CONCRETE PONDING EFFECTS IN COMPOSITE FLOOR SYSTEMS · 2020. 4. 2. · Order Number 1330532 Concrete ponding effects in composite floor systems Pena Ramos, Carlos Enrique, M.S. The

CONCRETE PONDING EFFECTSIN COMPOSITE FLOOR SYSTEMS

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Order Number 1330532

Concrete ponding effects in composite floor systems

Pena Ramos, Carlos Enrique, M.S.

The University of Arizona, 1987

U M I 300 N. Zeeb Rd. Ann Arbor, MI 48106

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International

CONCRETE PONDING EFFECTS IN

COMPOSITE FLOOR SYSTEMS

by

Carlos Enrique PeRa-Ramos

A Thesis Submitted to the Faculty of the

DEPARTMENT OF CIVIL ENGINEERING AND ENGINEERING MECHANICS

In Partial Fulfillment of the Requirements For the Degree of

MASTER OF SCIENCE WITH A MAJOR IN CIVIL ENGINEERING

In the Graduate College

THE UNIVERSITY OF ARIZONA

19 8 7

STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of the source is made. Requests for permission for extended quotations from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

Signature

APPROVAL BY THESIS DIRECTOR

This thesis has been approved on the date shown below:

: /

Reidar Bjor&ovde, Professor Department of Civil Engineering and Engineering Mechanics.

</Di Date

ACKNOWLEDGMENTS

I would like to express my sincere appreciation to

Dr. R. Bjorhovde for his generous time, effort and

invaluable advice throughout the development of this

thesis. Special thanks are due to Dr. M. R. Ehsani and Dr.

P. D. Kiousis for their review of the manuscript, as well

as their helpful comments and suggestions.

I would like to thank my parents for their moral

and financial support throughout my graduate and

undergraduate work. Their love and support have been a

driving force behind my academic career.

I would also like to thank my lovely fiance, Vicky

for her love, patience and understanding, as well as her

help in preparing the final drafts of the manuscript.

Last but not least, I would like to thank everyone

who supported me throughout my graduate study, especially

the CIAD Research Institute and the Ministry of Education

of Mexico for their financial assistance.

This thesis is dedicated to my parents, brother,

sisters, fiance, and especially to my son, Carlos, who has

been my inspiration and reason for wanting to become a

better engineer, a better person, and a better father.

iii

TABLE OF CONTENTS

Page L IST OF ILLUSTRATIONS v

LIST OF TABLES x

ABSTRACT xi

CHAPTER

1. INTRODUCTION I

2. ANALYTICAL MODEL FOR THE FLOOR STRUCTURE 5

2.1 Methods of Analysis for Floor Structures 5 2.2 Analytical Model for The Supporting Floor

Structure 6

3. ANALYTICAL LOADING MODEL 12

4. INCORPORATION OF ANALYTICAL MODEL INTO COMPUTER PROGRAM 20

5. RESULTS AND DESIGN RECOMMENDATIONS 30

6. SUMMARY AND CONCLUSIONS 40

6.1 Summary 40 6.2Conclusions 41

APPENDIX A: DESIGN CURVES 43

APPENDIX B: PROGRAM GRIDS 105

B.l Introduction ' 105 B.2 User Manual _ 107 B.3 Program Listing 112

NOTATION 121

LIST OF REFERENCES 123

i v

LIST OF ILLUSTRATIONS

Figure Page

1 Degrees of Freedom of a Typical Grid Element 7

2 Structural System Considered in this Study 11

3 Loading Stages of The Floor Structure 14

4 Analytical Loading Model 19

5 Floor Structure Computer Model 29

6 Total Interior Floor Beam Deflection 34

A1 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=5.5 in.; Sb=5.0 ft.) 45


A3 Maximum Momemt vs. Beam Stiffness (1=200 to 2000 in.4; Ts=5.5 in.; Sb=5.0 ft.) 47

A4 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=5.5 in.; Sb=5.0 ft.) 48







v

vi

LIST OF ILLUSTRATIONS - Continued

Figure . Page

All Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4; Ts=6.5 in.; Sb=5.0 ft.) 55















vii


Figure Page


A27 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4? Ts=6.0 in.; Sb=10.0 ft.) 71

A28 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=6.0 in.? Sb=10.0 ft.) 72


A3 0 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=6.5 in.? Sb=10.0 ft.) 74

A31 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.4? Ts=6.5 in.; Sb=10.0 ft.) 75










vlii


Figure Page

A41 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in. ; Ts=5.5 in.; Sb=15.0 ft.) 85

A42 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4? Ts=5.5 in.; Sb=15.0 ft.) 86














ix


Figure Page

A56 Maximum Moment vs. Beam Stiffness (1—2000 to 20000 in.4; Ts=7.0 in.; Sb=15.0 ft.) 100

A57 Maximum Deflection vs. Beam Stiffness (1-200 to 2000 in.4; Ts=7.5 in.; Sb=15.0 ft.) 101



A60 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.4; Ts=7.5in.; Sb=15.0 ft.) 104

LIST OF TABLES

Page

Table

4.1 Beam Size Classification 23

4.2 Concrete Weight as a Function of Slab Thickness and Metal Deck Geometry (Normal Weight Concrete) 25

4.3 (a) w-Loads (lbs/ft) for Floor Structure (5 ft. Beam Spacing) 27

(b) w-Loads (lbs/ft) for Floor Structure (10 ft. Beam Spacing) 27

(c) w-Loads (lbs/ft) for Floor Structure (15 ft. Beam Spacing) 28

x

ABSTRACT

During the first placement of concrete, the

typical floor structure will deflect under the concrete

load. This will produce an uneven slab surface that must

be leveled before curing and finishing procedures can be

performed. In order to avoid the uneven slab surfaces, the

slab is releveled with additional concrete until an

equilibrium position is reached where additional

deformations of the structure are negligible.

The increase in load and deformation resulting

from the releveling of the slab surface has usually been

neglected in the design of floor structures. The purpose

of this study is to predict the increase in load and

deflection caused by the ponding of the concrete during

the releveling procedure, and to prepare design

recommendations that incorporate the effects of these

i ncreases.

xi

CHAPTER 1

INTRODUCTION

The composite floor system consisting of a

concrete slab with metal deck reinforcement and supported

by an arrangement of steel beams and girders is commonly

used for steel structures. During construction, the first

permanent loading the steel framing and metal deck receive

is the one caused by the placement of the concrete slab.

If temporary shoring is not used during the placement of

the concrete slab, the steel framing and metal deck will

deflect under the additional loading. If the concrete

floor slab was placed to a uniform thickness, the result

would be a nonuniform slab surface whose shape would be

defined by the deflected shape of the structural system.

In order to avoid the above condition and assure an

acceptable slab surface, the following procedures are

normally used:

1. The entire floor system is shored during the concrete

placement.

1

2

2. Camber is given to floor beams to counteract the

deflections produced by the concrete placement.

3. The slab surface is releveled, and the additional

concrete that is poured results in a nonuniform slab

thickness.

Of the procedures given above, releveling the slab

surface is probably the most common. This is an iterative

procedure where the additional concrete used to relevel

the slab surface will cause an additional deflection of

the supporting floor structure, which in turn will require

additional releveling. The procedure is continued until

an equilibrium position is reached, presumably before any

local or overall failure of the floor structure occurs.

The equilibrium position is reached when the floor

structure undergoes negligible deflections with the

placement of additional concrete.

A previous study on the ponding of two-way roof

systems caused by rainwater was done by Marino (1).

Subsequent work by Ruddy (2) found this approach to be

suitable for the analysis of floor systems when releveling

is used. Since the structural compositions of roof and

floor systems are very similar, the only possible problem

was the assumption that concrete, like water, will seek a

horizontal level when placed on the deflected structure.

3

This assumption is the basis of the analytical loading

that is model developed in Chapter 3. However, it is

precisely the fact that the concrete will not seek a

constant level that causes the uneven slab surfaces.

Despite this, Ruddy found that the results obtained for

the deflection values using the rainwater ponding analogy

were reasonably accurate when compared to actual field

measurements. Therefore, given the reliability of the

analogy for the analysis of floor systems, this approach

was used to develop the complete analytical model of the

floor system, given in Chapters 2 and 3.

Chapter 4 treats all the assumptions that have

been made, and the load values used in implementing the

analytical model into program GRIDS, a finite element

computer program developed in this study. The listing of

the program along with its User Manual are given in

Appendix 6.

The purpose of this study is to investigate the

deflection and strength characteristics of the floor beams

when subjected to the loading associated with the

releveling of the slab surface. The beam spacing, span,

and required slab thickness are considered variables, in

order to incorporate as many floor structure compositions

4

as possible, and the effect of this variability has been

investigated.

. Finally, the results of the investigation are

given in Chapter 5, along with design recommendations that

are needed to avoid some of the problems that may arise

when the load and deflection increments associated with

releveling of the slab surface are neglected. From the

results obtained from program GRIDS, a series of design

curves giving maximum deflection and moment values as a

function of the beam spacing, beam span, slab thickness,

and beam stiffness have been constructed for the whole

range of beam sections indicated in the AISC Steel

Construction Manual (3). The complete set of design curves

are given in Appendix A.

CHAPTER 2

ANALYTICAL MODEL FOR THE FLOOR STRUCTURE

2.1 Methods of Analysis for Floor Structures

Since the theoretical methods of analysis given by

classical theory of elasticity have proven to be

impractical when analyzing large structural systems such

as a typical floor structure, engineers have been forced

to devise alternative approximate methods of analysis.

Various approximate methods of analysis have been

proposed and found to be reliable and useful. Before the

use of the computer in structural design, approximate

methods based on mor.ant distribution (4) and plate analogy

(5) theory were commonly used in the design office. As the

computer began to gain widespread use in the structural

design office, previous approximate methods of analysis

that seemed impractical due to the large set of

simultaneous equations that had to be solved became more

attractive to the engineer. Methods based on slope defle

5

6

ction (6) and finite element theory (7) began to be used

for the analysis of such system.

Since the finite element method represents the

most advanced and powerful tool of analysis available

today, it was chosen in this study for the development of

the complete analytical model of the floor system.

The purpose of the finite element method is to

simplify an otherwise complex structural system into a

series of smaller structures called finite elements.

Mathematical models are developed for the simplified

structures in such a way that these can be easily

reassembled. As a result, a more complex mathematical

model is created for the actual structural system from the

models developed for the simplified structures.

2.2 Analytical Model of The Supporting Floor Structure

The support system of a typical floor structure,

due the loading conditions to which the system is

subjected, can be characterized as a grid type structure.

A grid type structure defines a structural system where

all the loading the system is subjected to is applied in a

plane perpendicular to the plane of the structure. Due to

this loading condition, the individual elements of the

7

grid structure are subjected to torsional, bending, and

shear forces, which in turn cause two types of rotations

and a linear displacement at the ends of the grid

elements. Figure 1 illustrates the end displacements or

degrees of freedom that a typical grid element is

subjected to.

