Simplified Shear Design of Structural Concrete Members (2005)

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31 CHAPTER 3 PROPOSED CHANGES TO LRFD BRIDGE DESIGN SPECIFICATIONS Sections 3.1 and 3.2 present the two change proposals. The first proposal is the Proposed Simplified Provisions (in a form similar to the AASHTO standard provisions) and the second is a modification to the current LRFD Sectional Design Model. Section 3.3 presents an overview of design examples using these two methods. Section 3.4 presents a justification for the proposals using experimental test data and Section 3.5 presents a justification through a comparison of required strengths of shear reinforcement for many design cases. Section 3.6 examines how change proposals, if imple- mented, may affect design, and then in Section 3.7 their anticipated effect on safety and economy is presented. This effect is evaluated for users of both the AASHTO Standard Specifications and LRFD Sectional Design Model who choose to use either of the two proposed shear design meth- ods. Section 3.8 presents how design database was incorpo- rated in NCHRP Process 12-50. Two changes to the LRFD specifications are presented. Change Proposal 1 is the addition of alternative (or simpli- fied) shear design provisions that reintroduce the idea of bas- ing Vc on the lower of the calculated web-shear (Vcw) and flexure-shear (Vci) strength but where a new and more conservative relationship is used for Vcw and where a variable angle truss model is introduced for evaluating the contribution of the shear reinforcement based on the angle of diagonal cracking. This report refers to this alternative as either the “proposed simplified provisions” or the “Modified Standard Approach.” Change Proposal 2 is that the current tables for determin- ing β and θ, as well as the equation for evaluating
x in the Sectional Design Model (S5.8.3), be replaced by the rela- tionships for β, θ, and
x that have already been incorporated in the Canadian Concrete Design Code for Concrete Struc- tures (CSA A23.3-04). Therefore this change proposal is referred to as the CSA Method. 3.1 CHANGE PROPOSAL 1: PROPOSED SIMPLIFIED APPROACH (MODIFIED VCW AND VCI OR MODIFIED STANDARD) An alternative (or simplified) shear design method is pro- posed to overcome the limitations of the modified LRFD Sectional Design Model as presented in change proposal 2 (CSA Method). After considering the provisions in numerous codes of practice and of other suggested shear design approaches, the research team proposes to adopt a method that shares the approach taken in the current AASHTO Standard Specifica- tions and in ACI 318-02 where, for shear design, the struc- ture is considered to be divided into regions of web-shear and flexure-shear cracking. The ability to estimate a lower bound to the diagonal cracking load for the purpose of service eval- uations was considered important to include in the AASHTO LRFD Specifications, particularly because the AASHTO Standard Specifications will be discontinued in time. The proposed simplified Specifications differ from the current AASHTO Standard Specifications in the expression for Vcw, the assumed angle for θ, the maximum shear stress permit- ted for design, the minimum required amount of shear rein- forcement, and requirements for the amount of longitudinal tension reinforcement that must be developed at the face of the support. Furthermore, the values for Vcw are selected so that they are consistent with the contribution of the concrete to the ultimate shear capacity of a beam in accordance with the crack model with friction concept. The expressions for Vcw are developed so that they can be applied easily to beams with deformed bar reinforcement only, with prestressed rein- forcement only, and all combinations of those reinforce- ments. A need for seamlessness between reinforced and prestressed concrete provisions for shear was not recognized when the AASHTO Standard Specifications for shear in prestressed beams were developed because, at that time, prestressed and reinforced concrete were seen as separate materials. The basis for the proposed simplified provisions is sum- marized below, followed by the specific proposed relation- ships for the simplified (alternative) LRFD shear design specifications. The detailed explanation of the basis for the equations of the proposed simplified provision is given in Appendix F, which is included in NCHRP Web-Only Document 78. 3.1.1 Basis of Proposed Simplified Provisions Web-Shear Cracking Strength, Vcw The estimate of the web-shear cracking force follows directly from Mohr’s Circle of stress.

