SAP2000/Bridge
Bridge Superstructure Design Concrete Box Girder Bridges and Precast Concrete Composite Girder Bridges AASHTO LRFD 2002 and 2007
ISO SAP041709M21 Rev. 1 Berkeley, California, USA
Version 14 July 2009
COPYRIGHT Copyright Computers & Structures, Inc., 1978-2009 All rights reserved. ®
The CSI Logo® and SAP2000 are a registered trademarks of Computers & Structures, TM Inc. Watch & Learn is a trademark of Computers & Structures, Inc. Adobe and Acrobat are registered trademarks of Adobe Systems Incorported. AutoCAD is a registered trademark of Autodesk, Inc. The computer program SAP2000 and all associated documentation are proprietary and copyrighted products. Worldwide rights of ownership rest with Computers & Structures, Inc. Unlicensed use of these programs or reproduction of documentation in any form, without prior written authorization from Computers & Structures, Inc., is explicitly prohibited. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior explicit written permission of the publisher. Further information and copies of this documentation may be obtained from: Computers & Structures, Inc. 1995 University Avenue Berkeley, California 94704 USA Phone: (510) 649-2200 FAX: (510) 649-2299 e-mail:
[email protected] (for general questions) e-mail:
[email protected] (for technical support questions) web: www.csiberkeley.com
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
Introduction 1.1 Orga nization
1-1
1.2 Re commended Reading
1-2
Design Prerequisites 2-1
3
Load Pattern Types
2-1
2.2 De sign Load Combinations
2-3
2.3 Default
2-4
Load Combinations
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
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
5
ii
Name and Bridge Object
4-3
4.2 Che ck Type
4-3
4.3 Station
Range
4-4
4.4 De sign Range
4-4
4.5 Dema nd Sets
4-8
4.6
4-8
Live Load Distribution Factors
Design Concrete Box Girder Bridges 5.1
Stress Design AASHTO-STD-2002 5.1.1 Cap acity Parameters 5.1.2 Dema nd Parameters 5.1.3 Algorithm
5-1 5-1 5-2 5-2
5.2
Stress Design AASHTO-LFRD-2007 5.2.1 Cap acity Parameters 5.2.2 Algorithm 5.2.3 Stress Design Example
5-2 5-2 5-3 5-3
5.3 Flexure Design AASHTO-LRFD-2007 5-6 5.3.1 Cap acity Parameters 5.3.2 Variabl es 5.3.3 De sign Process
5-6 5-6 5-7
Contents
5.3.4 Algorithm 5.3.5 Flexure Design Example 5.4 Shear Design AASHTO-LRFD-2007 5-15 5.4.1 Cap acity Parameters 5.4.2 Variabl es 5.4.3 De sign Process 5.4.4 Algorithm 5.4.5 Shear Design Example 5.5
6
Principal Stress Design AASHTO-LRFD-2007 5.5.1 Cap acity Parameters 5.5.2 Dema nd Parameters 5.5.3 Algorithm
5-15 5-15 5-16 5-18 5-24 5-31 5-31 5-31 5-31
Design Multi-Cell Concrete Box Bridges using AMA 6.1 Stress
7
5-8 5-10
Design
6-2
6.2 Shear Design 6.2.1 Variables 6.2.2 De sign Process 6.2.3 Algorithm s
6-3 6-4 6-5 6-6
6.3 Flexure Design 6.3.1 Variables 6.3.2 De sign Process 6.3.3 Algorithm s
6-10 6-10 6-11 6-11
Design Precast Concrete Girder Bridges 7.1 Stress
Design
7.2 Shear Design 7.2.1 Variables 7.2.2 De sign Process 7.2.3 Algorithm s 7.2.4 Shear Design Example 7.3 Flexural Design 7.3.1 Variables 7.3.2 De sign Process 7.3.3 Algorithm s
7-1 7-2 7-3 7-5 7-5 7-8 7-14 7-15 7-16 7-16
iii
SAP2000/Bridge Superstructure Design Guide
7.3.4 Flexure Capacity Design Example
8
Run a Bridge Design Request 8.1
Description of Example Model
8-3
8.3 Load
8-3
8.5
Combinations Design Request
Start Design/Check of Structure
8-5 8-6
Design Output 9.1
Display Results as a Plot
9.2 Display 9.3
Data Tables
Advanced Report Writer
9.4 Verificatio n
References
iv
8-2
8.2 De sign Preferences
8.4 Bridge
9
7-19
9-1 9-7 9-8 9-11
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 Superstructure Design Request Parameters Form 4-5 Figure 5-1 LRFD 2007 Stress Design, AASHTO Box Beam, Type BIII-48
5-4
Figure 5-2 Reinforcement, LRFD 2007 Stress Design AASHTO Box Beam, Type BIII-48
5-4
Figure 5-3 LRFD 2007 Flexure Design Cross-Section, AASHTO Box Beam, Type BIII-48
5-10
Figure 5-4 Reinforcement, LRFD 2007 Flexure Design Cross-Section, AASHTO Box Beam, Type BIII-48 5-10 Figure 5-5 Shear Design Example, AASHTO Box Beam, Type BIII-48
5-23
Figure 5-5 Shear Design Example Reinforcement AASHTO Box Beam, Type BIII-48 5-24 Figure 7-1 Shear design example deck section
7-9
Figure 7-2 Shear design example beam section
7-9
Figure 7-3 Flexure capacity design example deck section
7-19
Figure 7-4 Flexure capacity design example beam section
7-19
Figure 8-1 3D view of example concrete box girder bridge model
8-2
v
SAP2000/Bridge Superstructure Design Guide
Figure 8-2 Elevation view of example bridge
8-2
Figure 8-3 Plan view of the example bridge
8-3
Figure 8-4 Bridge Design Preferences form
8-3
Figure 8-5 Code-Generated Load Combinations for Bridge Design form
8-4
Figure 8-6 Define Load Combinations form
8-4
Figure 8- 7 Define Load Combinations form
8-5
Figure 8-8 Perform Bridge Design - Superstructure
8-6
Figure 8-9 Plot of flexure check results
8-6
Figure 9-1 Plot of flexure check results for the example bridge design model 9-2 Figure 9-2 Select the location on the beam or slab for which results are to be displayed 9-3 Figure 9-3 Bridge Concrete Box Deck Section - External Girders Vertical 9-3 Figure 9-4 Bridge Concrete Box Deck Section - External Girders Sloped 9-4 Figure 9-5 Bridge Concrete Box Deck Section - External Girders Clipped 9 -4 Figure 9-6 Bridge Concrete Box Deck Section - External Girders and Radius 9-5 Figure 9-7 Bridge Concrete Box Deck Section - External Girders Sloped Max 9-5 Figure 9-8 Bridge Concrete Box Deck Section - Advanced
9-6
Figure 9-9 Bridge Concrete Box Deck Section - AASHTO - PCI ASBI Standard 9-6
vi
Figure 9-10 Choose Tables for Display form
9-7
Figure 9-11 Design database Table for AASHTO LRFD 2007 flexure check
9-8
Figure 9-12 Choose Tables for Export to Access form
9-9
Contents
Figure 9-13 Create Custom Report form
9-10
Figure 9-14 An example of the printed output
9-11
vii
Chapter 1 Introduction
This manual describes using SAP2000/Bridge to complete bridge design in accordance with the AASHTO STD 2002 or AASHTO LRFD 2007 code for concrete box girder bridges or the AASHTO 2007 LRFD code for bridges when the superstructure includes Precast Concrete Box bridges with a 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, 6, and 7 provide the algorithms used by SAP2000/Bridge in completing concrete box girder, cast-in-place multi-cell concrete box, and precast concrete bridge design in accordance with the AASHTO code. Chapter 8 explains how to run a Design Request. Chapter 9 describes design output, which can be presented graphically as plots, in data tables, and in reports generated using the Advanced Report Writer feature in SAP2000®.
