Bridge Superstructure Design

SAP2000/Bridge

Bridge Superstructure Design Concrete Box Girder Bridges and Precast Concrete Composite Girder Bridges AASHTO LRFD 2002 and 2007

ISO SAP041709M21 Rev. 0 Berkeley, California, USA

Version 14 April 2009

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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.

Contents

Bridge Superstructure Design 1

2

3

Introduction

1-1

1.1

Organization

1-1

1.2

Recommended Reading

1-2

Design Prerequisites 2-1

Load Pattern Types

2-1

2.2

Design Load Combinations

2-3

2.3

Default Load Combinations

2-4

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 3.3.1 User Specified

3-3 3-4

i

SAP2000/Bridge Superstructure Design Guide

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

4

5

ii

3.4

Generate Virtual Combinations 3.4.1 Stress Check 3.4.2 Shear or Moment Check

3-5 3-5 3-6

3.5

Read Forces/Stresses Directly from Girders 3.5.1 Stress Check 3.5.2 Shear or Moment Check

3-6 3-6 3-6

3.6

LLDF Design Example Using Method 2

3-7

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

Design Concrete Box Girder Bridges 5.1

Design Stress AASHTO-STD-2002 5.1.1 Capacity Parameters 5.1.2 Demand Parameters 5.1.3 Algorithm

5-1 5-1 5-2 5-2

5.2

Design Stress AASHTO-LFRD-2007 5.2.1 Capacity Parameters 5.2.2 Algorithm 5.2.3 Stress Design Example

5-2 5-2 5-3 5-3

5.3

Design of Flexure AASHTO-LRFD-2007 5.3.1 Capacity Parameters 5.3.2 Variables 5.3.3 Design Process

5-5 5-5 5-5 5-6

Contents

5.3.4 Algorithm 5.3.5 Flexure Design Example

6

7

5-7 5-9

5.4

Design Shear AASHTO-LRFD-2007 5.4.1 Capacity Parameters 5.4.2 Variables 5.4.3 Design Process 5.4.4 Algorithm 5.4.5 Shear Design Example

5-14 5-14 5-15 5-16 5-17 5-23

5.5

Design Principal Stress AASHTO-LRFD-2007 5.5.1 Capacity Parameters 5.5.2 Demand Parameters 5.5.3 Algorithm

5-30 5-30 5-30 5-31

Design Algorithms for Precast I and U-Girder Bridges 6.1

Design Stress

6-1

6.2

Design Shear 6.2.1 Variables 6.2.2 Design Process 6.2.3 Algorithms 6.2.4 Shear Design Example

6-2 6-3 6-5 6-5 6-8

6.3

Design of Flexural 6.3.1 Variables 6.3.2 Design Process 6.3.3 Algorithms 6.3.4 Flexure Design Capacity Example

6-14 6-14 6-15 6-16 6-18

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

iii

SAP2000/Bridge Superstructure Design Guide

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

iv

Contents

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

v

SAP2000/Bridge Superstructure Design Guide

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

vi

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

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 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.

1-1

SAP2000/Bridge Superstructure Design Guide Chapter 4 describes defining the design request, which includes the design request 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., loading 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 applicable “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.

1-2

Recommended Reading/Practice

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 superstructure. 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 important 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 abbreviation and the load pattern descriptions. Users may choose any name to identify a load pattern type.

Load Pattern Types

2-1

SAP2000/Bridge Superstructure Design Guide

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 nonstructural 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 AASHTO Load Pattern Type 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

2-2

Load Pattern Types

Chapter 2 - Define Loads and Load Combinations

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 combinations 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 Str I Str II Str III Str IV Str V Ext Ev I Ext Ev II

DC DD DW EH EV ES EL PS CR SH

P P P P P P P

LL IM CE BR PL LS

WA

WS

WL

FR

TU

TG

SE

EQ

IC

CT

CV

1.75

1.00

-

-

1.00

0.5/1.20

-

-

-

1.00

-

-

1.00

0.5/1.20

-

-

-

-

-

1.00

1.40

-

1.00

0.5/1.20

 SE  SE  SE

-

1.35

 TG  TG  TG

-

-

-

-

-

1.00

-

-

1.00

0.5/1.20

-

-

-

-

-

1.35

1.00

0.40

1.00

1.00

0.5/1.20

 TG

-

-

-

-

 EQ

1.00

-

-

1.00

-

-

1.00

-

-

-

0.5

1.00

-

-

1.00

-

-

-

1.00

1.00

1.00

-

-

-

-

-

-

-

-

 SE

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

-

-

-

-

FatigueLL, 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.

Design Load Combinations

2-3

SAP2000/Bridge Superstructure Design Guide

Table 2-4 Load Factors for Permanent Loads,

 P , Used in the AASHTO LRFD 2007 Code

Type of Load

Load Factor Maximum Minimum

DC DC: Strength IV only DD: Downdrag DW: Wearing Surfaces and Utilities EH: Horizontal Earth Pressure EL: Locked in Construction Stresses

1.25 1.50 1.40 1.50 1.50 1.00

0.90 0.90 0.25 0.65 0.90 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.

2-4

Default Load Combinations

Chapter 2 - Define Loads and Load Combinations

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.

Default Load Combinations

2-5

SAP2000/Bridge Superstructure Design Guide

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 combinations will allow for some efficiency later when the bridge design requests are defined (see Chapter 4).

2-6

Default Load Combinations

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 superstructures that have a deck that includes precast I or U girders with composite 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 procedures outlined in AASHTO LRFD Section 4.6.2.2.

Algorithm for Determining Live Load Distribution Factors (LLDF)

3-1

SAP2000/Bridge Superstructure Design Guide

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 combinations.  When SAP2000/Bridge reads the LLD factors directly from individual girders (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 accordance with code requirements.

3.2

Determine Live Load Distribution Factors At every section cut, the following geometric information is evaluated to determine the LLD factors.  span lengththe length of span for which moment or shear is being calculated  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  overhangconsists of the horizontal distance from the centerline of the exterior web of the left exterior beam at deck level to the interior edge of the curb or traffic barrier

3-2

Determine Live Load Distribution Factors

Chapter 3 - Determine Live Load Distribution Factors

 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 midway 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 accordance 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 onehalf 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.

Apply LLD Factors

3-3

SAP2000/Bridge Superstructure Design Guide

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 individual 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 parameters 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).