(Eq. 15) where d is suggested to be not less than 0.8h. The tensile strength of the concrete, ft, can be taken as somewhere between and where f ′c is in psi units. In the proposed provisions, stress is expressed in ksi but it is considered more useful to present the proposal with the stress given in psi units. A tensile cracking strength close to is believed to provide a more accurate estimate of the diagonal cracking strength in the design of the end regions of a fully prestressed member in which there is no effect of flexure while a value of is a better estimate of the diagonal cracking load in a reinforced concrete member or a prestressed member with a low level of prestressing. A transition between those two levels is a function of the level of the prestress and the axial load. Flexure-Shear Cracking Strength,Vci For flexure-shear cracking of prestressed beams, the expression used in the AASHTO Standard Specification is (Eq. 16) where the sum of the second and third terms is an estimate of the shear force at the time of flexural cracking while the first term is the increase in shear that has been observed in experiments for a flexural crack to propagate into a diag- onal crack. Although the concrete contribution, Vc, is taken as an esti- mate of the diagonal cracking load, it must also be a lower bound estimate of the concrete contribution to shear resis- tance at the ultimate limit state. At the ultimate limit state, the concrete contribution is the sum of the shear carried in the compression zone, the shear carried across diagonal crack due to shear-friction (aggregate interlock), direct tension across diagonal cracks, dowel action, and arch action. Many factors influence the contributions of each of these mecha- nisms and attempts to reasonably account for them lead to complicated expressions for Vc. Thus, the approach taken by the research team in developing this simplified approach has been to use a lower bound estimate of the diagonal cracking load that, when added to the calculated stirrup contribution to shear resistance, is shown to provide a conservative esti- mate of the capacity of test beams presented and discussed in Section 3.3. Contribution of Shear Reinforcement, Vs The contribution of the shear reinforcement to shear resis- tance is given in Equation 17. The angle of shear cracking can V f b d V V M M f b dci c w d i cr c w= ′ + + ≥ ′0 6 1 7. . max 2 ′fc 4 ′fc 4 ′fc2 ′fc V f ff b d Vcw t pc t w p= + +1 32 be directly calculated by Mohr’s circle of stress, as shown in Figures B-4 and F-2 in the appendixes; see Equation 18. (Eq. 17) where: (Eq. 18) When fpc = 0, the axial stress is zero, or if flexure-shear cracking governs, then cot θ = 1 (θ = 45 degrees). 3.1.2 Proposed Simplified Provisions The proposed simplified provisions are given here in both ksi and psi units. In order not to imply a greater level of pre- cision in the procedure than can be justified, the coefficients for the expressions in ksi units, as currently used in the LRFD specifications, are rounded off. Web-Shear Cracking Strength (where stress is in ksi units) (Eq. 19) which is equivalent to (where stress is in psi units) (Eq. 20) Flexure-Shear Cracking Strength (where stress is in ksi units) (Eq. 21) which is equivalent to (where stress is in psi units) (Eq. 22) The 0.06 coefficient in Equation 19 establishes a uniform minimum Vc contribution over the length of the member independent of whether a web-shear or flexure-shear region is being designed. The coefficient of 0.06 (ksi units) is also very close to the traditional coefficient of 1.7 (psi units) when it is considered that dv = 0.9d. (Eq. 23) V f b d f ci c v v c ,min . ( ) . / / = ′ ≡ ′ 1 7 1 7 1000 0 psi units . . ( )9 0 0597≡ ′f b dc v v ksi units V f b d V V M M f b dci c v v d i cr c v v= ′ + + ≥ ′0 632 1 9. . max V f b d V V M M f b dci c v v d i cr c v v= ′ + + ≥ ′0 02 0 06. . max V f f b d Vcw c pc v v p= ′+ +( . . )1 9 0 30 V f f b d Vcw c pc v v p= ′ + +( . . )0 06 0 30 cot θ = +1 f f pc t V A f d s s v y = cot θ