1.2
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, multicell, 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 effects
ICE
IC
Ice loads
-
IM
Vehicle Dynamic Load Allowance
BRIDGE LL
LL
Vehicular live load
LL SURCHARGE
LS
Live load surcharge
PEDESTRIAN LL
PL
Pedestrian live load
SETTLEMENT
SE
Force effects due to settlement
TEMP GRADIENT
TG
Temperature gradient loads
TEMPERATURE
TU
Uniform temperature effects
STEAM FLOW
WA
Water load and steam pressure
WIND–LIVE LOAD
WL
Wind on live load
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 0.25 1.50 0.65 1.50 0.90 1.00 1.00
EV: Vertical Earth Pressure
1.35 1.00
ES: Earth Surcharge
1.50 0.75
0.90 0.90
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 or concrete multi-cell box. Legend: Girder = beam + tributary area of composite slab for precast composite; web + tributary area of top and bottom slab for multi-cell box Section Cut = all girders present in the cross-section at the cut location
3.1
Algorithm for Determining Live Load Distribution Factors (LLDF) SAP2000/Bridge gives the user a choice of four methods to address distribution of live load to individual girders. Method 1 – The LLD factors are specified directly by the user.
Algorithm for Determining Live Load Distribution Factors (LLDF)
3-1
SAP2000/Bridge Superstructure Design Guide
Method 2 – SAP2000/Bridge calculates the LLD factors by following procedures outlined in AASHTO LRFD Section 4.6.2.2. Method 3 – SAP2000/Bridge reads the calculated live load demands directly from individual girders (available only for Area or Solid models). Method 4 – SAP2000/Bridge distributes the live load uniformly into all girders. It is important to note that to obtain relevant results, the definition of a Moving Load case must be adjusted depending on which method is selected. When the LLD factors are user specified or specified in accordance with the code (Method 1 or 2), only one lane with a MultiLane Scale Factor = 1 should be loaded into a Moving Load cases included in the demand set 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 lengththe length of span for which moment or shear is being calculated the number of girders girder designationthe first and last girder are designated as exterior girders and the other girders are classified as interior girders roadway widthmeasured as the distance between curbs/barriers; medians are ignored
3-2
Determine Live Load Distribution Factors
Chapter 3 - Determine Live Load Distribution Factors
overhangconsists 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 the beamsincludes the area, moment of inertia, torsion constant, center of gravity the thickness of the composite slab t1 and the thickness of concrete slab haunch t2 the tributary area of the composite slabwhich is bounded at the interior girder by the midway distances to neighboring girders and at the exterior girder; includes the entire overhang on one side, and is bounded by the midway distances to neighboring girder on the other side Young’s modulus for both the slab and the beamsangle of skew support. SAP2000/Bridge then evaluates the longitudinal stiffness parameter, Kg, in 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 parameter values are outside the range required by the code, the section cut is excluded from the Design Request. At every section cut, SAP2000/Bridge then evaluates the live load distribution factors for moment and shear for exterior and interior girders using formulas specified in the code (Tables 4.6.2.2.2b-1, 4.6.2.2.2d-1, 4.6.2.2.3a-1 and 4.6.2.2.3b-1). After evaluation, the LLDF values are assigned to individual girders based on their designation (exterior, interior). The value equal to the average of the LLDF calculated for the left and right girders is assigned to both exterior girders. Similarly, all interior girders use the LLDF equal to the 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 of the beams in the entire section cut and dividing the demands by the number of girders. Similarly, P and M3 forces in the composite slab are 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 Combinationsone for shear and one for momentgenerated 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 those 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/Bridge 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.” That provision was added to the specifications because the original study that developed the distribution factor equations did not consider intermediate diaphragms. Application of the provision requires the presence of a sufficient number of intermediate diaphragms whose stiffness is adequate to force the cross-section to act as a rigid section. For prestressed girders, different 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 “Yes” in the “DiaLLDF Design Example Using Method 2
3-7
SAP2000/Bridge Superstructure Design Guide
phragms Present” option, the program follows the procedure outlined in the provision 4.6.2.2.2d. For this example, one deep reinforced concrete diaphragm is located at the midspan of each span. The stiffness of the diaphragm was deemed sufficient to force the cross-section to act as a rigid section; therefore, the provisions of S4.6.2.2.2d apply. Required information: AASHTO Type I-Beam (28/72) Noncomposite beam area, Ag 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
= EB 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
2 = 1.225 733 320 1 085 39.62 2 984 704 in 4
3-8
LLDF Design Example Using Method 2
(4.6.2.2.1-1)
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 in accordance with 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 in accordance with Table 4.6.2.2.3b1. 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 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 Request
This chapter describes the Bridge Design Request, which is defined using the Define menu > Bridge Design > Define Design Request command. Each Bridge Design Request is unique and specifies which bridge object is to be designed, the type of check to be performed (e.g., concrete box stress, 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 the design of a concrete box girder or the AASHTO 2007 LRFD code is available for the design of a Precast I or U Beam with Composite Slab 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 Request
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 Multi-Cell Concrete Box Girder bridge, SAP2000/Bridge provides the following check type options: Concrete Box Stress Concrete Box Flexure Concrete Box Shear
Name and Bridge Object
4-3
SAP2000/Bridge Superstructure Design Guide
For bridge models with precast I or U Beams with Composite Slabs, SAP2000/Bridge provides three check type options, as followings: AASHTO LRFD 2007 Precast Comp Stress Precast Comp Shear Precast Comp Flexure The bold type denotes the name that appears in the check type drop-down list. A detailed description of the design algorithm can be found in Chapter 5 for concrete box girder bridges, in Chapter 6 for multi-cell box girder bridges, and in Chapter 7 for precast I or U beam with composite slabs.