3-4

Apply LLD Factors

Chapter 3 - Determine Live Load Distribution Factors

3.4

Generate Virtual Combinations When the method for determining the live load distribution factors is userspecified, 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 design type of each load case present in the user specified COMBO and multiplies 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 (exterior 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 integrated 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 module, the stresses are multiplied by n (where n is number of girders) to make up for the reduction applied in the Virtual Combinations.

Generate Virtual Combinations

3-5

SAP2000/Bridge Superstructure Design Guide

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 Combinationsone for shear and one for momentgenerated for exterior beam are assigned to both exterior beams, and similarly, Virtual Combinations for interior 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 Request. 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.

3-6

Read Forces/Stresses Directly from Girders

Chapter 3 - Determine Live Load Distribution Factors

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 designated as cross-section “K.” Other tables in 4.6.2.2.2 list the distribution factors 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 compared 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 computations 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 development 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 crosssections with diaphragms or cross-frames, the distribution factor for the exterior beam shall not be taken less than that which would be obtained by assuming that the cross-section deflects and rotates as a rigid cross-section.” This provision was added to the specifications because the original study that developed the distribution factor equations did not consider intermediate diaphragms. 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 jurisdictions 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 LLDF Design Example Using Method 2

3-7

SAP2000/Bridge Superstructure Design Guide

“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 Noncomposite beam moment of inertia, Ig Deck slab thickness, ts Span length, L Girder spacing, S Modulus of elasticity of the beam, EB Modulus of elasticity of the deck, ED C.G. to top of the basic beam C.G. to bottom of the basic beam 1.

2

= 1,085 in 4 = 733,320 in = 8 in. = 110 ft. = 9 ft.-8 in. = 4,696 ksi = 3,834 ksi = 35.62 in. = 36.38 in.

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

= E B ED (4.6.2.2.1-2) = 4696 3834 = 1.225

2.

Calculate eg, the distance between the center of gravity of the noncomposite beam and the deck. Ignore the thickness of the haunch in determining eg eg = NAYT + t s 2 = 35.62 + 8 2 = 39.62 in.

3.

Calculate Kg, the longitudinal stiffness parameter.





Kg = n I  Aeg2 (4.6.2.2.1-1) 2 = 1.225 733 320  1 085  39.62    2 984 704 in 4

3-8

LLDF Design Example Using Method 2

Chapter 3 - Determine Live Load Distribution Factors

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 = 0.075   S 9.5 

0.6

 S L 0.2  K g

 0.075   9.667 9.5 

0.6

12.0 Lt s 3



0.1

 9.667 110 0.2 2 984 704 





3 12 110  8   

= 0.796 lane 5.

0.1

(eq. 1)

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 = 0.06   S 14 

0.4

 S L 0.3  K g

= 0.06   9.667 14 

0.4

12.0 Lt s 3



0.1

 9.667 110 0.3 2984704 

12 100 8  3

= 0.542 lane

0.1

(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 determine 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.

LLDF Design Example Using Method 2

3-9

SAP2000/Bridge Superstructure Design Guide



SC = 1.0  0.20 12.0 Lt s3 K g





0.3

tan 

= 1.0  0.20 12.0 110  8  2 984 704 3



0.3

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. DV = 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 8.

(eq. 4)

Calculate the shear distribution factor for an interior beam with one design 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 9.

(eq. 5)

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.

3 - 10

LLDF Design Example Using Method 2

Chapter 3 - Determine Live Load Distribution Factors

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) 12.

= 0.772 lane

(eq. (7)

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 LLDF Design Example Using Method 2

3 - 11

SAP2000/Bridge Superstructure Design Guide

= 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 13.

(eq. 9) (Strength and Service)

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  de 10 = 0.6  1.83 10 = 0.783

DV = 0.783  0.973  = 0.762 lane 14.

(eq. 10)

Calculate the shear distribution factor for an exterior beam with one design 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 correction 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 calculation 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.

3 - 12

Additional check for rigidly connected girders (4.6.2.2.2d)

LLDF Design Example Using Method 2

Chapter 3 - Determine Live Load Distribution Factors

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  x

2

(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 pattern of girders to each girder (ft.)

Xext = horizontal distance from the center of gravity of the pattern to the exterior girder (ft.) See Figure 1 for dimensions. 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 

LLDF Design Example Using Method 2

3 - 13

SAP2000/Bridge Superstructure Design Guide

= 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

3 - 14

LLDF Design Example Using Method 2

Chapter 3 - Determine Live Load Distribution Factors

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

Distribution factors from Tables in 4.6.2.2.2

Multiple lanes loaded

0.796

0.772

0.973

0.762

Single lane loaded

0.542

0.806

0.782

0.845

Additional check for rigidly connected girders

Multiple lanes loaded

NA

0.776

NA

0.776

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

LLDF Design Example Using Method 2

3 - 15

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, precast 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., parameters 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 superstructure. 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.

Name and Bridge Object

4-1

SAP2000/Bridge Superstructure Design Guide

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

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

4-2

Name and Bridge Object

Chapter 4 - Define a Bridge Design Requests

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 available 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

Name and Bridge Object

4-3

SAP2000/Bridge Superstructure Design Guide

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 determines 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 Parameters form.

Figure 4-3 Bridge Design Request Form

4-4

Station Range

Chapter 4 - Define a Bridge Design Requests

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

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. 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 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 longitudinal torsional rebar required

Design Parameters

4-5

SAP2000/Bridge Superstructure Design Guide

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 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 factor may be specified using either MPa or ksi units for f c 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 multiplies nominal shear resistance to obtain factored resistance for light-weight concrete  Negative limit on strain in nonprestressed longitudinal reinforcement - per 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 reinforcement - 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 determine 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

4-6

Design Parameters

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

Chapter 4 - Define a Bridge Design Requests

4.5

Demand Sets A demand set name is required for each load combination that is to be considered 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.