Theta when Vcw < Vct (where stress is in ksi units) (Eq. 24) which is equivalent to: (where stress is in psi units) (Eq. 25) This expression was selected so that cot(θ) was equal to 1.0 (θ = 45 degrees) when fpc = 0 (i.e. non-prestressed member). The slope of the influence of fpc on θ provides a good correlation with test data. The complete design procedure is shown in Figure 21. 3.2 CHANGE PROPOSAL 2: MODIFICATION OF LRFD SECTIONAL DESIGN MODEL (S5.8.3) The shear design provisions in the 1994 Canadian Stan- dards Association code for the Design of Concrete Structures (6) were essentially the same as the Sectional Design Model in the first three editions of the LRFD Bridge Design Specifi- cations (1, 7, 17). In order to simplify the CSA shear design provisions, the 2004 code introduced equations for evaluating β and θ that replaced the tables. Furthermore, a new equation for
x was introduced by assuming that θ was 30 degrees when evaluating the influence of shear on the longitudinal strain,
x. Change proposal 2 is the adoption of the CSA relationships for β, θ, and
x. These provisions are herein referred to as the CSA Method. This method is presented below. (Eq. 26) where: (in., psi) : concrete contribution (Eq. 27) and : steel contribution (Eq. 28) As shown in Figure 1, the longitudinal strain, εx, is com- puted at mid-depth of the cross-section by: (Eq. 29a) When εx is negative, it is taken as either zero or recalcu- lated by changing the denominator of Equation 29a such that the equation becomes: (Eq. 29b) where Act is the area of concrete in tension.
x u v u u p ps po s s p ps c c M d N V V A f E A E A E A = + + − − + + 0 5 2 . ( t )
x u v u u p ps po s s p ps M d N V V A f E A E A = + + − − + 0 5 2 . ( ) V A f d s s v y v = +(cot cot )sinθ α α V f b dc c v v= ′β V V V V f b d Vn c s p c v v p= + + ≤ ′ +0 25. cot .θ = + ′ ≤. f f . pc c 1 0 0 095 1 8 cot θ = + ′ ≤. f f . pc c 1 0 3 1 8 33 However
x shall not be taken as less than −0.2 × 10−3. For members having less than minimum shear reinforce- ment, as required by Equation 32, the equivalent crack spac- ing parameter, sxe, is calculated as: (in. units) (Eq. 30) where ag is the maximum aggregate size (in.). Then, the fac- tor accounting for the shear resistance of cracked concrete, β, can be computed from: (in. units) (Eq. 31) The minimum area of shear reinforcement is: (in., psi) (Eq. 32) It should be noted that minimum shear reinforcement is required when the factored shear force exceeds Vc, rather than Vc/2 as required by the ACI 318-02 code. Furthermore, the minimum amount of shear reinforcement is greater than the minimum amount required by ACI 318-02 and the AASHTO Standard Specifications. For members having at least minimum transverse rein- forcement, the angle of the diagonal compression field, θ, is calculated as: (Eq. 33) and the coefficient, β, is obtained from Equation 31 with the equivalent crack spacing parameter, sze, set to 12 inches. In the modified LRFD design provisions presented in Appendix F, the contractor proposes that when the member is not continuous, or cast integrally with the support, the end region is designed by the strut-and-tie method in LRFD Arti- cle 5.6.3 (7) when the design shear stress exceeds 0.18f ′c at the first critical section from the support. This is to guard against a diagonal compression failure that could occur due to fun- neling of the diagonal compression above a simple support. The complete CSA design procedure is presented in Figure 22. 3.3 DISCUSSION OF DESIGN EXAMPLES To illustrate the use of these two proposed methods in dif- ferent design situations, eight design examples were pre- pared. These examples were selected from existing PCI examples, suggestions from project panel members, and new examples selected by the contractor. Each example begins with the completed flexural design at a section with specified θ = +29 7000
x A f b sfv c v y ,min = ′ β = + + 4 8 1 1500 51 39 . ( )( )
x xes s s a xe x g = + 1 38 0 63 . .

34 Figure 21. Flowchart for use of proposed simplified provisions.

35 Figure 22. Flowchart for shear design in accordance with CSA.

factored sectional design forces (Vu, Mu, and Nu). In these examples, the critical sections used are those already avail- able from the designers who provided the case studies on which these examples are based. Example 1: Precast, Pretensioned Noncomposite Box Beam This example demonstrates the shear design at a specific section of a 95-ft single-span AASHTO Type BIII-48 box beam bridge with no skew. The example is based on Exam- ple 9.2 of the PCI Bridge Design Manual [PCI, 1997]. Seven of the 29 0.5-inch diameter strands used for flexural tension reinforcement in the 39-inch-deep precast box beams are debonded. Example 2: Three-Span Continuous Precast, Pretensioned Girders This example is based on Example 9.6 of the PCI Bridge Design Manual [PCI, 1997]. The bridge uses 72-inch bulb tees with harped (draped) pretensioned strands on 110-foot end spans and 120-foot interior span. The beams are made contin- uous for live load by the addition of unstressed reinforcement in the deck in the negative moment region. This example illus- trates the shear design in the negative moment region of a beam made continuous with nonprestressed reinforcement. Example 3: Reinforced Concrete Cap Beam This design example demonstrates the shear design of a section of a non-prestressed 15-ft span cap beam supported on three circular columns of 3-ft diameter. The cap beam supports a 3-lane superstructure consisting of six AASHTO Type IV beams. Example 4: Reinforced Concrete Column and Footing This design example demonstrates shear design for two sections of a reinforced concrete column and footing, which are part of a pier designed by Modjeski and Masters, Inc. In the shear design of the footing, only one-way action is con- sidered for a demonstration of proposals. Example 5: Two-Span Continuous Post- Tensioned Box Bridges in Nevada BERGER/ABAM designed this two-span, cast-in-place, post-tensioned box girder bridge. Spans are 110 and 120 feet for the 5-foot deep box girder. Shear design for positive and negative moment regions, and in the vicinity of the inflection point, are illustrated. 36 Example 6: Multi-Post Bent Cap This design example is for a multi-post bent cap beam 86 feet wide. The beam is supported on four columns distributed at 22 ft centers below the beam. Figure J-20 shows the elevation of the multi-post bent cap beam. The design section is taken at the internal face of the first pier in the first bay. Bridge details were provided by the Tennessee Department of Transportation. Example 7: Type IV Girder This example demonstrates the shear design of a section of a 100-ft span AASHTO Type IV beam bridge. Bridge details were provided by the Texas Department of Trans- portation. The bridge consists of 3 spans with each span sim- ply supported. The composite pretensioned beams are 54 inch deep and have an 8 in. thick deck. Example 8: Segmental Girder This example gives the shear design calculations for a 5-span Precast Balanced Cantilever Bridge constructed using AASHTO-PCI-ASBI segmental box girders. The design sec- tion is taken from the second bay near the support. 3.4 EVALUATION OF SIMPLIFIED PROVISIONS WITH SELECTED TEST DATA Appendix G presents a detailed evaluation of the two pro- posed changes to the LRFD shear design provisions using the selected experimental database. The first of these is the proposed simplified provisions which are a significant mod- ification to the AASHTO standard approach and which intro- duces the use of a variable angle truss model. The second is the proposed modification to the LRFD Sec- tional Design in which the equations replace the tables for evaluating β and θ and a simplified relationship is used for evaluating the strain at mid-depth,
x, that eliminates the dependency on the angle θ. This is essentially the new CSA approach. The selected experimental database consists of 64 rein- forced concrete members and 83 prestressed concrete mem- bers. All of these members contain at least the traditional ACI level of minimum shear reinforcement (ρv fy > 50 psi), have an overall height of a least 20 inches, were cast with concretes that had cylinder compressive strengths of 4000 psi or greater, and had shear span-to-depth ratios at least 1.70 and usually considerably higher. The members were selected from the larger database described in Section 2.3. Appendix G also presents evaluations with the selected database of the current LRFD Specifications and AASHTO standard procedures, an examination of the proposal’s abil- ity to predict cracking strengths, and a more detailed discus-

