Bridge Superstructure Design

ISO SAP041709M21 Rev. 0 Berkeley, California, USA

Version 14April 2009

SAP2000/Bridge

Bridge Superstructure Design Concrete Box Girder Bridges and

Precast Concrete Composite Girder Bridges AASHTO LRFD 2002 and 2007

COPYRIGHT

Copyright Computers & Structures, Inc., 1978-2009 All rights reserved. The CSI Logo® is a registered trademark of Computers & Structures, Inc. SAP2000TM and Watch & LearnTM are trademarks of Computers & Structures, Inc. Adobe and Acrobat are registered trademarks of Adobe Systems Incorported. AutoCAD is a registered trademark of Autodesk, Inc. The computer program SAP2000TM and all associated documentation are proprietary and copyrighted products. Worldwide rights of ownership rest with Computers & Structures, Inc. Unlicensed use of these programs or reproduction of documentation in any form, without prior written authorization from Computers & Structures, Inc., is explicitly prohibited.

No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior explicit written permission of the publisher.

Further information and copies of this documentation may be obtained from:

Computers & Structures, Inc. 1995 University Avenue Berkeley, California 94704 USA Phone: (510) 649-2200 FAX: (510) 649-2299 e-mail: [email protected] (for general questions) e-mail: [email protected] (for technical support questions) web: www.csiberkeley.com

DISCLAIMER

CONSIDERABLE TIME, EFFORT AND EXPENSE HAVE GONE INTO THE DEVELOPMENT AND TESTING OF THIS SOFTWARE. HOWEVER, THE USER ACCEPTS AND UNDERSTANDS THAT NO WARRANTY IS EXPRESSED OR IMPLIED BY THE DEVELOPERS OR THE DISTRIBUTORS ON THE ACCURACY OR THE RELIABILITY OF THIS PRODUCT.

THIS PRODUCT IS A PRACTICAL AND POWERFUL TOOL FOR STRUCTURAL DESIGN. HOWEVER, THE USER MUST EXPLICITLY UNDERSTAND THE BASIC ASSUMPTIONS OF THE SOFTWARE MODELING, ANALYSIS, AND DESIGN ALGORITHMS AND COMPENSATE FOR THE ASPECTS THAT ARE NOT ADDRESSED.

THE INFORMATION PRODUCED BY THE SOFTWARE MUST BE CHECKED BY A QUALIFIED AND EXPERIENCED ENGINEER. THE ENGINEER MUST INDEPENDENTLY VERIFY THE RESULTS AND TAKE PROFESSIONAL RESPONSIBILITY FOR THE INFORMATION THAT IS USED.

i

Contents

Bridge Superstructure Design

1 Introduction 1-1

1.1 Organization 1-1

1.2 Recommended Reading 1-2

2 Design Prerequisites

2-1 Load Pattern Types 2-1

2.2 Design Load Combinations 2-3

2.3 Default Load Combinations 2-4

3 Determine Live Load Distribution Factors (LLDF)

3.1 Algorithm for Determining Live Load Distribution Factors (LLDF) 3-1

3.2 Determine Live Load Distribution Factors 3-2

3.3 Apply LLD Factors 3-3 3.3.1 User Specified 3-4

SAP2000/Bridge Superstructure Design Guide

ii

3.3.2 Calculated by SAP2000/Bridge in Accordance with Code 3-4

3.3.3 Read Directly from Girder 3-4 3.3.4 Uniformly Distribution to Girders 3-4

3.4 Generate Virtual Combinations 3-5

3.4.1 Stress Check 3-5 3.4.2 Shear or Moment Check 3-6

3.5 Read Forces/Stresses Directly from Girders 3-6

3.5.1 Stress Check 3-6 3.5.2 Shear or Moment Check 3-6

3.6 LLDF Design Example Using Method 2 3-7

4 Define a Bridge Design Request

4.1 Name and Bridge Object 4-3

4.2 Check Type 4-3

4.3 Station Range 4-4

4.4 Design Range 4-4

4.5 Demand Sets 4-7

4.6 Live Load Distribution Factors 4-7

5 Design Concrete Box Girder Bridges

5.1 Design Stress AASHTO-STD-2002 5-1

5.1.1 Capacity Parameters 5-1 5.1.2 Demand Parameters 5-2 5.1.3 Algorithm 5-2

5.2 Design Stress AASHTO-LFRD-2007 5-2 5.2.1 Capacity Parameters 5-2 5.2.2 Algorithm 5-3

5.2.3 Stress Design Example 5-3

5.3 Design of Flexure AASHTO-LRFD-2007 5-5 5.3.1 Capacity Parameters 5-5

5.3.2 Variables 5-5 5.3.3 Design Process 5-6

Contents

iii

5.3.4 Algorithm 5-7 5.3.5 Flexure Design Example 5-9

5.4 Design Shear AASHTO-LRFD-2007 5-14 5.4.1 Capacity Parameters 5-14 5.4.2 Variables 5-15

5.4.3 Design Process 5-16 5.4.4 Algorithm 5-17 5.4.5 Shear Design Example 5-23

5.5 Design Principal Stress AASHTO-LRFD-2007 5-30 5.5.1 Capacity Parameters 5-30 5.5.2 Demand Parameters 5-30

5.5.3 Algorithm 5-31

6 Design Algorithms for Precast I and U-Girder Bridges

6.1 Design Stress 6-1

6.2 Design Shear 6-2 6.2.1 Variables 6-3 6.2.2 Design Process 6-5

6.2.3 Algorithms 6-5 6.2.4 Shear Design Example 6-8

6.3 Design of Flexural 6-14

6.3.1 Variables 6-14 6.3.2 Design Process 6-15 6.3.3 Algorithms 6-16

6.3.4 Flexure Design Capacity Example 6-18

7 Run a Bridge Design Request

7.1 Description of Example Model 7-2

7.2 Design Preferences 7-3

7.3 Load Combinations 7-3

7.4 Bridge Design Request 7-5

7.5 Start Design/Check of Structure 7-6


iv

8 Design Output

8.1 Display Results as a Plot 8-1

8.2 Display Data Tables 8-2

8.3 Advanced Report Writer 8-4

8.4 Verification 8-6

References

Contents

v

List of Figures

Figure 2-1 Code-Generated Load Combinations for Bridge

Design form 2-5

Figure 2-2 Define Load Combinations form 2-6

Figure 3-1 Lever Rule 3-11

Figure 3-2 General Dimensions 3-14

Figure 4-1 Bridge Design Request - Concrete Box Girder Bridges 4-2

Figure 4-2 Bridge Design Request - Composite I or U Girder Bridges 4-2

Figure 4-3 Bridge Design Request Form 4-4

Figure 5-1 LRFD 2007 Stress Design, AASHTO Box Beam, Type BIII-48 5-4

Figure 5-2 Reinforcement, LRFD 2007 Stress Design

AASHTO Box Beam, Type BIII-48 5-4

Figure 5-3 LRFD 2007 Flexure Design Cross-Section, AASHTO Box Beam, Type BIII-48 5-10

Figure 5-4 Reinforcement, LRFD 2007 Flexure Design Cross-Section, AASHTO Box Beam, Type BIII-48 5-10

Figure 5-5 Shear Design Example, AASHTO Box Beam,

Type BIII-48 5-23

Figure 5-5 Shear Design Example Reinforcement AASHTO Box Beam, Type BIII-48 5-24

Figure 6-1 Shear design example deck section 6-9

Figure 6-2 Shear design example beam section 6-9

Figure 6-3 Flexure capacity design example deck section 6-19

Figure 6-4 Flexure capacity design example beam section 6-19

Figure 7-1 3D view of example concrete box girder bridge model 7-2


vi

Figure 7-2 Elevation view of example bridge 7-2

Figure 7-3 Plan view of the example bridge 7-3

Figure 7-4 Bridge Design Preferences form 7-3

Figure7-5 Code-Generated Load Combinations for Bridge Design form 7-4

Figure 7-6 Define Load Combinations form 7-4

Figure 7- 7 Define Load Combinations form 7-5

Figure 7-8 Perform Bridge Design - Superstructure 7-6

Figure 7-9 Plot of flexure check results 7-6

Figure 8-1 Plot of flexure check results for the example bridge design model 8-2

Figure 8-2 Choose Tables for Display form 8-3

Figure 8-3 Design database table for AASHTO LRFD 2007 flexure check 8-3

Figure 8-4 Choose Tables for Export to Access form 8-4

Figure 8-5 Create Custom Report form 8-5

Figure 8-6 An example of the printed output 8-6

1 - 1

Chapter 1 Introduction

This manual describes using SAP2000/Bridge to complete bridge design in ac-cordance with the AASHTO STD 2002 or AASHTO LRFD 2007 code for con-crete box girder bridges or the AASHTO 2007 LRFD code for bridges when the superstructure includes Precast I or U Beam with a composite slab. Design using SAP2000/Bridge is based on load patterns, load cases, load combinations and design requests. The design output can then be displayed graphically and printed using a customized reporting format.

It should be noted that the design of bridge superstructure is a complex subject and the design codes cover many aspects of this process. SAP2000/Bridge is a tool to help the user with that process. Only the aspects of design documented in this manual are automated by the SAP2000/Bridge design capabilities. The user must check the results produced and address other aspects not covered by SAP2000/Bridge.

1.1 Organization

This manual is designed to help you become productive using SAP2000/Bridge design in accordance with the available codes when modeling concrete box girder bridges and precast concrete girder bridges. Chapter 2 describes loading and load combinations. Chapter 3 describes Live Load Distribution Factors.


1 - 2 Recommended Reading/Practice

Chapter 4 describes defining the design request, which includes the design re-quest name, a bridge object name (i.e., the bridge model), check type (i.e., the type of design), station range (i.e., portion of the bridge to be designed), design parameters (i.e., overwrites for default parameters) and demand sets (i.e., load-ing combinations). Chapters 5 and 6 provide the algorithms used by SAP2000/Bridge in completing concrete box girder and precast concrete bridge design in accordance with the AASHTO code. Chapter 7 explains how to run a Design Request. Chapter 8 describes design output, which can be presented graphically as plots, in data tables, and in reports generated using the Advanced Report Writer feature in SAP2000.

1.2 Recommended Reading/Practice

It is strongly recommended that you read this manual and review any applica-ble “Watch & Learn” Series™ tutorials, which are found on our web site, http://www.csiberkeley.com, before attempting to design a concrete box girder or precast concrete bridge using SAP2000/Bridge. Additional information can be found in the on-line Help facility available from within the software’s main menu.

Load Pattern Types 2 - 1

Chapter 2 Define Loads and Load Combinations

This chapter describes the steps that are necessary to define the loads and load combinations that the user intends to use in the design of the bridge superstruc-ture. The user may define the load combinations manually or have SAP2000/Bridge automatically generate the code generated load combinations. The appropriate design code may be selected using the Design menu > Bridge Design > View/Revise Preference command. Currently, the AASHTO STD 2002 and AASHTO LRFD 2007 design codes are supported by SAP2000/ Bridge.

When the code generated load combinations are going to be used, it is impor-tant for users to define the load pattern type in accordance with the applicable code. The load pattern type can be defined using the Define menu > Load Patterns command. The user options for defining the load pattern types are summarized in the tables 2-1 and 2-2.

2.1 Load Pattern Types

Tables 2-1 and 2-2 show the permanent and transient load pattern types that can be defined in SAP2000/Bridge. The tables also show the AASHTO abbre-viation and the load pattern descriptions. Users may choose any name to iden-tify a load pattern type.


2 - 2 Load Pattern Types

Table 2-1 PERMANENT Load Pattern Types Used in the AASHTOLRFD 2007 Code

SAP2000 Load Pattern Type

AASHTO Reference Description of Load Pattern

CREEP CR Force effects due to creep

DOWNDRAG DD Downdrag force

DEAD DC Dead load of structural components and non-structural attachments

SUPERDEAD DW Superimposed dead load of wearing surfaces and utilities

BRAKING BR Vehicle braking force

HORIZ. EARTH PR EH Horizontal earth pressures

LOCKED IN EL Misc. locked-in force effects resulting from the construction process

EARTH SURCHARGE ES Earth surcharge loads

VERT. EARTH PR EV Vertical earth pressure

PRESTRESS PS Hyperstatic forces from post-tensioning

Table 2-2 TRANSIENT Load Pattern Types Used in the AASHTO LRFD 2007 Design Code

SAP2000 Load Pattern Type

AASHTO Reference Description of Load Pattern

BRAKING BR Vehicle braking force

CENTRIFUGAL CE Vehicular centrifugal loads

VEHICLE COLLISION CT Vehicular collision force

VESSEL COLLISION CV Vessel collision force

QUAKE EQ Earthquake

FRICTION FR Friction affects

ICE IC Ice loads

- IM Vehicle Dynamic Load Allowance

BRIDGE LL LL Vehicular live load

LL SURCHARGE LS Live load surcharge

PEDESTRIAN LL PL Pedestrian live load

SETTLEMENT SE Force effects due settlement

TEMP GRADIENT TG Temperature gradient loads

TEMPERATURE TU Uniform temperature effects

STEAM FLOW WA Water load and steam pressure

WIND–LIVE LOAD WL Wind on live load

WIND WS Wind loads on structure

Chapter 2 - Define Loads and Load Combinations

Design Load Combinations 2 - 3

2.2 Design Load Combinations

The code generated design load combinations make use of the load pattern types noted in Tables 2-1 and 2-2. Table 2-3 shows the load factors and com-binations that are required in accordance with the AASHTO LRFD 2007 code.

