274 21 W 40 x 327 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2647.40 0.00273 22 W 40 x 362 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2657.90 0.00271 23 W 40 x 431 30 K 12 2.5 2.5" Nwt 2.0 VL 22 2678.60 0.00269 24 W 40 x 199 30 K 12 2.5 3.5" Nwt 2.0 VL 22 2735.00 0.00264 25 W 40 x 264 30 K 12 2.5 3.5" Nwt 2.0 VL 22 2754.50 0.00259 26 W 40 x 297 30 K 12 2.5 3.5" Nwt 2.0 VL 22 2764.40 0.00257 27 W 40 x 362 30 K 12 2.5 3.5" Nwt 2.0 VL 22 2783.90 0.00254 28 W 40 x 362 30 K 12 2.5 3.5" Nwt 2.0 VL 21 3083.90 0.00254 29 W 36 x 798 30 K 12 2.5 3.0" Nwt 1.5 VL 16 3538.70 0.00250
183
Appendix C – Design Solution Checks
C.1 Purpose for Checks 184 C.2 Checks for Bay Configuration 2 and Load Case 1 184 C.3 Checks for Bay Configuration 3 and Load Case 2 186 C.4 Checks for Light Load Systems 187
184
C.1 Purpose for Checks
After running the algorithm multiple times, it becomes necessary verify the validity of the
results. To reduce the time consuming nature of the checking the results, only the results,
which approach the limits will be checked. This means that the lightest components of
the lightest systems will be examined. In addition to this, several special runs were
conducted at very light loadings similar to those found in roof systems. The lightest of
these systems will also be subjected to the same checks.
C.2 Checks for Bay Configuration 2 and Load Case 1
For the first bay and load configuration, the best results are provided in Chapter 5 (Table
5.1 repeated below). The lightest systems are shown below. This will be the first to be
evaluated, because the most is known about this system from the previous hand
calculations.
Run Girder Joist Spacing Concrete Deck Cost Ap 1 W 18 x 106 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1744.10 0.00465 2 W 16 x 89 30 K 10 4.3 2.0" Nwt 1.5 VL 22 1739.20 0.00480 3 W 24 x 76 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1735.70 0.00448 4 W 30 x 99 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1742.00 0.00461 5 W 24 x 94 30 K 12 4.3 2.0" Nwt 1.5 VL 22 1741.50 0.00446 6 W 24 x 68 26 K 12 4.3 2.0" Nwt 1.5 VL 22 1733.40 0.00472 7 W 18 x 86 28 K 10 4.3 2.0" Nwt 1.5 VL 22 1738.10 0.00480 8 W 30 x 90 30 K 11 4.3 2.0" Nwt 1.5 VL 22 1739.90 0.00446 9 W 18 x 86 24 K 12 4.3 2.0" Nwt 1.5 VL 22 1757.30 0.00554 10 W 27 x 84 26 K 12 4.3 2.0" Nwt 1.5 VL 22 1738.20 0.00469
Table 5.1: Configuration 2 (L = 40 ft.) Least Expensive Systems
The first set of components to be checked is the concrete decking. Loading for
the decking consists of a 15 psf superimposed dead load (SDL) and a 50-psf live load
(LL). Therefore the superimposed design load is 65 psf. Since the required span of all
the decks is less than five feet, the span value for five feet will be used.
185
Deck Type Max Clear Span Max Safe Live Load Weight of Deck 2.0” on 1.5 VL 22 7’ 0” (ok) 314 psf (ok) 33 psf
Table C.2.1: Configuration 2 (L = 40 ft.) Least Expensive Systems
All the Deck values prove to be adequate, so onto the next component. The joist
capacities are subject to the loading imposed by the decking. The required loading for
the capacity check is calculated below:
:(65 33) 4.28 419.4
:max((33 15),50) 4.28 214
Capacity
Deflection
+ ⋅ =
+ ⋅ =
Deflection load checks are subjected to live and dead loads separately.
Joist Capacity Load Deflection Load
Joist Weight
30 K 12 438 (ok) 315 (ok) 17.6 plf 30 K 11 438 (ok) 315 (ok) 16.4 plf 30 K 10 438 (ok) 315 (ok) 15.0 plf 28 K 10 424 (ok) 284 (ok) 14.3 plf 26 K 12 438 (ok) 269 (ok) 16.6 plf 24 K 12 438 (ok) 247 (ok) 16.0 plf
Table C.2.2: Joist Checks
Finally, the girders must be checked. To reduce the checks, number of checks, only three
critical systems will be checked. The lightest girder, heaviest girder, and the girder
carrying the heaviest joist system. Table C.2.3 provides the required information for these
girders.
Girder ΦMn (k-ft) ΦVn (k) Ixx (in4) δall (in) W 24 x 68 664 k-ft 266 k 1830 in4 1.0 in W 18 x 106 836 k-ft 298 k 1910 in4 1.0 in W 24 x 94 953 k-ft 338 k 2700 in4 1.0 in
Table C.2.3: Girder Information
The loadings for these girders are calculated using the check equations: 3.24,3.25,3.26.
186
Girder FL Mu Vu SL δ W 24 x 68 5.772 649 87 2.143 0.74 W 18 x 106 5.792 652 87 2.160 0.71 W 24 x 94 5.814 654 87 2.178 0.51
Table C.2.4: Applied Loads and Deflection
The three girders satisfy the shear moment and deflection requirements.
C.3 Checks for Bay Configuration 3 and Load Case 2
For the third bay and second load configuration, the best results are provided in Chapter 5
(Table 5.7 is repeated below). The lightest systems are shown below. This is the run,
which posed the most problems to the algorithm. With the smallest feasible search space,
it should show problems with the fitness code first.