sion of the accuracy of proposed changes. In this section, the strength ratios are evaluated briefly. Table 5 compares the strength ratios for the 64 selected reinforced concrete and 83 selected prestressed concrete test results. From this table, the following observations can be made about how well these four design provisions predicted the capacity of members in the selected databases: 1. The LRFD and CSA methods provided the most accu- rate and very similar estimates of the shear capacity of both the RC and PC members. With Mean strength ratios ranging from 1.1 to 1.24 and coefficients of variation (COV) for these ratios that range from 0.13 to 0.18, the fit with the test data is considered to be excellent. 2. For the 64 RC members, the proposed simplified pro- visions were slightly less accurate than the LRFD and CSA approaches but far more accurate than the AASHTO standard approach. The proposed simplified provisions and the STD approach had similar Mean values, but the STD approach had a significantly larger COV. This result implies that there are likely to be many more situations in which the STD provisions are less conservative than the proposed simplified provisions. 3. For the 83 PC members, the proposed simplified provi- sions had a larger Mean and a slighter larger COV than the STD specifications. Figure 23 presents the trends in the strength ratios for the proposed simplified provisions as a function of: (a) f ′c; (b) depth, d; (c) percentage of longitudinal reinforcement, ρl; and (d) strength of shear reinforcement provided, ρv fy. The results illustrate a slight trend in strength ratio with the per- centage of longitudinal reinforcement. Some of the most conservative results are for members that contain high lev- els of shear reinforcement. This result is to be expected as the proposed simplified provisions limit the shear strength conservatively to guard against brittle diagonal compressive failures. 37 Figure 24 presents the trends in the strength ratios for the CSA method as a function of: (a) f ′c; (b) depth, d; (c) per- centage of longitudinal reinforcement, ρl; and (d) strength of shear reinforcement provided, ρv fy. The results illustrate no trend in the strength ratio with any of these parameters for the range of values shown. The results presented in Table 5, Figure 23, and Figure 24 give the impression that the proposed simplified provisions are less accurate than the current AASHTO standard meth- ods for prestressed members and are significant less accurate than the CSA method. This may be misleading for, as previ- ously mentioned, the types of members tested in laboratories do not represent well the types of members built in the field. Additionally, most members tested in laboratories were designed to fail near a support while a member in the field must be designed for shear over its entire length. For this rea- son, the fit with experimental test data should only be viewed as one evaluation metric. Another is a comparison of the required amount of shear reinforcement by multiple meth- ods, including analytical results, for a large number of design sections that represent the types of sections designed in prac- tice. This comparison is presented in the next section. 3.5 COMPARISON OF REQUIRED STRENGTH OF SHEAR REINFORCEMENT IN DESIGN DATABASE To further assess the safety and efficiency of the two change proposals, the required amount of shear reinforce- ment (ρv fy) by these two methods, the LRFD and AASHTO standard methods, and program Response 2000 (Response 2000) are compared for a large number of design sections. A summary of these results is presented following a description of the design database. A more complete presentation of the results and the design database is presented in Appendix H, which is included in NCHRP Web-Only Document 78. The design database was developed to cover practical design sections. The database included prestressed and non- prestressed sections, composite and non-composite, as well TABLE 5 Evaluation of approaches for selected RC and PC members