Table 2-3 Load Combinations and Load Factors Used in the AASHTO LRFD 2007 Code

Load Combo Limit State

DC DD DW EH EV ES EL PS CR SH

LL IM CE BR PL LS

WA

WS

WL

FR

TU

TG

SE

EQ

IC

CT

CV

Str I P

1.75 1.00 - - 1.00 0.5/1.20 TG SE

- - - -

Str II P

1.35 1.00 - - 1.00 0.5/1.20 TG SE

- - - -

Str III P

- 1.00 1.40 - 1.00 0.5/1.20 TG SE

- - - -

Str IV P

- 1.00 - - 1.00 0.5/1.20 - - - - -

Str V P

1.35 1.00 0.40 1.00 1.00 0.5/1.20 TG SE

- - - -

Ext Ev I P EQ

1.00 - - 1.00 - - 1.00 - - -

Ext Ev II P

0.5 1.00 - - 1.00 - - - 1.00 1.00 1.00

Serv I 1.00 1.00 1.00 0.30 1.00 1.00 0.5/1.20 TG SE

- - - -

Serv II 1.00 1.00 1.00 - - 1.00 0.5/1.20 - - - - -

Serv III 1.00 1.00 1.00 - - 1.00 0.5/1.20 TG SE

- - - -

Serv IV 1.00 1.00 1.00 0.70 - 1.00 0.5/1.20 - 1.00 - - - -

Fatigue- LL, IM & CE Only

- 0.75 - - - - - - - - - - -

Table 2-4 shows the maximum and minimum factors for the permanent loads in accordance with the AASHTO LRFD 2007 code.


2 - 4 Default Load Combinations

Table 2-4 Load Factors for Permanent Loads, P , Used in the AASHTO LRFD 2007 Code

Load Factor Type of Load Maximum Minimum

DC

DC: Strength IV only

1.25

1.50

0.90

0.90

DD: Downdrag 1.40 0.25

DW: Wearing Surfaces and Utilities 1.50 0.65

EH: Horizontal Earth Pressure 1.50 0.90

EL: Locked in Construction Stresses 1.00 1.00

EV: Vertical Earth Pressure 1.35 1.00

ES: Earth Surcharge 1.50 0.75

Two combinations for each permanent load pattern are required because of the maximum and minimum factors. When the default load combinations are used, SAP2000/Bridge automatically creates both load combinations (one for the maximum and one for the minimum factor), and then automatically creates a third combination that represents an enveloped combination of the max/min combos.

2.3 Default Load Combinations

Default design load combinations can be activated using the Define menu > Load Combination command. Users can set the load combination data after the Add Default Design Combos button is selected and the “Bridge” option is chosen. The users may select the desired limit states and load cases using the Code Generated Load Combinations for Bridge Design form shown in Figure 2-1.

Chapter 2 - Define Loads and Load Combinations

Default Load Combinations 2 - 5

Figure 2-1 Code-Generated Load Combinations for Bridge Design form

After the desired limit states and load cases have been selected, SAP2000/Bridge will generate all of the code-required load combinations. These can be viewed using the Display menu > Show Tables command or by using the Show/Modify button on the Define Combinations form, which is shown in Figure 2-2.


2 - 6 Default Load Combinations

Figure 2-2 Define Load Combinations form

The load combinations denoted as Str-I1, Str-I2, and so forth refer to Strength I load combinations. The load case StrIGroup1 is the name given to enveloped load combination of all of the Strength I combinations. Enveloped load combi-nations will allow for some efficiency later when the bridge design requests are defined (see Chapter 4).

Algorithm for Determining Live Load Distribution Factors (LLDF) 3 - 1

Chapter 3 Determine Live Load Distribution Factors

This chapter describes the algorithms used by SAP2000/Bridge to determine the live load distribution factors used to assign live load demands to individual girders. An explanation is given with respect to how the distribution factors are applied in a shear, stress, and moment check in accordance with the AASHTO LRFD 2007 code. The live load distribution factors are applicable only to su-perstructures that have a deck that includes precast I or U girders with compos-ite slabs.

Legend: Girder = beam + tributary area of composite slab Section Cut = all girders present in the cross-section at the cut location

3.1 Algorithm for Determining Live Load Distribution Factors (LLDF)

SAP2000 gives the user a choice of four methods to address distribution of live load to individual girders.

Method 1 – The LLD factors are specified directly by the user.

Method 2 – SAP2000/Bridge calculates the LLD factors by following proce-dures outlined in AASHTO LRFD Section 4.6.2.2.


3 - 2 Determine Live Load Distribution Factors

Method 3 – SAP2000/Bridge reads the calculated live load demands directly from individual girders (available only for Area or Solid models).

Method 4 – SAP2000/Bridge distributes the live load uniformly into all girders.

It is important to note that to obtain relevant results, the definition of a Moving Load case must be adjusted depending on which method is selected.

When the LLD factors are user specified or specified in accordance with the code (Method 1 or 2), only one lane with a MultiLane Scale Factor = 1 should be loaded into a Moving Load cases included in the demand set com-binations.

When SAP2000/Bridge reads the LLD factors directly from individual gird-ers (Method 3, applicable to area and solid models only) or when SAP2000/Bridge applies the LLD factors uniformly (Method 4), multiple traffic lanes with relevant Multilane Scale Factors should be loaded in accor-dance with code requirements.

3.2 Determine Live Load Distribution Factors

At every section cut, the following geometric information is evaluated to de-termine the LLD factors.

span lengththe length of span for which moment or shear is being calcu-lated

the number of girders

girder designationthe first and last girder are designated as exterior girders and the other girders are classified as interior girders

roadway widthmeasured as the distance between curbs/barriers; medians are ignored

overhangconsists of the horizontal distance from the centerline of the exte-rior web of the left exterior beam at deck level to the interior edge of the curb or traffic barrier

Chapter 3 - Determine Live Load Distribution Factors

Apply LLD Factors 3 - 3

the beamsincludes the area, moment of inertia, torsion constant, center of gravity

the thickness of the composite slab t1 and the thickness of concrete slab haunch t2

the tributary area of the composite slabwhich is bounded at the interior girder by the midway distances to neighboring girders and at the exterior girder; includes the entire overhang on one side, and is bounded by the mid-way distances to neighboring girder on the other side

Young’s modulus for both the slab and the beamsangle of skew support.

SAP2000/Bridge then evaluates the longitudinal stiffness parameter, Kg, in ac-cordance with AASHTO LRFD 4.6.2.2 (eq. 4.6.2.2.1-1). The center of gravity of the composite slab measured from the bottom of the beam is calculated as the sum of the beam depth, thickness of the concrete slab haunch t2, and one-half the thickness of the composite slab t1. Spacing of the girders is calculated as the average distance between the centerlines of neighboring girders.

SAP2000/Bridge then verifies that the selected LLD factors are compatible with the type of model: spine, area, or solid. If the LLD factors are read by SAP2000/Bridge directly from the individual girders, the model type must be area or solid. This is the case because with the spine model option, SAP2000/Bridge models the entire cross section as one frame element and there is no way to extract forces on individual girders. All other model types and LLDF method permutations are allowed.

3.3 Apply LLD Factors

The application of live load distribution factors varies, depending on which method has been selected: user specified; in accordance with code; directly from individual girders; or uniformly distributed onto all girders.


3 - 4 Apply LLD Factors

3.3.1 User Specified When this method is selected, SAP2000/Bridge reads the girder designations (i.e., exterior and interior) and assigns live load distribution factors to the indi-vidual girders accordingly.

3.3.2 Calculated by SAP2000/Bridge in Accordance with Code When this method is selected, SAP2000/Bridge considers the data input by the user for truck wheel spacing, minimum distance from wheel to curb/barrier and multiple presence factor for one loaded lane.

Depending on the section type, SAP2000/Bridge validates several section pa-rameters against requirements specified in the code (Tables 4.6.2.2.2b-1, 4.6.2.2.2d-1, 4.6.2.2.3a-1 and 4.6.2.2.3b-1). When any of the parameters values are outside the range required by the code, the section cut is excluded from the Design Request.

At every section cut, SAP2000/Bridge then evaluates the live load distribution factors for moment and shear for exterior and interior girders using formulas specified in the code (Tables 4.6.2.2.2b-1, 4.6.2.2.2d-1, 4.6.2.2.3a-1 and 4.6.2.2.3b-1). After evaluation, the LLDF values are assigned to individual girders based on their designation (exterior, interior). The same value equal to the average of the LLDF calculated for the left and right girders is assigned to both exterior girders. Similarly, all interior girders use the same LLDF equal to the average of the LLDF of all of the individual interior girders.

3.3.3 Forces Read Directly from Girders When this method is selected, SAP2000/Bridge sets the live load distribution factor for all girders to 1.

3.3.4 Uniformly Distributed to Girders When this method is selected, the live load distribution factor is equal to 1/n where n is the number of girders in the section. All girders have identical LLD factors disregarding their designation (exterior, interior) and demand type (shear, moment).


Generate Virtual Combinations 3 - 5

3.4 Generate Virtual Combinations

When the method for determining the live load distribution factors is user-specified, code-specified, or uniformly distributed (Methods 1, 2 or 4), SAP2000/Bridge generates virtual load combination for every valid section cut selected for design. The virtual combinations are used during a stress check and check of the shear and moment to calculate the forces on the girders. After those forces have been calculated, the virtual combination are deleted. The process is repeated for all section cuts selected for design.

Four virtual COMBO cases are generated for each COMBO that the user has specified in the Design Request (see Chapter 4). The program analyzes the de-sign type of each load case present in the user specified COMBO and multi-plies all non-moving load case types by 1/n (where n is the number of girders) and the moving load case type by the section cut values of the LLD factors (ex-terior moment, exterior shear, interior moment and interior shear LLD factors). This ensures that dead load is shared evenly by all girders, while live load is distributed based on the LLD factors.

The program then completes a stress check and a check of the shear and the moment for each section cut selected for design.

3.4.1 Stress Check At the Section Cut being analyzed, the girder stresses at all stress output points are read from SAP2000/Bridge for every virtual COMBO generated. To ensure that live load demands are shared equally irrespective of lane eccentricity by all girders, SAP2000/Bridge uses averaging when calculating the girder stresses. It calculates the stresses on a beam by integrating axial and M3 moment demands on all the beams in the entire section cut and dividing the demands by the number of girders. Similarly, P and M3 forces in the composite slab are inte-grated and stresses are calculated in the individual tributary areas of the slab by dividing the total slab demand by the number of girders.

When stresses are read from SAP2000/Bridge into the SAPBridgeDesign mod-ule, the stresses are multiplied by n (where n is number of girders) to make up for the reduction applied in the Virtual Combinations.


3 - 6 Read Forces/Stresses Directly from Girders

3.4.2 Shear or Moment Check At the Section Cut being analyzed, the entire section cut forces are read from SAP2000/Bridge for every Virtual COMBO generated. The forces are assigned to individual girders based on their designation. (Forces from two virtual Com-binationsone for shear and one for momentgenerated for exterior beam are assigned to both exterior beams, and similarly, Virtual Combinations for inte-rior beams are assigned to interior beams.)

3.5 Read Forces/Stresses Directly from Girders

When the method for determining the live load distribution is based on forces read directly from the girders, the method varies based on which Design Check has been specified in the Design Request (see Chapter 4).

3.5.1 Stress Check At the Section Cut being analyzed, the girder stresses at all stress output points are read from SAP2000/Bridge for every COMBO specified in the Design Re-quest. SAP2000/Bridge calculates the stresses on a beam by integrating axial, M3 and M2 moment demands on the beam at the center of gravity of the beam. Similarly P, M3 and M2 demands in the composite slab are integrated at the center of gravity of the slab tributary area.

3.5.2 Shear or Moment Check At the Section Cut being analyzed, the girder forces are read from SAP2000/ Bridge for every COMBO specified in the Design Request. SAP2000/Bridge calculates the demands on a girder by integrating axial, M3 and M2 moment demands on the girder at the center of gravity of the girder.


LLDF Design Example Using Method 2 3 - 7

3.6 LLDF Design Example Using Method 2

The AASHTO-LRFD Specifications allow the use of advanced methods of analysis to determine the live load distribution factors. However, for typical bridges, the specifications list equations to calculate the distribution factors for different types of bridge superstructures. The types of superstructures covered by these equations are described in Table 4.6.2.2.1-1. From this table, bridges with concrete decks supported on precast concrete I or bulb-tee girders are des-ignated as cross-section “K.” Other tables in 4.6.2.2.2 list the distribution fac-tors for interior and exterior girders including cross-section “K.”

The distribution factor equations are largely based on work conducted in the NCHRP Project 12-26 and have been verified to give accurate results com-pared to 3-dimensional bridge analysis and field measurements. The multiple presence factors are already included in the distribution factor equations except when the tables call for the use of the lever rule. In these cases, the computa-tions need to account for the multiple presence factors. The user is providing those as part of the Design Request definition together with wheel spacing, curb to wheel distance and lane width.

Notice that the distribution factor tables include a column with the heading “range of applicability”. The ranges of applicability listed for each equation are based on the range for each parameter used in the study leading to the devel-opment of the equation. When any of the parameters exceeds the listed value in the “range of applicability” column, SAP2000 reports the incompliance and excludes the section from design.

Article 4.6.2.2.2d of the specifications states: “In beam-slab bridge cross-sections with diaphragms or cross-frames, the distribution factor for the exte-rior beam shall not be taken less than that which would be obtained by assum-ing that the cross-section deflects and rotates as a rigid cross-section.” This provision was added to the specifications because the original study that devel-oped the distribution factor equations did not consider intermediate dia-phragms. Application of this provision requires the presence of a sufficient number of intermediate diaphragms whose stiffness is adequate to force the cross section to act as a rigid section. For prestressed girders, different jurisdic-tions use different types and numbers of intermediate diaphragms. Depending on the number and stiffness of the intermediate diaphragms, the provisions of 4.6.2.2.2d may not be applicable. If the user specifies option “Yes” in the


3 - 8 LLDF Design Example Using Method 2

“Diaphragms Present” option the program follows the procedure outlined in the provision 4.6.2.2.2d.

For this example, one deep reinforced concrete diaphragm is located at the midspan of each span. The stiffness of the diaphragm was deemed sufficient to force the cross-section to act as a rigid section; therefore, the provisions of S4.6.2.2.2d apply.

Required information:

AASHTO Type I-Beam (28/72) Noncomposite beam area, Ag = 1,085 in2 Noncomposite beam moment of inertia, Ig = 733,320 in4 Deck slab thickness, ts = 8 in. Span length, L = 110 ft. Girder spacing, S = 9 ft.-8 in. Modulus of elasticity of the beam, EB = 4,696 ksi Modulus of elasticity of the deck, ED = 3,834 ksi C.G. to top of the basic beam = 35.62 in. C.G. to bottom of the basic beam = 36.38 in.