Figure 1 Degrees of Freedom of a Typical Grid Element

8

In Figure 1, the rotations along the x-axis are

caused by the torsional forces, and the rotations along

the y-axis, as well as the linear displacement

perpendicular to the plane of the structure are caused by

the bending forces. The assumption is made that axial

deformations, as well as the shear deformations of the

grid elements are negligible.

By recognizing that the floor beams and girders of

a typical floor structure can be modeled as grid elements,

an analytical model for the support system of the floor

structure may be developed from the mathematical model of

a grid element. Since the mathematical model for the grid

element has already been developed through a finite

element solution (7), the basis for the support system

model has already been established, and the development of

the analytical model becomes a simpler task.

The analytical model of the support system is

based on the structural system given in Fig. 2. The system

represents an interior bay of a typical floor structure,

consisting of equally spaced floor beams, simply supported

at the ends by girders. The perimeter members of the bay

are considered to be simply supported on columns at the

ends, and identical structural arrangements are considered

to occur on all sides of the bay. The floor beams are

assumed to have the same moment of inertia, and the moment

9

of inertia of the girders is considered to be related to

the moment of inertia of the floor beams by the following

formula:

Ig o lb + 500 (2.1)

Where:

Ig = Moment of Inertia of the Girder (in4)

lb = Moment of Inertia of the Beam (in4)

Equation (2.1) is simply an approximation that has been

used in order to avoid having to obtain Ig from standard

design procedures each time a new lb, slab thickness, or

beam spacing is used. The effect of using Eq (2.1) in the

model is explained in Chapter 5.

The modulus of elasticity for all of the members

of the bay is assumed to be the same, and remains constant

regardless of the load intensities the support system is

subjected to. In other words, the system is assumed to be

elastic. The assumption has been found to be reliable

(1,2), as will be explained in more detail in Chapter 5.

Since all of the members of the bay and therefore

the entire support system are considered to be simply

supported, the system is treated as having no torsional

rigidity. As a consequence, the torsional displacements

and forces are neglected in the analysis.

10

Finally, the analytical model of the support

system was developed by subdividing the members of the bay

into a series of grid elements. The number of grid

elements per member is directly related to how the load is

applied to the support system, as will be explained in

Chapter 3. The development of the load model is described

in detail in this chapter.

11

Lg

equal equal equal

METAL DECK ^ Lb

ORIENTATION

Figure 2 Structural System Considered In This Study

CHAPTER 3

ANALYTICAL LOADING MODEL

In order to simulate the loading conditions the

floor structure is subjected to, a loading model has to be

created. Since the floor will deflect under the concrete

placement, the loading model would have to be a function

of the deflected shape of the structure, and since another

concrete placement would be needed to relevel the slab

surface, the loading model would have to be able to

accommodate the corresponding increase in load.

It is important to note that the concrete is in

its fluid state during the construction stage. As a

consequence, the effect of the concrete on the supporting

structure can be modeled by considering its loading role

only. In other words, the concrete weight is the only

contribution of the slab to the structural system and any

composite action has not come into being.

12

13

Figure 3 shows the loading sequence for the floor

structure during the releveling procedure. The loading

condition illustrated in Fig. 3(a) represents the first

stage of the sequence, that is, the type of loading the

floor structure is subjected to when the first placement

of concrete is made. The next loading stage is illustrated

in Fig 3(b). Here, the floor has already deflected under

the load defined in the first stage, and the distribution

is now nonlinear, with its shape defined by the deflected

shape of the structure.

The third stage illustrated in Fig. 3(c) defines

the loading conditions of the stmacture when an additional

placement of concrete is made. The load is defined by a

combination of the loads of the first and second stages.

The linear portion of the load reflects the increased

loading given by the additional concrete placement. The

nonlinear portion reflects the loading conditions just

prior to the placement of the additional concrete. The

assumption that the concrete seeks a constant level is

represented by the straight line that defines the boundary

between the linear and nonlinear loading. (If this

assumption is neglected and the concrete is allowed to

behave in a realistic fashion, the result would be a

nonlinear boundary between the two loading conditions

14

w-Load

' \ 1 ' \ I \ 1 . 1 . 1 ' ' f 1 1

/J&T ~

(a)

(b)

(c)

Figure 3 Loading Stages of The Floor Structure

15

which will greatly complicate the development of a loading

model for the floor structure. The effects of the

simplification are negligible.)

As was explained in Chapter 1, releveling of the

slab surface is an iterative procedure. In the loading

model, the iterative procedure is incorporated by defining

a loading cycle in which the second and third stage

loading conditions are alternated. An increase in

deflection with a corresponding increase in loading is

then defined at the second loading stage each time the

next loading cycle is started.

Theoretically, the loading cycle could be repeated

until collapse of the floor structure occurs or the

deflection increases become acceptably small. However, it

was found that the increase in deflection from one cycle

to the next became smaller with the number of loading

cycles. As a consequence, the deflection converged to a

constant value and an equilibrium position was reached.

This meant that a leveled surface of the concrete slab can

be achieved given that the resulting loading and

deflection increments will be small enough, and thus will

not violate established design values.

The analytical model for the loading is

illustrated in Fig. 4. A typical floor beam is subdivided

16

into smaller grid elements. To accommodate the first

loading stage, a uniform load w is applied thoughout the

beam length. This load is a function of the floor beam

spacing, the design slab thickness, the unit weight of

concrete, and the beam self weight. The numerical value of

w can be computed using the following formula:

w =(Wc x Ts) x Atw + DL X Atw + Wb (3.1)

Where:

Wc = Unit Weight of Concrete.

Ts = Slab Thickness.

Atw= Tributary Area Width.

DL = Additional Dead Loads. (Construction

Loads, Partitions, etc.)

Wb = Beam Self-Weight.

In Fig. 4, the second and third loading stages are

considered in one single step. First, the deflections

caused by w are computed. Then the deflection values

defined at the nodal points are converted into load values

by adding their respective numerical values to w without

regard to the units involved„ In general, for the i-th

loading cycle:

17

W^+1 Wj_ + Wc X Atw X (d^+1 - d^) (3.2)

Where:

w = Uniform Load.

d •= Deflection.

Convergence to an equilibrium position is assured if the

second term on the right hand side of Eq. (3.2) converges

to zero.

It is important to note in Fig. 4 that the

nonlinear loading curve defined in the second loading

stage is approximated by a series of linear segments,

where each segment is defined by the length of a grid

element. Therefore, care must be exercised when choosing

the number of grid elements per floor beam. If too few

grid are used, a rough approximation of the loading curve

will result, and unconservative values for the forces and

deflections will be obtained. On the other hand, if too

many grid elements are used for each floor beam, the

loading model itself will be reliable, since a closer

approximation to the nonlinear loading curve will be

achieved. However, the complete analytical model of the

floor system will become impractical for programming

purposes, due to the large number of nodal points that

result. This is particularly true in the microcomputer

environment, but may not represent a problem if a computer

18

with greater computing power is used to perform the

analysis. In general, the reliability and efficiency of

the analytical loading model will depend on good

engineering judgement when choosing the number of grid

elements that will define a floor beam.

Finally, the complete analytical model for the

floor system is created if the model for the loading and

the support system are considered as a single unit. In

Chapter 4, this model is modified to account for the

different compositions of floor systems and programmed for

a computer using the program GRIDS.

19

w

1 3 ft Z

Figure 4 Analytical Loading Model

CHAPTER 4

INCORPORATION OF ANALYTICAL MODEL

INTO COMPUTER PROGRAM.

Once the complete analytical model for the floor

system was assembled, the next step consisted in

subdividing the model into grid elements and numbering the

resulting nodes and elements so that a computer model

could be created.

Figure 5 illustrates the computer model used in

this study, including node and element numbers. Although

the nodes can be numbered in an arbitrary fashion, it has

been found (8) that minimizing the node number separation

greatly enhances the efficiency of the computer model. The

node number separation is the mathematical difference

between the numbers of adjacent nodes. In Fig. 5, the

maximum node number separation is five. On the other hand,

the way in which the elements are numbered has not been

found to affect the performance of the computer model, but

20

21

may simplify the task of writing the data file represen

ting the model.

Boundary conditions were defined for the computer

model in the following fashion (refer to Fig. 5):

1. Nodes 1, 5, 18, and 22 are considered to be res

tricted against linear displacements, to simulate

the column supports.

2. All other nodes were free to displace on any al

lowable mode.

It is important to note that the interior floor

beams will be supported on free nodes instead of simply

supported nodes due to the assumed boundary conditions.

This was necessary to allow the floor structure to deflect

in a realistic fashion. However, once the results of a

computer analysis are obtained, the settlement of the beam

supports can be substracted from the maximum deflection at

the midspan, therefore, creating a simple support effect.

This effect is discused in more detail in Chapter 5.

In order to implement the analytical model into

program GRIDS, the following assumptions were made:

1. All members of the floor system have constant mo

ment of inertia and have no initial deflections

prior to the first placement of concrete.

22

2. The stiffness of the floor system is comprised of

the stiffnesses of the beams and girders only. In

other words, the metal deck and the concrete slab

have no contribution to the stiffness of the sys

tem.

3. The uniform load w is carried by the beams only.

The girders carry the beam load at the nodal

points commom to both, but carry no uniform load

4. All other assumptions made during the development

of the complete analytical model apply here too.

Once the computer model was created for a general floor

structure, specific cases representing a wide variety of

floor structure compositions could be easily created by

varying some specific characteristics of the floor

structure. For this study, close to 1800 variations of the

floor structure of Fig. 2 were created by varying the beam

size, the beam span, the beam spacing, and the slab

thickness.

The beam size was determined from the the

W-sections listed in the AISC Steel Construction Manual.

Since using all of the individual W-sections would be

impractical, the sections were divided into a total of

seven groups using the moment of inertia as the governing

criterion. The average weight of all the sections belon-

23

ing to each group was computed and that value was taken as

the self-weight of all the beams belonging to the group.

Table 4.1 gives the seven groups, the number of

sections belonging to each group, and the average weight

of the beams.

Table 4.1 Beam Size Classification

Moment of Inertia No. of W-sections Average Weight

(in4) (Lbs/ft)

200 32 21.0316

600 26 49.6923

1000 16 68.6872

2000 21 95.2852

4000 21 140.7148

10000 21 152.5720

20000 13 226.5388

The value of the moment of inertia defines the

maximum value that a section can have in order to belong

to the group. The minimum value of the moment of inertia

for the members of a group is defined by the value

categorizing the preceeding group. For instance, there are

24

21 W-sections whose moments of inertia fall between 2000

and 4000 in^.