4.3
Station Range The station range refers to the particular zone or portion of the bridge that is to be designed. The user may choose the entire length of the bridge, or specify specific zones using station ranges. Multiple zones (i.e., station ranges) may be specified as part of a single design request. When defining a station range, the user specifies the Location Type, which 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.
4-4
Station Range
Chapter 4 - Define a Bridge Design Request
Figure 4-3 Superstructure Design Request Parameters form Table 4-1 shows the parameters for concrete box girder bridges. Table 4-2 shows the parameters for multi-cell concrete box bridges. Table 4-3 shows the parameters applicable when the superstructure has a deck that includes precast I or U girders with composite slabs. Table 4-1 Design Request Parameters for Concrete Box Girders AASHTO STD 2002 Concrete Box Stress
Resistance Factor - multiplies both compression and tension stress limits Multiplier on f 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
Design Parameters
4-5
SAP2000/Bridge Superstructure Design Guide
Table 4-1 Design Request Parameters for Concrete Box Girders AASHTO LRFD 2007 Concrete Box Stress
Concrete Box Stress, PhiC, - Resistance Factor that multiplies both compression and tension stress limits Concrete Box Stress Factor Compression Limit - Multiplier on f c to calculate the compression stress limit Concrete Box Stress Factor Tension Limit Units - Multiplier on sqrt( f 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 in accordance with Article 5.8.6.2. Concrete Box Shear Rebar Material - A previously defined rebar material label that will be used to determine the area of shear rebar required Longitudinal Torsional Rebar Material - A previously defined rebar material that will be used to determine the area of longitudinal torsional rebar required
Concrete Box Flexure
Concrete Box Flexure, PhiC, - Resistance Factor that multiplies both compression and tension stress limits
Concrete Box Principal
See the Box Stress design parameter specifications
Table 4-2 Design Request Parameters for Multi-Cell Concrete Box AASHTO LRFD 2007 Multi-Cell Concrete Box Stress
Multi-Cell Concrete Box Stress, PhiC, - Resistance Factor that multiplies both compression and tension stress limits Multi-Cell Concrete Box Stress Factor Compression Limit - Multiplier on f c to calculate the compression stress limit Multi-Cell Concrete Box Stress Factor Tension Limit Units - Multiplier on sqrt( f c ) to calculate the tension stress limit, given in the units specified
4-6
Design Parameters
Chapter 4 - Define a Bridge Design Request
Table 4-2 Design Request Parameters for Multi-Cell Concrete Box Multi-Cell Concrete Box Stress Factor Tension Limit - The tension limit factor may be specified using either MPa or ksi units for f c and the resulting tension limit Multi-Cell Concrete Box Shear
Multi-Cell Concrete Box Shear, PhiC, - Resistance Factor that multiplies both compression and tension stress limits Multi-Cell 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 – in accordance with section 5.8.3.4.2; Default Value = -0.4x10-3, Typical value(s): 0 to -0.4x10-3 Positive limit on strain in nonprestressed longitudinal reinforcement - in accordance with section 5.8.3.4.2; Default Value = 6.0x10-3, Typical value(s): 6.0x10-3 PhiC for Nu - Resistance Factor used in equation 5.8.3.5-1; Default Value = 1.0, Typical value(s): 0.75 to 1.0 Phif for Mu - Resistance Factor used in equation 5.8.3.5-1; Default Value = 0.9, Typical value(s): 0.9 to 1.0 Specifies 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. 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
Multi-Cell Concrete Box Flexure
Multi-Cell Concrete Box Flexure, PhiC, - Resistance Factor that multiplies both compression and tension stress limits
Table 4-3 Design Request Parameters for Precast I or U Beams AASHTO Precast Comp Stress
Precast Comp Stress, PhiC, - Resistance Factor that multiplies both compression and tension stress limits Precast Comp Stress Factor Compression Limit - Multiplier on fc to calculate the compression stress limit Precast Comp Stress Factor Tension Limit Units - Multiplier on sqrt(fc) 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 fc and the resulting tension limit
Design Parameters
4-7
SAP2000/Bridge Superstructure Design Guide
Table 4-3 Design Request Parameters for Precast I or U Beams AASHTO Precast Comp Shear
PhiC, - Resistance Factor that multiplies both compression and tension stress limits PhiC, Lightweight Resistance Factor that multiplies nominal shear resistance to obtain factored resistance for light-weight concrete Negative limit on strain in nonprestressed longitudinal reinforcement – in accordance with section 5.8.3.4.2; Default Value = 0.4x10-3, Typical value(s): 0 to -0.4x10-3 Positive limit on strain in nonprestressed longitudinal reinforcement - in accordance with section 5.8.3.4.2; Default Value = 6.0x10-3, Typical value(s): 6.0x10-3 PhiC for Nu - Resistance Factor used in equation 5.8.3.5-1; Default Value = 1.0, Typical value(s): 0.75 to 1.0 Phif for Mu - Resistance Factor used in equation 5.8.3.5-1; Default Value = 0.9, Typical value(s): 0.9 to 1.0 Specifies what method for shear design will be used - either Modified Compression Field Theory (MCFT) in accordance with 5.8.3.4.2 or Vci Vcw method in accordance with 5.8.3.4.3 Currently only the MCFT option is available. A previously defined rebar material label that will be used to 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
4.5
Precast Comp Flexure, PhiC, - Resistance Factor that multiplies both compression and tension stress limits
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 or multi-cell boxes, Live Load Distribution Factors can be specified. LLD factors are described in Chapter 3.
4-8
Demand Sets
Chapter 5 Design Concrete Box Girder Bridges
This chapter describes the algorithms applied in accordance with the AASHTO STD-2002, LRFD 07 code for design and stress check of the superstructure of a concrete box type bridge deck section. In SAP2000, ConcBox design differs from MulticellConcBox design (Chapter 6) with respect to the distribution of loads and the method for checking shear and torsion. In distributing loads for concrete box design, the section is always treated as one beam, all load demands (permanent and transient) are distributed evenly to the webs for stress and flexure and proportionally to the slope of the web for shear. Torsion effects are always considered and assigned to the outer webs and the top and bottom slab. With respect to shear and torsion check, in accordance with Article 5.8.6 of the code, torsion is considered.