1. Calculate n, the modular ratio between the beam and the deck.

n = B DE E (4.6.2.2.1-2)

= 4696 3834 = 1.225

2. Calculate eg, the distance between the center of gravity of the noncompo-site beam and the deck. Ignore the thickness of the haunch in determin-ing eg

eg = NAYT + 2st = 35.62 + 8 2 = 39.62 in.

3. Calculate Kg, the longitudinal stiffness parameter.

Kg = 2gn I Ae (4.6.2.2.1-1)

= 2 41.225 733 320 1 085 39.62 2 984 704 in



4. Interior girder. Calculate the moment distribution factor for an interior beam with two or more design lanes loaded using Table S4.6.2.2.2b-1.

DM = 0.10.6 0.2 30.075 9.5 12.0g sS S L K Lt

0.10.6 0.2 3

0.075 9.667 9.5 9.667 110 2 984 704 12 110 8

= 0.796 lane (eq. 1)

5. In accordance with 4.6.2.2.2e, a skew correction factor for moment may be applied for bridge skews greater than 30 degrees. The bridge in this example is skewed 20 degrees, and therefore, no skew correction factor for moment is allowed.

Calculate the moment distribution factor for an interior beam with one design lane loaded using Table 4.6.2.2.2b-1.

DM = 0.10.4 0.3 30.06 14 12.0g sS S L K Lt

= 0.10.4 0.3 3

0.06 9.667 14 9.667 110 2984704 12 100 8

= 0.542 lane (eq. 2)

Notice that the distribution factor calculated above for a single lane loaded already includes the 1.2 multiple presence factor for a single lane, therefore, this value may be used for the service and strength limit states. However, multiple presence factors should not be used for the fatigue limit state. Therefore, the multiple presence factor of 1.2 for the single lane is required to be removed from the value calculated above to deter-mine the factor used for the fatigue limit state.

6. Skew correction factor for shear.

In accordance with 4.6.2.2.3c, a skew correction factor for support shear at the obtuse corner must be applied to the distribution factor of all skewed bridges. The value of the correction factor is calculated using Table 4.6.2.2.3c-1.



SC = 0.331.0 0.20 12.0 tans gLt K

= 0.331.0 0.20 12.0 110 8 2 984 704 tan20

= 1.047

7. Calculate the shear distribution factor for an interior beam with two or more design lanes loaded using Table S4.6.2.2.3a-1.

DV = 20.2 12 35S S

= 20.2 9.667 12 9.667 35

= 0.929 lane

Apply the skew correction factor:

DV = 1.047 0.929 0.973 lane (eq. 4)

8. Calculate the shear distribution factor for an interior beam with one de-sign lane loaded using Table S4.6.2.2.3a-1.

DV = 0.36 25.0S

= 0.36 9.667 25.0

= 0.747 lane

Apply the skew correction factor:

DV = 1.047 0.747

= 0.782 lane (eq. 5)

9. From (1) and (2), the service and strength limit state moment distribution factor for the interior girder is equal to the larger of 0.796 and 0.542 lane. Therefore, the moment distribution factor is 0.796 lane.

From (4) and (5), the service and strength limit state shear distribution factor for the interior girder is equal to the larger of 0.973 and 0.782 lane. Therefore, the shear distribution factor is 0.973 lane.



10. Exterior girder

Figure 3-1 Lever Rule

11. Calculate the moment distribution factor for an exterior beam with two or more design lanes using Table 4.6.2.2.2d-1.

DM = eDVinterior

e = 0.77 9.1de

where de is the distance from the centerline of the exterior girder to the inside face of the curb or barrier.

e = 0.77 + 1.83/9.1 = 0.97

DM = 0.97(0.796) = 0.772 lane (eq. (7)

12. Calculate the moment distribution factor for an exterior beam with one design lane using the lever rule as per Table 4.6.2.2.2d-1.

DM = 3.5 6 3.5 9.667 1.344 wheels 2



= 0.672 lane (eq. 8)

Notice that this value does not include the multiple presence factor, therefore, it is adequate for use with the fatigue limit state. For service and strength limit states, the multiple presence factor for a single lane loaded needs to be included.

DM = 0.672 1.2

= 0.806 lane (eq. 9) (Strength and Service)

13. Calculate the shear distribution factor for an exterior beam with two or more design lanes loaded using Table 4.6.2.2.3b-1.

DV = eDVinterior

where:

e = 0.6 10de

= 0.6 1.83 10

= 0.783

DV = 0.783 0.973

= 0.762 lane (eq. 10)

14. Calculate the shear distribution factor for an exterior beam with one de-sign lane loaded using the lever rule as per Table 4.6.2.2.3b-1. This value will be the same as the moment distribution factor with the skew correc-tion factor applied.

DV = 1.047 0.806

= 0.845 lane (eq. 12) (Strength and Service)

Notice that 4.6.2.2.2d includes additional requirements for the calcula-tion of the distribution factors for exterior girders when the girders are connected with relatively stiff cross-frames that force the cross-section to act as a rigid section. As indicated in the introduction, these provisions are applied to this example; the calculations are shown below.

15. Additional check for rigidly connected girders (4.6.2.2.2d)



The multiple presence factor, m, is applied to the reaction of the exterior beam (Table 3.6.1.1.2-1)

m1 = 1.20

m2 = 1.00

m3 = 0.85

R = 2L b extN N X e x (4.6.2.2.2d-1)

where:

R = reaction on exterior beam in terms of lanes

NL = number of loaded lanes under consideration

e = eccentricity of a design truck or a design land load from the center of gravity of the pattern of girders (ft.)

x = horizontal distance from the center of gravity of the pat-tern of girders to each girder (ft.)

Xext = horizontal distance from the center of gravity of the pat-tern to the exterior girder (ft.) See Figure 1 for dimen-sions.

One lane loaded (only the leftmost lane applied):

R = 1 6 24.167 21 2 24.1672 14.52 4.8332

= 0.1667 + 0.310

= 0.477 (Fatigue)

Add the multiple presence factor of 1.2 for a single lane:

R = 1.2 0.477

= 0.572 (Strength)

Two lanes loaded:

R = 2 6 24.167 21 9 2 24.1672 14.52 4.8332



= 0.333 + 0.443

= 0.776

Add the multiple presence factor of 1.0 for two lanes loaded:

R = 1.0 0.776

= 0.776 (Strength)

Three lanes loaded:

R = 3 6 24.167 21 9 3 2 24.1672 14.52 4.8332

= 0.5 + 0.399

= 0.899

Add the multiple presence factor of 0.85 for three or more lanes loaded:

R = 0.85 0.899

= 0.764 (Strength)

These values do not control over the distribution factors summarized in Design Step 16.

Figure 3-2 General Dimensions



16. From (7) and (9), the service and strength limit state moment distribution factor for the exterior girder is equal to the larger of 0.772 and 0.806 lane. Therefore, the moment distribution factor is 0.806 lane.

From (10) and (12), the service and strength limit state shear distribution factor for the exterior girder is equal to the larger of 0.762 and 0.845 lane. Therefore, the shear distribution factor is 0.845 lane.

Table 3.1 Summary of Service and Strength Limit State Distribution Factors

Load Case

Moment interior beams

Moment exterior beams

Shear interior beams

Shear exterior beams

Multiple lanes loaded 0.796 0.772 0.973 0.762 Distribution factors from Tables in 4.6.2.2.2

Single lane loaded 0.542 0.806 0.782 0.845

Multiple lanes loaded NA 0.776 NA 0.776 Additional check for rigidly connected girders Single lane loaded NA 0.572 NA 0.572

Design Value 0.796 0.806 0.973 0.845

Value reported by SAP2000 0.796 0.807 0.973 0.845

Name and Bridge Object 4 - 1

Chapter 4 Define a Bridge Design Requests

This chapter describes the Bridge Design Request, which is defined using the Define menu > Bridge Design > Define Design Request command.

Each Bridge Design Request is unique and specifies which bridge object is to be designed, the type of check to be performed (e.g., concrete box stress, pre-cast composite stress, and so on), the station range (i.e., the particular zone or portion of the bridge that is to be designed), the design parameters (i.e., pa-rameters that may be used to overwrite the default values automatically set by the program) and demand sets (i.e., the load combination[s] to be considered). Multiple Bridge Design Requests may be defined for the same bridge object.

Before defining a design request, the applicable code should be specified using the Design menu > Bridge Design > View/Revise Preferences command. Currently, the AASHTO STD 2002 or AASHTO LRFD 2007 code is available for a design of a concrete box girder or the AASHTO 2007 LRFD code is available for the design of a Precast I or U Beam with Composite Slab super-structure.

Figure 4-1 shows the Bridge Design Request form when the bridge object is for a concrete box girder bridge, and the check type is concrete box stress. Figure 4-2 shows the Bridge Design Request form when the bridge object is for a Composite I or U girder bridge and the check type is precast composite stress.


4 - 2 Name and Bridge Object

Figure 4-1 Bridge Design Request - Concrete Box Girder Bridges

Figure 4-2 Bridge Design Request - Composite I or U Girder Bridges

Chapter 4 - Define a Bridge Design Requests

Name and Bridge Object 4 - 3

4.1 Name and Bridge Object

Each Bridge Design Request must have unique name. Any name can be used.

If multiple Bridge Objects are used to define a bridge model, select the bridge object to be designed for the Design Request. If a bridge model contains only a single bridge object, the name of that bridge object will be the only item avail-able from the Bridge Object drop-down list.

4.2 Check Type

The Check Type refers to the type of design to be performed and the available options depend on the type of bridge deck being modeled.

For a Concrete Box Girder bridge, SAP2000/Bridge provides the following check type options:

AASHTO STD 2002

Concrete Box Stress

AASHTO LRFD 2007

Concrete Box Stress

Concrete Box Flexure

Concrete Box Shear and Torsion

Concrete Box Principal

For bridge models with precast I or U Beams with Composite Slabs, SAP2000/Bridge provides three check type options, as followings:

AASHTO LRFD 2007

Precast Comp Stress

Precast Comp Shear

Precast Comp Flexure


4 - 4 Station Range

The bold type denotes the name that appears in the check type drop-down list. A detailed description of the design algorithm can be found in Chapter 5 for Concrete Box Girder Bridges and in Chapter 6 for Precast I or U Beam with Composite Slabs.

4.3 Station Range

The station range refers to the particular zone or portion of the bridge that is to be designed. The user may choose the entire length of the bridge, or specify specific zones using station ranges. Multiple zones (i.e., station ranges) may be specified as part of a single design request.

When defining a station range, the user specifies the Location Type, which de-termines if the superstructure forces are to be considered before or at a station point. The user may choose the location type as before the point, after the point or both.

4.4 Design Parameters

Design parameters are overwrites that can be used to change the default values set automatically by the program. The parameters are specific to each code, deck type, and check type. Figure 4-3 shows the Superstructure Design Pa-rameters form.

Figure 4-3 Bridge Design Request Form


Design Parameters 4 - 5

Table 4-1 shows the parameters for concrete box girder. Table 4-2 shows the parameters applicable when the superstructure has a deck that includes precast I or U girders with composite slabs.

Table 4-1 Design Request Parameters for Concrete Box Girders AASHTO STD 2002

Concrete Box Stress Resistance Factor - multiplies both compression and tension

stress limits

Multiplier on cf to calculate the compression stress limit

Multiplier on sqrt( cf ) to calculate the tension stress limit, given

in the units specified

The tension limit factor may be specified using either MPa or ksi units for cf and the resulting tension limit

AASHTO LRFD 2007 Concrete Box Stress

Concrete Box Stress, PhiC, - Resistance Factor that multiplies both compression and tension stress limits

Concrete Box Stress Factor Compression Limit - Multiplier on cf

to calculate the compression stress limit

Concrete Box Stress Factor Tension Limit Units - Multiplier on sqrt( cf ) to calculate the tension stress limit, given in the units

specified

Concrete Box Stress Factor Tension Limit - The tension limit fac-tor may be specified using either MPa or ksi units for cf and the

resulting tension limit

Concrete Box Shear Concrete Box Shear, PhiC, - Resistance Factor that multiplies both compression and tension stress limits

Concrete Box Shear, PhiC, Lightweight Resistance Factor that multiplies nominal shear resistance to obtain factored resistance for light-weight concrete

Include Resal (Hunching-girder) shear effects – Yes or No. Speci-fies whether the component of inclined flexural compression or tension, in the direction of the applied shear, in variable depth members shall or shall not be considered when determining the design factored shear force per Article 5.8.6.2.

Concrete Box Shear Rebar Material - A previously defined rebar material label that will be used to determine the area of shear rebar required

Longitudinal Torsional Rebar Material - A previously defined rebar material that will be used to determine the area of longi-tudinal torsional rebar required


4 - 6 Design Parameters

Table 4-1 Design Request Parameters for Concrete Box Girders Concrete Box Flexure

Concrete Box Flexure, PhiC, - Resistance Factor that multiplies both compression and tension stress limits

Concrete Box Principal See the Box Stress design parameter specifications

Table 4-2 Design Request Parameters for Precast I or U Beams AASHTO STD 2002

Precast Comp Stress Precast Comp Stress, PhiC, - Resistance Factor that multiplies both compression and tension stress limits

Precast Comp Stress Factor Compression Limit - Multiplier on cf

to calculate the compression stress limit

Precast Comp Stress Factor Tension Limit Units - Multiplier on sqrt( cf ) to calculate the tension stress limit, given in the units

specified

Precast Comp Stress Factor Tension Limit - The tension limit fac-tor may be specified using either MPa or ksi units for cf and the

resulting tension limit

Precast Comp Shear Concrete Box Shear, PhiC, - Resistance Factor that multiplies both compression and tension stress limits

Concrete Box Shear, PhiC, Lightweight Resistance Factor that mul-tiplies nominal shear resistance to obtain factored resistance for light-weight concrete

Negative limit on strain in nonprestressed longitudinal rein-forcement - per section 5.8.3.4.2; Default Value = -0.4x10-3, Typi-cal value(s): 0 to -0.4x10-3

Positive limit on strain in nonprestressed longitudinal reinforce-ment - per section 5.8.3.4.2; Default Value = 6.0x10-3, Typical value(s): 6.0x10-3

PhiC for Nu - Resistance Factor used in equation 5.8.3.5-1; Default Value = 1.0, Typical value(s): 0.75 to 1.0

Phif for Mu - Resistance Factor used in equation 5.8.3.5-1; Default Value = 0.9, Typical value(s): 0.9 to 1.0

Specifies what method for shear design will be used - either Modified Compression Field Theory (MCFT) per 5.8.3.4.2 or Vci Vcw method per 5.8.3.4.3 Currently only MCFT option is available.