In choosing the group ranges, it was attempted to

keep the number of shapes in each category close to the

same as that of adjoining groups. By doing this, sparsely

populated groups next to highly populated ones could be

avoided. The advantage of this procedure becomes clear as

the design curves are developed, as shown in Appendix A.

The design curves are more or less evenly spaced which

allows the user to perform a better interpolation with

specific "l"-value in between the regions defined by the

various curves.

The beam span was varied in five foot increments,

starting with a beam span of five feet, and continuing

until the resulting deflections or bending moment values

became too large for practical purposes. Since the

deflection is a function of the moment of inertia of the

beams, the lighter sections, such as the ones belonging to

the first two groups, required less increments in the span

length to reach large deflection values than the heavier

sections. As a result, the design curves for the lighter

sections were defined using fewer data points than curves

for the heavier sections. Since the curves were computer

drawn, the distances between data points were approximated

25

by straight lines. This resulted in better curve approxi

mations for the heavier sections than for the lighter

sections. The inconvenience can easily be avoided by

drawing the best curve by other means before actually

using the curves.

For the beam spacing, values of five, ten, and

fifteen feet were chosen. The reason for using these

particular values was that the most commomly used spacings

will either fall on or in between these values.

Interpolation can therefore be performed from the values

obtained from the design curves defining the beam spacing

range of the user.

The final variable considered in this study was

the slab thickness. Slab thicknesses of 5.5 in, 6.0 in,

6.5 in, 7.0 in, and 7.5 in were considered. The concrete

was assumed to be of normal weight (145 pcf). The load

that the concrete exerts on the floor has been computed as

a function of the slab thickness taking into account the

metal deck geometry (9), and is given in Table 4.2.

Table 4.2 Concrete Weight as a Function

of Slab Thickness and Metal Deck Geometry

(Normal Weight Concrete)

Slab Thickness (in.) 5.5 6.0 6.5 7.0 7.5

Concrete Wt. (psf) 40 46 52 58 64

26

Values similar to those of Table 4.2 can be

obtained from the term in the paranthesis in Eq.(3.1).

However, this term predicts the weight of concrete of a

solid slab, and these values are therefore somewhat

higher. Despite this, Eq.(3.1) is still valid for

computing the w-load if the term in parenthesis is

subtituted by one of the values given in Table 4.2,

depending on which slab thictaiess is being considered.

To make the total loading more realistic, some the

of the most common dead loads a floor structure can be

subjected to during construction and its service life were

also included in the model. Typically assumed values for

these loads were chosen as given below:

Construction Load: 5 psf

Carpets : 5 psf

Ceiling Load : 10 psf

Partitions : 20 psf

Total Load : 40 psf

This extra dead load is incorporated in Eq.(3.1) by the

"DL" term.

Given all of the information needed to compute the

w loads form Eq.(3.1), a set of loads were prepared for

the different floor structures of the study. These are

given in Tables 4.3(a), 4.3(b), and 4.3(c).

Table 4.3(a) w Loads (lbs/ft) for Floor Structure

(5 ft. Beam Spacing)

"I" Value Slab Thickness (in. )

(in.4) 5. 5 6. 0 6. 5 7. 0 7. 5

200 421. 0316 451. 0316 481. 0316 511. 0316 541. 0316

600 449. 6923 479. 6923 509. 6923 539. 6923 569. 6923

1000 468. 6872 498. 6872 528. 6872 558. 6872 588. 6872

2000 495. 2852 525. 2852 555. 2852 585. 2852 615. 2852

4000 540. 7148 570. 7148 600. 7148 630. 7148 660. 7148

10000 552. 5720 582. 5720 612. 5720 642. 5720 672. 5720

20000 626. 5388 656. 5388 686. 5388 716. 5388 746. 5388

Table 4.3(b) w Loads (lbs/ft) for Floor Structure

(10 ft. Beam spacing)

"I" Value Slab Thickness (in.)

(in.4) 5. 5 6. 0 6. 5 7. 0 7. 5

200 821. 0316 881. 0316 941. 0316 1001. 0316 1061. 0316

600 849. 6923 909. 6923 969. 6923 1029. 6923 1089. 6923

1000 868. 6872 928. 6872 988. 6872 1048. 6872 1108. 6872

2000 895. 2852 955. 2852 1015. 2852 1075. 2852 1135. 2852

4000 940. 7148 1000. 7148 1060. 7148 1120. 7148 1180. 7148

10000 952. 5720 1012. 5720 1072. 5720 1132. 5720 1192. 5720

20000 1026. 5388 1086. 5388 1146. 5388 1206. 5388 1266. 5388

28

Table 4.3(c) w Loads (lbs/ft) for Floor Structure

(15 ft. Beam Spacing)

"I" Value Slab Thickness (in.)

(in.4) 5. 5 6. 0 6. 5 7. 0 7. 5

200 1221. 0316 1311. 0316 1401. 0316 1491. 0316 1581. 0316

600 1249. 6923 1339. 6923 1429. 6923 1519. 6923 1609. 6923

1000 1268. 6872 1358. 6872 1448. 6872 1538. 6872 1628. 6872

2000 1295. 2852 1385. 2852 1475. 2852 1565. 2852 1655. 2852

4000 1340. 7148 1430. 7148 1520. 7148 1610. 7148 1700. 7148

10000 1352. 5720 1442. 5720 1532. 5720 1622. 5720 1712. 5720

20000 1426. 5388 1516. 5388 1606. 5388 1696. 5388 1786. 5388

Given the different possible combinations of

loading and structural arrangements, exactly 1785

different floor compositions were created and analyzed by

program GRIDS. The end result is the set of design curves

that is presented in Appendix A.

29

Figure 5 Floor Structure Computer Model

CHAPTER 5

RESULTS AND DESIGN RECOMMENDATIONS

From the computer analysis performed for the range

of floor structures considered, the first conclusion that

can be drawn is that all of the structures converged to an

equilibrium position. As a result, a leveled slab surface

could be assured in all cases.

The rate of convergence was found to be high, and

only three to five iterations were required to achieve

converged deflection values for up to four significant

figures. In general, the heavier the initial load and the

longer the span length of the floor beams, the more

iterations were needed to arrive at an equilibrium

position. However, only a two iteration difference was

found between the lightest and heaviest load as well as

for the shortest and longest span considered in the

analysis. Therefore, under typical dead loads and span

lengths, only three to five relevelings of the slab surfa

30

31

ce will be needed in order -to reach an equilibrium

position.

Probably the most important finding of this study

is that only a maximum increase of about 3% was found to

occur for the maximum moment and deflection values from

the initial to the equilibrium state of the structure. The

3% increase was found to apply for maximum dead load

deflections of about 3 in. and below, as well as for

maximum dead load moments of about 4000 kip-in. and below,

for groups having moments of inertia less than 4000 in4,

and 9000 kip-in for groups having moments of inertia above

4000 in4. Converged deflection and moment values falling

above these limits were found to have greater than 3%

increases when compared to the initial values. However,

these increases rarely exceeded 10%. In the design curves,

this can be observed by the larger discrepancy that exists

between initial and converged values after the approximate

limits have been reached. Before reaching these limits,

the initial and converged data points almost coincide at a

same point, since due to graph accuracy limitations, the

maximum 3% difference is almost undetectable.

The 3 in. limit on maximum dead load deflection

seems to be of no concern for typical span lengths, since

deflections approaching this limit will have violated

32

design values during the first placement of concrete and

the slab releveling procedure therefore would be

meaningless. On the other hand, it may be possible that

the dead load moment capacity of a beam exceeds the limits

for the maximum dead load moment given above. If this is

the case, the design curves may provide the projected

increase above 3% that the beam will have after

termination of the slab releveling procedure.

Since the increases in moment and deflection

values due to slab releveling were found to be rather

small, the assumption that the system is perfectly elastic

(made during the development of the floor analytical model

in Chapter 2) will not be violated if the system was

within the elastic range before the slab releveling is

performed. Also, due to the small increases involved, slab

releveling should be considered a feasible alternative to

achieve a leveled slab surface, if the system is not

overstressed to begin with. This is also correct if the

system just barely complies with maximum design values for

which the associated small increases become significantly

large to cause the system to violate maximum design values

after the slab releveling is terminated.

During slab releveling, the maximum moments were

found to occur at the girder midspans for the short floor

33

beam spans (40 ft. and below), but as the spans were

increased, a critical span length was reached where the

maximum moment was found to occur simultaneously at the

girder and the floor beam midspans. As the span length

continued to increase, the maximum moments occurred at the

floor beam midspans only. The critical span length was

found to be independent of the loading conditions and

influenced primarily by the geometrical arrangement of the

structural members within the bay.

The maximum deflection values were always found to

occur at the interior floor beam midspans. These were

measured with respect to the beam supports, in order to

comply with the assumption that the beams were simply

supported at the girders. Since the supports will deflect

along with the girders, the total deflection calculated

with respect to the true simple supports (column supports)

would have to be modified to account for the simply

supported beam assumption. Equation (5.1) was used to

perform the modification, and Fig. 6 illustrates how this

modification was made.

Db = Dt - Dg (5.1)

34

Column Support Level

Figure 6 Total Interior Floor Beam Deflection

The maximum floor beam deflections calculated by Eg.(5.1)

are the ones used in the design curves.

Using Eg (5.1) actually decouples the girder and

beam deflections, even though this is done following the

analysis considering the girder iteration. Therefore,

using Eq.(2.1), which relates the girder and beam moments

of inertia, should have no effect on the validity of the

design curves which consider the floor beams only. If Eq

(5.1) was not used, and the deflections of the beams were

measured with respect to the column supports, the girder

iteration would have to be considered, and Eq (2.1) should

not be used. Instead, the girders should be designed using

standard procedures each time the characteristics given in

Chapter 4 are changed. It is obvious that the amount of

time involved in producing each floor composition would

35

have increased substantially, to the point of making this

study impractical if the girders had to be redesigned each

time the floor composition was varied. This is the reason

why Eg (2.1) and Eg (5.1) had to be developed.

Using Eg.(5.1) also allows the perimeter floor

beams to be incorporated into the design curves, since the

only difference between the deflections of the interior

and perimeter beams is the value of the support deflection

of the interior beams.

Finally, the results have been used to develop the

following recommendations for achieving a leveled slab

surface:

1. If the floor structure has already been built and

the first placement of concrete has been made,

the midspan deflections of the structural members

must be checked. If the midspan deflections are

within 3% of the maximum allowable value, then

other alternative to achieve a leveled slab sur

face should be used (such as the ones given in

Chapter 1). Otherwise, the slab releveling proce

dure can be used without violated allowable de

flection values. Next, maximum dead load moments

must be checked. Enter the corresponding design

curve with the beam "I" value and the computed

36

maximum moment. If the actual maximum moment

falls above the point where the increase on the

maximum moment due to slab releveling is above

3%, compute the actual increase by the following

formula:

1% = (Cv - Iv) x 100 / Cv (5.2)

Where:

Cv = Converged Data Point Value.