Demand Sets

4-7

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

Design Stress AASHTO-STD-2002

5-1

SAP2000/Bridge Superstructure Design Guide

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 compression 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 section 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

5-2

Design Stress AASHTO-LRFD-2007

Chapter 5 - Design Concrete Box Girder Bridges

FactorCompLim – f c multiplier; Default Value = 0.4; Typical value(s): 0.4 to 0.6. The f c 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

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 recorded. 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 = 0.150 kcf Concrete unit weight, wc Concrete strength at 28 days, f c = 5.0 ksi Design span = 95.0 ft Prestressing strands: ½ in. dia., seven wire, low relaxation 2 Area of one strand = 0.153 in Ultimate strength fpu = 270.0 ksi = 0.9 ksi Yield strength fpy fpu = 243 ksi Modulus of elasticity, Ep = 28 500 ksi

Design Stress AASHTO-LRFD-2007

5-3

SAP2000/Bridge Superstructure Design Guide

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

5-4

Design Stress AASHTO-LRFD-2007

Chapter 5 - Design Concrete Box Girder Bridges

Reinforcing bars: yield strength, fy

= 60.0 ksi

Section Properties A = area of cross-section of beam h = overall depth of precast beam I = moment of inertia about centroid of the beam yb,yt = distance from centroid to the extreme bottom (top) fiber of the beam

2

= 826 in = 39 in 4 = 170812 in = 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 =  top 

P M3 856.51 897.599 ytop    19.5  0.9344 ksi A I 826 170812

Bottom fiber stress =  top 

P M3 856.51 897.599 ybot    19.5  1.139 ksi A I 826 170812

Stresses reported by SAP2000: top fiber stress envelope bottom fiber stress envelope

5.3

= 0.9345 ksi = 1.13945 ksi

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

Design of Flexure AASHTO-LRFD-2007

5-5

SAP2000/Bridge Superstructure Design Guide

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 f c 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

5-6

Design of Flexure AASHTO-LRFD-2007

Chapter 5 - Design Concrete Box Girder Bridges

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 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.

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. tslabeq 

Aslab bslab

 The equivalent web thickness is evaluated as summation of all web horizontal thicknesses

bwebeq 

nweb

b

web

1

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

Design of Flexure AASHTO-LRFD-2007

5-7

SAP2000/Bridge Superstructure Design Guide

if f c > 28 MPa then 1  max(0.85 

f c  28 0.05; 0.65) 7

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 moment 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 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. 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) k  2(1.04 

f py f pu

)

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

c

APT f pu 0.85 f c 1bslab  kAPT

f pu y pt

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

5-8

Design of Flexure AASHTO-LRFD-2007

Chapter 5 - Design Concrete Box Girder Bridges

If c1  tslabeq , 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). c

APT f pu  0.85 f c (bslab  bwebeq )tslabeq f pu 0.85 f c 1bwebeq  kAPT y pt

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

f ps  f pu (1  k

c ) y pt

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

If the section is a T-section,

 c tslabeq  c   M n  APT f ps  yPT  1   0.85 f c  bslab  bwebeq  tslabeq  1   2  2    2 else

c   M n  APT f ps  yPT  1  2    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.

Design of Flexure AASHTO-LRFD-2007

5-9

SAP2000/Bridge Superstructure Design Guide

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

5 - 10

Design of Flexure AASHTO-LRFD-2007

Chapter 5 - Design Concrete Box Girder Bridges

= 0.150 kcf Concrete unit weight, wc Concrete strength at 28 days, f c = 5.0 ksi (~34.473 MPa) Design span = 95.0 ft Prestressing strands: ½ in. dia., seven wire, low relaxation 2 Area of one strand = 0.153 in = 270.0 ksi Ultimate strength fpu Yield strength fpy = 0.9 ksi = 243 ksi fpu = 28 500 ksi Modulus of elasticity, Ep Reinforcing bars: yield strength, fy

= 60.0 ksi

Section Properties A = area of cross-section of beam h = overall depth of precast beam I = moment of inertia about centroid of the beam yb, yt = distance from centroid to the extreme bottom (top) fiber of the beam

2

= 826 in = 39 in 4 = 170812 in = 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. tslabeq 

Aslab 48  5.5   5.5in bslab 48

Value reported by SAP2000/Bridge = 5.5 in  The equivalent web thickness is evaluated as summation of all web horizontal thicknesses

bwebeq 

nweb

b

web

 5  5  10 in

1

Value reported by SAP2000/Bridge = 10.0 in

Design of Flexure AASHTO-LRFD-2007

5 - 11

SAP2000/Bridge Superstructure Design Guide

Tendons are split into two groups depending on which sign of moment they resistnegative 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 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 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. For each tendon group, an area weighted average of the following values is determined: - sum of tendon areas APTbottom  0.153  6  23   4.437 in 2 2

Value reported by SAP2000/Bridge = 4.437 in

- distance from center of gravity of tendons to extreme compression fiber 23  2  6  4 yPTbottom  39   36.586 in 23  6 Value reported by SAP2000/Bridge = 19.5 + 17.0862 = 36.586 in - specified tensile strength of prestressing steel f pu  270 kip Value reported by SAP2000/Bridge = 270 kip - constant k (eq. 5.7.3.1.1-2) f py   243   k  2  1.04    0.28   2  1.04  270  f pu   

Value reported by SAP2000/Bridge = 0.28 

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

tion f c

If f c > 28 MPa, then f c  28   0.05;0.65  7   34.473  28   0.05;0.65   0.80376  max  0.85  7   Value calculated by SAP2000/Bridge = 0.8037 (not reported)

1  max  0.85 

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Design of Flexure AASHTO-LRFD-2007

Chapter 5 - Design Concrete Box Girder Bridges

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

c

APT f pu 0.85 f c 1bslab  kAPT



f pu y pt

4.437  270  6.91in 270 0.85  5  0.8037  48  0.28  4.437 36.586

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 c1  tslabeq  6.91  0.80376  5.56 in  5.5in , 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). c

APT f pu  0.85 f c (bslab  bwebeq )tslabeq  f pu  0.85 f c 1bwebeq  kAPT y pt 4.437  270  0.85  5(48  10)5.5  7.149 in 270 0.85  5  0.8037  10  0.28  4.437 36.586

Value reported by SAP2000/Bridge = 7.1487 in  Average stress in prestressing steel fps is calculated in accordance with (eq. 5.7.3.1.1-1). c  7.149    f ps  f pu  1  k  270  1  0.28   255.23 ksi  y pt  36.586    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

Design of Flexure AASHTO-LRFD-2007

5 - 13

SAP2000/Bridge Superstructure Design Guide

 c tslabeq  c   M n  APT f ps  yPT  1   0.85 f c  bslab  bwebeq  tslabeq  1   2  2    2 7.149  0.80376    4.437  255.228   36.586   2    7.149  0.80376 5.5   0.85  5  48  10  5.5   2 2    38287.42 kip-in Value calculated by SAP2000/Bridge = 38287.721 kip-in (not reported) Factored flexural resistance is obtained by multiplying Mn by . Mr   M n  1.0  38287.42  38287.42 kip-in 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.

