Bridge Superstructure Design

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ISO SAP041709M21 Rev. 1 Berkeley, California, USA

Version 14 July 2009

SAP2000



/Bridge

Bridge Superstructure Design

Concrete Box Girder Bridges and

Precast Concrete Composite Girder Bridges

AASHTO LRFD 2002 and 2007

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COPYRIGHT

The CSI Logo® and SAP2000®

are a registered trademarks of Computers & Structures, Inc. Watch & LearnTM is a trademark 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 SAP2000 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

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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.

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i

Bridge Superstructure Design

1 Introduction

1.1 Orga nization 1-1

1.2 Re commended Reading 1-2

2 Design Prerequisites

2-1 Load Pattern Types 2-1

2.2 De sign 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

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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 Che ck Type 4-3

4.3 Station Range 4-4

4.4 De sign Range 4-4

4.5 Dema nd Sets 4-8

4.6 Live Load Distribution Factors 4-8

5 Design Concrete Box Girder Bridges

5.1 Stress Design AASHTO-STD-2002 5-1

5.1.1 Cap acity Parameters 5-1

5.1.2 Dema nd Parameters 5-2

5.1.3 Algorithm 5-2

5.2 Stress Design AASHTO-LFRD-2007 5-2

5.2.2 Algorithm 5-3

5.2.3 Stress Design Example 5-3

5.3 Flexure Design AASHTO-LRFD-2007 5-6

5.3.2 Variabl es 5-6

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Contents

iii

5.3.4 Algorithm 5-8

5.3.5 Flexure Design Example 5-10 5.4 Shear Design AASHTO-LRFD-2007 5-15

5.4.2 Variabl es 5-15

5.4.3 De sign Process 5-16

5.4.4 Algorithm 5-18

5.4.5 Shear Design Example 5-24

5.5 Principal Stress Design AASHTO-LRFD-2007 5-31

5.5.2 Dema nd Parameters 5-31

5.5.3 Algorithm 5-31

6 Design Multi-Cell Concrete Box Bridges using AMA

6.1 Stress Design 6-2 6.2 Shear Design 6-3 6.2.1 Variables 6-4 6.2.2 De sign Process 6-5 6.2.3 Algorithm s 6-6 6.3 Flexure Design 6-10 6.3.1 Variables 6-10 6.3.2 De sign Process 6-11 6.3.3 Algorithm s 6-11

7 Design Precast Concrete Girder Bridges

7.1 Stress Design 7-1

7.2 Shear Design 7-2

7.2.1 Variables 7-3

7.2.3 Algorithm s 7-5

7.2.4 Shear Design Example 7-8

7.3 Flexural Design 7-14

7.3.1 Variables 7-15

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iv

7.3.4 Flexure Capacity Design Example 7-19

8 Run a Bridge Design Request

8.1 Description of Example Model 8-2

8.2 De sign Preferences 8-3

8.3 Load Combinations 8-3

8.4 Bridge Design Request 8-5

8.5 Start Design/Check of Structure 8-6

9 Design Output

9.1 Display Results as a Plot 9-1

9.2 Display Data Tables 9-7

9.3 Advanced Report Writer 9-8

9.4 Verificatio n 9-11

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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 Superstructure Design Request Parameters Form 4-5 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 7-1 Shear design example deck section 7-9 Figure 7-2 Shear design example beam section 7-9 Figure 7-3 Flexure capacity design example deck section 7-19 Figure 7-4 Flexure capacity design example beam section 7-19 Figure 8-1 3D view of example concrete box girder

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vi

Figure 8-2 Elevation view of example bridge 8-2 Figure 8-3 Plan view of the example bridge 8-3 Figure 8-4 Bridge Design Preferences form 8-3 Figure 8-5 Code-Generated Load Combinations for Bridge

Design form 8-4

Figure 8-6 Define Load Combinations form 8-4 Figure 8- 7 Define Load Combinations form 8-5 Figure 8-8 Perform Bridge Design - Superstructure 8-6 Figure 8-9 Plot of flexure check results 8-6 Figure 9-1 Plot of flexure check results for the example bridge

design model 9-2

Figure 9-2 Select the location on the beam or slab for which results

are to be displayed 9-3

Figure 9-3 Bridge Concrete Box Deck Section - External Girders Vertical 9-3

Figure 9-4 Bridge Concrete Box Deck Section - External Girders Sloped 9-4

Figure 9-5 Bridge Concrete Box Deck Section - External Girders

Clipped 9 -4

and Radius 9-5

Sloped Max 9-5

Figure 9-8 Bridge Concrete Box Deck Section - Advanced 9-6 Figure 9-9 Bridge Concrete Box Deck Section - AASHTO - PCI -