Iv <= Initial Data Point Value.

If the increase obtained from Eg (5.2) is more

than the percentage difference between the actual

maximum moment and the maximum allowable dead

load moment, then the slab releveling procedure

must not be used and other alternatives must be

considered. Otherwise, using the releveling pro

cedure should not violate maximum allowable dead

load moment values. On the other hand, if the ac

tual moment value falls below the point where in

creases under 3% are predicted, slab releveling

can also be used, provided that the percentage

difference between the actual and allowable mom

ents is not below 3%.

must be equal). Then a region between the "I"-va-

lue curves can be chosen and floor beam sections

within the region can be picked. The coorespon-

ding maximum moment and deflection values can be

determined from the curves and checked to see if

they violate established design values. If the

chosen sections violate the design values, then

the beam span can be changed or a new section can

be chosen until all beams are adequate. The sec

tions can also be checked for the slab releveling

procedure, along with the standard checks an the

deflection and moment values. The sections that

are chosen must also be tested agaist live load

moments and deflections before the girders are

designed. Once the design of the girders is com

pleted, the whole floor structure can be reche-

cked with program GRIDS for the final design.

When using the design curves to see if the slab

releveling procedure is a feasible alternative for the

floor structure, it is important to recognize that these

curves were developed on the basis of specific loading

conditions as given by the numerical values of the

different w-loads (see Tables 4.3 (a), (b), and (c) ). If

the actual w-load is different from the assumed value that

39

pertains to the case under consideration, then the nature

of this difference will define the applicability ot the

curves. If the actual value is lower, then the resulting

structure may be on the conservative side. If the actual

value is higher, then the resulting structure may be

unconservative. If the values differ significantly in any

way, the use of the design curves is not recommended.

However, program GRIDS will still provide the means of

testing the structure for the feasibilty of the slab

releveling procedure, since it allows the user to use the

actual loading conditions and the actual properties of the

structural members.

CHAPTER 6

SUMMARY AND CONCLUSIONS

6.1 Summary

Slab releveling is one of the commonly used

procedures for floor structures for achieving a leveled

slab surface. The procedure is iterative in nature and

consists of adding concrete to level the uneven surface

caused by the deflecting floor. This is continued until an

equilibrium position is reached, and further deflection

increases are negligible.

The purpose of this study was to investigate the

strength and behavior of the floor beams when subjected to

the load and deflection increments caused by the slab

releveling procedure.

An analytical model was developed for the floor

and the loading, and the model configuration was varied to

accommodate many structural configurations. The analyses

were carried out by a computer program that was developed

40

41

for this specific purpose. The results of the analysis are

given in the form of a set of design curves.

Finally, from the results of the study,

recommendations are given for achieving a leveled slab

surface.

6.2 Conclusions

The following conclusions can be drawn:

1. All structural configurations converged to an

equilibrium position, which assures the possibi

lity of obtaining a leveled slab surface.

2. The convergence rate was found to be good, and

only 3 to 5 iterations were needed to reach an

equilibrium position.

3. Only a maximum increase of 3% was found for the

dead load deflections and moments when the equi

librium position was reached. This was true for

deflection values of 3 in. and less, and for mo

ment values of about 4000 kip-in for moment of

inertia of 4000 in.4 and below, and 9000 kip-in

for moments of inertia above 4000 in.4. After

reaching these approximate limits, the percentage

42

increase of the moment values can be predicted by

using the design curves.

Due to the small increments in load and defle

ction, overstressing of the structure will rarely

occur, and the slab releveling procedure is

therefore a feasible approach to obtaining a le

veled slab surface.

APPENDIX A

DESIGN CURVES

Included in this appendix is the set of design

curves developed from the results of the computer analysis

of the different floor structure compositions considered

in this study.

The curves define the relationship between the

floor beam stiffness and the maximum dead load moments and

deflections. Although a modulus of elasticity of 30,000

ksi was used in the analysis, a value of unity (E = 1.0)

was used to construct the curves. Therefore, only the

moment of inertia and the span length are needed to enter

the curves. However, these values must be entered in

inches since the maximum dead load momemts and deflections

are in Lbs.-in. and in. respectively. There are 15 sets of

curves, each being defined by the beam spacing and slab

thickness given on the upper right corner of the curve

sets. Each set consists of 4 graphs, which include the

curves for the different "I"-value groups. The solid

43

44

curves represent the behavior of the floor beans after the

first placement of concrete, and the dashed curves

represent the behavior of the beams after an equilibrium

position has been reached.

Due to the assumption that the system is perfectly

elastic, the curves are asymptotic to the vertical axis.

In reality, the curves should have a cutoff point where

the plastic moment is reached for the maximum moment

curves, and at the corresponding deflection value for the

maximum deflection curves. To avoid any confusion, the

user is adviced to draw a cutoff line at the maximum

allowable dead load moments and deflections. In general,

the curves are asymptotic to the axes as long as the

structure is within the elastic range.

Finally, it is important to understand the limitations

and applicability of these curves. Therefore, the user is

advised to read Chapters 4 and 5 before using these

curves.

9.00

5.5 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value

o o i o jz

o c

6.00

a a> CD O

3.00 E u

E X a 2

0.00 0.00 3.00

Stiffness (El/L) Bea

Figure A1 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.*; Ts=5.5 in.; Sb=5.0 ft.)

9.00 - o o o o

o o 9o i o

i o | o oJ o


CM

6.00 -

o 3.00 -

0.00 0.00 5.00 10.00 15.00

Beam Stiffness 20.00

(El A) 25.00 30.00

Figure A2 Maximum Deflection vs Beam Stiffness C1=2000 to 20000 in.*; Ts=5.5 in.; Sb=5.0 ft.)

9.00 -i

CD * * O .


x 6.00 - o

o o o

o o

<v> CO 3.00 -o o

<? CM

X

0.00 0.00 3.00

Beam Stiffness (El/L) 6.00

Figure A3 Maximum Moment vs Beam Stiffness CI=200 to 2000 in.4*-; Ts=5.5 in.; Sb=5.0 ft.)

25.00 -i

20.00 -j * O

5.5 in. Slab 5.0 ft. Beam Sp.

Initial Value o Converged Value

M5.00 -

10.00 -

5.00 h

0.00 0.00

M i i i i 5.00 10.00

Beam

i M 11 1 5.00

Stiffness

i M 11 11 M 20.00

(ei A) I I II | I I M I J 1 I 1 | 25.00 30.00

Figure A4 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.1*-; Ts=5.5 in.; Sb=5.0 ft.)

9.00 -

o o

o o 6.0 in. Slab

5.0 ft. Beam Sp. * Initial Value o Converged Value

o o CM

O _ O Q O /n T

© — I c

6.00 -

O " 3.00 -

x

0.00 0.00 3.00


Figure A5 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.*; Ts=6.0 in.; Sb=5.0 ft,)

o o o o eg

o o o

9.00 -

5.0 ft. Beam Sp. * Initial Value o Con-verged Value

6.00 -

a 3.00 -

x

0.00 0.00 5.00 10.00 15.00 20.00

Beam Stiffness (El/L) 25.00 30.00


9.00

to * * o

6.0 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value o x 6.00 -

o o

o JL 9 to

_Q

3.00 -

0.00 0.00 3.00


Figure A7 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.t-; Ts=6.0 in.; Sb=5.0 ft.)

25.00

^ 20.00 * o

o o o

? ° . CSJ :Q JL


o o o o

CO

10.00 o Q Ti

cs O O CM

5.00

0.00 0.00 5.00 10.00 15.00

Beam Stiffness (El/L) 20.00 25.00 30.00

Figure A8 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=6.0 in.; Sb=5.0 ft.)

Ul to

9.00 -

CO CD J= (J CZ

6.00 -

c o CJ w CJ O

3.00 -E

E X o

0.00 0.00



Beam 3.00

Stiffness (EI/L) 6.00


•9.00 -


o o o o

o 1 o -.o t o ?01j O 1 CM Td-

o o

c

6.00 -

Q 3.00 -

x

o.oo 4t-.0.00 5.00 10.00 15.00 20.00 25.00 30.00

Beam Stiffness (El/L)

Figure AlO Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.*-; Ts=6.5 in.; Sb=5.0 ft.)

9.00

CD 6.5 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value

O o o o x 6.00

c:

o o o

3.00 £Z a>

x o

0.00 6.00 0.00 3.00


Figure All Maximum Moment vs, Beam Stiffness (1=200 to 2000 in.*; Ts=6.5 in.; Sb=5.0 ft.)

25.00

o o o

• 8 il

6.5 in. Slob 5.0 ft. Beam Sp. * Initial Value o Converged Value

* 20.00 * o

o o o o

>15.00

_Q

10.00

o o

5.00

x

0.00 25.00 30.00 0.00 5.00 15.00 20.00 10.00



9.00 -

CO CD

sz o

6.00 -

c o o _QJ

a.i 3.00 -

E Z3

E X o

0.00 0.00


Beam 3.00

Stiffness (EL/L) 6.00

Figure A13 Maximum Deflection vs. Beam Stiffness CI=200 to 2000 in.*; Ts=7.0 in.; Sb=5.0 ft.)

25.00

^ 20.00 * o


o o o >15.00

o o

9 5 10.00 o o CM

5.00

X

0.00 0.00 5.00 10.00 15.00


Figure A14 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.4-; Ts=7.0 in,; Sb=5.0 ft.)

9.00 n

to * * o

X 6.00

d * I (A -O

c <u E o

3.00 -

x a



0.00 6.00 0.00

t r 3.00


Figure A15 Maximum Moment vs. Beam Stiffness CI=200 to 2000 in.1*-; Ts=7.0 in.; Sb=5.0 ft.)

o o o

9.00 - o o a o

Oa o o O 1 o CM , Tj-

-M Ji 7.0 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value

to <u

6.00 -c o a <D QJ O

3.00 -E

£ X a

0.00 0.00 5.00 10.00 30.00 15.00 20.00 25.00


Figure A16 Maximum Moment vs. Beam Stiffness CI=2000 to 20000 in.*; Ts=7.0 in.; Sb=5.0 ft.)

o o o CM

O O O

to 7.5 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value

CM to

6.00 -

3.00 -

0.00 0.00 3.00



9.00 -o o o o

o a 7.5 in. Slab


6.00 -

a 3.00 -

x

0.00 5.00 10.00 Beam Stiffness (El/L)

15.00 20.00 25.00 30.00

Figure A Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=7.5 in.; Sb=5.0 ft.)

9.00 n

7.5 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value o

o © o . oj

x 6.00 -

o o o

jD o

3.00 -

X

0.00 3.00 Beam Stiffness (El/L)

6.00

Figure A19 Maximum Moment vs, Beam Stiffness (1=200 to 2000 in.*; Ts=7.5 in.; Sb=5.0 ft.)