5 - 14

Design Shear AASHTO-LRFD-2007

Chapter 5 - Design Concrete Box Girder Bridges

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

Pu ,Vu 2 , M u 3 , Tu 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) = b cos  web 

bv

Minimum effective horizontal width of web adjusted for presence ducts

be

Minimum effective normal width of shear flow path adjusted to account for presence of ducts

tv

Minimum effective normal width of web = bv cos  web 

web

Distribution factor for web

h

Vertical height of section

dv

Effective vertical height of section = max(0.8h, distance from extreme compression fiber to CG of tensile PT)

Design Shear AASHTO-LRFD-2007

5 - 15

SAP2000/Bridge Superstructure Design Guide



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 applicable 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 variable 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 calculated is the actual, normal cross-section of the bars. The rebar ratio is calculated 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 presence in the web or flange and the minimum controlling effective web and flange thicknesses are evaluated.

5 - 16

Design Shear AASHTO-LRFD-2007

Chapter 5 - Design Concrete Box Girder Bridges

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 element 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 thickness. 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 variable depth members is evaluated as follows:

Inclination angles of the top and bottom slabs are determined  yslab top2  yslab top1    Stat2  Stat1 

 slab top  arctan 

 yslab bot2  yslab bot1    Stat2  Stat1 

 slab bot  arctan  where

Design Shear AASHTO-LRFD-2007

5 - 17

SAP2000/Bridge Superstructure Design Guide

yslab top2 , yslab top1  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. Stat1 , Stat2  stations of adjacent sections. When section being analyzed is “Before” the current section station is Stat2, when section being analyzed is “After” the current section station is Stat1. Therefore statement Stat1  Stat2 is always valid.  The magnitudes of normal forces in slabs are determined as follows:

P M  Pslab top  Aslab top  u  u 3 dslab top  I3  A  P M  Pslab bot  Aslab bot  u  u 3 dslab bot  I3  A  where dslab top , dslab bot 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: Presal top  Pslab top tan  slab top Presal bot  Pslab bot tan  slab bot

 On the basis of the location and inclination of each web, the per-web demand values are evaluated Location Shear and Torsion Check

Outer Web Vuweb abs(Vu2  Presal top  Presal bot )  

cos  |  web |



nweb 1

Inner Web Vuweb abs(Vu2  Presal top  Presal bot )  

cos web

cos web

where  web 

5 - 18

Tuweb Abs(Tu)

cos  |  web |

Design Shear AASHTO-LRFD-2007

Tuweb 0

Chapter 5 - Design Concrete Box Girder Bridges

 Evaluate effective thicknesses

Evaluate dv bv be tv – If bv  0, then

D  0; Avsweb  0; Avtweb  0; Avsflag  2; Avtflag  2 C proceed to report web results WebPassFlag  2,

– If be < 0 then SectionPassFlag = 2 f c

 Evaluate design f c min(

f c , 8.3 MPa)

 Evaluate stress variable K

Calculate extreme fiber stress

 bot 

P M3 P M3 CGbot  top   CGtop  tens  max  top , bot   A I 33 A I 33

– If  tens

|P| A  0.5 f c , then K = 1 else K  1  0.166  f c

where K < 2  Evaluate Vc per web (shear capacity of concrete)

Vcweb  0.1663K  f c bv dv

(5.8.6.5-3)

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

Vsweb 

Vuweb  Vcweb



– If Vsweb  0 then Avsweb  0

Design Shear AASHTO-LRFD-2007

5 - 19

SAP2000/Bridge Superstructure Design Guide

else Avsweb 

Vsweb f y dv

 Verify minimum reinforcement requirement – If Avsweb  0.35t f y (eq. 5.8.2.5-2), then

Avsweb  0.35t f y and Aswebflag  0

else Avswebflag  1  Evaluate nominal capacities

Vsweb  Avsweb f y dv

Vnweb  Vcweb  Vsweb  Evaluate shear D/C for web

Vuweb

 D     C sweb bv dv f c  Evaluate Tcr (eq. 5.8.6.3-2)

Tcr  0.166 K f c 2 A0 be  Evaluate torsion rebar

1 If Tuweb  Tcr then: 3 – Avtflag  0 – Avtweb  0 – Al  0

TorsionEffectsFlag=0 else: 5 - 20

Design Shear AASHTO-LRFD-2007

Chapter 5 - Design Concrete Box Girder Bridges

Avtflag  1 Avtweb 

Al 

Tuweb  A0 2 f y

Tuweb ph  A0 2 f ylong

TorsionEffectsFlag=1  Evaluate combined shear and torsion D/C for web

D    C tweb

Vuweb T  uweb  b d  2 A0 be  v v 1.25 f c

 Evaluate controlling D/C for web

D D then RatioFlag = 0 If      C  sweb  C tweb else RatioFlag=1  D  D D   max    ,   C   C sweb  C tweb  If

D  1 then WebPassFlag=1 C

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

Design Shear AASHTO-LRFD-2007

5 - 21

SAP2000/Bridge Superstructure Design Guide

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

V  V  V A A

Vcsection 

cweb

Vssection

sweb

Vnsection

nweb

Avssection

vsweb

Avtsection

vtweb

Alsection  Al  Evaluate entire section D/C



nweb 1



D     C  ssection

tv

Vuweb  bv dv

nweb 1

tv

f c

This is equivalent to: | Vu |

 D     C  s sec tion



nweb 1

t v dv

f c

and | Vu |

 D     C tsection



nweb 1

 t v dv

| Tu |  2 A0 be

1.25 f c

 Evaluate controlling D/C for section D D   If   then RatioFlag = 0 else RatioFlag = 1  C  ssection  C tsection

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Design Shear AASHTO-LRFD-2007

Chapter 5 - Design Concrete Box Girder Bridges

 D   D D ,   max     C   C  ssection  C tsection  If

D  1 then SectionPassFlag=1 C

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

Design Shear AASHTO-LRFD-2007

5 - 23

SAP2000/Bridge Superstructure Design Guide

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, f c = 5.0 ksi (~34.473 MPa)

Design span = 95.0 ft Prestressing strands: ½ in. dia., seven wire, low relaxation 2 Area of one strand = 0.153 in Ultimate strength fpu = 270.0 ksi Yield strength fpy = 0.9 = 243 ksi fpu Modulus of elasticity, Ep = 28 500 ksi Reinforcing bars: yield strength, fy Section Properties A = area of cross-section of beam h = overall depth of precast beam I = moment of inertia about centroid of the beam

5 - 24

Design Shear AASHTO-LRFD-2007

= 60.0 ksi (~413.68 MPa) 2

2

= 826 in (~532902 mm ) = 39 in (~990.6 mm) 4

4

= 170812 in (~71097322269 mm )