ASBI Standard 9-6

Figure 9-10 Choose Tables for Display form 9-7 Figure 9-11 Design database Table for AASHTO LRFD 2007

flexure check 9-8

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Contents

vii Figure 9-13 Create Custom Report form 9-10 Figure 9-14 An example of the printed output 9-11

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1 - 1

Chapter 1 Introduction

This manual describes using SAP2000/Bridge to complete bridge design in accordance with the AASHTO STD 2002 or AASHTO LRFD 2007 code for concrete box girder bridges or the AASHTO 2007 LRFD code for bridges when the superstructure includes Precast Concrete Box bridges with a compos-ite slab. Design using SAP2000/Bridge is based on load patterns, load cases, load combinations and design requests. The design output can then be dis-played 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.

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SAP2000/Bridge Superstructure Design Guide

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, 6, and 7 provide the algorithms used by SAP2000/Bridge in completing concrete box girder, cast-in-place multi-cell concrete box, and precast concrete bridge design in accordance with the AASHTO code. Chapter 8 explains how to run a Design Request. Chapter 9 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

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.

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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 effects

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 to 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

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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.

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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.

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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.

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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).

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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 or concrete multi-cell box.

Legend:

Girder = beam + tributary area of composite slab for precast composite; web + tributary area of top and bottom slab for multi-cell box

Section Cut = all girders present in the cross-section at the cut location

3.1 Algorithm for Determining Live Load Distribution

Factors (LLDF)

SAP2000/Bridge gives the user a choice of four methods to address distribu-tion of live load to individual girders.

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3 - 2 Determine Live Load Distribution Factors

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

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

 the number of girders

 girder designationthe first and last girder are designated as exterior girders and the other girders are classified as interior girders

 roadway widthmeasured as the distance between curbs/barriers; medians are ignored

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Chapter 3 - Determine Live Load Distribution Factors

Apply LLD Factors 3 - 3  overhangconsists of the horizontal distance from the centerline of the

exte-rior web of the left exteexte-rior beam at deck level to the inteexte-rior edge of the curb or traffic barrier

 the beamsincludes 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 slabwhich 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 beamsangle 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.

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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 parameter 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 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 LLDF equal to the aver-age 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).

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Generate Virtual Combinations 3 - 5

3.4 Generate Virtual Combinations

When the method for determining the live load distribution factors is user-specified, code-user-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, ex(ex-terior shear, in(ex-terior moment, and in(ex-terior 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 of 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.

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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-binationsone for shear and one for momentgenerated for exterior beam are assigned to both exterior beams, and similarly, Virtual Combinations for inte-rior beams are assigned to inteinte-rior 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.

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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, ex-cept when the tables call for the use of the lever rule. In those cases, the com-putations need to account for the multiple presence factors. The user is provid-ing those as part of the Design Request definition together with wheel spacprovid-ing, 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/Bridge 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.” That 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 the 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 “Yes” in the

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“Dia-SAP2000/Bridge Superstructure Design Guide

3 - 8 LLDF Design Example Using Method 2

phragms 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 in 2

Noncomposite beam moment of inertia, Ig = 733,320 in 4

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 = E_B E _D (4.6.2.2.1-2)

= 4 696 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 + ts 2 = 35.62 + 8 2 = 39.62 in.

3. Calculate Kg, the longitudinal stiffness parameter.

K_g = n I



Ae2_g



(4.6.2.2.1-1) = 1.225 733 320 1 085 39.62_   2_2 984 704 in4

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LLDF Design Example Using Method 2 3 - 9 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 =



 







0.1 0.6 0.2 ₃ 0.075 S 9.5 S L Kg 12.0Lts



 

0.6



0.2



  3



0.1 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 =



 







0.1 0.4 0.3 ₃ 0.06 S 14 S L Kg 12.0Lts =



 





  



0.1 0.4 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.

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SC =





0.3 3 1.00.20 12.0Lts Kg tan =



  



0.3 3 1.00.20 12.0 110 8 2 984 704 tan 20 = 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.