5.1
Stress Design AASHTO-STD-2002
5.1.1 Capacity Parameters PhiC – Resistance Factor; Default Value = 1.0, Typical value(s): 1.0 The compression and tension limits are multiplied by the C factor.
Stress Design AASHTO-STD-2002
5-1
SAP2000/Bridge Superstructure Design Guide
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.
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
Stress Design AASHTO-STD-2002
Chapter 5 - Design Concrete Box Girder Bridges
5.2
Stress Design AASHTO-LRFD-2007
5.2.1 Capacity Parameters PhiC – Resistance Factor; Default Value = 1.0, Typical value(s): 1.0 The compression and tension limits are multiplied by the C factor FactorCompLim – f 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 Section 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
Stress Design AASHTO-LRFD-2007
5-3
SAP2000/Bridge Superstructure Design Guide
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 ksi fpu = 243 ksi Modulus of elasticity, Ep = 28 500 ksi
Figure 5-1 LRFD 2007 Stress Design, AASHTO Box Beam, Type BIII-48
5-4
Stress Design AASHTO-LRFD-2007
Chapter 5 - Design Concrete Box Girder Bridges
Figure 5-2 Reinforcement, LRFD 2007 Stress Design AASHTO Box Beam, Type BIII-48
Reinforcing bars: yield strength, fy
= 60.0 ksi
Section Properties A = area of cross-section of beam 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
Stress Design AASHTO-LRFD-2007
5-5
SAP2000/Bridge Superstructure Design Guide
Bottom fiber stress = top
P M3 856.51 897.599 ybot 19.5 1.139 ksi A I 826 170812
Stresses reported by SAP2000/Bridge: top fiber stress envelope = 0.9345 ksi bottom fiber stress envelope = 1.13945 ksi
5.3
Flexure Design AASHTO-LRFD-2007
5.3.1 Capacity Parameters PhiC – Resistance Factor; Default Value = 1.0, Typical value(s): 1.0 The nominal flexural capacity is multiplied by the resistance factor to obtain factored resistance.
5.3.2 Variables
5-6
Resistance factor for flexure
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)
Flexure Design AASHTO-LRFD-2007
Chapter 5 - Design Concrete Box Girder Bridges
fpy
Yield tensile strength of prestressing steel (area weighted average of all tendons in tensile zone)
fps
Average stress in prestressing steel (eq. 5.7.3.1.1-1)
k
PT material constant (eq. 5.7.3.1.1-2)
1
Stress block factor is as specified in Section 5.7.2.2.
5.3.3 Design Process The derivation of the moment resistance of the section is based on approximate stress distribution specified in Article 5.7.2.2. The natural relationship between concrete stress and strain is considered satisfied by an equivalent rectangular concrete compressive stress block of 0.85 f 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 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 the section local axis. It is assumed that the effective width of the flange (slab) in compression is equal to the width of the slab.
Flexure Design AASHTO-LRFD-2007
5-7
SAP2000/Bridge Superstructure Design Guide
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 the 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
f 28 if f c > 28 MPa, then 1 max 0.85 c 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 resistnegative 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.
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Flexure Design AASHTO-LRFD-2007
Chapter 5 - Design Concrete Box Girder Bridges
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.
If c1 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)
Flexure Design AASHTO-LRFD-2007
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SAP2000/Bridge Superstructure Design Guide
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.
Figure 5-3 LRFD 2007 Flexure Design Cross-Section, AASHTO Box Beam, Type BIII-48
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Flexure Design AASHTO-LRFD-2007
Chapter 5 - Design Concrete Box Girder Bridges
Figure 5-4 Reinforcement, LRFD 2007 Flexure Design Cross-Section, AASHTO Box Beam, Type BIII-48
= 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 = 0.9 ksi Yield strength fpy fpu = 243 ksi = 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: Flexure Design AASHTO-LRFD-2007
5 - 11
SAP2000/Bridge Superstructure Design Guide
P M3
= 856.51 kip = 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 Tendons are split into two groups depending on which sign of moment they resistnegative 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
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Flexure Design AASHTO-LRFD-2007
Chapter 5 - Design Concrete Box Girder Bridges
- 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
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 c1 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).
Flexure Design AASHTO-LRFD-2007
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SAP2000/Bridge Superstructure Design Guide
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 7.149 270 1 0.28 f ps f pu 1 k 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
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
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Flexure Design AASHTO-LRFD-2007
Chapter 5 - Design Concrete Box Girder Bridges
5.4
Shear Design 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 light-weight 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 (haunched girder) 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. 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 the center of gravity of the section to the top and bottom fiber Shear Design AASHTO-LRFD-2007
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SAP2000/Bridge Superstructure Design Guide
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(0.8h, distance from extreme compression fiber to center of gravity of tensile PT)
Normal or light-weight concrete factor
Avsweb
Area of shear reinforcement in web per unit length
Avtweb
Area of transverse torsion reinforcement in web per unit length
Al
Area of longitudinal torsion reinforcement
5.4.3 Design Process The shear resistance is determined in accordance with Paragraph 5.8.6 (Shear and Torsion for Segmental Box Girder Bridges). The procedure is not 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.
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Shear Design AASHTO-LRFD-2007
Chapter 5 - Design Concrete Box Girder Bridges
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. 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).
Shear Design AASHTO-LRFD-2007
5 - 17
SAP2000/Bridge Superstructure Design Guide
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
yslab top2 , yslab top1 vertical coordinate of the center of gravity of the top slab at stations 1 and 2. The Y origin is assumed to be at the top of the section and the + direction is up.