25.00

^ 20.00 * o

7.5 in. Slab 5.0 ft. Beam Sp. * Initial Value o Converged Value o o o

9 o

x

>15.00 t=

CO -O

o o

9 ° 10.00

o o o CM

CD

5.00

X

0.00 0.00 5.00 10.00 25.00 30.00 15.00 20.00



9.00 -

o o o

i 8 CM


to

9 -!!•

6.00 -

O 3.00 -

0.00 0.00 3.00

Stiffness (El/L) 6.00

Bea


3.00 -

to CD

SZ o c

6.00 -

c: Q

O _QJ

O 3.00 -

£

E "x o

5.5 in. Slob

* Initial Value o Converged Value

0.00 0.00 5.00 10,00 15.00

Beam Stiffness

11 pi 11 20.00

(El/L)

I fl I 1 I I I I 25.00

T-n 30.00


I 9.00 -i i 0 L ° CD o


CD * * o o

9 £ x 6.00

o o , o 1 CO

9 o i ° ' CM

3.00

0.00 6.00 3.00

Beam Stiffness (El/L) o.oo

Figure A23 Maximum Moment vs. Beam Stiffness (1 = 200 to 2000 in,1*-; Ts=5.5 in.; Sb=J0.0 ft.)

-j

<?

25.00 o o o o . CN o

o

o ** 20.00 * o


>15.00

CO a o

<?c3 10.00

5.00

x

0.00 0.00 5.00 10.00 15.00


Figure A24 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.*-; Ts=5.5 in.; Sb=10.0 ft.)

o o o

o o o

CO (f> II

CM 6.0 in. Slob 10.0 ft. Beam Sp. * Initial Value o Converged Value

6.00 -

tr

a 3.00 -

ZJ

0.00 r* i—i— ii—r P i n=* 3.00


6.00

Figure A25 Maximum Deflection vs. Beam Stiffness (1 = 200 to 2000 in.f-; Ts=G,0 in.; Sb=10.0 ft.)

I 1

o o o

o o

o o o 9.00 - o

CN

6.0 in. Slab 10.0 ft. Beam Sp. • Initial Value o Converged Value

6.00 -

a 3.00 -

0.00 0.00 5.00 10.00 15.00 20.00

Stiffness (El/L) 25.00 30.00

Bea

Figure A26 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.*; Ts=6.0 in.; Sb=10.Q ft.)

o o 9.00

CSJ

CO * * O


o o

x 6.00 9 o i ° 1 to 1 II *i —

<?o 1 O . CN

3.00

0.00 3.00 Beam Stiffness (El/L)

6.00

Figure A27 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.*; Ts=6.0 in.; Sb=10.0 ft,)

25.00 o o o

o o

20.00 * o

6.0 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value o o o >15.00

o o CM 10.00

5.00

0.00 0.00 5.00 10.00

Beam Stiffness ' (El/L) 15.00 20.00 25.00 30.00

\

Figure A28 Maximum Moment vs. Beam Stiffness (1 = 2000 to 20000 in.f-; Ts=6.0 in.; Sb=10.0 ft.)

9

o o o

o o o CM

O o to 9 — I Ji.


CO

6.00 -

£Z

Q 3.00 -

x

0.00 0.00 3.00


Figure A29 Maximum Deflection vs. Beam Stiffness (1 = 200 to 2000 in.*-; Ts=6.5 in.; Sb=10.0 ft.)

-4 U

o o o o o

o o o o o o CM _ -51-

9 A 6.5 in. Slob 10.0 ft. Beam Sp. • Initial Value o Converged Value

6.00

a 3.00

x

0.00 0,00 5.00 10.00 15.00


Figure A3 0 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in>; Ts=6.5 in.; Sb=10.0 ft.)

9.00 -i

o o o

CO 6.5 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value

O

** o l o , CO

x 6.00 -

;1 —

IS -Q

3.00 -

0.00 0.00 3.00



25.00 o Q O Y O I O

q> JJ_

o o CM


* 20.00 * o

o Q O T r-s x

>15.00

CO _Q O o o CSI 10.00

5.00

x

0.00 0.00 5.00 10.00 15.00 20.00


Figure A32 Maximum Moment vs. Beam Stiffness (1 = 2000 to 20000 in.t-; Ts=6.5 in.; Sb=10.0 ft.)

9.00 - o o o

o o CM

O o CO

o o CM

<? JL 7.0 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value

JL CO a)

6.00

c o u (D m— CD o

3.00 -E

E X a

0.00 0.00 3.00



© o 1 § H o

'9.00 - o 0*0 O T O CM I Tf

> O 1 CM

-i 7.0 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value

to

6.00 -

o 3.00 -

x

0.00 0.00 5.00 10.00 15.00


Figure A34 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.*-; Ts=7.0 in.; Sb=10.0 ft.)

9.00 -i o

o CM

?8 t o CO


O ( § \ j J!

x 6.00 -

o o

3.00 -

0.00 0.00 3.00



25.00 o o o o

9 R o

^ 20.00 # o

7.0 in. Slab 10.0 ft. Beom Sp. * Initial Value o Converged Value

« ° 9 o , o x

>15.00

o o o CN

0.00

5.00

0.00 0.00 5.00 10.00

Beam Stiffness (El/L) 15.00 20.00 25.00 30.00

Figure A36 Maximum Moment vs. Beam Stiffness (1 = 2000 to 20000 in.*-; Ts=7.0 in.; Sb=10.0 ft.)

9.00 -


to o o o o to CN

9 ~ 6.00 -

o 3.00 -

x

0.00 H— 0.00 3.00



9.00 - o o

o o o o

o o

<?CM o o


6.00 -

O 3.00 -

x

0.00 0.00 5.00 10.00 15.00 20.00


Figure A38 Maximum Deflection vs. Beam Stiffness (1=2000 to 20000 in.f; Ts=7.5 in.; Sb=10.0 ft.)

9.00 -i o o

Q O » I *

CD 7.5 in. Slab 10.0 ft. Beam Sp. * Initial Value o Converged Value

i JL

O i to

X 6.00 -

00 _Q

3.00 -

0.00 0.00 6.00 3.00


Figure A39 Maximum Moment vs. Beam Stiffness (1=200 to 2000 in.*; Ts=7.5 in.; Sb=10,0 ft.)

25.00

o o CM

O

: V

^ 20.00 * o 10.0 ft. Beam Sp.


9 o

x

>15.00

o o o CN 10.00

5.00

0.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00


Figure A40 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in,*; Ts=7.5 in.; Sb=10,0 ft.)*

9.00 -

o o o

o o o

o o CM 15.0 ft. Beam Sp.


CO CD _c o d

6.00 -

c: o CJ QJ

a> Q

3.00 -E ZD .i "x a

0.00 0.00 3.00


Figure A41 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.*;Ts=5.5 in,; Sb=15.0 ft.)

o o o o CNI

o o o o

o o o 9.00 -


6.00

O 3.00

x

0.00 0.00 5.00 10.00

Beam Stiffness (El/L) 15.00 20.00 25.00 30.00


9.00 n o o

o Q O T /—i

V — 16 to

* * o

5.5 in. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value x 6.00 - 9 o

i ° 1 CN

CO -Q

3.00 -

0.00 0.00 3.00

Beam Stiffness (Ef/L) 6.00

Figure A43 Maximum Moment vs. Beam Stiffness CI=200 to 2000 in.*; Ts=5.5 in.; Sb=15.0 ft.)

25.00 o o o o CM

O O o

^ 20.00 * O


o

>15.00

10.00

J 5.00

x

0.00 0,00 10.00 20.00


Figure A44 Maximum Moment vs. Beam Stiffness (1=2000 to 20000 in.1*-; Ts=5.5 in.; Sb=15.0 ft.)

CO 09

9.00 -


o o o

o o CM

9 -c

6.00 -

3.00 -

x

0.00 0.00 3.00


Figure A45 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.4; Ts=5.0 in.; Sb=15.0 ft.)

' 9.00 o o o o

o o o CM

o o o

o o 6.0 in. Slab


9

6.00 -

O 3.00 -

x

0.00 0.00 5.00 10.00 15.00



9.00 o o o

o o o C\|

o


O o

x 6.00 -

3.00 -

x

0.00 0.00 3.00



u> H

25.00 o o o o CM

O

O _ o <? o * "51-

^ 20.00 * o


>15.00

o o o CM

to -Q

10.00

0.00 0.00 10.00 20.00



9.00 -

o o o o o o CM

O o CN

O „ O 9 co 6.5 in. Slab


co

9 ~

6.00

a

3.00 -

0.00 0.00 3.00


Figure A49 Maximum Deflection vs. Beam Stiffness [1=200 to 2000 in.*; Ts=6.5 in.; Sb=15.0 ft.)

9.00 - o o o CN

O O O o o o

o o


CO a> _ir o c

6.00 -

u a) CD o

3.00 -E 13

E X o 2

0.00 0.00 5.00 10.00 15.00

Beam Stiffness 20.00

(El/L) 25.00 30.00

Figure A50 Maximum Deflection ys, Beam Stiffpess (1=2000 to 20000 in.*; Ts=6.5 in.; Sb=15.0 ft.)

9.00 o o

o CM


* O o

o x 6.00

CO

3.00

0.00 0.00 3.00

Beam Stiffness .(El/L) 6.00


25.00 o o o o

o o o o CM

o

^ 20.00 * o


>15.00

10.00

5.00 -

x

0.00 0.00 10.00 20.00


Figure A52 Maximum Moment vs, Beam Stiffness (1=2000 vs. 20000 in.*; Ts=6.5 in.; Sb=15.0 ft.)

9.00 -

CO CD

sz o d

6.00 -

c: o

u CD

"cd Q

3.00 -

E

E X o

0.00 0.00

Beam 3.00

Stiffness

7.0 in. Slab 15.0 ft. Beam Sp-• Initial Value o Converged Value

(El/L) 6.00


9:oo -

6.00 -

3.00 -


0.00 0.00 5.00 10.00


T !"l I j TT rTI I I I I iT I A I M 1 I I I I I l i I I I I 15.00 20.00 25.00 30.00


9.00 -i

o o o

o o CM

9 - 7.0 in. Slob 15.0 ft. Beam Sp. * Initial Volue o Converged Value

O

x 6.00 -

to JO

3.00 -

0.00 0.00 3.00


Figure A55 Maximum Moment vs, Beam Stiffness (1=200 to 2000 in.t-; Ts=7.0 in.; Sb=15,0 ft.)

25.00 o o o

o o o o

o o

92

^ 20.00 * o


x

>15.00

10.00

5.00

x

0.00 0.00 10.00 20.00 40.00 30.00



o o o CM

o o o CO

9.00 - o CM

o

7.5 In. Slab 15.0 ft. Beam Sp. * Initial Value o Converged Value

to CD

_£=

O c

6.00

c; o

u QJ CD o

3.00

E c X O

0.00 6.00 3.00 0.00


Figure A57 Maximum Deflection vs. Beam Stiffness (1=200 to 2000 in.*; Ts=7.5in.; Sb=15.0 ft.)