A previously defined rebar material label that will be used to de-termine the required area of transverse rebar in the girder

A previously defined rebar material that will be used to determine the required area of longitudinal rebar in the girder

Precast Comp Flexure Precast Comp Flexure, PhiC, - Resistance Factor that multiplies both compression and tension stress limits


Demand Sets 4 - 7

4.5 Demand Sets

A demand set name is required for each load combination that is to be consid-ered in a design request. The load combinations may be selected from a list of user defined or default load combinations that are program determined (See Chapter 2).

4.6 Live Load Distribution Factors

When the superstructure has a deck that includes precast I or U girders with composite slabs, Live Load Distribution Factors can be specified. LLD factors are described in Chapter 3.

Design Stress AASHTO-STD-2002 5 - 1

Chapter 5 Design Concrete Box Girder Bridges

This chapter describes the algorithms applied in accordance with the AASHTO STD-2002, LRFD 07 code for design and stress check of the superstructure of a concrete box type bridge deck section.

5.1 Design Stress AASHTO-STD-2002

5.1.1 Capacity Parameters PhiC – Resistance Factor; Default Value = 1.0, Typical value(s): 1.0 The compression and tension limits are multiplied by the C factor.

FactorCompLim – cf multiplier; Default Value = 0.4; Typical value(s): 0.4 to

0.6. The cf is multiplied by the FactorCompLim to obtain the compression limit.

FactorTensLim – cf multiplier; Default Value = 0.19 (ksi) 0.5(MPa);

Typical value(s): 0 to 0.24 (ksi) 0 to 0.63 (MPa)

The cf is multiplied by the FactorTensLim to obtain the tension limit.


5 - 2 Design Stress AASHTO-LRFD-2007

5.1.2 Demand Parameters FactorCompLim – percentage of the basic unit stress for compression service design; Default value = 1.0; Typical values 1.0 to 1.5 The demand compressive stresses are divided by the FactorCompLim factor. This way the controlling stress can be selected and compared against one com-pression limit.

FactorTensLim – percentage of the basic unit stress for tension service design; Default value = 1.0; Typical values 1.0 to 1.5 The demand tensile stresses are divided by the FactorCompLim factor. This way the controlling stress can be selected and compared against one tension limit.

5.1.3 Algorithm The stresses are evaluated at three points at the top fiber and three points at the bottom fiber. The location of the points are extreme left, Bridge Layout Line and extreme right. The stresses assume linear distribution and take into account axial (P) and both bending moments (M2 and M3).

The stresses are evaluated for each demand set. If the demand set contains live load, the program positions the load to capture extreme stress at each of the evaluation points.

The stresses are divided by the appropriate demand parameter. Then extremes are found for each point and the controlling demand set name is recorded.

The stress limits are evaluated by applying the Capacity Parameters (see sec-tion 5.1.1).

5.2 Design Stress AASHTO-LRFD-2007

5.2.1 Capacity Parameters PhiC – Resistance Factor; Default Value = 1.0, Typical value(s): 1.0 The compression and tension limits are multiplied by the C factor

Chapter 5 - Design Concrete Box Girder Bridges

Design Stress AASHTO-LRFD-2007 5 - 3


0.6. The cf is multiplied by the FactorCompLim to obtain compression limit.

FactorTensLim – cf multiplier; Default Value = 0.19 (ksi) 0.5(MPa); Typi-

cal value(s): 0 to 0.24 (ksi) 0 to 0.63 (MPa)

The cf is multiplied by the FactorTensLim to obtain tension limit

5.2.2 Algorithm The stresses are evaluated at three points at the top fiber and three points at the bottom fiber. The location of the points are extreme left, Bridge Layout Line and extreme right. The stresses assume linear distribution and take into account axial (P) and both bending moments (M2 and M3).

The stresses are evaluated for each demand set. If the demand set contains live load, the program positions the load to capture extreme stress at each of the evaluation points.

Extremes are found for each point and the controlling demand set name is re-corded.

The stress limits are evaluated by applying the Capacity Parameters (see 5.2.1).

5.2.3 Stress Design Example Cross Section: AASHTO Box Beam, Type BIII-48 as shown in Figure 5-1

Concrete unit weight, wc = 0.150 kcf Concrete strength at 28 days, cf = 5.0 ksi

Design span = 95.0 ft Prestressing strands: ½ in. dia., seven wire, low relaxation Area of one strand = 0.153 in2 Ultimate strength fpu = 270.0 ksi Yield strength fpy = 0.9 ksi fpu = 243 ksi Modulus of elasticity, Ep = 28 500 ksi


5 - 4 Design Stress AASHTO-LRFD-2007

Figure 5-1 LRFD 2007 Stress Design, AASHTO Box Beam, Type BIII-48

Figure 5-2 Reinforcement, LRFD 2007 Stress Design

AASHTO Box Beam, Type BIII-48


Design of Flexure AASHTO-LRFD-2007 5 - 5

Reinforcing bars: yield strength, fy = 60.0 ksi

Section Properties A = area of cross-section of beam = 826 in2 h = overall depth of precast beam = 39 in I = moment of inertia about centroid of the beam = 170812 in4 yb,yt = distance from centroid to the extreme

bottom (top) fiber of the beam = 19.5 in

Demand forces from Dead and PT (COMB1) at station 570: P = 856.51 kip M3 = 897.599 kip-in

Top fiber stress =

3top top

856 51 897 59919 5 0 9344 ksi

826 170812

P M . .y . .

A I

Bottom fiber stress =

3top bot

856 51 897 59919 5 1 139 ksi

826 170812

P M . .y . .

A I

Stresses reported by SAP2000: top fiber stress envelope = 0.9345 ksi

bottom fiber stress envelope = 1.13945 ksi

5.3 Design of Flexure AASHTO-LRFD-2007

5.3.1 Capacity Parameters PhiC – Resistance Factor; Default Value = 1.0, Typical value(s): 1.0 The nominal flexural capacity is multiplied by the resistance factor to obtain factored resistance

5.3.2 Variables Resistance factor for flexure


5 - 6 Design of Flexure AASHTO-LRFD-2007

Mn Nominal flexural resistance

Mr Factored flexural resistance

tslabeq Equivalent thickness of slab

bslab Effective flange width = horizontal width of slab, measured from out to out

bwebeq Equivalent thickness of all webs in section

Aslab Area of slab

APT Area of PT in tension zone

yPT Distance from extreme compression fiber to the centroid of the prestressing tendons

fpu Specified tensile strength of prestressing steel (area weighted average of all tendons in tensile zone)

fpy Yield tensile strength of prestressing steel (area weighted average if all tendons in tensile zone)

fps Average stress in prestressing steel (eq. 5.7.3.1.1-1)

k PT material constant (eq. 5.7.3.1.1-2)

1 Stress block factor is (eq. 5.7.2.2)

5.3.3 Design Process The derivation of the moment resistance of the section is based on approximate stress distribution specified in Article 5.7.2.2. The natural relationship between concrete stress and strain is considered satisfied by an equivalent rectangular concrete compressive stress block of 0.85 cf over a zone bounded by the

edges of the cross-section and a straight line located parallel to the neutral axis at the distance a = β1c from the extreme compression fiber. The distance c is measured perpendicular to the neutral axis. The factor β1 is taken as 0.85 for concrete strengths not exceeding 4.0 ksi. For concrete strengths exceeding 4.0



ksi, β1 is reduced at a rate of 0.05 for each 1.0 ksi of strength in excess of 4.0 ksi, except that β1 is not to be taken to be less than 0.65.

The flexural resistance is determined in accordance with Paragraph 5.7.3.2. The resistance is evaluated only for bending about horizontal axis 3. Separate capacity is calculated for positive and negative moment. The capacity is based only on bonded tendons defined in the Bridge Object. Mild steel reinforcement is not considered. If there is no prestressing in the tension zone of the section, the capacity is reported as zero. It is assumed that all defined tendons in a sec-tion, stressed or not, have fpe (effective stress after loses) larger than 0.5 fpu

(specified tensile strength). If a certain tendon should not be considered for the flexural capacity calculation, its area must be set to zero.

The section properties are calculated for the section before skew, grade, and superelevation are applied. This is consistent with the demands being reported in section local axis. It is assumed that the effective width of the flange (slab) in compression is equal to the width of the slab.

5.3.4 Algorithm At each section:

All section properties and demands are converted from SAP2000 model units to N, mm.

The equivalent slab thickness is evaluated based on slab area and slab width assuming rectangular shape.

slabslabeq

slab

At

b

The equivalent web thickness is evaluated as summation of all web horizon-tal thicknesses

web

webeq web1

n

b b

1 stress block factor is evaluated in accordance with 5.7.2.2 based on sec-

tion cf



if cf > 28 MPa then 128

max 0 85 0 05 0 657

cf( . . ; . )

else 1 0 85.

The tendon location, area and material are read. Only bonded tendons are processed; unbonded tendons are ignored.

Tendons are split into two groups depending on what sign of moment they resistnegative or positive. A tendon is considered to resist a positive mo-ment when it is located outside of the top fiber compression stress block and is considered to resist a negative moment when it is located outside of the bottom fiber compression stress block. The compression stress block extends over a zone bounded by the edges of the cross-section and a straight line lo-cated parallel to the neutral axis at the distance a = β1c from the extreme compression fiber. The distance c is measured perpendicular to the neutral axis.

For each tendon group, an area weighted average of the following values is determined:

- sum of tendon areas APT

- center of gravity of tendons yPT

- specified tensile strength of prestressing steel fpu

- constant k (eq. 5.7.3.1.1-2)

2(1.04 )py

pu

fk

f

The distance c between neutral axis and the compressive face is evaluated in accordance with (eq. 5.7.3.1.1-4).

1 slab0.85

PT pu

puc PT

pt

A fc

ff b kA

y

The distance c is compared to the equivalent slab thickness to determine if the section is a T-section or rectangular section.



If 1 slabeq ,c t the section is a T-section.

If the section is a T-section, the distance c is recalculated in accordance with (eq. 5.7.3.1.1-3).

slab webeq slabeq

1 webeq

0.85 ( )

0.85

PT pu c

puc PT

pt

A f f b b tc

ff b kA

y

Average stress in prestressing steel fps is calculated in accordance with (eq. 5.7.3.1.1-1).

(1 )ps pupt

cf f k

y

Nominal flexural resistance Mn is calculated in accordance with (eq. 5.7.3.2.2-1)

If the section is a T-section,

slabeq1 1slab webeq slabeq0.85

2 2 2n PT ps PT c

tc cM A f y f b b t

else

1

2n PT ps PT

cM A f y

Factored flexural resistance is obtained by multiplying Mn by .

Mr = Mn

Extreme moment M3 demands are found from the specified demand sets and the controlling demand set name is recorded.

5.3.5 Flexure Design Example Cross Section: AASHTO Box Beam, Type BIII-48 as shown in Figure 5-3.



Figure 5-3 LRFD 2007 Flexure Design Cross-Section, AASHTO Box Beam, Type BIII-48

Figure 5-4 Reinforcement, LRFD 2007 Flexure Design

Cross-Section, AASHTO Box Beam, Type BIII-48



Concrete unit weight, wc = 0.150 kcf Concrete strength at 28 days, cf = 5.0 ksi (~34.473 MPa)

Design span = 95.0 ft Prestressing strands: ½ in. dia., seven wire, low relaxation Area of one strand = 0.153 in2 Ultimate strength fpu = 270.0 ksi Yield strength fpy = 0.9 ksi fpu = 243 ksi Modulus of elasticity, Ep = 28 500 ksi

Reinforcing bars: yield strength, fy = 60.0 ksi

Section Properties A = area of cross-section of beam = 826 in2 h = overall depth of precast beam = 39 in I = moment of inertia about centroid of the beam = 170812 in4 yb, yt = distance from centroid to the extreme

bottom (top) fiber of the beam = 19.5 in

Demand forces from Dead and PT (COMB1) at station 570: P = 856.51 kip M3 = 897.599 kip-in

The equivalent slab thickness is evaluated based on slab area and slab width assuming rectangular shape.

slabslabeq

slab

48 5.55.5in

48

At

b

Value reported by SAP2000/Bridge = 5.5 in

The equivalent web thickness is evaluated as summation of all web horizon-tal thicknesses

web

webeq web1

5 5 10 inn

b b




Tendons are split into two groups depending on which sign of moment they resistnegative or positive. A tendon is considered to resist a positive mo-ment when it is located outside of the top fiber compression stress block and is considered to resist a negative moment when it is located outside of the bottom fiber compression stress block. The compression stress block extends over a zone bounded by the edges of the cross-section and a straight line lo-cated parallel to the neutral axis at the distance a = β1c from the extreme compression fiber. The distance c is measured perpendicular to the neutral axis.


- sum of tendon areas 2bottom 0.153 6 23 4.437inPTA

Value reported by SAP2000/Bridge = 4.437 in2

- distance from center of gravity of tendons to extreme compression fiber

bottom23 2 6 4

39 36.586 in23 6

PTy

Value reported by SAP2000/Bridge = 19.5 + 17.0862 = 36.586 in

- specified tensile strength of prestressing steel 270 kippuf

Value reported by SAP2000/Bridge = 270 kip

- constant k (eq. 5.7.3.1.1-2)

243

2 1.04 2 1.04 0.28270

py

pu

fk

f

Value reported by SAP2000/Bridge = 0.28

1 stress block factor is evaluated in accordance with 5.7.2.2 based on sec-

tion cf

If cf > 28 MPa, then

128

max 0.85 0.05;0.657

34.473 28max 0.85 0.05;0.65 0.80376

7

cf

Value calculated by SAP2000/Bridge = 0.8037 (not reported)



The distance c between neutral axis and the compressive face is evaluated in accordance with (eq. 5.7.3.1.1-4).