38 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 5000 10000 15000 20000 f'c (psi) V te st /V n, Pr op 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 5000 10000 15000 20000 f'c (psi) V te st /V n, Pr op 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 10 20 30 40 d (in) V te st /V n, Pr op 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 20 40 60 80 100 d (in) V te st /V n, Pr op 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 5.0 10.0 15.0 20.0 ρ l (%) V te st /V n, Pr op 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 2.0 4.0 6.0 8.0 ρ pl (%) V te st /V n, Pr op 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 500 1000 1500 2000 2500 3000 ρ v fy (psi) V te st /V n, ST D 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 500 1000 1500 2000 2500 3000 ρ v fy (psi) V te st /V n, Pr op (a) 64 RCmembers (b) 83 PCmembers as simply-supported and continuous members. All members supported a uniformly distributed load and were designed for flexure to satisfy the requirements of the LRFD specifi- cations. The sections selected for shear design of simply- supported members were at “d”, 0.1L, 0.2L, 0.3L, and 0.4L from the support. The sections selected for shear design of continuous members were at “d”, 0.1L, 0.2L, 0.3L, 0.4L, 0.8L, 0.9L and L-d from the simple support. In order to obtain a range of shear design stress levels and M/V ratios at each of these sections, each member was designed for Figure 23. Comparison of simplified approach predictions and test results.

multiple lengths and to support loads that required different levels of flexural reinforcement (50%, 75%, or 100% of the maximum allowable flexural reinforcement). This led to some shear design stress levels larger than those commonly seen in current design but are still admissible by the LRFD 39 specifications. The six different types of members from which the sections were selected are: 1. A 36-inch Deep Simply-Supported Prestressed I-Beam with 7.5-inch Thick Composite Slab 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 5000 10000 15000 20000 f'c (psi) V te st /V n ,C S A 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 5000 10000 15000 20000 f'c (psi) V te st /V n ,C S A 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 10 20 30 40 d (in) V te st /V n ,C S A 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 20 40 60 80 100 d (in) V te st /V n ,C S A 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 5.0 10.0 15.0 20.0 (%) V te st /V n ,C S A 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 2.0 4.0 6.0 8.0 (%) V te st /V n ,C S A 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 500 1000 1500 2000 2500 3000 ρv ρ I ρ pI fy (psi) V te st /V n ,C S A 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 500 1000 1500 2000 2500 3000 V te st /V n ,C S A (a) 64 RCmembers (b) 83 PCmembers ρv fy (psi) Figure 24. Comparison of CSA 2004 predictions and test results.

2. A 72-inch Deep Simply-Supported Prestressed Bulb- Tee Girder with 7.5 inch Thick Composite Slab 3. A 78-inch Deep Two-Span Continuous Post-Tensioned Box Girder 4. A 36-inch Deep Simply-Supported Rectangular Rein- forced Concrete Beam 5. A 42-inch Deep Simply-Supported T-Shaped Rein- forced Concrete Beam 6. A 36-inch Deep Two-Span Continuous Reinforced Concrete Beam In comparing required amounts of shear reinforcement by the five methods, it is considered particularly important to con- sider the differences between the amounts required by each design method and the amounts required by using program Response 2000. While the required amount by Response 2000 is not a replacement for experimental test data, this program has successfully proven to provide accurate predictions of the shear capacity for members in the experimental database. Since this program is based on a general behavioral model (MCFT) and not calibrated by this beam test data, it is reasonable to expect that the program will provide similarly accurate estimates of the capacity of members in this design database as it did for the members in the experimental test database. In this use of Response 2000, the appropriate ratio of M/V and level of pre- stressing was input and then the amount of shear reinforcement was adjusted until the predicted capacity was equal to Vu/φ. As stated in the previous section, most experimental test data is from small-sized member tests and nearly all failures have occurred near simple supports. Thus, an evaluation of the proposed changes in sections away from the support is 40 required. For this evaluation, the required amount of rein- forcement, ρv fy, required by 5 methods (AASHTO Standard Specifications), the LRFD Sectional Design Model (LRFD), the proposed simplified provisions (p, change proposal 1), the CSA Method (CSA, change proposal 2) and Response 2000 (Response 2000)). This required amount of shear rein- forcement by each design method is given by Equation 34. where Acv is the area of concrete resisting shear. These required amounts are compared for all 473 design cases in a series of tables, charts and plots in Appendix H. All of the design sections in this database are representative of the types of situations to which the proposed simplified pro- visions would be applicable. In reviewing the results, the researchers were particularly interested in identifying those conditions under which any of the methods were either unconservative or particularly different than other provi- sions. For each of the prestressed members, the total number of design sections is further divided into web-shear regions, transition regions, and flexure-shear regions. A transition region is where both web-shear cracking and flexure shear cracking could be expected to occur and which was numeri- cally considered to be when Vcw was greater than Vci but Mu > Mcr. In many of the design cases, minimum shear rein- forcement was required. In these cases, the ratio is shown by a hollow symbol. Otherwise a solid symbol is used. Table 6 summarizes the results of the comparisons by pre- senting the mean and COV of the ratios of the requirement strength of shear reinforcement by each of the four design ρv y u s cvf V V A= −( ) (Eq. 34) TABLE 6 Comparisons of ratios of required strength of shear reinforcement by four design methods to required strength determined by program R2K