9.00 -

CO 0) _C u a.

6.00 -t= o

<_> _QJ M— QJ o

3.00 -

E 13 E X O


0.00 7^ I I I I [

's!6o 10.00 15.00 20.00


11 In I I I |

25.00 30.00


o to

i q

9.00 o o o CN

O O

9 —


o

x 6.00

3.00 -

0.00 0.00 3.00


Figure A59 Maximum Moment vs, Beam Stiffness (1=200 to 2000 in.*; Ts=7.5 in.; Sb=15.0 ft.)

25.00

o o o o o CN

CD O O O O O CM Q

o

V 20.00 * o


x

>15.00 c:

CO -O

10.00

a>

5.00

0.00 0.00 10.00 20.00 30.00 40.00



APPENDIX B

PROGRAM GRIDS

B.l Introduction

Program GRIDS is a finite element computer program

which performs a structural analysis on a floor system

whose members are modeled by grid elements.

The program was written for an IBM-XT

microcomputer, using a Microsoft FORTRAN compiler. Care

was taken to assure maximum portability of the program.

Therefore, the program should run on any IBM compatible

microcomputer as well as any mini or main frame computer.

Comments are provided throughout the program to make it

more readable and understandable to the user.

Program GRIDS uses an iterative procedure which

accounts for the increase in load and deflection due to

the ponding of the concrete during the slab releveling

procedure. The purpose of the iterative procedure is to

investigate the convergence of the structural system to an

equilibrium position so that a leveled slab surface can be

achieved.

105

106

The program computes the linear and rotational

nodal displacements as well as the bending and torsional

end moments of all grid elements of the model under

consideration, each time an iteration is completed. In

this fashion, initial and converged values for the force

and deflection values can be obtained.

Finally, it must be understood that the solutions

obtained from program GRIDS are not guaranteed to be the

"optimum" solutions for the problem under consideration.

In finite element analyses, the solutions of a given

problem are only as good as the finite element model

representing such a problem.

107

B.2 User Manual

Program GRIDS prompts to the terminal screen for

the name of the input and output file. The input file

under the name given by the user must be in existence at a

memory location readibly available for the executable

program. The output file is created by the program under

the name given by the user.

The input file must be written in free format

style, with all data entries separated by commas. All data

entries must be filled. If a data entry is not relevant

for a particular problem to be analyzed, then a "zero"

must be entered at that location.

The input file must be created in the following

format:

PART I: Problem Identification Number. Any integer

may be entered

PART II:General Parameters for the Problem.

Enter in order: Number of nodes, Number

of elements, Number of element property

types, Number of concentrated loads, Num

ber of Iterations, Concrete Unit Weight.

108

PART III:Structure Geometry.

Enter the X and Y nodal coordinates. Use

one line per node starting with the

coordinates corresponding to the first

node and continuing in sequence until

the last nodal coordinates are reached.

The node number is automatically assig

ned by the program.

Note: A right handed coordinate system

must be used.

Maximum number of nodes allowed: 100

PART IV: Element Property Types.

Enter : Elastic Modulus, Rigidity Modu

lus, Torsional Constant, Moment of Iner

tia. Enter this information for each

element, starting with the first proper

ty type and proceeding in sequence until

the last property type is reached. The

property type number is automatically

assigned by the program.

Maximum number of property types allo

wed: 100.

109

PART V: Element Connectivity and Distributed Load

Position.

Enter : Node i, Node j, Element Property

Type, DLV^, DLVj, Atw. Enter this

information for each element in the mo

del, starting with the first element and

proceeding in sequence until the last

element is reached. The element number

is automatically assigned by the program.

The i-th and j-th nodal positions must

be assigned from left to right.

DLV^ stands for Distributed Load Value

at the i-th node.

Atw stands for Tributary Area Width of

the element.

z

j

Figure B1 Load Distribution on the Element

110

PART VI:Concentrated Loads.

Enter : Node Number, Load Intensity. Do

this for all concentrated loads applied

to the model. If no concentrated loads

are present, ignore this part. Do not

fill the data entry corresponding to this

part with "zeros".

PART VII:Support Conditions.

Enter : Support in the 2-direction, rota

tional support in the x-direction and y-

direction. Enter this information for

each node in the model, starting with the

first node and proceeding in sequence un

til the last node is reached. The program

automatically assigns the node number.

Support Coding:

Support = 0 : Mode is free to move in the

specified direction.

Support = 1 : Node is restricted to move

in the specified direction.

Ill

PART VIII:End of File Flag.

Enter a "zero" followed by a comma to

signal the end of the file. Program GRIDS

can handle more than one input file in a

single run. To achieve this, replace the

"zero" with the problem ID number of the

next file to be run. Continue this proce

dure until the last file is typed in. The

"zero" must still appear on the last line

of the last file to be run.

112

B.3 Program Listing c********************************************************************** C*************************** PROGRAM GRIDS ***************************

C PROGRAMER: CARLOS E. PEflA RAMOS C UNIVERSITY OF ARIZONA C CIVIL ENGINEERING DEPT.

C THIS IS A FINITE ELSMENT PROGRAM THAT PERFORMS A STRUCTURAL ANALYSIS C ON A FLOOR SYSTEM WHOSE MEMBERS ARE MODELED BY GRID ELEMENTS. C THE PROGRAM HAS AN ITERATIVE PROCEDURE WHICH ACCOUNTS FOR THE INCREA C SE IN LOAD AMD DEFLECTION CAUSED BY PONDING OF THE CONCRETE DECK DU-C RING SLAB RELEVELING PROCEDURES. THE PURPOSE OF THE ITERATIVE PROCE-C DURE IS TO INVESTIGATE THE CONVERGENCE OF THE FLOOR STRUCTURE TO AN C EQUILIBRIUM POSITION SO THAT A LEVELED SLAB SURFACE CAN BE ACHEXVED.

PROGRAM GftlDS IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION X(100),Y(100),NODI(100)/NODJ(IOO).EMOD(IOO),RG(100), STC(IOO),XMI(100),MAT(100),T(6,6),SL(6,6),SE(6,6),SC(6,6),R(300) , &U(300),C(300),8(300,160),UI(100),UJ(100),BX(2,6),SG(6),SD(6), &ST(2),P(2),SP(6),13(300),UP(100),B(300),ATW(100) CHARACTER*15 IFNAME CHARACTER*15 OFNAME

C WRITE NAME OF INPUT FILE TO THE SCREEN

WRITE(*,10) 10 FORMAT(IX,"NAME OF INPUT FILE :•)

R£AD(*,30) IFNAME

C WRITE NAME OF OUTPUT FILE TO THE SCREEN

WRITE(*,20) 20 FORMAT(IX,'NAME OF OUTPUT FILE :')

READ(*,30) OFNAME 30 FORMAT(A)

C OPEN THE INPUT AND OUTPUT FILES UNDER THE NAMES GIVEN BY THE USER

OPEN(10,FILE-IFNAME) OPEN(11,FILE-OFNAME,STATUS-•NEW')

C OPEN FILES FOR DISPLACEMENT VECTOR STORAGE

OPEN(12,FILE-'CVEC.DAT',STATUS-1NEW•,FORM- • UNFORMATTED•) OPEN(13,FILE-1BVEC.DAT',STATUS-•NEW•,FORM-•UNFORMATTED')

C READ PROBLEM NUMBER

READ(10,*) NPROB IF(NPROB.EQ.O) GO TO 1000 WRITE(11,40) NPROB

40 FORMAT(//,5X,'PROBLEM NO.',13,//)

C ENTER GENERAL PARAMETERS FOR THE PROBLEM

C NODEN - NUMBER OF NODES. C NUMEL - NUMBER OF ELEMENTS. C NMAT - NUMBAR OF ELEMENT PROPERTY TYPES. C NC - NUMBER OF CONCENTRATED LOADS. C NIT - NUMBER OF ITERATIONS. C WC - CONCRETE UNIT WEIGHT.

READ(10,*) NODEN,NUMEL,NMAT,NC,NIT,WC WRITE(11,50) NODEN,NUMEL,NMAT,NC,NIT,WC

50 FORMAT(>//,2IX,'GENERAL PARAMETERS',//,2X,*NO. OF NODES -',13,/, &2X,'NO. OF ELEMS -',I3,/,2X,'NO. OF Ellf TYPS -*,13,/,2X,'NO. OF SCON. LOADS ,I3,/,2X,'NO. OF ITERATIONS -',I3,/,2X, £'CONCRETE UNIT WEIGHT -',F4.3,/)

113 WRITE(11,195) HEQ.HBAND

195 FORMAT(///,10X,•NUMBER OF EQUATIONS «•,14,//,10X,'BANDWIDTH «3,//) WRITE(11,240)

240 FORMAT (////72 ('*') ,/,31(1 *1) , 1 RESULTS 32 (1 *'),/, 72 (•*'),///)

C******************** START ITERATIVE PROCEDURE *******************

DO 500 N-1,NIT IF(N.EQ.l) GO TO 225 DO 215 K-1,NEQ

215 R(K)-0.

C NOTE:IF INITIALIZATION OF THE R(I) VECTOR IS MADE WHEN CONCENTRATED C LOADS ARE PRESENT, THE POSITIONS OF THESE LOADS MUST BE FILLED C AGAIN WITH THE VALUES GIVEN IN SUBROUTINE "INPUT". THIS MUST BE C DONE INSIDE LOOP 500.

C INITIALIZE THE SYSTEM MATRIX OF THE STRUCTURAL MODEL.

225 DO 210 J~1,NEQ DO 200 K«1,MBAND

200 S(J,K)-0. 210 CONTINUE

C CALCULATE THE GLOBAL STIFFNESS MATRIX FOR EACH GRID ELEMENT AND C STORE IT IN THE SYSTEM MATRIX. C MODIFY ALSO THE LOAD VECTOR TO REFLECT DISTRIBUTED LOADING CONDI-C TIONS.

CALL GRID(NUMEL,X,Y,NODI,NODJ,EHOD,RG,TC,MAT,UI,UJ,XMI,T,SC,SL, £SE,S,SP,SQ,R,MBAND,NEQ)

C MODIFY THE SYSTEM MATRIX AND LOAD VECTOR TO REFLECT SUPPORT CONDI-C TIONS.

DO 230 1-1,NEQ IF(IS(I).EQ.O) GO TO 230 S(I,1)»(S(I,1)+1.0)*1.0E+20 R(I)-0.

230 CONTINUE

C CHECK TO SEE IF THE MAXIMUM NUMBER OF NODES ALLIED BY THE PROGRAM C IS NOT EXCEEDED.

IF(NODEN.GT.IOO) GO TO 998

C NUMBER OF SIMULTANEOUS EQUATIONS REPRESENTING THE STRUCTURAL MODEL.