Stat1 , Stat2 stations of adjacent sections. When the section being analyzed is “Before,” the current section station is Stat2; when the section being analyzed is “After,” the current section station is Stat1. Therefore, the 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)
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Shear Design AASHTO-LRFD-2007
Chapter 5 - Design Concrete Box Girder Bridges
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 Outer Web Vuweb
Location Shear and Torsion Check
abs(Vu 2 Presal top Presal bot )
Inner Web Vuweb
Tuweb
abs(Vu 2 Presal top Presal bot )
Abs(Tu)
cos web
where web
Tuweb
0
cos web
cos | web |
nweb
1
cos | web |
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 Evaluate design f c min(
f c 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
Shear Design AASHTO-LRFD-2007
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SAP2000/Bridge Superstructure Design Guide
– 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 Evaluate Vs per web (shear force that is left to be carried by rebar)
Vsweb
Vuweb Vcweb
– If Vsweb 0 then Avsweb 0
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
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Shear Design AASHTO-LRFD-2007
(5.8.6.5-3)
Chapter 5 - Design Concrete Box Girder Bridges
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:
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 If then RatioFlag = 0 C sweb C tweb
Shear Design AASHTO-LRFD-2007
5 - 21
SAP2000/Bridge Superstructure Design Guide
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 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
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Shear Design AASHTO-LRFD-2007
Chapter 5 - Design Concrete Box Girder Bridges
nweb 1
D C ssection
Vuweb bv dv
tv
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 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)
Shear Design AASHTO-LRFD-2007
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SAP2000/Bridge Superstructure Design Guide
5.4.5 Shear Design Example Cross Section: AASHTO Box Beam, Type BIII-48, as shown in Figure 5-5.
Figure 5-5 Shear Design Example, AASHTO Box Beam, Type BIII-48
Figure 5-6 Shear Design Example Reinforcement AASHTO Box Beam, Type BIII-48
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Shear Design AASHTO-LRFD-2007
Chapter 5 - Design Concrete Box Girder Bridges
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 yb,yt = distance from centroid to the extreme bottom (top) fiber of the beam Aslabtop = Aslabbot = 485.5 Ao = (48 5) (39 5.5) = 2 (48 5 + 39 5.5) Ph
= 60.0 ksi (~413.68 MPa) 2
2
= 826 in (~532902 mm ) = 39 in (~990.6 mm) 4
4
= 170812 in (~71097322269 mm )
= 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.
Shear Design AASHTO-LRFD-2007
5 - 25
SAP2000/Bridge Superstructure Design Guide
Outer Web Location
Vuweb
Tuweb
abs(Vu 2 Presal top Presal bot )
Shear and Torsion Check
cos web
Abs(Tu)=515.2E+06
Inner Web Vuweb Tuweb N/A
0 N/A
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
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 min
f c
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. A I 33 532902 71097322269 P M3 3560 E 03 852 E 06 495.3 0.745MPa CGtop 532902 71097322269 A I 33
bot top
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Shear Design AASHTO-LRFD-2007
Chapter 5 - Design Concrete Box Girder Bridges
tens max( top , bot ) max(12.61, 0.745) 0.745MPa If tens 0.5 f c then K = 1 false
| 3560 E 03 | |P| 532902 A 1 2.8 else K 1 5.871 0.166 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 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
Shear Design AASHTO-LRFD-2007
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SAP2000/Bridge Superstructure Design Guide
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 127 460 147 419 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
TorsionEffectsFlag=1 Evaluate combined shear and torsion D/C for web
Vuweb T 74151.9 515.2 E 6 uweb D b d A b 2 0 e v v 0.9 127 914.4 0.9 2 929352 127 1.25 5.871 C tweb 1.25 f c 0.427 Evaluate controlling D/C for web
D D If , then RatioFlag = 0 false C sweb C tweb
else RatioFlag =1 true
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Shear Design AASHTO-LRFD-2007
Chapter 5 - Design Concrete Box Girder Bridges
D D D max , max 0.1208, 0.427 0.427 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 the web is not part of the shear flow path for torsion Evaluate entire section values
V V V A A
Vcsection
cweb
2 226 781 453562 N
Vssection
sweb
2 40645 81290 N
nweb
2 267 426 534852 N
Vnsection
Avssection
vsweb
2 0.10 745 0.2149 mm 2 / mm
Avtsection
vtweb
2 0.7444887 1.48898mm 2 / mm
Alsection Al 2893mm 2 Evaluate entire section D/C
D C ssection
nweb 1
tv
Vuweb bv dv
nweb 1
f c
tv
This is equivalent to:
Shear Design AASHTO-LRFD-2007
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SAP2000/Bridge Superstructure Design Guide
| Vu |
D C ssection
nweb 1
148.3E 3
t v dv
0.9
f c
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 If , then RatioFlag = 0 false C ssection C tsection else RatioFlag = 1 true
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)
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Shear Design AASHTO-LRFD-2007
Chapter 5 - Design Concrete Box Girder Bridges
5.5
Principal Stress Design, 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. 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
Principal Stress Design, AASHTO LRFD 2007
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SAP2000/Bridge Superstructure Design Guide
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 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).
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Principal Stress Design, AASHTO LRFD 2007
Chapter 6 Design Multi-Cell Concrete Box Bridges using AMA
This chapter describes the algorithms applied in accordance with the AASHTO-LRFD-07 code for design checks when the superstructure has a deck that includes cast-in-place multi-cell concrete box design and uses the Approximate Method of Analysis, as described in Section 4.6.2.2 of the code. In SAP2000, MulticellConcBox design differs from ConcBox design (Chapter 5) with respect to the distribution of loads and the method for checking shear and torsion. In distributing loads for cast-in-place multi-cell concrete box design, each web and its tributary slabs are designed separately, and live loads are distributed to webs using the Approximate Methods of Analysis in accordance with AASHTO Article 4.6.2.2. Torsion effects are always ignored. When SAP2000 calculates the Live Load Distribution Factors (LLDFs), the section and span qualification criteria stated in AASHTO 4.6.2.2 are verified and noncompliant sections are not designed. With respect to shear and torsion check, in accordance with Article 5.8.3.4.2 of the code, torsion is ignored When the multi-cell concrete box design option is used, moments and shears due to live load are distributed to individual webs in accordance with the factors specified in Articles 4.6.2.2.2 and 4.6.2.2.3 of the code. Torsion effects are
Stress Design
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SAP200/Bridge Superstructure Design Guide
ignored. The user can control if the section is designed as “a whole-width structure” in accordance with Article 4.6.2.2.1 of the code by selecting “Yes” for the “Diaphragms Present” option.
6.1
Stress Design 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 top slab and three points at the bottom fiber of the bottom slab: the left corner, the centerline web and the right corner of the relevant slab tributary area. The location is labeled in the output plots and tables. See Chapter 9, Section 9.1.1. 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 parameters.
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Stress Design
Chapter 6 - Design Multi-Cell Concrete Box Bridges using AMA
6.2
Shear Design 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 light-weight 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 Section 5.8.3.4.2 of the code; or the Vci/Vcw method in accordance with Section 5.8.3.4.3 of the code. Currently only the MCFT option is available.