methods to the required strength determined by program Response 2000. In this table, only the results from design cases in which all methods required greater than minimum shear reinforcement are considered. From this table, the fol- lowing observations are made: 1. The proposed simplified provisions provided the most conservative estimate of the required amount of shear reinforcement with a mean ratio to the Response 2000 requirements of 1.57. It also had the smallest coefficient of variation of the four design methods of 0.23. If a nor- mal distribution of data is assumed and a strength reduc- tion factor of 0.9 is applied, then it would be expected that in only 6% of cases would sections be under-reinforced relative to the amount of shear reinforcement required by Response 2000. For each of the six design cases, the pro- posed simplified provisions were conservative. 2. The AASHTO Standard Specifications had the lowest mean reinforcement requirement ratio of 1.26. When coupled with a COV of 0.31, this suggests that in 25% of the cases sections would be under-reinforced relative to the amount of shear reinforcement required by Response 2000. The Standard specifications were found to be par- ticularly unconservative for the design of the continuous box beam and somewhat unconservative for the design of the continuous RC beam and Bulb-Tee girders. 3. The LRFD Sectional Design Model and CSA methods had similar mean reinforcement requirement ratios for most of the design cases. This was somewhat expected given that the relationships for β and θ were also derived from the MCFT using the longitudinal strain at mid-depth and similar assumptions as described in Sec- tion 2.1.3 and justified in the paper by Bentz (51). The CSA method was somewhat less conservative for con- tinuous members. For the LRFD Sectional Design Model, the mean reinforcement ratio for all members is 1.42 with a COV of 0.37 suggesting that in 21% of cases sections would be under-reinforced relative to the amount of shear reinforcement required by Response 2000. For the CSA method, the mean reinforcement ratio for all members is 1.40 with a COV of 0.48 sug- gesting that in 28% of cases sections would be under- reinforcement relative to the amount of shear rein- forcement required by Response 2000. If the results from Response 2000 are perfectly correct, then only the proposed simplified provisions come close to satisfying the general design philosophy that less than 5% of designs are unconservative. 3.6 EFFECT OF CHANGE PROPOSALS ON DESIGN PROCESS This section is concerned solely with the effect of the change provisions on design for engineers. It addresses the effort required by the designer, the range of structures that 41 can designed by the provisions, and the use of provisions for evaluating the condition or strength of existing structures in the field. Two change proposals were presented and current design- ers may be using the shear design provisions of either the AASHTO Standard or LRFD specifications. Thus, there are four possible changes that a designer can make: • AASHTO Standard Specifications → LRFD Modified Sectional Design Method (CSA Method) • AASHTO Standard Specifications → LRFD Proposed Simplified Provisions (Modified Standard) • LRFD Sectional Design Model → LRFD Modified Sec- tional Design Model (CSA Method) • LRFD Sectional Design Model → LRFD Proposed Simplified Provisions (Modified Standard) The effect of each of these changes on shear design is dis- cussed below. 3.6.1 AASHTO Standard Specifications → LRFD Modified Sectional Design Method (CSA Method) Differences in the design by the AASHTO Standard Spec- ifications and LRFD Sectional Design Model were presented in Section 1.1.3. Because the CSA Method is only a simpli- fication to the LRFD Sectional Design Model, all of these differences remain the same except item vii as the design procedure by the CSA method is non-iterative. Based on the use of the CSA and AASHTO standard method in preparing the design examples and in calculating the required amounts of shear reinforcement in the design database, it is considered that the CSA method is slightly easier to execute than the AASHTO standard method. It should be reemphasized that the CSA Method enables a section to be designed for a much higher shear design stress than permitted in the AASHTO standard Method. Furthermore, the CSA Method is a com- prehensive design approach capable of designing a section for shear also subjected to the actions of axial load, moment, and prestressing and that it is derived from a complete behav- ior model for shear. By contrast the AASHTO standard method is an empirical approach for the design of prestressed and non-prestressed flexural members justified by a fit of design equations with experimental test data. 3.6.2 AASHTO-Standard Specifications → LRFD Proposed Simplified Provisions (Modified Standard) Given that many states have not yet switched to using the LRFD Bridge Design Specifications, many designers proba- bly will choose to use the proposed simplified provisions because they are more similar in structure to the AASHTO Standard Specifications than is the Modified LRFD Sectional Design Model (CSA Method). The equation for Vcw in the