1 slab

27036.586

0.85

4.437 2706.91in

0.85 5 0.8037 48 0.28 4.437

PT pu

puc PT

pt

A fc

ff b kA

y

Value calculated by SAP2000/Bridge = 6.919 in (not reported)

The distance c is compared to the equivalent slab thickness to determine if the section is a T-section or a rectangular section.

If 1 slabeq 6.91 0.80376 5.56 in 5.5inc t , the section is a

T-section. Value reported by SAP2000/Bridge, section = T-section

If the section is a T-section, the distance c is recalculated in accordance with (eq. 5.7.3.1.1-3).

slab webeq slabeq

1 webeq

27036.586

0.85 ( )

0.85

4.437 270 0.85 5(48 10)5.57.149in

0.85 5 0.8037 10 0.28 4.437

PT pu c

puc PT

pt

A f f b b tc

ff b kA

y


Average stress in prestressing steel fps is calculated in accordance with (eq. 5.7.3.1.1-1).

7.149

1 270 1 0.28 255.23ksi36.586

ps pupt

cf f k

y

Value reported by SAP2000/Bridge = 255.228 ksi

Nominal flexural resistance Mn is calculated in accordance with (5.7.3.2.2-1)

If the section is a T-section, then


5 - 14 Design Shear AASHTO-LRFD-2007


2 2 2

7.149 0.803764.437 255.228 36.586

2

7.149 0.80376 5.50.85 5 48 10 5.5

2 238287.42 kip-in

n PT ps PT c

tc cM A f y f b b t

Value calculated by SAP2000/Bridge = 38287.721 kip-in (not reported)


1.0 38287.42 38287.42 kip-inr nM M

Value reported by SAP2000/Bridge = 38287.721 kip-in

5.4 Design Shear AASHTO-LRFD-2007

5.4.1 Capacity Parameters PhiC – Resistance Factor; Default Value = 0.9, Typical value(s): 0.7 to 0.9 The nominal shear capacity of normal weight concrete sections is multiplied by the resistance factor to obtain factored resistance.

PhiC (Lightweight) – Resistance Factor for lightweight concrete; Default Value = 0.7, Typical value(s): 0.7 to 0.9 The nominal shear capacity of light-weight concrete sections is multiplied by the resistance factor to obtain factored resistance

Include Resal (haunchedgirder) Shear Effect – Typical value: Yes Specifies whether the component of inclined flexural compression or tension, in the direction of the applied shear, in variable depth members shall or shall not be considered when determining the design factored shear force.

Shear Rebar Material A previously defined rebar material label that will be used to determine the area of shear rebar required.


Design Shear AASHTO-LRFD-2007 5 - 15

Longitudinal Torsional Rebar Material A previously defined rebar material label that will be used to determine the area of longitudinal torsional rebar required.

5.4.2 Variables Resistance factor for shear

uuuu TMVP ,, 32, Factored demand forces and moments per section

web Web angle of inclination from the vertical

A Gross area of section

AO Area enclosed by shear flow path, including area of holes if any

CGtop, CGbot Distance from c.g. of section to top and bottom fiber

ph Perimeter of the polygon defined by the centroids of the longitudinal chords of the space truss resisting torsion

b Minimum horizontal gross width of web (not adjusted for ducts)

t Minimum normal gross width of web (not adjusted for ducts) = webcosb

bv Minimum effective horizontal width of web adjusted for presence ducts

be Minimum effective normal width of shear flow path adjusted to ac-count for presence of ducts

tv Minimum effective normal width of web = webcosvb

web Distribution factor for web

h Vertical height of section

dv Effective vertical height of section = max(0.8h, distance from ex-treme compression fiber to CG of tensile PT)



Normal or light weight concrete factor

Avsweb Area of shear reinforcement in web per unit length

Avtweb Area of transverse torsion reinforcement in web per unit length

Al Area of longitudinal torsion reinforcement

5.4.3 Design Process The shear resistance is determined in accordance with Paragraph 5.8.6 (Shear and Torsion for Segmental Box Girder Bridges). The procedure is not applica-ble to discontinuity regions and applies only to sections where it is reasonable to assume that plane sections remain plane after loading. The user should select for design only those sections that comply with the preceding assumptions by defining appropriate station ranges in the Bridge Design Request.

If the option to consider resal effects is activated, the component of inclined flexural compression or tension, in the direction of the demand shear, in vari-able depth members is considered when determining the design section shear force (paragraph 5.8.6.1).

The section design shear force is distributed into individual webs assuming that the vertical shear that is carried by a web decreases with increased inclination of the web from vertical. Section torsion moments are assigned to external webs and slabs.

The rebar area and ratio are calculated using measurements normal to the web. Thus, vertical shear forces are divided by cos(alpha_web). The rebar area cal-culated is the actual, normal cross-section of the bars. The rebar ratio is calcu-lated using the normal width of the web, tweb = bweb cos(alpha_web).

The effects of ducts in members are considered in accordance with paragraph 5.8.6.1. In determining the web or flange effective thickness, be, one-half of the diameters of ducts is subtracted. All defined tendons in a section, stressed or not, are assumed to be grouted. Each tendon at a section is checked for pres-ence in the web or flange and the minimum controlling effective web and flange thicknesses are evaluated.



The tendon duct is considered as having effect on the web or flange effective thickness even if only part of the duct is within the element boundaries. In such cases, the entire one-half of the tendon duct diameter is subtracted from the ele-ment thickness

If several tendon ducts overlap in one flange or web (when projected on the horizontal axis for flange, or when projected on vertical axis for the web), the diameters of ducts are added for the sake of evaluation of the effective thick-ness. In the web, the effective web thickness is calculated at the top and bottom of each duct; in the flange, the effective thickness is evaluated at the left and right side of the duct.

The Shear and Torsion Design is completed first on a per web basis. Rebar needed for individual webs is then summed and reported for the entire section. The D/C ratio is calculated for each web. Then the shear area of all webs is summed and the entire section D/C is calculated. Therefore, the controlling section D/C does not have to necessarily match the controlling web D/C (in other words, other webs can make up the capacity for a “weak” web).

5.4.4 Algorithm All section properties and demands are converted from SAP2000 model units

to N, mm.

If the option to consider resal effects is activated, the component of inclined flexural compression or tension, in the direction of the demand shear, in vari-able depth members is evaluated as follows:

Inclination angles of the top and bottom slabs are determined

slab top2 slab top1slab top

2 1

arctany y

Stat Stat

slabbot2 slabbot1slabbot

2 1

arctany y

Stat Stat

where



slab top2 slab top1,y y vertical coordinate of center of gravity of top slab at

stations 1 and 2. Y origin assumed at the top of the section and + direction is up.

1 2,Stat Stat stations of adjacent sections. When section being ana-lyzed is “Before” the current section station is Stat2, when sec-tion being analyzed is “After” the current section station is Stat1. Therefore statement 1 2Stat Stat is always valid.

The magnitudes of normal forces in slabs are determined as follows:

3slab top slab top slab top

3

u uP MP A d

A I

3slabbot slabbot slabbot

3

u uP MP A d

A I

where slab top slabbot,d d are distances from center of gravity of the section

to center of gravity of the slab (positive) The magnitudes of vertical components of slab normal forces are determined

as follows:

resal top slab top slab toptanP P

resalbot slabbot slabbottanP P

On the basis of the location and inclination of each web, the per-web demand

values are evaluated

Outer Web Inner Web Location Vuweb Tuweb Vuweb Tuweb Shear and Torsion Check

2 resal top resalbot

web

abs( )

cosuV P P

Abs(T

u) 2 resal top resalbot

web

abs( )

cosuV P P

0

where

web

web webweb1

cos | |

cos | |n



Evaluate effective thicknesses

Evaluate dv bv be tv

– If bv 0, then

web web flag flag2, 0; 0; 0; 2; 2vs vt vs vtD

WebPassFlag A A A AC

proceed to report web results

– If be < 0 then SectionPassFlag = 2

Evaluate design cf

cf min( cf , 8.3 MPa)

Evaluate stress variable K

Calculate extreme fiber stress

bot bot3

33

P MCG

A I top top

3

33

P MCG

A I tens top botmax ,

– If tens 0.5 ccf , then K = 1 else

| |

10.166 c

P

AKf

where K < 2

Evaluate Vc per web (shear capacity of concrete)

web 0.1663c c v vV K f b d (5.8.6.5-3)

Evaluate Vs per web (shear force that is left to be carried by rebar)

web webweb

u cs

V VV

– If web 0sV then web 0vsA



else webweb

svs

y v

VA

f d

Verify minimum reinforcement requirement

– If web 0.35vs yA t f (eq. 5.8.2.5-2), then

web 0.35vs yA t f and webflag 0sA

else webflag 1vsA

Evaluate nominal capacities

web webs vs y vV A f d

web web webn c sV V V

Evaluate shear D/C for web

web

web

u

s v v c

VD

C b d f

Evaluate Tcr (eq. 5.8.6.3-2)

00.166 2cr c eT K f A b

Evaluate torsion rebar

If web1

3u crT T then:

– flag 0vtA

– web 0vtA

– 0lA

TorsionEffectsFlag=0

else:



flag 1vtA

webweb

0 2u

vty

TA

A f

web

0 long2u h

ly

T pA

A f


Evaluate combined shear and torsion D/C for web

web web

0

web

2

1.25

u u

v v e

t c

V TD b d A b

C f

Evaluate controlling D/C for web

If web webs t

D D

C C

then RatioFlag = 0

else

RatioFlag=1

web web

max ,s t

D D D

C C C

If 1C

D then WebPassFlag=1

else

WebPassFlag = 0

Assign web rebar flags where rebar flag convention is:

Flag = 0 – rebar governed by minimum code requirement

Flag = 1 – rebar governed by demand



Flag = 2 – rebar not calculated since web bv< 0

Flag = 3 – rebar not calculated since web not part of shear flow path for torsion

Evaluate entire section values

section webc cV V

section webs sV V

section webn nV V

section webvs vsA A

section webvt vtA A

sectionl lA A Evaluate entire section D/C

web web1

web

1

section

n uv

v vn

v

s c

Vt

b d

tD

C f

This is equivalent to:

web

1

sec

| |un

v v

s tion c

V

t dD

C f

and

web0

1

section

| | | |2

1.25

u un

ev v

t c

V T

A bt dD

C f

Evaluate controlling D/C for section

If section sections t

D D

C C

then RatioFlag = 0 else RatioFlag = 1



section section

max ,s t

D D D

C C C

If 1D

C then SectionPassFlag=1

else

SectionPassFlag = 0

Assign section design flags where flag convention is:

Flag = 0 – Section Passed all code checks

Flag = 1 – Section D/C >1

Flag = 2 – Section be < 0 (section invalid)

5.4.5 Shear Design Example Cross Section: AASHTO Box Beam, Type BIII-48 as shown in Figure 5-5.

Figure 5-5 Shear Design Example, AASHTO Box Beam, Type BIII-48



Figure 5-6 Shear Design Example Reinforcement

AASHTO Box Beam, Type BIII-48

φ = 0.9 Concrete unit weight, wc = 0.150 kcf λ =1.0 Concrete strength at 28 days, cf = 5.0 ksi (~34.473 MPa)

Design span = 95.0 ft Prestressing strands: ½ in. dia., seven wire, low relaxation Area of one strand = 0.153 in2 Ultimate strength fpu = 270.0 ksi Yield strength fpy = 0.9 fpu = 243 ksi Modulus of elasticity, Ep = 28 500 ksi

Reinforcing bars: yield strength, fy = 60.0 ksi (~413.68 MPa) Section Properties

A = area of cross-section of beam = 826 in2 (~532902 mm2) h = overall depth of precast beam = 39 in (~990.6 mm) I = moment of inertia about

centroid of the beam = 170812 in4 (~71097322269 mm4)



yb,yt = distance from centroid to the extreme bottom (top) fiber of the beam = 19.5 in (~495.3 mm)

Aslabtop = Aslabbot = 485.5 = 264 in2 (~170322 mm2) Ao = (48 5) (39 5.5) = 1440.5 in2 (~929353 mm2) Ph = 2 (48 5 + 39 5.5) = 153 in (~3886.2 mm)

Demand forces from Dead and PT (COMB1) at station 114 before: P = 800 kip (~ 3560 E+03 N) M3 = 7541 kip-in (~ 852 E+06 Nmm) V2 = 33 kip (~ 148.3 E+03 N) T = 4560 kip-in (515.2 E+06 Nmm)


On the basis of the location and inclination of each web, the per-web demand values are evaluated.