D_V = 0.2



S 12

 

 S 35



2

= 0.2



9.667 12

 

 9.667 35



2 = 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



S 25.0



= 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.

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LLDF Design Example Using Method 2 3 - 11 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de 9.1

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 in accordance with Table 4.6.2.2.2d-1.

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= 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.

D_M = 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de10 = 0.6 1.83 10 = 0.783 D_V = 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 in accordance with Table 4.6.2.2.3b-1. This value will be the same as the moment distribution factor with the skew correction factor applied.

D_V = 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 distribucalcula-tion 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.

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LLDF Design Example Using Method 2 3 - 13 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 = N_L N_bX_ext

 

 

e x2 (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:

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3 - 14 LLDF Design Example Using Method 2 = 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 the distribution factors summarized in Design Step 16.

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LLDF Design Example Using Method 2 3 - 15 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

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Name and Bridge Object 4 - 1

Chapter 4 Define a Bridge Design Request

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 the 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.

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4 - 2 Name and Bridge Object

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

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Chapter 4 - Define a Bridge Design Request

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 Multi-Cell Concrete Box Girder bridge, SAP2000/Bridge provides the fol-lowing check type options:

 Concrete Box Stress  Concrete Box Flexure  Concrete Box Shear

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4 - 4 Station Range

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

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, in Chapter 6 for multi-cell box girder bridges, and in Chapter 7 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.

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Design Parameters 4 - 5

Figure 4-3 Superstructure Design Request Parameters form

Table 4-1 shows the parameters for concrete box girder bridges. Table 4-2 shows the parameters for multi-cell concrete box bridges. Table 4-3 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 f to calculate the compression stress limit c

 Multiplier on sqrt( f ) to calculate the tension stress limit, given c

in the units specified

 The tension limit factor may be specified using either MPa or ksi units for f and the resulting tension limit _c

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4 - 6 Design Parameters

Table 4-1 Design Request Parameters for Concrete Box Girders

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 f c

to calculate the compression stress limit

 Concrete Box Stress Factor Tension Limit Units - Multiplier on sqrt( f ) to calculate the tension stress limit, given in the units _c specified

 Concrete Box Stress Factor Tension Limit - The tension limit fac-tor may be specified using either MPa or ksi units for f and the _c 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 in accordance with 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

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 Multi-Cell Concrete Box

AASHTO LRFD 2007

Multi-Cell Concrete Box

Stress  Multi-Cell Concrete Box Stress, PhiC, - Resistance Factor that _{multiplies both compression and tension stress limits}  Cell Concrete Box Stress Factor Compression Limit -

Multi-plier on f to calculate the compression stress limit _c

 Multi-Cell Concrete Box Stress Factor Tension Limit Units - Mul-tiplier on sqrt( f ) to calculate the tension stress limit, given in _c the units specified

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Design Parameters 4 - 7 Table 4-2 Design Request Parameters for Multi-Cell Concrete Box

 Multi-Cell Concrete Box Stress Factor Tension Limit - The tension limit factor may be specified using either MPa or ksi units for f _c and the resulting tension limit

Multi-Cell Concrete Box

Shear  Multi-Cell Concrete Box Shear, PhiC, - Resistance Factor that mul-_{tiplies both compression and tension stress limits}  Multi-Cell Concrete Box Shear, PhiC, Lightweight Resistance

Fac-tor that multiplies nominal shear resistance to obtain facFac-tored resistance for light-weight concrete

 Negative limit on strain in nonprestressed longitudinal rein-forcement – in accordance with section 5.8.3.4.2; Default Value = -0.4x10-3_{, Typical value(s): 0 to -0.4x10}-3

 Positive limit on strain in nonprestressed longitudinal reinforce-ment - in accordance 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; De-fault 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 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. Cur-rently only the MCFT option is available.

 A previously defined rebar material label that will be used to determine the required area of transverse rebar in the girder.  A previously defined rebar material that will be used to

deter-mine the required area of longitudinal rebar in the girder Multi-Cell Concrete Box

Flexure  Multi-Cell Concrete Box Flexure, PhiC, - Resistance Factor that _{multiplies both compression and tension stress limits}

Table 4-3 Design Request Parameters for Precast I or U Beams

AASHTO

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 fc

to calculate the compression stress limit

 Precast Comp Stress Factor Tension Limit Units - Multiplier on sqrt(fc) 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 fc and the

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4 - 8 Demand Sets

Table 4-3 Design Request Parameters for Precast I or U Beams

AASHTO

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

 PhiC, Lightweight Resistance Factor that multiplies nominal shear resistance to obtain factored resistance for light-weight concrete  Negative limit on strain in nonprestressed longitudinal

reinforcement – in accordance with section 5.8.3.4.2; Default Value = -0.4x10-3_{, Typical value(s): 0 to -0.4x10}-3

 Positive limit on strain in nonprestressed longitudinal reinforce-ment - in accordance 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

 Specifies what 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.