Chapter 5 - Design Concrete Box Girder Bridges

yb,yt = distance from centroid to the extreme bottom (top) fiber of the beam Aslabtop = Aslabbot = 485.5 Ao = (48  5)  (39  5.5) = 2  (48  5 + 39  5.5) Ph

= 19.5 in (~495.3 mm) 2 2 = 264 in (~170322 mm ) 2 2 = 1440.5 in (~929353 mm ) = 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)  All section properties and demands are converted from SAP2000 model units to N, mm.  On the basis of the location and inclination of each web, the per-web demand values are evaluated. Location Shear and Torsion Check

Outer Web Vuweb abs(Vu 2  Presal top  Presal bot )    cos  web

Tuweb Abs(Tu)=515.2E+06

Inner Web Vuweb Tuweb N/A

0 N/A

Design Shear AASHTO-LRFD-2007

5 - 25

abs(148.3E  03  0  0)  1  74151.9 N cos0

where  web 

cos  |  web |



nweb 1

cos  |  web |



cos  | 0 |



cos  | 0 | 1 2

 0.5

 Evaluate effective shear flow path thicknesses

be  min(tfirstweb , t lastweb , t topslabv , t botslabv )  min(127,127,139.7,139.7)  127mm  Evaluate effective web width and normal thickness

Since the web is vertical, bv = tv = 127 mm  Evaluate effective depth

SAP2000/Bridge Superstructure Design Guide

Since M3 < 0 then dv  max(0.8h, ybot  yPTtop )  max(0.8  990.6,495.3  419.1)  914.4mm  Evaluate design

f c

f c  min( f c ,8.3 MPa)  min( 34.473,8.3 MPa)  5.871

 Evaluate stress variable K

Calculate extreme fiber stress P M3 3560 E  03 852 E  06  CGbot   495.3  12.616 MPa. 532902 71097322269 A I 33 P M3 3560 E  03 852 E  06 CGtop   495.3  0.745MPa  top   A I 33 532902 71097322269  tens  max( top , bot )  max( 12.61, 0.745)  0.745MPa

 bot 

 If  tens  0.5 f c then K = 1 false

| 3560 E  03 | |P| 532902 A  1  2.8 else K  1  0.166  5.871 0.166  f c where K < 2 therefore K = 2  Evaluate Vc per web (shear capacity of concrete) (5.8.6.5-3)

Vcweb  0.1663K  f c bv dv  0.1663  2  1.0  5.871  127  914.4  226781N.  Evaluate Vs per web (shear force that is left to be carried by rebar)

Vsweb 

Vuweb  Vcweb





74151.9  0.9  226781  144392 N 0.9

If Vsweb  0 then Avsweb  0 True

5 - 26

Design Shear AASHTO-LRFD-2007

Chapter 5 - Design Concrete Box Girder Bridges

else Avsweb 

Vsweb f y dv

 Verify minimum reinforcement requirement

If Avsweb  0.35t f y (eq. 5.8.2.5-2) then  true Avsweb  0.35t f y 

0.35  127  0.10745mm 2 / mm and Aswebflag  0 413.68

Else Avswebflag  1  Evaluate nominal capacities Vsweb  Avsweb f y dv  0.10745  413.68  914.4  40645N

Vn web  Vcweb  Vsweb  226781  40645  267426 N  Evaluate shear D/C for web

Vuweb 74151.9 D    0.9    0.1208    C sweb bv dv f c 127  914.4  5.871  Evaluate Tcr (eq. 5.8.6.3-2)

Tcr  0.166 K f c 2 A0 be  0.166  2  5.871  2  929353  127  460147 419 Nmm  Evaluate torsion rebar

1 1 If Tuweb  Tcr  515.2 E 6  0.9  460 E 6  false, then: 3 3 Avtflag  1 Avtweb 

Al 

Tuweb 515.2 E 6   0.7444mm 2 / mm  A0 2 f y 0.9  929352  2  413.68

Tuweb ph 515.2e6  3886.2   2893mm 2  A0 2 f ylong 0.9  929352  2  413.68

Design Shear AASHTO-LRFD-2007

5 - 27

SAP2000/Bridge Superstructure Design Guide

TorsionEffectsFlag=1  Evaluate combined shear and torsion D/C for web

Vuweb T 74151.9 515.2 E 6  uweb   bv dv  2 A0 be 0.9  127  914.4 0.9  2  929352  127   1.25  5.871 1.25 f c

D    C tweb

 0.427  Evaluate controlling D/C for web D D If   then RatioFlag = 0  false    C  sweb  C tweb

else RatioFlag =1  true  D  D D  ,     max  0.1208,0.427   0.427  max    C   C  sweb  C tweb  If

D  1 , then WebPassFlag =1  true C

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

Vcsection 

5 - 28

V

cweb

 2  226 781  453562 N

Design Shear AASHTO-LRFD-2007

Chapter 5 - Design Concrete Box Girder Bridges

V  V A A

Vssection 

sweb

 2  40645  81290 N

Vnsection

nweb

 2  267 426  534852 N

Avssection

Avtsection

vsweb

 2  0.10 745  0.2149 mm 2 / mm

vtweb

 2  0.7444887  1.48898mm 2 / mm

Alsection  Al  2893mm 2  Evaluate entire section D/C



nweb 1



D     C  ssection

tv

Vuweb  bv dv

nweb 1

tv

This is equivalent to:

f c | Vu |

 D     C ssection



nweb 1

148.3E 3

t v dv

f c

0.9



 127  914.4  0.1208 2

1

5.871

and | Vu |

 D     C tsection



nweb 1

 t v dv

| Tu |  2 A0 be

1.25 f c 148.3E 3



0.9

 127  914.4 2



515.2 E 6 0.9  2  929352  127

1

1.25  5.871

 0.427

 Evaluate controlling D/C for section D D then RatioFlag = 0  false If      C  ssection  C tsection

else RatioFlag = 1 true

Design Shear AASHTO-LRFD-2007

5 - 29

SAP2000/Bridge Superstructure Design Guide

 D   D D ,   max      max  0.1208,0.427   0.427 C   C ssection  C tsection  If

D  1 , then SectionPassFlag = 1  true C

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

f c multiplier; Default Value = 0.19 (ksi) 0.5(MPa); Typi-

cal value(s): 0 to 0.24 (ksi) 0 to 0.63 (MPa). The FactorTensLim to obtain tension limit

f c is multiplied by the

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 compressive stresses are divided by the FactorCompLim factor. This way the controlling stress can be selected and compared against one compression limit.