NEQ-0 NEQ-NODEN*3

C INITIALIZE THE PARAMETERS TO BE USED IN THE PROGRAM.

CALL INITIAL(X,Y,NODI,NODJ,MAT,EMOD,RG,TC,XMI,UI,UJiUP,IS,C,R,U, &B,NODEN,NEQ,ATW)

C READ INFORMATION FROM INPUT FILE.

CALL INPUT(NODEN,X,Y,NMAT,EMODfRG,TC,XMX,NUMEL,NODI,NODJ,MAT,UI, «UJ,NC,R,IS,ATH)

C CALCULATE THE BANDWIDTH.

MHAND-0 DO 190 I-1,NUMEL MB»(IABS(NOD(I)-NODJ(1))+l)*3 IF(KB.GT.MBAND) MBAND-MB

190 CONTINUE

c***************** RESULTS OF THE PROBLEM SET FOLLOWS 114

C SOLVE FOR THE MODAL DISPLACEMENTS.

DO 250 I-1,NEQ C(I)-RCI)

250 CONTINUE CALL GEQSOL(NEQ,MBAND,NODEN, S, C) REWIND 12 WRITE(12) (C(I),I-1,NEQ)

C CALCULATE TORQUE AND BENDING MOMENT FOR EACH GRID ELEMENT.

DO 280 I-1,NEQ U(I)-C(I)

2BO CONTINUE CALL GREAC(NUMEL,X,Y,NODI,NODJ,EMOD,RG,TC,XMI,MAT,UI,UJ,SG,T,SD, &BX,ST,U,P)

C MODIFY DISTRIBUTED LOADING CONDITIONS TO ACCOUNT FOR THE INCREASE C LOAD DUE TO THE SLAB RELEVELING PROCEDURE.

REWIND 12 READ(12) (C(I),I-1,NEQ) IF(N.GT.l) GO TO 310 DO 300 J-l,NODEN UP(J)-C(3*J-2)

. 300 CONTINUE GO TO 325

310 REWIND 13 READ(13) (B(I),1-1,NEQ) DO 320 J-l,NODEN UP(J)-C(3*J-2)-B(3*J-2)

320 CONTINUE 325 DO 330 I-1,NUMEL

UI(I)-UP(NODI(I))*ATW(I)*WC + UI(I) UJ(I)-UP(NODJ(I))*ATW(I)*WC + UJ(I)

330 CONTINUE DO 340 I—1,NEQ BCI)-C(I)

340 CONTINUE REWIND 13 WRITE(13) (B(I),I-1,NEQ)

C SET UI(X)-0. t UJ(I)-0. FOR ALL GIRDER ELEMENTS FOR CONSISTENCY IN C ASSUMED LOADING CONDITIONS. IF GIRDERS ARE ASSUMED TO CARRY ANY C DISTRIBUTED LOADING, THEN UI(I) AND UJ(I) MUST NOT BE INITIALIZED C AND THE FOLLOWING TEN FORTRAN LINES MUST BE DELETED:

UI(5)-0. UJ(5)-0. DO 345 J—10,14 UI(J)-0. UJ(J)-0.

345 CONTINUE UI(19)-0. UJ(19)-0. UI(24)-0. UJ(24)-0.

C THE POSITIONS IN THE ARRAYS UI(I) AND UJ(I) GIVEN ABOVE (5,10 TO C 14,19,24) DEFINE NODAL POSITIONS WITHIN A GIRDER LENGTH. THEREFORE, C IF A MODEL DIFFERENT THAT THE ONE GIVEN IN THE THESIS STUDY IS USED C THESE NODAL POSITIONS MAY CHANGE AND THE NODAL POSITIONS GIVEN C ABOVE MAY HAVE TO BE CHANGED ACCORDING TO THE USER'S OWN MODEL.

500 CONTINUE GO TO 35

C FLAGS

998 HRITE(*,999) 999 FORMAT(//,XX,'MAXIMUM NUMBER OF MODES EXCEEDED.',/,IX,

6'MUST MODIFY THE PROGRAM TO INCREASE CAPABILITY TO HANDLE MORE (NODES')

1000 STOP END

C******************************************************************** C************************** SUBROUTINES *****************************

SUBROUTINE INITIAL(X,Y,NODI,NODJ,HATtEMOD,RG,TC,XHI,UI,UJ,UP(IS, fiC,R,U,B,NODEN,NEQ) IMPLICIT DOUBLE PRECISION (A-H,0-2) DIMENSION X(100),Y(100),NODI(100).NODJ(IOO),MAT(100),EMOD(100) , tRG(lOO),TC(100),XMI(100),UI(100),UJ(100),UP(100),IS(300),C(300), &R(300),U(300),B(300)

C THIS SUBROUTINE INITIALIZES THE PARAMETERS USED UN THE MAIN PROGRAM

DO 60 1-1,NODEN X(I)-0. Y(I)-0. NODI(I)-0. NODJ(I)-0. MAT(I)-0. EMOD(I)"0. RG(I)-0. TC(I)-0. XMI(I)»0. UI(I)-0. UJ(I)-0. UP(I)-0. ATW(I)«0.

60 CONTINUE DO 70 J-1,NEQ IS(J)-0. C(J)»0. R(J)-0. U(J)-0. B(J)-0.

70 CONTINUE RETURN END

SUBROUTINE INPUT(NODEN,X,Y,NMAT,EMOD,RG,TC,XMI,NUMEL,NODI,NODJ, £MAT,UI,UJ,NC,R,IS) IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION X(100),Y(100),EMOD(100),RG(100),TC(100),XMI(IOO),

(NODI(100),NODJ(100),MAT(100),UI(100),UJ(100),R(300),IS(300)

C THIS SUBROUTINE READS THE INFORMATION FROM THE INPUT FILE.

C READ NODAL POINT COORDINATES.

WRITE(11,B0) 80 FORMAT(////,21X,'NODAL POINT COORDINATES',//)

WRIT£(11,90) 90 FORMAT(5X,'NODE NO. ' ,9X,'COORDINATES' ,//,20X,'XM3X,'Y»,//)

DO 100 1-1,NODEN READ(10,*) X(X),Y(X) WRITE(11,95) X,X(I),Y(I)

95 FORMAT(8X,I2,2(SX,F8.2)) 100 CONTINUE

C READ ELEMENT PROPERTY TYPES.

WRITE(11,110) 1X0 FORHAT(///,30X,'ELEMENT PROPERTIES',//,IX,'ELTYP',5X,'ELASTIC

SHOD',6X,'RIGID MOD*,8X,'TORSIONAL C.',5X,*H. OF INERTIA',//) DO 120 I-1,NMAT READ(10,*) EHOD(I),RG(I),TC(I),XHI(I)

. WRITE(11,115) I,EMOD(I),RG(I),TC(I),XMI{I) 115 F0RKAT(1X,I2,5X,F15.2,1X,F15.8,3X,F15.8,6X,FB.2,/) 120 CONTINUE

C READ ELEMENT CONNECTIVITY AND DISTRIBUTED LOADING CONDITIONS.

WRITE(11,130) 130 FORMAT(////,3OX,'ELEMENT CONNECTIVITY1,//,IX,'ELEMENT',6X,

S'NQDEI',6X,'NODEJ',6X,'ELTYP',7X,'DLV I1,9X,1DLV J',/) DO 140 I-1,NUMEL READ(10,*) NODI(I),NODtf(I),MAT(I),UI(I),UJ(I),ATW(I) WRITE(11,135) I,NODI(I),NODJ(I),KAT(I),UX(I),UJ(I)

135 FORMAT(IX,15,IX,110,1X,I10,9X,I2,6X,F9.4,6X,F9>4) 140 CONTINUE

C READ CONCENTRATED LOADING CONDITIONS.

IF(NC.EQ.O) GO TO 165 WRITE(11,150)

150 FORMAT(///,30X,'CONCENTRATED LOADING CONDITIONS',//,5X,'NODE &NO. MIX, *Z LOAD',/) DO 160 I-1,NC KEAD(10,*) NODE,ZLOAD K-NODE*3-2 R(K)-R{K)+ZLOAD WRITE(11,155) NODE,ZLOAD

155 FORMAT(4X,I5,12X,E10.2) 160 CONTINUE

C READ SUPPORT CONDITIONS.

165 WRITE(11,170) 170 FORMAT(///,2IX,'SUPPORT CONDITIONS',//,12X,'UNSUPPORTED-O',/,12X

ft' SUPPORTED" 1', //, 10X, ' NODE NO.',3X,'Z SUPP',5X,'ROTX SUPP',5X, S'ROTY SUPP',//) DO 180 J-l,NODEN READ(10,*) IS(3*(J-1)+1),IS(3*(J-l)+2),IS(3*(J—l)+3) WRITE(11,175) J,IS(3*(J-l)+1),IS(3*(J-l)+2),IS(3*(J-l)+3)

175 FORMAT(11X,13,3(9X,12)) RETURN END

117

SUBROUTINE GRID(NUMEL,X,Y,NODI,NODJ,EMOD,RG,TC,MAT,UI,UJ,XMI,T,SC 6,SL,SE,SP,SQ,R,MBAHD,NEQ) IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION X(100),Y(100),NODI(100),NODJ(100),EMOD(1CIO),RG(IOO), fiTC(lOO),MAT(100),UI(100),UJ(100),XMI(100),T(6,6),SC(6,6),SL(6,6) , 6SE(6,6),8(300,160),SP(6),SQ(6),R(300),ID0F(12)

C THIS SUBROUTINE CALCULATES THE GLOBAL STIFFNESS MATRIX FOR THE GRID C ELEMENTS AND STORES IT IN THE SYSTEM MATRIX. C THE LOAD VECTOR IS MODIFIED TO REFLECT DISTRIBUTED LOADING CONDI-C TIONS.

DO 200 I-1,NUMEL XL-X(N0DJ(I))-X(NODI(X)) YL-Y(NODJ(I))-Y(NODI(I)) EL-SQRT(XL**2+YL**2) COSX-XL/EL SINX-YL/EL

C POPULATE THE TRANSFORMATION MATRIX.

DO 20 J-1,6 DO 10 K-1,6

10 T(J,K)-0. 20 CONTINUE

T(l,l)-1.0 T(4,4)-1.0 T(2,3)-SINX T(5,6)-SINX T(3,2)— SINX T(6,5) — SINX T(2,2)-COSX T(3,3)-C0SX T(5,5)—COSX T(6,6)-C0SX

C POPULATE THE LOCAL STIFFNESS MATRIX.