simplified proposed provisions is somewhat different from that in the Standard specifications in that it can also be used for partially prestressed members and it produces somewhat lower estimates of Vc. The larger difference between the methods is that the proposed simplified provisions introduce the use of a variable angle truss model. 3.6.3 LRFD Sectional Design Model → LRFD Modified Sectional Design Model (CSA Method) The CSA Method was developed to provide a simplified design approach for the entire class of members for which the LRFD Sectional Design Model was developed. The equations for β and θ replace the use of more complicated tables and the new expression for
x reduces the dependence on the angle Theta making the design process non-iterative. These changes greatly simplify the design process. Although evaluating the capacity of a structure is simplified by the CSA method, it is still iterative as
x is a function of the angle θ. 3.6.4 LRFD Sectional Design Model → LRFD Proposed Simplified Provisions (Modified Standard) This will be a more significant switch than going from the LRFD Sectional Design Model to CSA Method because it is returning to the more traditional approach of calculating Vc from the diagonal cracking strength and using an approach justified purely by experimental test data. The proposed sim- plified provisions are not a comprehensive design approach and thus more limited in what they can be used to design. Since the CSA Method and the proposed simplified provi- sions are similarly easy design procedures, the designer is more likely to use the simplified proposed provisions only if the outcome leads to more acceptable levels of required shear reinforcement. 3.7 SAFETY AND ECONOMY OF STRUCTURES DESIGNED BY SIMPLIFIED PROVISIONS In this section the influences of changes in the minimum shear reinforcement requirements, the maximum shear stress limit, and the required amounts of shear reinforcement are examined and evaluated using both experimental test data and design case examples. 3.7.1 Minimum Shear Reinforcement Requirements Both change proposals require the use of the same minimum shear reinforcement requirements as does the 42 current LRFD Sectional Design Model. Therefore, design- ers and owners switching from the AASHTO standard method to the LRFD specifications may be required to include additional shear reinforcement. The justifica- tion for this additional required reinforcement is now discussed. Minimum shear reinforcement is provided to ensure that a member will be able to continue to provide the calculated concrete contribution to shear resistance after diagonal cracking has developed and progressed. The larger the amount and the closer the spacing of the shear reinforcement, the smaller are the crack spacings and crack widths. Provided crack widths are kept sufficiently small, it is considered that shear stresses can be transmitted across cracks. It is this inter- face shear transfer (or aggregate interlock) that contributes significantly to the concrete contribution to shear resistance at the ultimate limit state and effectively eliminates the depth effect on shear strength that occurs for beams without shear reinforcement. Minimum shear reinforcement requirements consist of three components: (i) A minimum required strength (ρv fy) of shear rein- forcement, (ii) Rules for the spacing of the shear reinforcement, and (iii) Rules for when it is necessary to use minimum shear reinforcement In traditional U.S. design practice and in the AASHTO Standard Specifications, the minimum required strength of shear reinforcement is ρv fy = 50 psi, and the maximum spac- ing of reinforcement is d/2. This reinforcement is required for most bridge members when . The exception is wide members, including footings and one-way slabs, where minimum shear reinforcement is not required until . In recent ACI codes and in the AASHTO LRFD specifi- cation, the amount of required minimum shear reinforcement has been increased above traditional levels, as shown in Fig- ure 7, and made a function of concrete strength. This increase was made over the concern that with higher strength con- cretes both the calculated concrete contribution to shear resistance is larger and that cracks become smoother, providing less interface shear transfer resistance. The exper- imental test data presented in Figure 20 illustrated that addi- tional reinforcement was required. Because the minimum reinforcement requirement governs over a large percentage of the span in prestressed concrete bridge members, this increased requirement is a significant improvement in safety at a cost in economy. 3.7.2 Maximum Shear Design Stress Limit As presented in Section 1.1.3, the LRFD Sectional Design Model allows the design of members with shear stresses as V Vu c> φ V Vu c> φ / 2