5 - 30

Design Principal Stress, AASHTO LRFD 2007

Chapter 5 - Design Concrete Box Girder Bridges

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 centerline 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 (ycoordinate) 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. ShearScaleFactor 

where  web 



 web bweb cos  web

cos(|  web |) nweb 1

cos(|  web |)

and bweb 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.

TorsionScaleFactor 

1 Wt

where Wt  2 A0 t min

A0 = area enclosed by shear flow path, including area of holes if any

Design Principal Stress, AASHTO LRFD 2007

5 - 31

SAP2000/Bridge Superstructure Design Guide

tmin = minimum normal width of shear flow path  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 the 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 Section 5.6.1).

5 - 32

Design Principal Stress, AASHTO LRFD 2007

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 – f c multiplier; Default Value = 0.4; Typical value(s): 0.4 to 0.6. The f c 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

Design Stress

6-1

SAP200/Bridge Superstructure Design Guide

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 f c is read at every point and compression and tension limits are evaluated using the FactorCompLim - f c multiplier and FactorTensLim f ' c multiplier.

The stresses assume linear distribution and take into account axial (P) and either 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-3 cordance with section 5.8.3.4.2; Default Value = 0.4x10 , Typical value(s): -3 0 to 0.4x10

6-2

Design Shear

Chapter 6 - Design Precast Concrete Girder Bridges

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

Vu

Factored shear demand per girder excluding force in tendons

Nu

Applied factored axial force taken as positive if tensile

Mu

Factored moment at the section

V2 c

Shear in Section Cut excluding force in tendons

V2Tot

Shear in Section Cut including force in tendons

Vp

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. Design Shear

6-3

SAP200/Bridge Superstructure Design Guide

dv

Effective shear depth per 5.8.2.9

d girder

Depth of girder

dcompslab Depth of composite slab (includes concrete haunch t2)

d PTBot

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

b

Minimum web width of beam

Aps

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

f pu

Specified tensile strength of prestressing steel

Ep

Pestressing steel Young’s modulus

Avl

Area of nonprestressed steel on the flexural tension side of the member at the section under consideration

Es

Reinforcement Young’s modulus

s

Strain in nonprestressed longitudinal tension reinforcement – eq. 5.8.3.4.2-4

 sLimitPos ,  sLimitNeg = Max and min value of strain in nonprestressed longitudinal tension reinforcement as specified in the Design request

6-4

Ec

Young’s modulus of concrete

Ac

Area of concrete on the flexural tension side of the member

AVS

Area of transverse shear reinforcement per unit length

AVS min

Minimum area of transverse shear reinforcement per unit length per eq. 5.8.2.5

Design Shear

Chapter 6 - Design Precast Concrete Girder Bridges

6.2.2 Design Process The shear resistance is determined in accordance with paragrah 5.8.3.4.2 (derived 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 tensile 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 COMBOMax, Min and ABSand the controlling StepType is reported.  The component in the direction of the applied shear of the effective prestressing force, positive if resisting the applied shear, is evaluated:

Design Shear

6-5

SAP200/Bridge Superstructure Design Guide



Vp 

V2 c  V2Tot n girders

 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 dv  max(0.72  dgirder ,0.9  dPTBot ,dPTBot  0.5  a)  If Mu < 0 then  dv  max  0.72  dgirder ,0.9  (dgirder  0.5  dcompslab ),(dgirder  0.5  dcompslab )  0.5  a 

 If Mu  Vu  Vp  dv then Mu  (Vu  Vp )  dv  The demand/capacity ratio (D/C) is calculated based on the maximum permissible shear capacity at a section in accordance with 5.8.3.2-2

Vu

 Vp V D  C 0.25  f 'c  b  dv  Evaluate numerator and denominator of (eq. 5.8.3.4.2-4)

 snumerator 

Mu dV

 0.5  N u  Vu  Vp  Aps  0.7  f pu

 sdenominator  E p  Aps  Es  Avl  Adjust denominator values as follows

If  sdenominator  0 and  snumerator  0 then  s   sLimitPos and Avl 

 snumerator  E p  Aps s Es

If  snumerator  0 then  sdenominator  E p  Aps  Es  Avl  Ec  Ac

6-6

Design Shear

(5.8.3.2-2)

Chapter 6 - Design Precast Concrete Girder Bridges

 Evaluate (eq. 5.8.3.4.2-4)

s 

 snumerator  sdenominator

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

If

Nu  0.52  f 'c then  s  2   s Agirder

 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

 s  max( s ,  sLimitNeg ) and  s  min( s ,  sLimitPos )  Evaluate the angle  of inclination of diagonal compressive stresses as determined in Article 5.8.3.4

18  29  3500   s  45

(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

Vc  0.083      f 'c  b  dv  Evaluate how much shear demand is left to be carried by rebar

VS 

Vu

s

 V p  Vc

Design Shear

6-7

SAP200/Bridge Superstructure Design Guide

if VS  0 then AVS  0 else AVS 

Vs 1 f y  dv  tan

(per eq. 5.8.3.3-4)

 Check against minimum transverse shear reinforcement

If Vu  0.5  s  Vc  Vp then AVSmin 

0.083   f 'c  b in accordance fy

with (eq. 5.8.2.5-1), else AVS min  0 If VS  0 then AVS  AVSmin else AVS  max( AVSmin , AVS )  Recalculate Vs in accordance with (eq. 5.8.3.3-4)

VS  AVS  f y  dv 

1 tan

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

VU ASLreq  (

MU dv   f

 0. 5 

NU

P



S

 VP  0.5  min(VS ,

Vu



)

tan 

 E p  Aps ) 

AVL  max( AVL , ASLreq )  Assign longitudinal rebar to top or bottom side of girder based on moment sign

If M U  0 then AVLCompSlabU  AVL and AVLBeamBotFlange  0 Else AVLCompSlabU  0 and AVLBeamBotFlange  AVL

6.2.4 Shear Design Example The girder spacing is 9’-8”. The girder type is AASHTO Type VI Girders, 72inch-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.