Outer Web Inner Web Location Vuweb Tuweb Vuweb Tuweb

Shear and Torsion Check

2 resal top resalbot

web

abs( )

cos

abs(148.3 03 0 0) 174151.9

cos0

uV P P

EN

Abs(Tu)=515.2E+06 N/A 0 N/A

where

webweb web 2

web1 1

cos | | cos | 0 |0.5

cos | | cos | 0 |n

Evaluate effective shear flow path thicknesses

firstweb lastweb topslab botslabmin( , , , )

min(127,127,139.7,139.7) 127mm

e v vb t t t t

Evaluate effective web width and normal thickness

Since the web is vertical, bv = tv = 127 mm

Evaluate effective depth



Since M3 < 0 then

bot topmax(0.8 , )

max(0.8 990.6,495.3 419.1) 914.4mm

v PTd h y y

Evaluate design cf

min( ,8.3MPa) min( 34.473,8.3MPa) 5.871c cf f

Evaluate stress variable K

Calculate extreme fiber stress

bot bot3 3560 03 852 E 06

495.3 12.616MPa.33 532902 71097322269

P M ECG

A I

3 3560 03 852 E 06495.3 0.745MPa

33 532902 71097322269top top

P M ECG

A I

tens top botmax( , ) max( 12.61, 0.745) 0.745MPa

If tens 0.5 cf then K = 1 false

else

| 3560 03 || |5329021 1 2.8

0.166 5.8710.166 c

EP

AKf

where K < 2 therefore K = 2

Evaluate Vc per web (shear capacity of concrete) (5.8.6.5-3)

web 0.1663 0.1663 2 1.0 5.871 127 914.4

226781N.c c v vV K f b d

Evaluate Vs per web (shear force that is left to be carried by rebar)

web webweb

74151.9 0.9 226781144392N

0.9u c

sV V

V

If web 0sV then web 0vsA True



else webweb

svs

y v

VA

f d

Verify minimum reinforcement requirement

If web 0.35vs yA t f (eq. 5.8.2.5-2) then true

2web

0.35 1270.35 0.10745mm / mm

413.68vs yA t f

and webflag 0sA

Else webflag 1vsA

Evaluate nominal capacities

web web 0.10745 413.68 914.4 40645Ns vs y vV A f d

web web web 226781 40645 267426Nn c sV V V

Evaluate shear D/C for web

web

web

74151.90.9 0.1208

127 914.4 5.871

u

s v v c

VD

C b d f

Evaluate Tcr (eq. 5.8.6.3-2)

00.166 2 0.166 2 5.871 2 929353 127

460147419 Nmmcr c eT K f A b

Evaluate torsion rebar

If web1 1

515.2 6 0.9 460 63 3

u crT T E E false, then:

flag 1vtA

2webweb

0

515.2 60.7444mm / mm

2 0.9 929352 2 413.68u

vty

T EA

A f

2web

0 long

515.2 6 3886.22893mm

2 0.9 929352 2 413.68u h

ly

T p eA

A f




Evaluate combined shear and torsion D/C for web

web web

0

web

74151.9 515.2 62 0.9 127 914.4 0.9 2 929352 127

1.25 5.8711.25

0.427

u u

v v e

t c

V T ED b d A b

C f

Evaluate controlling D/C for web

If web webs t

D D

C C

then RatioFlag = 0 false

else

RatioFlag =1 true

web web

max , max 0.1208,0.427 0.427s t

D D D

C C C

If 1D

C , then WebPassFlag =1 true

else

WebPassFlag = 0

Assign web rebar flags where rebar flag convention is:

Flag = 0 – rebar governed by minimum code requirement

Flag = 1 – rebar governed by demand => true

Flag = 2 – rebar not calculated since web bv< 0

Flag = 3 – rebar not calculated since web not part of shear flow path for torsion

Evaluate entire section values

section web 2 226781 453562 Nc cV V



section web 2 40645 81290 Ns sV V

section web 2 267426 534852 Nn nV V

2section web 2 0.10745 0.2149 mm / mmvs vsA A

2section web 2 0.7444887 1.48898mm / mmvt vtA A

2section 2893mml lA A

Evaluate entire section D/C

web web1

web

1

section

n uv

v vn

v

s c

Vt

b d

tD

C f

This is equivalent to:

web 2

1 1

section

| | 148.3 3

0.9 127 914.40.1208

5.871

un

v v

s c

V E

t dD

C f

and

web0

1

section

2

1

| | | |2

1.25

148.3 3 515.2 60.9 2 929352 1270.9 127 914.4

0.4271.25 5.871

u un

ev v

t c

V T

A bt dD

C f

E E

Evaluate controlling D/C for section

If section sections t

D D

C C

then RatioFlag = 0 false

else RatioFlag = 1 true


5 - 30 Design Principal Stress, AASHTO LRFD 2007

section section

max , max 0.1208,0.427 0.427s t

D D D

C C C

If 1D

C , then SectionPassFlag = 1 true

else

SectionPassFlag = 0

Assign section design flags where flag convention is:

Flag = 0 – Section Passed all code checks true

Flag = 1 – Section D/C >1

Flag = 2 – Section be < 0 (section invalid)

5.5 Design Principal Stress, AASHTO LRFD 2007

5.5.1 Capacity Parameters PhiC – Resistance Factor; Default Value = 1.0, Typical value(s): 1.0. The compression and tension limits are multiplied by the C factor


0.6. The cf is multiplied by the FactorCompLim to obtain compression limit

FactorTensLim – cf multiplier; Default Value = 0.19 (ksi) 0.5(MPa); Typi-

cal value(s): 0 to 0.24 (ksi) 0 to 0.63 (MPa). The cf is multiplied by the

FactorTensLim to obtain tension limit

5.5.2 Demand Parameters FactorCompLim – Percentage of the basic unit stress for compression service design; Default value = 1.0; Typical values 1.0 to 1.5. The demand compres-sive stresses are divided by the FactorCompLim factor. This way the control-ling stress can be selected and compared against one compression limit.


Design Principal Stress, AASHTO LRFD 2007 5 - 31

FactorTensLim – Percentage of the basic unit stress for tension service design; Default value = 1.0; Typical values 1.0 to 1.5. The demand tensile stresses are divided by the FactorCompLim factor. This way the controlling stress can be selected and compared against one tension limit.

5.5.3 Algorithm The principal stresses are evaluated at three points at each web: the web cen-terline at the bottom of the top slab; web centerline at the top of the bottom slab; and web centerline at the section neutral axis.

The principal stresses are evaluated for each demand set using the Mohr circle to combine bending, shear, and torsion stresses. The bending stresses assume linear distribution and take into account axial (P) and both bending moments (M2 and M3). The shear flow is calculated internally by the program taking into account section properties at the elevation of the stress point. A shear scale factor is used to convert the total shear flow acting at an elevation (y-coordinate) to tangential shear stress in the web. The scale factor is equal to the web shear-distribution factor divided by the cosine of the angle of inclination of the web from vertical, and divided again by the design width of the web.

web

web web

ShearScaleFactorcosb

where webweb web

web1

cos(| |)

cos(| |)n

and webb is the horizontal width of web

A torsion scale factor is used to convert the total torque acting on the section to tangential shear stress in the web. For interior webs, this is equal to zero. For exterior webs, this is equal to one divided by the plastic torsional modulus.

1TorsionScaleFactor

tW

where min02 tAWt

0A = area enclosed by shear flow path, including area of holes if any


5 - 32 Design Principal Stress, AASHTO LRFD 2007

tmin = minimum normal width of shear flow path

If the demand set contains live load, the program positions the load to cap-ture extreme stress at each of the evaluation points.

The stresses are divided by the appropriate demand parameter. Then the ex-tremes are found for each point and the controlling demand set name is re-corded.

The stress limits are evaluated by applying the Capacity Parameters (see Sec-tion 5.6.1).

Design Stress 6 - 1

Chapter 6 Design Precast Concrete Girder Bridges

This chapter describes the algorithms applied in accordance with the AASHTO-LRFD-07 code for design and stress check when the superstructure has a deck that includes precast I or U girders with composite slabs.

6.1 Design Stress The following parameters are considered during stress design: PhiC – Resistance Factor; Default Value = 1.0, Typical value(s): 1.0. The

compression and tension limits are multiplied by the C factor

FactorCompLim – cf multiplier; Default Value = 0.4; Typical value(s): 0.4

to 0.6. The cf is multiplied by the FactorCompLim to obtain compression

limit

FactorTensLim – f ' c multiplier; Default Value = 0.19 (ksi) 0.5(MPa);

Typical value(s): 0 to 0.24 (ksi) 0 to 0.63 (MPa). The f ' c is multiplied by

the FactorTensLim to obtain tension limit

The stresses are evaluated at three points at the top fiber of the composite slab: the left corner, the centerline beam and the right corner of the composite slab


6 - 2 Design Shear

tributary area. The location of stress output points at the slab bottom fiber and beam top and bottom fiber depends on the type of precast beam present in the section cut. The location is labeled in the output plots and tables.

Concrete strength cf is read at every point and compression and tension limits

are evaluated using the FactorCompLim - cf multiplier and FactorTensLim -

f ' c multiplier.

The stresses assume linear distribution and take into account axial (P) and ei-ther both bending moments (M2 and M3) or only P and M3, depending on which method for determining LLDF has been specified in the design request (see Chapters 3 and 4).

The stresses are evaluated for each demand set. Extremes are found for each point and the controlling demand set name is recorded.

The stress limits are evaluated by applying the preceding Capacity Parameters.

6.2 Design Shear

The following parameters are considered during shear design:

PhiC – Resistance Factor; Default Value = 0.9, Typical value(s): 0.7 to 0.9. The nominal shear capacity of normal weight concrete sections is multiplied by the resistance factor to obtain factored resistance.

PhiC (Lightweight) – Resistance Factor for lightweight concrete; Default Value = 0.7, Typical value(s): 0.7 to 0.9. The nominal shear capacity of light-weight concrete sections is multiplied by the resistance factor to obtain factored resistance.

Check Sub Type – Typical value: MCFT. Specifies which method for shear design will be used: either Modified Compression Field Theory (MCFT) in accordance with 5.8.3.4.2; or Vci/Vcw method in accordance with 5.8.3.4.3 Currently only the MCFT option is available.

Negative limit on strain in nonprestressed longitudinal reinforcement in ac-cordance with section 5.8.3.4.2; Default Value = 0.4x10-3, Typical value(s): 0 to 0.4x10-3

Chapter 6 - Design Precast Concrete Girder Bridges

Design Shear 6 - 3

Positive limit on strain in nonprestressed longitudinal reinforcement in ac-cordance with section 5.8.3.4.2; Default Value = 6.0x10-3, Typical value(s): 6.0x10-3

PhiC for Nu – Resistance Factor used in equation 5.8.3.5-1; Default Value = 1.0, Typical value(s): 0.75 to 1.0

Phif for Mu – Resistance Factor used in equation 5.8.3.5-1; Default Value = 0.9, Typical value(s): 0.9 to 1.0. Shear Rebar Material- A previously defined rebar material label that will be used to determine the required area of trans-verse rebar in the girder

Longitudinal Rebar Material - A previously defined rebar material label that will be used to determine the required area of longitudinal rebar in the girder

6.2.1 Variables

V Resistance factor for shear

P Resistance factor for axial load

F Resistance factor for moment

uV Factored shear demand per girder excluding force in tendons

uN Applied factored axial force taken as positive if tensile

uM Factored moment at the section

cV2 Shear in Section Cut excluding force in tendons

TotV2 Shear in Section Cut including force in tendons

pV Component in the direction of the applied shear of the effective

prestressing force; if Vp has the same sign as Vu, then the component is resisting the applied shear

a Depth of equivalent stress block per 5.7.3.2.2. Varies for positive and negative moment.


6 - 4 Design Shear

vd Effective shear depth per 5.8.2.9

girderd Depth of girder

compslabd Depth of composite slab (includes concrete haunch t2)

PTBotd Distance from top of composite slab to cg of tendons in the bottom

of the precast beam

b Minimum web width of beam

psA Area of prestressing steel on the flexural tension side of the member,

puf Specified tensile strength of prestressing steel

pE Pestressing steel Young’s modulus

vlA Area of nonprestressed steel on the flexural tension side of the mem-

ber at the section under consideration

sE Reinforcement Young’s modulus

s Strain in nonprestressed longitudinal tension reinforcement – eq.

5.8.3.4.2-4

sLimitNegsLimitPos , = Max and min value of strain in nonprestressed longitudi-

nal tension reinforcement as specified in the Design request

cE Young’s modulus of concrete

cA Area of concrete on the flexural tension side of the member

VSA Area of transverse shear reinforcement per unit length

minVSA Minimum area of transverse shear reinforcement per unit length per

eq. 5.8.2.5


Design Shear 6 - 5

6.2.2 Design Process The shear resistance is determined in accordance with paragrah 5.8.3.4.2 (de-rived from Modified Compression Field Theory). The procedure assumes that the concrete shear stresses are distributed uniformly over an area bv wide and dv deep, that the direction of principal compressive stresses (defined by angle θ and shown as D) remains constant over dv, and that the shear strength of the section can be determined by considering the biaxial stress conditions at just one location in the web. The user should select for design only those sections that comply with these assumptions by defining appropriate station ranges in the design request (see Chapter 4).

It is assumed that the precast beams are pre-tensioned, and therefore, no ducts are present in webs. The effective web width is taken as the minimum web width, measured parallel to the neutral axis, between the resultants of the ten-sile and compressive forces as a result of flexure.

Shear design is completed on a per-girder basis. Please refer to Chapter 3 for a description of the live load distribution to individual girders.

6.2.3 Algorithms All section properties and demands are converted from SAP2000 model units

to N, mm.

For every COMBO specified in the Design Request that contains envelopes, a new force demand set is generated. The new force demand set is built up from the maximum tension values of P and the maximum absolute values of V2 and M3 of the two StepTypes (Max and Min) present in the envelope COMBO case. The StepType of this new force demand set is named ABS and the signs of the P, V2 and M3 are preserved. The ABS case follows the industry practice where sections are designed for extreme shear and moments that are not necessarily corresponding to the same design vehicle position. The section cut is designed for all three StepTypes in the COMBOMax, Min and ABSand the controlling StepType is reported.

The component in the direction of the applied shear of the effective prestress-ing force, positive if resisting the applied shear, is evaluated:


6 - 6 Design Shear

girders

Totcp n

VVV 22

Depth of equivalent stress block ‘a’ for both positive and negative moment is evaluated in accordance with eq. 5.7.3.1.1

Effective shear depth is evaluated.

If Mu > 0 then girder Bot Botmax 0 72 0 9 0 5v PT PTd ( . d , . d ,d . a)

If Mu < 0 then

girder girder compslab girder compslabmax 0.72 ,0.9 ( 0.5 ),( 0.5 ) 0.5vd d d d d d a

If u u p vM V V d then u u p vM (V V ) d

The demand/capacity ratio (D/C) is calculated based on the maximum per-missible shear capacity at a section in accordance with 5.8.3.2-2

0 25

up

V

c v

VV

D

C . f ' b d

(5.8.3.2-2)

Evaluate numerator and denominator of (eq. 5.8.3.4.2-4)

numerator 0 5 0 7us u u p ps pu

V

M. N V V A . f

d

denominators p ps s vlE A E A

Adjust denominator values as follows

If denominator 0s and numerator 0s then LimitPoss s and

numeratorsp ps

svl

s

E A

AE

If numerator 0s then denominators p ps s vl c cE A E A E A


Design Shear 6 - 7

Evaluate (eq. 5.8.3.4.2-4)

numerator

denominator

ss

s

Check if axial tension is large enough to crack the flexural compression face of the section.