large as 0.25f ′c which is a very substantial increase over the maximum shear stress permitted in the AASHTO Standard Specifications. Both change proposals suggest imposing a maximum design stress limit of 0.18 f ′c. A maximum design stress lower than that in the current LRFD Sectional Design Method is recommended due to the results of shear tests on large bulb-tee girders conducted in NCHRP Project 12-56 in which the funneling of diagonal compressive stresses to the support was found to substantially magnify the diagonal compressive stresses which then led to compressive failures at loads lower than LRFD predicted capacities. For designers and owners still using the AASHTO Stan- dard Specifications, adoption of either change proposal method will result in a significant increase in the permis- sible shear design stress. This increase enables the same size section to be used to span longer distances or carry heavier loads and can result in significant improvements in economy. For designers and owners using the LRFD Sectional Design Model, this change leads to a significant decrease in the design shear stress limit. Since this change has been shown by testing to be required, it is a significant improve- ment in safety. 3.7.3 Evaluation of Change Proposals Using Experimental Test Results The evaluation of the design provisions using experi- mental test data, as summarized in Table 5, and more fully presented in Appendix G, can be used to draw observations on the changes in safety and economy resulting from the adoption of these change proposals. Due to the limitations of experimental test data, this evaluation is restricted to commenting on the safety and economy of the regions near supports for a limited range of member types. If the test data were representative of bridge girders in practice, then a comparison of the means of the strength ratios, as shown in the first row of Table 5, illustrates that the only potentially significant effects are slight decreases in the required amount of shear reinforcement for non-prestressed members if the CSA method is adopted and a modest increase in the amount of shear reinforcement required for prestressed concrete members if the proposed simplified pro- visions are adopted. The relative safety of the provisions can be evaluated by using the means and standard deviations of the strength ratios shown in Table 5 to calculate the percentage of cases for which the measured capacity is expected to be less than the design strength. A resistance factor of 0.9 and a normal dis- tribution are used in calculating the percentages given in the bottom row of Table 5. The results illustrate that switching from the AASHTO standard method results in a modest increase in safety for reinforced concrete members. All four methods had a very similar and very acceptable level of safety for prestressed concrete members. 43 3.7.4 Evaluation of Change Proposals Using Design Cases Examples In order to evaluate the expected safety and economy for regions away from supports and for members not well repre- sented in the experimental test database, it is useful to com- pare the required strengths of shear reinforcement for a large number of design cases by the four design methods and Response 2000. These design cases covered some design sections over the length of prestressed and non-prestressed members, simple and continuous structures, members with rectangular and I- or T-shaped cross-sections, and members designed to a different percentage of feasible flexural capacity. The required amounts of shear reinforcement are summarized in Table 6. If the predictions of Response 2000 are reasonably accu- rate and the design database representative, then a compari- son of the relative amounts of required reinforcement by each design procedure and Response 2000 is a good measure of the accuracy of each design procedure. The relative values of the means of the requirement ratios in Table 6 illustrate that the procedures, in order of increasing economy, are the AASHTO Standard Specifications, the CSA (change pro- posal 2), the LRFD Sectional Design Model, and the pro- posed simplified provisions (change proposal 1). Of course economy and safety must be examined together. For all methods but the proposed simplified provisions, the actual capacity would be expected to be less than the design strength in about 25 percent of cases. As discussed in Section 3.5, all methods but the simplified proposed specifications were particularly unconservative for continuous members. In only about 6% of the cases are the proposed simplified pro- visions found to be unconservative. It is also useful to examine where the largest differences are between required amounts of shear reinforcement by the different design methods and to assess whether or not these differences are justified. For this examination, the results for a selection of the design cases is shown in Table 7 and plot- ted in Figure 25. Additionally, a comparison of the required amounts of shear reinforcement for the eight complete design examples is summarized in Table 8. These results illustrate that some of the largest differences, particularly as a fraction of each other, are in transition zones (between web and flexure-shear regions) and in flexure-shear regions; see cases 3, 4, 7 and 8. In these cases, the AASHTO standard method is the least conservative of the approaches and sometimes quite unconservative if the predictions of Response 2000 are accurate. The proposed simplified provisions are always conservative while the CSA method is usually conservative relative to the Response 2000 values. Another area of significant differences is near inflection points in continuous members. Inflection point regions are a special transition region and the flexural and shear rein- forcement detailing requirements for that region as a function of the inclined cracking that can develop in the region have not been adequately researched. The wide variation in shear

44 TABLE 7 Comparison of required transverse reinforcement Figure 25. Selected design database.

reinforcement requirements for inflection point regions, as shown by the results for Number 5 in Table 8, effectively illustrates this point. 3.8 UTILIZATION OF NCHRP PROCESS 12-50 Code developers wish to be able to accurately assess the effect of changes in specifications on the safety and economy of the transportation infrastructure. NCHRP Process 12-50 is 45 an infrastructure of tools for making this type of evaluation by providing access to databases of bridge structures. It con- sists of three main components: (1) Generating input data for various design programs; (2) Collecting and Displaying the output on a common viewer (post-processing); and (3) Cre- ating access to the archived data through the World Wide Web. The NCHRP report on Process 12-50 provides sample codes written in Visual Basic, Visual C++, and FORTRAN that developers can use to generate input data. It also presents a common viewer program that enables the developers to find TABLE 8 Comparisons of selected design database

problems with their codes or programs by comparing results. Process 12-50 uses XML format, in which output data can be distributed. Process 12-50 was used in this project to input the design database. The existing NCHRP bridge database was reviewed. However, it was found to not contain the range in member types and shear design stress levels suitable for com- paring required amounts of shear reinforcement by different design code provisions. In addition, the information in the NCHRP database was not sufficient for shear design calcu- lations in accordance with other than AASHTO LRFD specifications. Thus, a new set of members, suitable for shear 46 calculations, was developed. The Design Database intro- duced in Appendix H includes those newly created members. In addition, creating the database was achieved either man- ually or by using spreadsheets to encompass the large range of design stress levels or sectional shapes being used in practice. The NCHRP Viewer program can be used to display the out- put including the required amount of shear reinforcement. This post-process program allows display of basic member data and comparison of the results obtained by various design methods. The Design Database is available in the accompanying CD. Directions for viewing the data, as well as the description of the database, are provided in Appendix I.