6-8

Design Shear

1 fy

Chapter 6 - Design Precast Concrete Girder Bridges

Figure 6-1 Shear design example deck section

Figure 6-2 Shear design example beam section

Design Shear

6-9

SAP200/Bridge Superstructure Design Guide

Materials

Concrete strength Prestressed girders 28-day strength, f c

= 6 ksi,

Girder final elastic modulus, Ec = 4,415 ksi Deck slab: 4.0 ksi, Deck slab elastic modulus, Es = 3,834 ksi = 60 ksi Reinforcing steel Yield strength, fy Prestressing strands 0.5-inch-diameter low relaxation strands Grade 270 2 Strand area, Aps = 0.153 in = 243 ksi Steel yield strength, fpy = 270 ksi Steel ultimate strength, fpu Prestressing steel modulus, Ep = 28,500 ksi Basic beam section properties

Depth Thickness of web Area, Ag Ac = Area of concrete on the flexural tension side of the member (bordered at mid depth of the beam + slab height) Moment of inertia, Ig N.A. to top, yt N.A. to bottom, yb P/S force eccentricity e

= 72 in. = 8 in. 2 = 1,085 in

2

= 551 in 4 = 733,320 in = 35.62 in. = 36.38 in. = 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 bslab  9'8"  116in . 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 prestressing force, positive if resisting the applied shear, is evaluated: 

6 - 10

Vp 

Design Shear

V2 c  V2Tot Vp = 0 since no inclined tendons are present. n girders

Chapter 6 - Design Precast Concrete Girder Bridges

 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)

dv  max(0.72  dgirder , 0.9  dPTBot , dPTBot  0.5  a)  max(0.72  80", 0.9  75", 75" 0.5  5.314  0.85)

d v  max(57.6" ,67.5" ,72.74" )  72.74" Value reported by SAP2000/Bridge = 72.74”  Check if Mu  Vu  Vp  dv

Mu  3678  12  44136 kip-in  (319  0)  72.74  23204 kip-in  D/C is calculated based on maximum permissible shear capacity at a section in accordance with 5.8.3.2-2

Vu

319  Vp 0 V D 0.9    0.406 C 0.25  f 'c  b  dv 0.25  6  8  72.74 Value reported by SAP2000/Bridge = 0.406  Evaluate the numerator and denominator of (eq. 5.8.3.4.2-4)

 snumerator  

Mu dV

 0.5  N u  Vu  Vp  Aps  0.7  f pu

3678  12  0.5  0  319  0  6.73  0.7  270  346.2 kip 72.74

 sdenominator  E p  Aps  Es  Avl  28500 ksi  6.73 in 2  191805 kip  Adjust denominator values as follows

Design Shear

6 - 11

SAP200/Bridge Superstructure Design Guide

If  sdenominator  0 and  snumerator  0 then  s   sLimitPos and Avl 

 snumerator  E p  Aps s Es

not applicable

If  snumerator  0 then  sdenominator  E p  Aps  Es  Avl  Ec  Ac  28500  6.73  4415  551.4  26 263461 kip  Evaluate (eq. 5.8.3.4.2-4)

s 

 snumerator 346.2   1.318e-4  sdenominator 2626346

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

Nu  0.52  f 'c then  s  2   s ; not applicable since Nu = 0 Agirder

 Check against limit on strain in nonprestressed longitudinal tension reinforcement as specified in the Design Request and recalculate Avl

 s  max( s ,  sLimitPos )  max( 1.318e-4,  1.318e-4  4)  1.318e-4  Evaluate angle  of inclination of diagonal compressive stresses as determined in Article 5.8.3.4

18    29  3500   s  45   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 4.8   5.3265 1  750   s 1  750  1.318e-4 Value reported by SAP2000/Bridge = 5.3267



6 - 12

Design Shear

Chapter 6 - Design Precast Concrete Girder Bridges

 Evaluate nominal shear resistance provided by tensile stresses in the concrete (eq. 5.8.3.3-3) Vc  0.0316      f 'c  b  dv  0.0316  5.32  1.0  6  8  72.74  239.92 kip Value reported by SAP2000/Bridge = 240.00 kip

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

Vu

s

 Vp  Vc 

319  0  239.6  114.8 kip 0.9

Value reported by SAP2000/Bridge = 114.64 kip If VS  0 then AVS  0 else Vs

AVS 



114.8

 1.43e-2 in 2 /in (eq. 5.8.3.3-4) 1 1 f y  dv  60  72.74  tan  tan 28.5  Check against minimum transverse shear reinforcement

If Vu  0.5  s  Vc  Vp   319.1 kip  0.5  239.6  119.8 kip is true, AVS min 

0.0316   f 'c  b 0.0316  1.0 6  8   0.01032in 2 /in (eq. fy 60

5.8.2.5-1) If VS  0 then AVS  AVS min else AVS  max( AVS min , AVS )  1.43e-2in 2 /2 2

Value reported by SAP2000/Bridge = 1.43e-2in /in  Recalculate Vs in accordance with (eq. 5.8.3.3-4)

1 1  0.0143  60  72.74   114.9 kip tan  tan 28.5 Value reported by SAP2000/Bridge = 114.6 kip VS  AVS  f y  dv 

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

Design Shear

6 - 13

SAP200/Bridge Superstructure Design Guide

ASLreq

   Vu  VU  VP  0.5  min  VS ,    MU  S  1 N     0.5  U  S  E p  Aps   tan  P  dv   f  fy   319  0  0.5  114.9  3678  12  1 0 0.9    0.5    28500  6.73    3176.3 in 2 1.0 tan 28.5  72.74  0.9  60 2 Value reported by SAP2000/Bridge = 0.00 in  no additional longitudinal 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 factored resistance

6.3.1 Variables 

Resistance factor for flexure

Mn

Nominal flexural resistance

Mr

Factored flexural resistance

t slabeq

Thickness of composite slab

bslab

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

6 - 14

bwebeq

Thickness of beam web

Aslab

Tributary area of slab

a

Depth of equivalent stress block in accordance with 5.7.3.2.2.

APT

Area of PT in tension zone

Design of Flexure

Chapter 6 - Design Precast Concrete Girder Bridges

yPT

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

f pu

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

f py

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

f ps

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 fc 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 be taken to be less than 0.65. The flexural resistance is determined according to 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 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.

Design of Flexure

6 - 15

SAP200/Bridge Superstructure Design Guide

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:  All section properties and demands are converted from SAP2000 model units to N, mm.  1 stress block factor is evaluated per 5.7.2.2 based on section fc

If f c >28 MPa, then 1  max(0.85 

f c  28 0.05;0.65) 7

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 moment 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 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. 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 f pu

6 - 16

Design of Flexure

Chapter 6 - Design Precast Concrete Girder Bridges

- constant k (eq. 5.7.3.1.1-2)

f py   k  2  1.04   f pu    Positive moment resistance – first it is assumed that the equivalent compression 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)

c

APT f pu 0.85 f c 1bslab  kAPT

f pu y pt

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 reevaluated. 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).

c

APT f pu  0.85 f c (bslab  bwebeq )tslabeq f pu 0.85 f c 1bwebeq  kAPT y pt

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 f c  b  bw  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 c   f ps  f pu  1  k y pt  

Design of Flexure

6 - 17

SAP200/Bridge Superstructure Design Guide

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

If the section is a T-section, then  c tslabeq  c   M n  APT f ps  yPT  1   0.85 f c  bslab  bwebeq  tslabeq  1   2  2    2

else

c   M n  APT f ps  yPT  1  2    Factored flexural resistance is obtained by multiplying Mn by  .