If girder

0 52uc

N. f '

A then 2s s

Check against the limit on the strain in nonprestressed longitudinal tension reinforcement specified in the Design Request, and if necessary, recalculate how much longitudinal rebar is needed to reach the EpsSpos tension limit

LimitNegmax( , )s s s and LimitPosmin( , )s s s

Evaluate the angle of inclination of diagonal compressive stresses as de-termined in Article 5.8.3.4

18 29 3500 45s (5.8.3.4)

Evaluate the factor indicating the ability of diagonally cracked concrete to transmit tension and shear, as specified in Article 5.8.3.4

4 8

1 750 s

.

(5.8.3.4)

Evaluate nominal shear resistance provided by tensile stresses in the concrete eq. 5.8.3.3-3

0 083c c vV . f ' b d

Evaluate how much shear demand is left to be carried by rebar

cps

uS VV

VV


6 - 8 Design Shear

if 0SV then 0VSA else 1

sVS

y v

VA

f dtan

(per eq. 5.8.3.3-4)

Check against minimum transverse shear reinforcement

If 0 5u s c pV . V V then min

0 083 cVS

y

. f ' bA

f

in accordance

with (eq. 5.8.2.5-1), else 0min VSA

If 0SV then minVS VSA A else minmaxVS VS VSA (A ,A )

Recalculate Vs in accordance with (eq. 5.8.3.3-4)

1S VS y vV A f d

tan

Evaluate longitudinal rebar on flexure tension side in accordance with (eq. 5.8.3.5-1)

0 5 min1

0 5

uUP S

U SUSLreq p ps

v f P y

VVV . (V , )

M NA ( . E A )

d tan f

),max( SLreqVLVL AAA

Assign longitudinal rebar to top or bottom side of girder based on moment sign

If 0UM then VLUVLCompSlab AA and 0langeVLBeamBotFA

Else 0UVLCompSlabA and VLlangeVLBeamBotF AA

6.2.4 Shear Design Example The girder spacing is 9’-8”. The girder type is AASHTO Type VI Girders, 72-inch-deep, 42-inch-wide top flange and 28-inch-wide bottom flange (AASHTO 28/72 Girders). The concrete deck is 8 inches thick, with the haunch thickness assumed = 0.


Design Shear 6 - 9

Figure 6-1 Shear design example deck section

Figure 6-2 Shear design example beam section


6 - 10 Design Shear

Materials

Concrete strength Prestressed girders 28-day strength, cf = 6 ksi,

Girder final elastic modulus, Ec = 4,415 ksi Deck slab: 4.0 ksi, Deck slab elastic modulus, Es = 3,834 ksi Reinforcing steel Yield strength, fy = 60 ksi Prestressing strands 0.5-inch-diameter low relaxation strands Grade 270 Strand area, Aps = 0.153 in2 Steel yield strength, fpy = 243 ksi Steel ultimate strength, fpu = 270 ksi Prestressing steel modulus, Ep = 28,500 ksi

Basic beam section properties

Depth = 72 in. Thickness of web = 8 in. Area, Ag = 1,085 in2

Ac = Area of concrete on the flexural tension side of the member (bordered at mid depth of the beam + slab height) = 551 in2

Moment of inertia, Ig = 733,320 in4

N.A. to top, yt = 35.62 in. N.A. to bottom, yb = 36.38 in. P/S force eccentricity e = 31.380 in.

In accordance with AASHTO LRFD 2007 4.6.2.6, the effective flange width of concrete deck slab is taken as the tributary width. For the interior beam, the inbslab 116"8'9 .

Demands at interior girder Section 2 = station 10’, after girder Section 2, Vu = 319.1 kip; Mu = 3678 kip-ft

The component in the direction of the applied shear of the effective prestress-ing force, positive if resisting the applied shear, is evaluated:

girders

Totcp n

VVV 22

Vp = 0 since no inclined tendons are present.


Design Shear 6 - 11

Depth of equivalent stress block ‘a’ for both positive and negative moment is evaluated in accordance with (eq. 5.7.3.1.1).

Effective shear depth is evaluated

Since Mu > 0, then (for calculation of the depth of the compression block, refer to the Ultimate Flexure example in Section 6.3.4 of this manual)

girder Bot Bot max(0.72 , 0.9 , 0.5 )

max(0.72 80", 0.9 75", 75" 0.5 5.314 0.85)

v PT PTd d d d a

"74.72)"74.72,"5.67,"6.57max( vd

Value reported by SAP2000/Bridge = 72.74”

Check if u u p vM V V d

3678 12 44136 kip-in (319 0) 72.74 23204 kip-inuM

D/C is calculated based on maximum permissible shear capacity at a section in accordance with 5.8.3.2-2

3190

0.90.406

0.25 ' 0.25 6 8 72.74

up

V

c v

VV

D

C f b d


Evaluate the numerator and denominator of (eq. 5.8.3.4.2-4)

numerator 0.5 0.7

3678 12 0.5 0 319 0 6.73 0.7 270 346.2 kip

72.74

us u u p ps pu

V

MN V V A f

d

2denominator 28500 ksi 6.73 in 191805 kips p ps s vlE A E A

Adjust denominator values as follows


6 - 12 Design Shear

If denominator 0s and numerator 0s then LimitPoss s and

numeratorsp ps

svl

s

E A

AE

not applicable

If numerator 0s then

denominator

28500 6.73 4415 551.4 26263461 kip

s p ps s vl c cE A E A E A

Evaluate (eq. 5.8.3.4.2-4)

numerator

denominator

346.21.318e-4

2626346s

ss

Value reported by SAP2000/Bridge = 1.318e-4

Check if axial tension is large enough to crack the flexural compression face of the section

If 0.52 'uc

girder

Nf

A then 2s s ; not applicable since Nu = 0

Check against limit on strain in nonprestressed longitudinal tension rein-forcement as specified in the Design Request and recalculate Avl

LimitPosmax( , ) max( 1.318e-4, 1.318e-4 4) 1.318e-4s s s

Evaluate angle of inclination of diagonal compressive stresses as deter-mined in Article 5.8.3.4

18 29 3500 45s 29 3500 1.318e-4 28.5deg Value reported by SAP2000/Bridge = 28.5 deg

Evaluate factor indicating ability of diagonally cracked concrete to transmit tension and shear as specified in Article 5.8.3.4

4.8 4.85.3265

1 750 1 750 1.318e-4s



Design Shear 6 - 13

Evaluate nominal shear resistance provided by tensile stresses in the concrete (eq. 5.8.3.3-3)

0.0316 '

0.0316 5.32 1.0 6 8 72.74 239.92 kip

c c vV f b d

Value reported by SAP2000/Bridge = 240.00 kip

Evaluate how much shear demand is left to be carried by rebar

3190 239.6 114.8 kip

0.9u

S p cs

VV V V


If 0SV then 0VSA else

2114.81.43e-2 in /in

1 160 72.74

tan tan28.5

sVS

y v

VA

f d

(eq. 5.8.3.3-4)

Check against minimum transverse shear reinforcement

If 0.5 319.1 kip 0.5 239.6 119.8 kipu s c pV V V is true,

2min

0.0316 ' 0.0316 1.0 6 80.01032in /in

60c

VSy

f bA

f

(eq.

5.8.2.5-1)

If 0SV then minVSVS AA else 2minmax( , ) 1.43e-2in /2VS VS VSA A A

Value reported by SAP2000/Bridge = 1.43e-2in2/in

Recalculate Vs in accordance with (eq. 5.8.3.3-4)

1 10.0143 60 72.74 114.9kip

tan tan28.5S VS y vV A f d


Evaluate longitudinal rebar on flexure tension side in accordance with eq. 5.8.3.5-1


6 - 14 Design of Flexure

req

2

0.5 min ,1

0.5tan

3190 0.5 114.9

3678 12 0 10.9 0.5 28500 6.73 3176.3 in

72.74 0.9 1.0 tan28.5 60

uUP S

U S SUSL p ps

v f P y

VVV V

M NA E A

d f

Value reported by SAP2000/Bridge = 0.00 in2 no additional longitu-dinal rebar required in beam bottom flange

6.3 Design of Flexure The following parameters are used in the design of flexure: PhiC – Resistance Factor; Default Value = 1.0, Typical value(s): 1.0. The nominal flexural capacity is multiplied by the resistance factor to obtain fac-tored resistance

6.3.1 Variables Resistance factor for flexure

nM Nominal flexural resistance

rM Factored flexural resistance

slabeqt Thickness of composite slab

slabb Effective flange width = horizontal width of slab tributary area,

measured from out to out

webeqb Thickness of beam web

slabA Tributary area of slab

a Depth of equivalent stress block in accordance with 5.7.3.2.2.

PTA Area of PT in tension zone


Design of Flexure 6 - 15

PTy Distance from extreme compression fiber to the centroid of the

prestressing tendons

puf Specified tensile strength of prestressing steel (area weighted aver-

age of all tendons in tensile zone)

pyf Yield tensile strength of prestressing steel (area weighted average if

all tendons in tensile zone)

psf Average stress in prestressing steel (eq. 5.7.3.1.1-1)

k PT material constant (eq. 5.7.3.1.1-2)

1 Stress block factor is (eq. 5.7.2.2)

6.3.2 Design Process The derivation of the moment resistance of the section is based on approximate stress distribution specified in Article 5.7.2.2. The natural relationship between concrete stress and strain is considered satisfied by an equivalent rectangular concrete compressive stress block of 0.85 cf over a zone bounded by the edges

of the cross-section and a straight line located parallel to the neutral axis at the distance a = β1c from the extreme compression fiber. The distance c is meas-ured perpendicular to the neutral axis. The factor β1 is taken as 0.85 for con-crete strengths not exceeding 4.0 ksi. For concrete strengths exceeding 4.0 ksi, β1 is reduced at a rate of 0.05 for each 1.0 ksi of strength in excess of 4.0 ksi, except that β1 is not be taken to be less than 0.65.

The flexural resistance is determined according to paragraph 5.7.3.2. The resis-tance is evaluated only for bending about horizontal axis 3. Separate capacity is calculated for positive and negative moment. The capacity is based only on bonded tendons defined in the Bridge Object. Mild steel reinforcement is not considered. If there is no prestressing in the tension zone of the section, the ca-pacity is reported as zero. It is assumed that all defined tendons in a section, stressed or not, have fpe (effective stress after loses) larger than 0.5 fpu (specified tensile strength). If a certain tendon should not be considered for the flexural capacity calculation, its area must be set to zero.



The section properties are calculated for the section before skew, grade, and superelevation are applied. This is consistent with the demands being reported in section local axis. It is assumed that the effective width of the flange (slab) in compression is equal to the width of the slab.

6.3.3 Algorithms At each section:


1 stress block factor is evaluated per 5.7.2.2 based on section cf

If cf >28 MPa, then 128

max(0.85 0.05;0.65)7

cf

else 1 0 85 .

The tendon location, area and material are read. Only bonded tendons are processed, unbonded tendons are ignored.

Tendons are split into two groups depending on what sign of moment they resistnegative or positive. A tendon is considered to resist a positive mo-ment when it is located outside of the top fiber compression stress block and is considered to resist a negative moment when it is located outside of the bottom fiber compression stress block. The compression stress block extends over a zone bounded by the edges of the cross-section and a straight line lo-cated parallel to the neutral axis at the distance a = β1c from the extreme compression fiber. The distance c is measured perpendicular to the neutral axis.


- sum of tendon areas PTA

- center of gravity of tendons PTy

- specified tensile strength of prestressing steel puf



- constant k (eq. 5.7.3.1.1-2)

2 1.04 py

pu

fk

f

Positive moment resistance – first it is assumed that the equivalent compres-sion stress block is within the top slab. Distance c between the neutral axis and the compressive face is calculated in accordance with (eq. 5.7.3.1.1-4)

10.85

PT pu

puc slab PT

pt

A fc

ff b kA

y

The distance c is compared to the slab thickness. If the distance to the neutral axis c is larger than the composite slab thickness, the distance c is re-evaluated. For this calculation, the beam flange width and area are converted to their equivalents in slab concrete by multiplying the beam flange width by the modular ratio between the precast girder concrete and the slab concrete. The web width in the equation for c is substituted for the effective converted girder flange width. The distance c is recalculated in accordance with (eq. 5.7.3.1.1-3).

slab webeq slabeq

1 webeq

0.85 ( )

0.85

PT pu c

puc PT

pt

A f f b b tc

ff b kA

y

If the calculated value of c exceeds the sum of the deck thickness and the equivalent precast girder flange thickness, the program assumes the neutral axis is below the flange of the precast girder and recalculates c. The term

0.85 c wf b b in the calculation is broken into two terms, one refers to the

contribution of the deck to the composite section flange and the second refers to the contribution of the precast girder flange to the composite girder flange.

Average stress in prestressing steel fps is calculated in accordance with 5.7.3.1.1-1

1ps pupt

cf f k

y



Nominal flexural resistance Mn is calculated in accordance with 5.7.3.2.2-1

If the section is a T-section, then


2 2 2n PT ps PT c

tc cM A f y f b b t

else

1

2n PT ps PT

cM A f y


nr MM

Extreme moment M3 demands are found from the specified demand sets and the controlling demand set name is recorded.

The process for evaluating negative moment resistance is analogous, except that calculation of positive moment resistance is not applicable.

6.3.4 Flexure Capacity Design Example Girder spacing: 9’-8”

Girder type: AASHTO Type VI Girders, 72 inches deep, 42-inch-wide top flange and 28-inch-wide bottom flange (AASHTO 28/72 Girders)

Concrete deck: 8 inches thick, haunch thickness assumed = 0

Materials

Concrete strength Prestressed girders 28-day strength, cf = 6 ksi,

Girder final elastic modulus, Ec = 4,696 ksi Deck slab = 4.0 ksi, Deck slab elastic modulus, Es = 3,834 ksi Reinforcing steel Yield strength, fy = 60 ksi Prestressing strands 0.5-inch-diameter low relaxation strands Grade 270



Figure 6-3 Flexure capacity design example deck section

Figure 6-4 Flexure capacity design example beam section



Strand area, Aps = 0.153 in2

Steel yield strength, fpy = 243 ksi Steel ultimate strength, fpu = 270 ksi Prestressing steel modulus, Ep = 28,500 ksi

Basic beam section properties

Depth = 72 in. Thickness of web = 8 in. Area, Ag = 1,085 in2

Moment of inertia, Ig = 733,320 in4

N.A. to top, yt = 35.62 in. N.A. to bottom, yb = 36.38 in. P/S force eccentricity e = 31.380 in. In accordance with AASHTO LRFD 2007 paragraph 4.6.2.6, the effec-tive flange width of concrete deck slab is taken as the tributary width. For the interior beam, the slab 9'8" 116 inb .