Negative limit on strain in nonprestressed longitudinal reinforcement in -3 accordance with Section 5.8.3.4.2 of the code; Default Value = 0.4x10 , -3 Typical value(s): 0 to 0.4x10
Positive limit on strain in nonprestressed longitudinal reinforcement in ac-3 cordance with Section 5.8.3.4.2 of the code; Default Value = 6.0x10 , -3 Typical value(s): 6.0x10
PhiC for Nu – Resistance Factor used in Equation 5.8.3.5-1 of the code; 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 of the code; 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.
Shear Design
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SAP200/Bridge Superstructure Design Guide
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, the component is resisting the applied shear
a
Depth of equivalent stress block in accordance with Section 5.7.3.2.2 of the code. Varies for positive and negative moment.
dv
Effective shear depth in accordance with 5.8.2.9 of the code.
dgirder
Depth of girder
dPTBot
Distance from top of top slab to center of gravity of tendons in the bottom of the precast beam
6-4
b
Minimum web width
bv
Effective web width adjusted for presence of prestressing ducts in accordance with Section 5.8.2.9 of the code
Aps
Area of prestressing steel on the flexural tension side of the member
Shear Design
Chapter 6 - Design Multi-Cell Concrete Box Bridges using AMA
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 of the code)
sLimitPos , sLimitNeg = Max and min value of strain in nonprestressed longitudinal tension reinforcement as specified in the Design Request
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 in accordance with Equation 5.8.2.5 of the code
6.2.2 Design Process The shear resistance is determined in accordance with paragraph 5.8.3.4.2 of the code (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. For design, the user should select only those sections that comply with these assumptions by defining appropriate station ranges in the Design Request (see Chapter 4). 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
Shear Design
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SAP200/Bridge Superstructure Design Guide
as a result of flexure. In determining the effective web width at a particular level, one-quarter the diameter of grouted ducts at that level is subtracted from the web width. 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 and the minimum controlling effective web thicknesses are evaluated. The tendon duct is considered as having effect on the web effective thickness even if only part of the duct is within the web boundaries. In such cases, the entire one-quarter of the tendon duct diameter is subtracted from the element thickness. If several tendon ducts overlap in one web (when projected on the vertical axis), the diameters of the ducts are added for the sake of evaluation of the effective thickness. The effective web thickness is calculated at the top and bottom of each duct. Shear design is completed on a per-web basis. Please refer to Chapter 3 for a description of the live load distribution to individual girders.
6.2.3 Algorithms All section properties and demands are converted from SAP2000 model units to N, mm. For every COMBO specified in the Design Request that contains envelopes, a new force demand set is generated. The new force demand set is built up from the maximum tension values of P and the maximum absolute values of V2 and M3 of the two StepTypes (Max and Min) present in the envelope COMBO case. The StepType of this new force demand set is named ABS and the signs of the P, V2 and M3 are preserved. The ABS case follows the industry practice where sections are designed for extreme shear and moments that are not necessarily corresponding to the same design vehicle position. The section cut is designed for all three StepTypes in the COMBOMax, Min and ABSand the controlling StepType is reported. In cases where the demand moment Mu Vu Vp dv , two new force demand sets are generated where Mupos Vu Vp dvpos and Muneg Vu Vp dvneg . The ac-
6-6
Shear Design
Chapter 6 - Design Multi-Cell Concrete Box Bridges using AMA
ronyms “-CodeMinMuPos” and “-CodeMinMuNeg” are added to the end of the StepType name. The signs of the P and V2 are preserved. The component in the direction of the applied shear of the effective prestressing force, positive if resisting the applied shear, is evaluated:
Vp
V2 c V2Tot ngirders
The depth of the equivalent stress block ‘a’ for both positive and negative moment is evaluated in accordance with Equation 5.7.3.1.1 of the code. 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 The demand/capacity ratio (D/C) is calculated based on the maximum permissible shear capacity at a section in accordance with Section 5.8.3.2-2 of the code
Vu
Vp V D C 0.25 f 'c b dv
(5.8.3.2-2)
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
Shear Design
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SAP200/Bridge Superstructure Design Guide
Avl
snumerator E p Aps s Es
If snumerator 0 then sdenominator E p Aps Es Avl Ec Ac 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 the 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.
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Shear Design
Chapter 6 - Design Multi-Cell Concrete Box Bridges using AMA
VS
Vu
s
V p Vc
if VS 0 , then AVS 0 else AVS
Vs 1 f y dv tan
(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 accorfy
dance 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 the longitudinal rebar on the flexure tension side in accordance with (eq. 5.8.3.5-1).
Vu VU VP 0.5 min VS , MU S 1 NU 0.5 E p Aps ASLreq P tan dv f fy AVL max( AVL , ASLreq ) Assign longitudinal rebar to the top or bottom side of the girder based on the moment sign.
If MU 0, then AVLCompSlabU AVL and AVLBeamBotFlange 0, else AVLCompSlabU 0 and AVLBeamBotFlange AVL .
Shear Design
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SAP200/Bridge Superstructure Design Guide
6.3
Flexure Design The following parameter is 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
tslabeq
Thickness of composite slab
bslab
Effective flange width = horizontal width of slab tributary area, measured from out to out
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
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
6 - 10
Average stress in prestressing steel (eq. 5.7.3.1.1-1)
Flexure Design
Chapter 6 - Design Multi-Cell Concrete Box Bridges using AMA
k
PT material constant (eq. 5.7.3.1.1-2)
1
Stress block factor is as specified in Section 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 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 the section local axis. It is assumed that the effective width of the flange (slab) in compression is equal to the width of the slab.
6.3.3 Algorithms At each section:
All section properties and demands are converted from SAP2000 model units to N, mm.
Flexure Design
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SAP200/Bridge Superstructure Design Guide
The equivalent slab thickness is evaluated based on the tributary slab area and the slab width assuming a rectangular shape.
tslabeq
Aslab bslab
1 stress block factor is evaluated in accordance with 5.7.2.2 based on section f c
f 28 If f c > 28 MPa, then 1 max 0.85 c 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 the sign of moment they resistnegative 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 – constant k (eq. 5.7.3.1.1-2)
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Flexure Design
Chapter 6 - Design Multi-Cell Concrete Box Bridges using AMA
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 equivalent slab thickness to determine if the section is a T-section or rectangular section.
If c1 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) c f ps f pu 1 k 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, then c tslabeq c M n APT f ps yPT 1 0.85 f c bslab bwebeq tslabeq 1 2 2 2
else
Flexure Design
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SAP200/Bridge Superstructure Design Guide
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.
6 - 14
Flexure Design
Chapter 7 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.
7.1
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
7-1
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 Parameters.