M r  M n  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, fc

= 6 ksi,

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

6 - 18

Design of Flexure

Chapter 6 - Design Precast Concrete Girder Bridges

Figure 6-3 Flexure capacity design example deck section

Figure 6-4 Flexure capacity design example beam section Design of Flexure

6 - 19

SAP200/Bridge Superstructure Design Guide

Strand area, Aps Steel yield strength, fpy Steel ultimate strength, fpu Prestressing steel modulus, Ep

= 0.153 in2 = 243 ksi = 270 ksi = 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 effective flange width of concrete deck slab is taken as the tributary width. For the interior beam, the bslab  9'8"  116 in . Tendons are split into two groups depending on which sign of moment they resistnegative 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 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 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.  For each tendon group, an area weighted average of the following values is determined: - sum of tendon areas APTBottom  44  0.153  6.732 in 2 2 Value reported by SAP2000/Bridge = 6.732 in - distance from center of gravity of tendons to extreme compression fiber 12  2  12  4  10  6  6  8  4  10 yPTBottom  (72  8)   75 in 12  12  10  6  4 - specified tensile strength of prestressing steel f pu  270 kip Value reported by SAP2000/Bridge = 270 kip

6 - 20

Design of Flexure

Chapter 6 - Design Precast Concrete Girder Bridges

- constant k (eq. 5.7.3.1.1-2) f py   243   k  2  1.04    0.28   2  1.04  f pu  270   

Value reported by SAP2000/Bridge = 0.28 

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

composite slab f c

1 shall be taken as 0.85 for concrete strength not exceeding 4.0 ksi. If f c > 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 fc = 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

c

APTBottom  f pu 0.85  f ' c  1  bslab  k  APTBottom 



f pu yPTBottom

6.732 * 270

 5.314 in 270 75 Value calculated by SAP2000/Bridge = 5.314 in 0.85  4  0.85  116  0.28  6.732 

 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

c 5.314     f ps  f pu  1  k  270   1  0.28    264.64 ksi  yPTBottom  75    Value reported by SAP2000/Bridge = 264.643 ksi  Nominal flexural resistance Mn is calculated in accordance with 5.7.3.2.2-1 Design of Flexure

6 - 21

SAP200/Bridge Superstructure Design Guide

Since the section is rectangular, c  5.314  0.85    M n  APTBottom f ps  yPTBottom  1   6.732  264.64   75   2  2     129 593.17 / 12  10 799.4 kip-ft

Value calculated by SAP2000/Bridge = 107 99 kip-ft (not reported)  Factored flexural resistance is obtained by multiplying Mn by  . Mr   M n  0.9  10 799.4  9 719.5 kip-ft

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

6 - 22

Design of Flexure

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 codethe Preferences  Type of design to be performedthe check type (Section 4.2.1)  Portion of the bridge to be designedthe station ranges (Section 4.1.3)  Overwrites of the Preferencesthe 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 assistance is needed to create the model, a 30-minute Watch and Learn video entitled, ”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.

Description of Example Model

7- 1

SAP2000/Bridge Superstructure Design Guide

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-in tendons (one in each girder) and a jacking force of 2160 kips per tendon.

2

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

7-2

Description of Example Model

Chapter 7 - Run a Bridge Design Request

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 Preferences 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-

Design Preference

7-3

SAP2000/Bridge Superstructure Design Guide

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

7-4

Load Combinations

Chapter 7 - Run a Bridge Design Request

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 combination as StrII (Strength Level II).

Bridge Design Request

7-5

SAP2000/Bridge Superstructure Design Guide

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 command 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 individual 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

7-6

Start Design/Check of the Bridge

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 enveloped 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.

Display Results as a Plot

8-1

Bridge Superstructure Design Guide

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 Tables 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.

8-2

Display Data Tables

Chapter 8 - Display Bridge Design Results

Figure 8-2 Choose Tables for Display form

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

Display Data Tables

8- 3

Bridge Superstructure Design Guide

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 document 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 using one of the File menu > Export commands: SAP2000 MS Access Database.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 Figure 8-4 displays.

Figure 8-4 Choose Tables for Export to Access form

8-4

Advanced Report Writer

Chapter 8 - Display Bridge Design Results

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 possible 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 existing 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

Advanced Report Writer

8- 5

Bridge Superstructure Design Guide

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 appear 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 formatting 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 Report Writer.

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 examined. The following output for negative bending has been pulled from the ConBoxFlexure data table, a portion of which is shown in Figure 8-3:

8-6

Verification

Chapter 8 - Display Bridge Design Results

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,

 f k  2  1.04  PY fPU 

 245.1    0.26444 (Result match)   2  1.04  270   

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

c





APT fPU  0.85 f c bslab  bwebeq tslabeq 0.85 f c 1bwebeq  kAPT

c

APT fPU 0.85 f c 1bwebeq  kAPT

c

fPU YPT

fPU YPT

, for a T-section

, when not a T-section

20.0(270)  270  0.85(4)(0.85)(360)  0.26444(20)    114 

 5.1286 (Results match)

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

 c  5.1286    266.788 (Results match) fPS  fPU  1  k   270  1  0.26444 YPT  144   

Verification

8- 7

Bridge Superstructure Design Guide

 For resisting moment, from eq. 5.7.3.2.2-1,

 c t  c   M N  APT f PS  YPT  1   0.85 f ' c  bSLAB  bwebeq  tslabeq  1  slabeq  2  2    2

c   M N  APT f PS  YPT  1  , when the box section is not a T-section 2  

5.1286(0.85)   M N  20.0(266.788)  144    596646.5 kip-in 2   M R   M N  0.85(596646.5)  536981.8 kip-in (Results match) The preceding calculations are a check of the flexure design output. Other design results for concrete box stress, concrete box shear and concrete box principal 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.

8-8

Verification

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 Officials, 444 North Capital Street, NW Suite 249, Washington, DC 2001

R-1