Tendons are split into two groups depending on which sign of moment they re-sistnegative or positive. A tendon is considered to resist a positive moment when it is located outside of the top fiber compression stress block and is con-sidered to resist a negative moment when it is located outside of the bottom fi-ber compression stress block. The compression stress block extends over a zone bounded by the edges of the cross-section and a straight line located par-allel to the neutral axis at the distance a = β1c from the extreme compression fi-ber. The distance c is measured perpendicular to the neutral axis.


- sum of tendon areas 2

Bottom 44 0.153 6.732 inPTA Value reported by SAP2000/Bridge = 6.732 in2

- distance from center of gravity of tendons to extreme compression fiber

Bottom12 2 12 4 10 6 6 8 4 10

(72 8) 75 in12 12 10 6 4

PTy

- specified tensile strength of prestressing steel 270 kippuf

Value reported by SAP2000/Bridge = 270 kip



- constant k (eq. 5.7.3.1.1-2)

2432 1.04 2 1.04 0.28

270py

pu

fk

f


1 stress block factor is evaluated in accordance with 5.7.2.2 based on the

composite slab cf

1 shall be taken as 0.85 for concrete strength not exceeding 4.0 ksi. If cf

> 4 ksi, then 1 shall be reduced at a rate of 0.05 for each 1.0 ksi of

strength in excess of 4.0 ksi. Since cf = 4 ksi, 1 = 0.85

Value calculated by SAP2000/Bridge = 0.85 (not reported)

The distance c between neutral axis and the compressive face is evaluated in accordance with 5.7.3.1.1-4

Bottom

1 slab BottomBottom

0 85

6 732 270 5 314 in

2700 85 4 0 85 116 0 28 6 732

75

PT pu

puc PT

PT

A fc

f. f ' b k A

y

. *.

. . . .

Value calculated by SAP2000/Bridge = 5.314 in

The distance c is compared to the composite slab thickness to determine if the c needs to be re-evaluated to include the precast beam flange in the equivalent compression block.

Since c = 5.314 in < 8", the c is valid

Average stress in prestressing steel fps is calculated in accordance with 5.7.3.1.1-1

Bottom

5.3141 270 1 0.28 264.64 ksi

75ps pu

PT

cf f k

y

Value reported by SAP2000/Bridge = 264.643 ksi

Nominal flexural resistance Mn is calculated in accordance with 5.7.3.2.2-1



Since the section is rectangular,

1Bottom Bottom

5 314 0 856 732 264 64 75

2 2 129593 17 12 10799 4 kip-ft

n PT ps PT

c . .M A f y . .

. / .

Value calculated by SAP2000/Bridge = 107 99 kip-ft (not reported)


0.9 10 799.4 9719.5 kip-ftr nM M

Value reported by SAP2000/Bridge = 9719.5 kip-ft (116633.5 kip-in)

Description of Example Model 7- 1

Chapter 7 Run a Bridge Design Request

This chapter identifies the steps involved in running a Bridge Design Request. (Chapter 4 explains how to define the Request.) Running the Request applies the following to the specified Bridge Object:

Program defaults in accordance with the selected codethe Preferences

Type of design to be performedthe check type (Section 4.2.1)

Portion of the bridge to be designedthe station ranges (Section 4.1.3)

Overwrites of the Preferencesthe design request parameters (Section 4.1.4)

Load combinations the demand sets (Chapter 2)

Live Load Distribution factors, where applicable (Chapter 3)

For this example, the AASHTO LRFD 2007 code is applied to the model of a concrete box-girder bridge shown in Figure 7-1.

It is assumed that the user is familiar with the steps that are necessary to create a SAP2000/Bridge model of a concrete box girder bridge. If additional assis-tance is needed to create the model, a 30-minute Watch and Learn video enti-tled, ”Bridge – Bridge Information Modeler” is available at the CSI website www.csiberkeley.com. The tutorial video guides the user through the creation of the bridge model referenced in this chapter.

http://www.csiberkeley.com/�


7 - 2 Description of Example Model

Figure 7-1 3D view of example concrete box girder bridge model

7.1 Description of Example Model

The example bridge is a two-span prestressed concrete box girder bridge with the following features:

Abutments: The abutments are skewed by 15 degrees and connected to the bottom of the box girder only.

Prestress: The concrete box girder bridge is prestressed with four 10-in2 tendons (one in each girder) and a jacking force of 2160 kips per tendon.

Bents: The one interior bent has three 5-foot-square columns.

Deck: The concrete box girder has a nominal depth of 5 feet. The deck has a parabolic variation in depth from 5 feet at the abutments to a maximum of 10 feet at the interior bent support.

Spans: The two spans are each approximately 100 feet long.

Figure 7-2 Elevation view of example bridge

Chapter 7 - Run a Bridge Design Request

Design Preference 7 - 3

Figure 7-3 Plan view of the example bridge

7.2 Design Preferences

Use the Design menu > Bridge Design > View/Revise Preferences command to select the AASHTO LRFD 2007 design code. The Bridge Design Prefer-ences form shown in Figure 7-4 displays.

Figure 7-4 Bridge Design Preferences form

7.3 Load Combinations

For this example, the default design load combinations were activated using the Define menu > Load Combination command. After the Add Default Design Combos button is used and the Bridge option has been selected, the Code-


7 - 4 Load Combinations

Generated Load Combinations for Bridge Design form shown in Figure 7-5 displays. The form is used to specify the desired limit states. Only the Strength II limit state was selected for this example. Normally, several limit states would be selected.

Figure7-5 Code-Generated Load Combinations for Bridge Design form

The defined load combination for this example are shown in Figure 7-6.

Figure 7-6 Define Load Combinations form

Chapter 7 - Run a Bridge Design Request

Bridge Design Request 7 - 5

The Str-II1, Str-II2 and StrIIGroup1 designations for the load combinations are specified by the program and indicate that the limit state for the combinations is Strength Level II.

7.4 Bridge Design Request

After the Design menu > Bridge Design > Define Design Request command has been used, the Bridge Design Request form shown in Figure 7-7 displays.

Figure 7- 7 Define Load Combinations form

The name given to this example design request is FLEX_1, the Check Type is for Concrete Box Flexure and the Demand Set, DSet1, specifies the combina-tion as StrII (Strength Level II).


7 - 6 Start Design/Check of the Bridge

The only Design Request Parameter option for a Concrete Box Flexural check type is for PhiC. A value of 0.9 for PhiC is used.

7.5 Start Design/Check of the Bridge

After an analysis has been run, the bridge model is ready for a design/check. Use the Design menu > Bridge Design > Start Design/Check of Bridge com-mand to start the design process. Select the design to be run using the Perform Bridge Design form shown in Figure 7-8:

Figure 7-8 Perform Bridge Design - Superstructure

The user may select the desired Design Request(s) and click on the Design Now button. A plot of the bridge model, similar to that shown in Figure 7-9, will display.

If several design requests have been run, the indi-vidual Design Requests can be selected from the Design Check options drop-down list. This plot is described further in Chapter 8.

Figure 7-9 Plot of flexure check results

Display Results as a Plot 8 - 1

Chapter 8 Display Bridge Design Results

Bridge design results can be displayed on screen and as printed output. The on-screen display can depict the bridge response graphically as a plot or in data tables. SAP2000's Advanced Report Writer can be used to create the printed output, which can include the graphical display as well as the database tables.

8.1 Display Results as a Plot

To view the forces, stresses, and design results graphically, click the Display menu > Show Bridge Forces/Stresses command, which will display the Bridge Object Response Display form shown in Figure 8-1.

The plot shows the design results for the FLEX_1 design request created using the process described in the preceding chapters. The demand moments are en-veloped and shown in the blue region, and the negative capacity moments are shown with a brown line. If the demand moments do not exceed the capacity moments, the superstructure may be deemed adequate in response to the flexure design request. Move the mouse pointer onto the demand or capacity plot to view the values for each nodal point. Move the pointer to the capacity moment at station 1200 and 536981.722 kip-in is shown. A verification calculation that shows agreement with this SAP2000/Bridge result is provided in Section 8.4.

Bridge Superstructure Design Guide

8 - 2 Display Data Tables

Figure 8-1 Plot of flexure check results for the example bridge design model

8.2 Display Data Tables

To view design results on screen in tables, click the Display menu > Show Ta-bles command, which will display the Choose Tables for Display form shown in Figure 8-2. Use the options on that form to select which data results are to be viewed. Multiple selection may be made. When all selections have been made, click the OK button and a database table similar to that shown in Figure 8-3 will display. Note the drop-down list in the upper right-hand corner of the table. That drop-down list will include the various data tables that match the selections made on the Choose Tables for Display form. Select from that list to change to a different database table.

Chapter 8 - Display Bridge Design Results

Display Data Tables 8- 3

Figure 8-2 Choose Tables for Display form

Figure 8-3 Design database table for AASHTO LRFD 2007 flexure check


8 - 4 Advanced Report Writer

The scroll bar along the bottom of the form can be used to scroll to the right to view additional data columns.

8.3 Advanced Report Writer

The File menu > Create Report command is a single button click output option but it may not be suitable for bridge structures because of the size of the docu-ment that is generated. Instead, the Advanced Report Writer feature within SAP2000 is a simply and easy way to produce a custom output report.

To create a custom report that includes input and output, first export the files us-ing one of the File menu > Export commands: SAP2000 MS Access Data-base.mdb File; SAP2000 MS Excel Spreadsheet .xls File; or SAP2000 .s2k Text File. When this command is executed, a form similar to that shown in Fig-ure 8-4 displays.

Figure 8-4 Choose Tables for Export to Access form


Advanced Report Writer 8- 5

This important step allows control over the size of the report to be generated. Export only those tables to be included in the final report. However, it is possi-ble to export larger quantities of data and then use the Advanced Report Writer to select only specific data sets for individual reports, thus creating multiple smaller reports. For this example, only the Bridge Data (input) and Concrete Box Flexure design (output) are exported.

After the data tables have been exported and saved to an appropriate location, click the File menu > Advanced Report Writer command to display a form similar to that show in Figure 8-5. Click the appropriate button (e.g., Find exist-ing DB File, Convert Excel File, Convert Text File) and locate the exported data tables. The tables within that Database, Excel, or Text file will be listed in the List of Tables in Current Database File display box.

Figure 8-5 Create Custom Report form


8 - 6 Verification

Select the tables to be included in the report from that display box. The selected items will then display in the Items Included in Report display box. Use the various options on the form to control the order in which the selected tables ap-pear in the report as well as the headers (i.e., Section), page breaks, pictures, and blanks required for final output in .rft, .txt, or .html format.

After the tables have been selected and the headers, pictures, and other format-ting items have been addressed, click the Create Report button to generate the report. The program will request a filename and the path to be used to store the report. Figure 8-6 shows an example of the printed output generated by the Re-port Writer.

Values used in the verification calculations.Values used in the verification calculations.

Figure 8-6 An example of the printed output

8.4 Verification

As a verification check of the design results, the output at station 1200 is exam-ined. The following output for negative bending has been pulled from the Con-BoxFlexure data table, a portion of which is shown in Figure 8-3:


Verification 8- 7

Demand moment, “DemandMax” (kip-in) = 245973.481

Resisting moment, “ResistingNeg” (kip-in) = 536981.722

Total area of prestressing steel, “AreaPTTop” (in2) = 20.0

Top k factor, “kFactorTop”, = 0.2644444

Neutral axis depth, c, “CDistForNeg” (in) = 5.1286

Effective stress in prestressing, fps, “EqFpsForNeg” (kip/in2) = 266.7879

A hand calculation verifies the results as follows:

For top k factor, from eq. 5.7.3.1.1-2,

245 12 1 04 2 1 04 0 26444

270PY

PU

f .k . . .

f

(Result match)

For neutral axis depth, from eq. 5.7.3.1.1-4,

slab webeq slabeq

1 webeq

0.85,

0.85

PT PU c

PUc PT

PT

A f f b b tc

ff b kA

Y

for a T-section

1 webeq

,0.85

PT PU

PUc PT

PT

A fc

ff b kA

Y

when not a T-section

20.0(270)5.1286

2700.85(4)(0.85)(360) 0.26444(20)

114

c

(Results match)

For effective stress in prestressing, from eq. 5.7.3.1.1-1,

5 12861 270 1 0 26444 266 788

144PS PUPT

c .f f k . .

Y

(Results match)


8 - 8 Verification

For resisting moment, from eq. 5.7.3.2.2-1,

1 10 852 2 2

slabeqN PT PS PT SLAB webeq slabeq

tc cM A f Y . f ' c b b t

1

2N PT PS PT

cM A f Y

, when the box section is not a T-section

NM

5.1286(0.85)20.0(266.788) 144 596646.5

2 kip-in

R NM M 0.85(596646.5) 536981.8 kip-in (Results match)

The preceding calculations are a check of the flexure design output. Other de-sign results for concrete box stress, concrete box shear and concrete box princi-pal have not been included. The user is encouraged to perform a similar check of these designs and to review Chapters 5 and 6 for a detailed description of the design algorithms.

R - 1

References

ACI, 2008. Building Code Requirements for Structural Concrete (ACI 318-08) and Commentary (ACI 318R-08), American Concrete Institute, P.O. Box 9094, Farmington Hills, Michigan.

AASHTO, 2009. AASHTO Guide Specifications for LRFD Seismic Bridge Design. American Association of Highway and Transportation Offi-cials, 444 North Capital Street, NW Suite 249, Washington, DC 2001

Documents

Bridge Superstructure Design