7.2
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 light-weight 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: 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
7-2
Design Shear
Chapter 7 - 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
7.2.1 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, the component is resisting the applied shear
a
Depth of equivalent stress block in accordance with 5.7.3.2.2. Varies for positive and negative moment.
Design Shear
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SAP200/Bridge Superstructure Design Guide
dv
Effective shear depth in accordance with 5.8.2.9
dgirder
Depth of girder
dcompslab Depth of composite slab (includes concrete haunch t2)
dPTBot
Distance from top of composite slab to center of gravity 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
7-4
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 in accordance with (eq. 5.8.2.5)
Design Shear
Chapter 7 - Design Precast Concrete Girder Bridges
7.2.2 Design Process The shear resistance is determined in accordance with paragraph 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.
7.2.3 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, two new force demand sets are generated. The new force demand sets are built up from the maximum tension values of P and the maximum and minimum values of V2 and minimum values of M3 of the two StepTypes (Max and Min) present in the envelope COMBO case. The StepType of these new force demand sets are named MaxM3MinV2 and MinM3MaxV2, respectively. The signs of all force components are preserved. The two new cases are added to comply with 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 four StepTypes in the COMBOMax, Min, MaxM3MinV2, and MinM3MaxV2and the controlling StepType is reported.
Design Shear
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SAP200/Bridge Superstructure Design Guide
In cases where the demand moment Mu Vu Vp dv , two new force demand
M
V V d
M
V V d
upos u p vpos uneg u p vneg sets are generated where and . The acronyms “-CodeMinMuPos” and “-CodeMinMuNeg” are added to the end of the StepType name. The signs of the P and V2 are preserved.The component in the direction of the applied shear of the effective prestressing force, positive if resisting the applied shear, is evaluated:
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
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Design Shear
(5.8.3.2-2)
Chapter 7 - Design Precast Concrete Girder Bridges
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 . 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).
Design Shear
7-7
SAP200/Bridge Superstructure Design Guide
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
If VS 0, then AVS 0; else AVS
Vs f y dv
1 tan
.
(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 accorfy
dance 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 VU VP 0.5 min VS , MU S 1 NU 0. 5 E p Aps ASLreq P tan dv f fy AVL max( AVL , ASLreq ) Assign longitudinal rebar to top or bottom side of girder based on moment sign.
If MU 0, then AVLCompSlabU AVL and AVLBeamBotFlange 0; else AVLCompSlabU 0 and AVLBeamBotFlange AVL .
7-8
Design Shear
Chapter 7 - Design Precast Concrete Girder Bridges
7.2.4 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. 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 Steel ultimate strength, fpu = 270 ksi = 28,500 ksi Prestressing steel modulus, Ep 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" 116 in. Demands at interior girder Section 2 = station 10’, after girder Section 2, Vu = 319.1 kip; Mu = 3678 kip-ft
Design Shear
7-9
SAP200/Bridge Superstructure Design Guide
Figure 7-1 Shear design example deck section
Figure 7-2 Shear design example beam section
7 - 10
Design Shear
Chapter 7 - Design Precast Concrete Girder Bridges
The component in the direction of the applied shear of the effective prestressing force, positive if resisting the applied shear, is evaluated: Vp
V2 c V2Tot Vp = 0 since no inclined tendons are present. ngirders
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 0 V D 0.9 0.406 C 0.25 f 'c b dv 0.25 6 8 72.74 Vp
Value reported by SAP2000/Bridge = 0.406 Evaluate the numerator and denominator of (eq. 5.8.3.4.2-4)
Design Shear
7 - 11
SAP200/Bridge Superstructure Design Guide
Mu
snumerator
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.
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(1.3184 , 1.318e -4 4) 1.3184 Evaluate angle of inclination of diagonal compressive stresses as determined in Article 5.8.3.4.
7 - 12
Design Shear
Chapter 7 - Design Precast Concrete Girder Bridges
18 29 3500 s 45 29 3500 1.3184 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 1.3184 Value reported by SAP2000/Bridge = 5.3267
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
114.8 1.432 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. AVS
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 ) 1.432 in 2 /2 -2
2
Value reported by SAP2000/Bridge = 1.43 in /in Design Shear
7 - 13
SAP200/Bridge Superstructure Design Guide
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).
ASLreq
Vu VU VP 0.5 min VS , MU S S 1 NU 0.5 E p Aps P tan 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
7.3
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
7.3.1 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
7 - 14
Design of Flexure
Chapter 7 - Design Precast Concrete Girder Bridges
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
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 as specified in Section 5.7.2.2
7.3.2 Design Process The derivation of the moment resistance of the section is based on the 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 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
Design of Flexure
7 - 15
SAP200/Bridge Superstructure Design Guide
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.
7.3.3 Algorithms At each section: All section properties and demands are converted from SAP2000 model units to N, mm. 1 stress block factor is evaluated in accordance with 5.7.2.2 based on section fc
f 28 If f c > 28 MPa, then 1 max 0.85 c 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 resistnegative or positive. A tendon is considered to resist a positive moment when it is located outside of the top fiber compression stress block, and it 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.
7 - 16
Design of Flexure
Chapter 7 - Design Precast Concrete Girder Bridges
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 - 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
Design of Flexure
7 - 17
SAP200/Bridge Superstructure Design Guide
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 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.
7.3.4 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
7 - 18
Design of Flexure
Chapter 7 - Design Precast Concrete Girder Bridges
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 = 0.153 in2 Strand area, Aps Steel yield strength, fpy = 243 ksi Steel ultimate strength, fpu = 270 ksi = 28,500 ksi Prestressing steel modulus, Ep Basic beam section properties
Depth Thickness of web Area, Ag Moment of inertia, Ig N.A. to top, yt N.A. to bottom, yb P/S force eccentricity e
= 72 in. = 8 in. = 1,085 in2 = 733,320 in4 = 35.62 in. = 36.38 in. = 31.380 in.
In accordance with AASHTO LRFD 2007 paragraph 4.6.2.6, the effective flange width of the concrete deck slab is taken as the tributary width. For the interior beam, the bslab 9'8" 116 in .
Design of Flexure
7 - 19
SAP200/Bridge Superstructure Design Guide
Figure 7-3 Flexure capacity design example deck section
Figure 7-4 Flexure capacity design example beam section
7 - 20
Design of Flexure
Chapter 7 - Design Precast Concrete Girder Bridges
Tendons are split into two groups depending on which sign of moment they resistnegative 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 75 in yPTBottom (72 8) 12 12 10 6 4 - 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 f 270 pu
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)
Design of Flexure
7 - 21
SAP200/Bridge Superstructure Design Guide
The distance c between neutral axis and the compressive face is evaluated in accordance with 5.7.3.1.1-4.