DO 40 J-1,6 DO 30 K-1,6

30 SL(J,K)-0. 40 CONTINUE

K-MAT(I) SL1-(12.0*EMOD(K)*XMI(K))/(EL**3) SL2-(RG(K)*TC(K))/EL SL3-(4.0*EM0D(K)*XHI(K))/EL SL4-(6.0*EMOD(K)*XMI(K))/(EL**2) SL(1,1 -SL1 SL(4,4 -SL1 SL(1,4 —SL1 SL(4,1 —SL1 SL(2,2 -SL2 SL(5,5 -SL2 SL(5,2 —SL2 SL(2,S —SL2 SL(3,3 -SL3 SL(6,6 -SL3 SL(6,3 -SL3/2.0 SL(3,6 -SL3/2.0 SL(4,3 -SL4 SL(3,4 -SL4 SL(6,4 -SL4 SL(4,6 -SL4 SL(3,1 —SL4 SL(1,3 —SL4 SL(6,1 —SL4 SL(1,6 —SL4

C CALCULATE THE GLOBAL STIFFNESS MATRIX.

DO 70 N-1,6 DO 60 M-1,6 SC(N,H)-0.

60 CONTINUE DO 65 H-1,6 DO 64 ]>1,6

64 SC(N,M) ",SC(N,H)+SL{N,L) *T(L,H) 65 CONTINUE 70 CONTINUE

DO 100 N-1,6 DO 90 M-1,6 SE(N,H)-0.

90 CONTINUE DO 95 M-1,6 DO 94 L-1,6

94 SE(N,M)-SE(N,M)+T(L,N)*SC(L,M) 95 CONTINUE 100 CONTINUE

C STORE THE GLOBAL STIFFNESS MATRIX IN THE SYSTEM MATRIX.

DO 300 J-1,2 NR-3*N0DJ(I)-3 IF(J.EQ.l) NR-3*NODI(I)-3 DO 290 M-1,3 NR-NR+1 IX-(J-1)*3+H DO 260 K-1,2 N9-3*NODJ(I)-3 IF(K.EQ.l) N9-3*NODI(I)-3 DO 270 N-1,3 LL-(K-1)*3+N KK-N9+N+1-NR IF(NK.LE.O) GO TO 270 S(NRfNK)-S{NR,NK)+SE(II,LL)

270 CONTINUE 280 CONTINUE 290 CONTINUE 300 CONTINUE

C MODIFY THE LOAD VECTOR TO REFLECT DISTRIBUTED LOADING CONDITIONS.

DO 140 J-1,6 SP(J)-0.

140 CONTINUE SP(1)—7 »0*UI (I) *EL/20.0 + 3.0*TJJ (I) * EL/20.0 SP(3J«—(UI(I)/20.0 + UJ(I)/30.0)*EL*EL SP(4)—3.0*UI(I)*EL/20.0 + 7.0*UJ(I)*EL/20.0 SP(6)-(UI(I)/30.0 + UJ(I)/20.0)*EL*EL DO 160 K-1,6 SQ(K}»0. DO 150 J-1,6

150 SQ(K)-SQ(X)+T(J,K)*SP(J) 160 CONTINUE

IN-3*NODI(I)-3 JN-3*NODJ(I)-3 R(IN+1)-R(IN+1)+SQ(1) R(IH+2)-R(IN+2)+SQ(2) R(IN+3)-R(IN+3)+SQ(3) R(JN+1)-R(JN+1)+SQ(4) R(JN+2)-R(JN+2)+SQ(5) R(JN+3)-R(JM+3)+SQ(6)

200 CONTINUE RETURN END

SUBROUTINE GEQSOL(NEQ,HBAHD,NODEN, S , C ) IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION S(300,160),C(300)

C THIS SUBROUTINE SOLVES FOR THE NODAL DISPLACEMENTS FROM THE SYSTEM C MATRIX AND THE LOAD VECTOR.

WRITE(11i200) 200 FORMAT(////,SOX,'DISPLACEMENTS',//,5X,'NODE NO.',10X,'Z-DISPL'

ft,'X-ROT',8X,'Y-ROT',/) DO 100 I-1,NEQ IK-X DO 90 J»2,MBAND IK-IK+1 CN-S(I,J)/S(I,1) JK-0 DO BO K-J,HBAND JK-JK+1 S(IK,JK)-S(IK,JK)-CN*S(I,K)

80 CONTINUE S(I,J)-CH C(IK)-C(IK)-CN*C(I)

90 CONTINUE C(I)-C(I)/S(I,1)

100 CONTINUE DO 120 N-2,NEQ I-NEQ-N+1 DO 110 K-2,MBAND J-I+K-l C(I)-C(I)-S(X,K)*C(J)

110 CONTINUE 120 CONTINUE

DO 260 K-l,NODEN XC-1.0E-10 IF(ABS(C(3*(K-1)+1).LE.XC) C{3*(K-l)+l)-0. IF(ABS(C(3*(K-l)+2).LE.XC) C(3*(K-1J+2)-0. IF(ABS(C(3*(K-l)+3).LE.XC) C(3*(K-l)+3)-0. WRITE(11,255) K,C(3*(K-1)+1),C(3*(K-l)+2),C(3*(K-l)+3)

255 FORMAT(7X,12,12X,3(E10.4,3X)) 260 CONTINUE

RETURN END

SUBROUTINE GREAC(NUMEL,X,Y,NODI,NODJ,EMOD,RG,TC,XMI,MAT,UI,UJ,SG, &T,SD,BX,ST,U,P) IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION X(100),Y(100),NODI(100),NODtf(100),EMOD(100),RG(100) ,

HC(IOO),XMI(100),MAT(100),UI(100),UJ(100),SG(6),T(6,6),SD(6),P(2) , ftBX(2,fi),ST(2),U(300)

C THIS SUBROUTINE CALCULATES THE TORQUE AND BENDING MOMENTS OF EACH C GRID ELEMENT ON THE FLOOR STRUCTURE.

WRITE(11,270) 270 FORMAT(///,3OX,'GRID REACTIONS',//,5X,'ELEMENT',12X,'TORQUE',6X,

&'MOMENT(I)',4X,'MOMENT(J)',/)

DO 200 I-1,NUMEL XI>X(N0DJ(I))-X(N0DI(Z)) YI Y(HODJ(IJ)-Y(HODI(I)) EL-SQRT(XL**2+YL**2) cosx-xl/el sihx-yl/el IN-3*NODI(I)-3 JN~3*N0DJ(I)-3 DO 10 K-1,3 MM-IN+K NM-JN+K SGCK)-U(HM) SG(K+3)-U(NM)

10 CONTINUE T(l,l)«1.0 T(4,4)-1.0 T(2,3)-SINX T(5,6)-SINX T(3,2)—SIHX T(6,5)— SIHX T(2,2)-COSX T(3,3)-COSX T(5,5)-COSX T(6,6)-COSX DO 50 J-1,6 SD(J)-0. DO 40 K-1,6

40 SD{J)-SD(J)+T(J,K)*SG(K) 50 CONTINUE

DO 110 H-1,2 XI-1.0 IF(H.EQ.l) XI-0. DO 70 J-1,2 DO 60 K-1,6

60 BX(J,K)-0. 70 CONTINUE

K-MAT(I) BX(1,2)—RG(K) *TC(K)/EL BX(1,5)—BX(1,2) BX(2,1)—(EMOD(K)*XMI(K)*(6*0 - 12.0*XI)/<EJ>*2) BX(2,3)—(EMOD(K)*XHI(K)*(-4•0 + 6.0*XI)/EL BX(2,6)—(EMOD(K)*XHI(K)*(-2.0 + 6.0*XI)/EL BX(2,4)--BX(2,1) DO 90 J-1,2 ST(J)-0. DO 80 K-1,6

80 ST(J)-ST(J)+BX(J,K)*SD(K) 90 CONTINUE

IF(XI.EQ.O) P(1)-ST(2) - (UI(I)/20.0 + UJ(IJ/30.0)*EL*EL IF(XI.EQ.l) P(2)-ST(2) - (UI(I)/30.0 + UJ(I)/20.0)*EL*EL

110 CONTINUE XC-1.0E-5 IF(ABS(ST(1)).LE.XC) ST(l)-0. IF(ABS(P(l)).LE.XC) P(l)-0. IF(ABS(P(2)).LE.XC) P(2}-0. WRITE(11,100) I,ST(1),P(1),P(2)

100 FORM&T(7X,I2,12X,3(E10.4,3X)) 200 CONTINUE

RETURN END

NOTATION

Atw = Tributary Area Width

Cv = Converged Data Point Value

d = Deflection of the Floor Beam

Db = Deflection of the Floor Beam Measured with Respect

the Its Supports

Dg = Deflection of the Floor Beam Supports

DL = Additional Dead Loads (Construction Loads, Parti

tions, etc.)

DLV = Distributed Load Value

Dt .= Total Deflection of the Floor Beams Measured with

Respect to the Column Supports

1% = Percentage Difference Between Initial and Converged

Dead Load Moments

lb = Moment of Inertia of the Floor Beams (in.4)

Ig = Moment of Inertia of the Girders (in.4)

Iv = Initia Data Point Value

Lb = Span Length of Beams

Lg = Span Length of Girders

Sb = Beam Spacing

Ts = Slab Thickness

121

122

w = Uniform Load Applied on the Floor Beam

Wb = Beam Self-Weight

Wc = Unit Weight of Concrete

LIST OF REFERENCES

(1) Marino, Frank, "Ponding of Two-Way Roof Systems", AISC

Engineering Journal, Vol. 3, July, 1966, (pp.93-100).

(2) Ruddy, John L., "Ponding of Concrete Decks", AISC En

gineering Journal, Third Quarter 1986, (pp.107-115).

(3) American Institue of Steel Construction, "AISC Steel

Construction Manual", Eighth Edition, American Insti

tute of Steel Construction Inc., Chicago 111., 1980.

(4) Ewell W.W., Okubo S., and Abrams J.I., "Deflections in

Gridworks and Slabs", Journal of the American Society

of Civil Engineers, Paper No.2520, 1952, (pp.872-913).

(5) Taraporewalla, K.Y., "Design of Grid and Diagrid Sys

tems on the Analogy of Design of Plates", Journal of

the Institution of Structual Engineers, Vol. XXXVI,

No. 4, April, 1958, (pp.121-128)

(6) Martin, I., and Hernandez, J., "Orthogonal Gridworks

Loaded Normally to Their Planes", Journal of the Stru

ctural Division, ASCE, Vol. 106, No. STI, January,

1980, (pp.1-12).

(7) Kardestuncer, H., "Elementary Matrix Analysis Of Stru

ctures", McGraw-Hill, 1974.

123

124

(8) Cook, R.D., "Concepts and Applications of Finite Ele

ment Analysis", John Wiley and Sons, 1981.

(9) Elwin G. Smith Division, Cyclops Corporation, Bowman

Construction Products, "Catalog of Steel Decks for

Floor and Roofs", Pittsburgh, PA, 1976.

(10)Ross, C.T.F., "Computational Methods in Structural and

Continuum Mechanics", John Wiley and Sons, 1982.

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