Run Girder Joist Spacing Concrete Deck Cost Ap
1 W 30 x 124 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2502.5 0.0037581 2 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495 0.0038858 3 W 27 x 102 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495.9 0.0039413 4 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495 0.0038858 5 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495 0.0038858 6 W 30 x 108 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2497.7 0.0038345 7 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495 0.0038858 8 W 27 x 102 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2519.2 0.0045334 9 W 30 x 99 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2495 0.0038858 10 W 30 x 108 30 K 12 2.5 2.0" Nwt 1.5 VL 22 2497.7 0.0038345
Table 5.7: Configuration 3 (L = 50 ft.) Least Expensive Systems
The same process from section C.2 will be used for evaluating the systems. The first
component to be checked is the concrete deck. The loading consists of a 15 psf SDL and
an 80 psf LL. The first component to be evaluated is the concrete deck. Since the
required span is less than five feet, the span value of five feet will be used.
187
Deck Type Max. Clear Span Max Safe Live Load Weight of Deck 2.0” on 1.5VL22 7’ 0” (ok) 314 psf (ok) 33 psf
Table C.3.1: Configuration 3 (L=50) Concrete Decks
All the deck values proved to be adequate. The next component on the checklist is the
joist. The required loading for the capacity check is calculated below:
:(95 33) 2.5 320
: max((33 15),80) 2.5 200
Capacity
Deflection
+ ⋅ =
+ ⋅ =
The capacity check included dead and live load, whereas the deflection check is subjected
to only the maximum of the live and dead loads.
Joist Capacity Load Deflection Load Joist Weight 30 K 12 438 (ok) 315 (ok) 17.6 plf
Table C.3.2: Joist Checks
The last portion to be checked is the girder. Table C.3.3 provides girder information.
Girder ΦMn (k-ft) ΦVn (k) Ixx (in4) δall (in) W 30 x 99 1170 417 3990 1.67 in W 27 x 102 1140 377 3620 1.67 in W 30 x 108 1300 439 4470 1.67 in W 30 x 124 1530 477 5360 1.67 in
Table C.3.3: Girder Information
The loadings for these girders are calculated using the check equations: 3.24, 3.25, 3.26.
Girder FL (klf) Mu (k-ft) Vu (k) SL (klf) δ (in) W 30 x 99 9.821 1105 147 4.00 0.63 W 27 x 102 9.825 1105 147 4.00 0.69 W 30 x 108 9.832 1106 147 4.00 0.56 W 30 x 124 9.851 1108 148 4.00 0.47
Table C.3.4: Applied Loads and Deflection
188
The girders satisfy all three criteria, which validate the ability of the algorithm to yield
feasible solutions.
C.4 Checks for Light Load Systems
Having examined the designs for heavy systems and average weight systems, it seems
logical that lightweight systems should be examined. This will ensure cover the full
range of the design options so as to ensure that no bugs exist in the finished code. This
was tackled by performing three runs of a lightweight system design. The loads for this
design consist of a 30-psf live load and no superimposed dead load. This run will
approximate the design of a snow-loaded roof. It will not be exact, but the purpose of
this is to show that the algorithm is still effective for light loadings. Table C.4.1 provides
the three least expensive of each of the three runs.
Run Girder Joist Spacing Concrete Deck Cost Ap 1 W 18 x 55 28 K 10 6 2.0" Nwt 1.5 VL 22 1528.80 0.00651 2 W 21 x 62 26 K 10 6 2.0" Nwt 1.5 VL 22 1530.70 0.00643 3 W 18 x 60 30 K 9 6 2.0" Nwt 1.5 VL 22 1530.00 0.00648
Table C.4.1: Least Expensive Systems
The first component to be checked is the composite concrete deck.
Deck Type Max Clear Span Max. Safe Live Load Weight of Deck 2.0” on 1.5VL 22 7’ 0” (ok) 314 psf (ok) 33 psf
Table C.4.2: Concrete Deck Specifications
The only deck specification is adequate; the next component to be evaluated is the joists.
:(30 33) 6.0 378
:max(33,30) 6.0 198
Capacity
Deflection
+ ⋅ =
⋅ =
Joist Capacity Load (klf) Deflection Load (klf) Joist Weight (plf) 26 K 10 393 (ok) 243 (ok) 13.8 28 K 10 424 (ok) 284 (ok) 14.3 30 K 9 384 (ok) 278 (ok) 13.4
Table C.4.3: Joist Checks
189
The joist selections are all adequate, and the next step is to check the girder using
equations 3.24,3.25, and 3.26. Before these equations can be used, the girder information
must be obtained.
Girder Mcap (k-ft) Vcap (k) Ixx(in4) δall (in) W 18 x 55 420 191 890 1.33 W 18 x 60 461 204 984 1.33 W 21 x 62 540 227 1330 1.33
Table C.4.4: Girder Information
Girder FL (klf) Mu (k-ft) Vu (k) SL (klf) δ (in) W 18 x 55 3.684 414 55 1.492 1.05 W 18 x 60 3.683 414 55 1.497 0.96 W 21 x 62 3.689 415 55 1.499 0.71
Table C.4.5: Applied Loads and Deflection
All the girders meet the requirements. So the algorithm appears to be functioning
properly for the light loading cases.
190
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Marquette University
This is to certify that we have examined this copy of the
master’s thesis by
Christopher W. Erwin
and have found that it is complete and satisfactory in all respects.
The thesis has been approved by: Dr. Christopher M. Foley, Department of Civil Engineering Dr. Sriramulu Vinnakota, Department of Civil Engineering Dr. Stephen Heinrich, Department of Civil Engineering
Approved on