APTBottom f pu
c
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 in, 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.
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)
7 - 22
Design of Flexure
Chapter 8 Run a Bridge Design Request
This chapter identifies the steps involved in running a Bridge Design Request. (Chapter 4 explains how to define the Request.) Running the Request applies the following to the specified Bridge Object: Program defaults in accordance with the selected codethe Preferences Type of design to be performedthe check type (Section 4.2.1) Portion of the bridge to be designedthe station ranges (Section 4.1.3) Overwrites of the Preferencesthe design request parameters (Section 4.1.4) Load combinations the demand sets (Chapter 2) Live Load Distribution factors, where applicable (Chapter 3) For this example, the AASHTO LRFD 2007 code is applied to the model of a concrete box-girder bridge shown in Figure 8-1. 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. 0
Description of Example Model
8- 1
SAP2000/Bridge Superstructure Design Guide
Figure 8-1 3D view of example concrete box girder bridge model
8.1
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 2,160 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 8-2 Elevation view of example bridge
8-2
Description of Example Model
Chapter 8 - Run a Bridge Design Request
Figure 8-3 Plan view of the example bridge
8.2
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 8-4 displays.
Figure 8-4 Bridge Design Preferences form
8.3
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 Preferences
8-3
SAP2000/Bridge Superstructure Design Guide
Generated Load Combinations for Bridge Design form shown in Figure 8-5 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.
Figure 8-5 Code-Generated Load Combinations for Bridge Design form
The defined load combinations for this example are shown in Figure 8-6.
Figure 8-6 Define Load Combinations form
8-4
Load Combinations
Chapter 8 - 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.
8.4
Bridge Design Request After the Design menu > Bridge Design > Define Design Request command has been used, the Bridge Design Request form shown in Figure 8-7 displays.
Figure 8-7 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
8-5
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.
8.5
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 8-8:
Figure 8-8 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 8-9, 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 9.
Figure 8-9 Plot of flexure check results
8-6
Start Design/Check of the Bridge
Chapter 9 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.
9.1
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 9-1. 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 9.4.
Display Results as a Plot
9-1
Bridge Superstructure Design Guide
Figure 9-1 Plot of flexure check results for the example bridge design model
9.1.1 Additional Display Examples Use the Display menu > Show Bridge Forces/Stresses command to select, on the example form shown in Figure 9-2, the location along the top or bottom portions of a beam or slab for which stresses are to be displayed. Figures 9-3 through 9-9 illustrate the left, middle, and right portions as they apply to Multicell Concrete Box Sections. Location 1, as an example, refers to the top left selection option while location 5 would refer to the bottom center selection option. Locations 1, 2, and 3 refer to the top left, top center, and top right selection option while locations 4, 5, and 6 refer to the bottom left, bottom center, and bottom right selection options.
9-2
Display Results as a Plot
Chapter 9 - Display Bridge Design Results
Figure 9-2 Select the location on the beam or slab for which results are to be displayed
2
1
3
1
2
3
5
6
Top slab cut line
Bottom slab cut line 4 5 Centerline of the web
6
4
Centerline of the web
Figure 9-3 Bridge Concrete Box Deck Section - External Girders Vertical
Display Results as a Plot
9- 3
Bridge Superstructure Design Guide
2
1 Top
slab
3
1
2
3
5
6
cut
Bottom slab cut line
4 5
4
6 Centerline of the web
Centerline of the web
Figure 9-4 Bridge Concrete Box Deck Section - External Girders Sloped
1 Top
slab
2
3
1
2
cut
Bottom slab cut line
4 5 Centerline of the web
6
4
5 Centerline of the web
Figure 9-5 Bridge Concrete Box Deck Section - External Girders Clipped
9-4
3
Display Results as a Plot
6
Chapter 9 - Display Bridge Design Results
1 Top
slab
2
1
3
2
3
5
6
cut
Bottom slab cut line
4 6
5
4
Centerline of the web
Centerline of the web
Figure 9-6 Bridge Concrete Box Deck Section - External Girders and Radius
2
1
3
1
2
3 1
2
3
5
6
5
6
Top slab cut line
Bottom slab cut line 4, 5
6
4
Centerline of the web Centerline of the web
4
Centerline of the web
Figure 9-7 Bridge Concrete Box Deck Section - External Girders Sloped Max
Display Results as a Plot
9- 5
Bridge Superstructure Design Guide
1
2
3
1
2
6
4
5
3
Top slab cut line
4
Bottom slab cut line
5 Centerline of the web
6
Centerline of the web
Figure 9-8 Bridge Concrete Box Deck Section - Advanced
2
1
3
Top slab cut line
Bottom slab cut line
4
5
6 Centerline of the web
Figure 9-9 Bridge Concrete Box Deck Section - AASHTO - PCI - ASBI Standard
9-6
Display Results as a Plot
Chapter 9 - Display Bridge Design Results
9.2
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 9-10. 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 9-11 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.
Figure 9-10 Choose Tables for Display form
Display Data Tables
9- 7
Bridge Superstructure Design Guide
Figure 9-11 Design database table for AASHTO LRFD 2007 flexure check
The scroll bar along the bottom of the form can be used to scroll to the right to view additional data columns.
9.3
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 9-12 displays.
9-8
Advanced Report Writer
Chapter 9 - Display Bridge Design Results
Figure 9-12 Choose Tables for Export to Access form
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 9-13. 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.
Advanced Report Writer
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Bridge Superstructure Design Guide
Figure 9-13 Create Custom Report form
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 names), 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 9-14 shows an example of the printed output generated by the Report Writer.
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Advanced Report Writer
Chapter 9 - Display Bridge Design Results
Values used in the verification calculations.
Figure 9-14 An example of the printed output
9.4
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 9-10: 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 245.1 k 2 1.04 PY 2 1.04 0.26444 (Results match) 270 fPU
Verification
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Bridge Superstructure Design Guide
For neutral axis depth, from (eq. 5.7.3.1.1-4),
c
f 0.85 f c 1bwebeq kAPT PU YPT APT fPU
c
0.85 f c 1bwebeq kAPT c
APT fPU 0.85 f c bslab bwebeq tslabeq
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 fPS fPU 1 k YPT
5.1286 266.788 (Results match) 270 1 0.26444 144
For resisting moment, from (eq. 5.7.3.2.2-1),
c tslabeq c M N APT f PS YPT 1 0.85 f c bSLAB bwebeq tslabeq 1 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, 6, and 7 for a detailed descriptions of the design algorithms.
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Verification