 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

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 or multi-cell boxes, Live Load Distribution Factors can be specified. LLD factors are described in Chapter 3.

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Stress Design 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.

In SAP2000, ConcBox design differs from MulticellConcBox design (Chapter 6) with respect to the distribution of loads and the method for checking shear and torsion. In distributing loads for concrete box design, the section is always treated as one beam, all load demands (permanent and transient) are distributed evenly to the webs for stress and flexure and proportionally to the slope of the web for shear. Torsion effects are always considered and assigned to the outer webs and the top and bottom slab.

With respect to shear and torsion check, in accordance with Article 5.8.6 of the code, torsion is considered.

5.1 Stress Design 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.

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5 - 2 Stress Design AASHTO-STD-2002

FactorCompLim – f  multiplier; Default Value = 0.4; Typical value(s): 0.4 to _c 0.6. The f  is multiplied by the FactorCompLim to obtain the compression _c 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  is multiplied by the FactorTensLim to obtain the tension limit. _c

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).

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Chapter 5 - Design Concrete Box Girder Bridges

Stress Design AASHTO-LRFD-2007 5 - 3

5.2 Stress Design 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

FactorCompLim – f  multiplier; Default Value = 0.4; Typical value(s): 0.4 to _c 0.6. The f  is multiplied by the FactorCompLim to obtain compression limit. _c FactorTensLim – f  multiplier; Default Value = 0.19 (ksi) 0.5(MPa); _c Typical value(s): 0 to 0.24 (ksi) 0 to 0.63 (MPa)

The f  is multiplied by the FactorTensLim to obtain tension limit _c

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 Sec-tion 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

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5 - 4 Stress Design AASHTO-LRFD-2007

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

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Stress Design AASHTO-LRFD-2007 5 - 5

Figure 5-2 Reinforcement, LRFD 2007 Stress Design AASHTO Box Beam, Type BIII-48

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 =

3 top top 856 51 897 599 19 5 0 9344 ksi 826 170812 P M . . y . . A I       

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5 - 6 Flexure Design AASHTO-LRFD-2007 Bottom fiber stress =

3 top bot 856 51 897 599 19 5 1 139 ksi 826 170812 P M . . y . . A I        

Stresses reported by SAP2000/Bridge:

top fiber stress envelope = 0.9345 ksi bottom fiber stress envelope = 1.13945 ksi

5.3 Flexure Design 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 M_n Nominal flexural resistance M_r Factored flexural resistance t_slabeq Equivalent thickness of slab

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

b_webeq Equivalent thickness of all webs in section A_slab Area of slab

A_PT Area of PT in tension zone

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

fpu Specified tensile strength of prestressing steel (area weighted average

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Flexure Design AASHTO-LRFD-2007 5 - 7

fpy Yield tensile strength of prestressing steel (area weighted average of 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 as specified in Section 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 f  over a zone bounded by the _c 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 the section local axis. It is assumed that the effective width of the flange (slab) in compression is equal to the width of the slab.

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5 - 8 Flexure Design AASHTO-LRFD-2007

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.

slab slabeq slab A t b 

 The equivalent web thickness is evaluated as the summation of all web hori-zontal thicknesses web webeq web 1 n b 



b

 ₁ stress block factor is evaluated in accordance with 5.7.2.2 based on sec-tion f  c

if f  > 28 MPa, then _c ₁ max 0.85 280.05; 0.65 7 c f   _    _   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 resistnegative 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.

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Flexure Design AASHTO-LRFD-2007 5 - 9 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 f k f  

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

1 slab 0.85 PT pu pu c PT pt A f c f f 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 c₁t_slabeq, 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 pu c PT pt A f f b b t c f f b kA y       

 Average stress in prestressing steel fps is calculated in accordance with (eq.

5.7.3.1.1-1). (1 ) ps pu pt c f f k y  

 Nominal flexural resistance Mn is calculated in accordance with (eq.