Chapter 3 Prestressing with Post-Tensioning
7.4 Strength Limit Verification—Shear
7.4.4 Shear Resistance from Concrete (V c )
The LRFD Specifications provide three different methods for determining the concrete contribution of shear resistance in reinforced and prestressed concrete members in flexural regions. Method 1 and Method 2 are based on the Modified Compression Field Theory.
Method 3, included in AASHTO LRFD specifications for many years, is more empirically derived from shear testing of prestressed girders. The applicability of the three methods is:
• Method 1 (LRFD Article 5.8.3.1)—for non-prestressed members.
• Method 2 (LRFD Article 5.8.3.4.2 or LRFD Appendix B5)—for both prestressed and non- prestressed members, with or without shear reinforcing.
• Method 3 (LRFD Article 5.8.3.4.3)—for both prestressed and non-prestressed concrete members where there is no net axial tensile load and minimum shear reinforcing is provided.
Since the concern of this manual is post-tensioned concrete box girder bridges, this section presents the LRFD requirements for Methods 2 and 3. (Method 1, though not a focus of this manual, can be viewed as a special case within Method 2.)
7.4.4.1 Method 2 (Simplified MCFT)
Method 2 for computing the contribution of the concrete to cross section nominal shear resistance is based on design procedures derived from the Modified Compression Field Theory.
Three primary parameters are used to evaluate concrete shear resistance in this method:
θ = angle, from horizontal, of the inclination of the diagonal compressive strut β = factor indicating ability of diagonally cracked concrete subjected to in-plane
shear and normal stresses to transmit tension.
εx = longitudinal strain in the web of the member
These three parameters are interdependent in the MCFT, with no direct solution available for the wide range of possible girder variables. As a result, the first implementation of MCFT in LRFD presented an iterative approach to evaluating these parameters. This iterative approach is now provided in LRFD Appendix B5 of LRFD. Beginning in 2008, LRFD offered a simplified, non-iterative solution for these parameters. This simplified approach is presented in the Manual.
The LRFD application of MCFT is based on a set of assumptions in modeling actual girder shear behavior. The top of figure 7.34 depicts a portion of a box girder cross section and
Chapter 7 – Longitudinal Analysis & Design 170 of 369 shows shear stress distributions, longitudinal strains and principal compressive stress trajectories. The bottom of figure 7.34 shows the LRFD behavior assumed in design procedures using MCFT. Three significant assumptions are:
• The distribution of shear stress over the depth of the section is taken as the value at mid-depth of the girder using MCFT methods and the computed longitudinal strain at mid-depth. Shear stresses are uniformly distributed over the rectangular area that has a height of dv and width of bv, as calculated in the previous sections.
• The direction of principal compressive stresses remains constant over the height dv.
• The web is modeled by one biaxial member.
Figure 7.34 –Actual vs. MCFT Girders
In the LRFD approach to MCFT shear design, the sectional forces shown in the bottom right of figure 7.34 are placed in equilibrium with forces in an idealized model comprised of compression and tension flanges and the web. The forces in these three elements are calculated as follows:
• The sectional bending moment, Mu, is the resisted force in the flanges multiplied by the lever arm, dv, between them.
• The sectional axial force, Nu, is resisted by forces in the flanges, one-half assumed to be carried by each flange.
• The sectional shear force, Vu, is resisted by the inclined compression strut force, D, which is resolved into horizontal and vertical components. The vertical component of the force is equal to the factored sectional shear force less the value of any vertical component of prestressing acting at the section. The horizontal component is resisted by tension in the flanges.
Chapter 7 – Longitudinal Analysis & Design 171 of 369 Figure 7.35 – MCFT Forces and Longitudinal Strain
The assumed resistance equilibrium for Method 2 is shown above in figure 7.35. Evaluating these forces acting at the center of the tensile force, allows the computation of the strain in the tensile element, εs, which is needed to evaluate the beta and theta terms. The strain is predicted by LRFD Equation 5.8.3.4.2-4:
(Eqn. 7.59)
Where, fpo = average stress in prestressing steel or 0.7fpu
An important difference between the simplified and iterative MCFT approaches in Method 2 is seen in equation 7.59. The longitudinal strain, εs, at the level of the tensile force is used for the simplified approach. The iterative approach presented in LRFD Appendix B5 uses the average longitudinal strain in the section. A longitudinal strain of εs/2 is used, which assumes that the strain at the level of the center of compression is small.
Equation 7.59 includes another adjustment to produce the simplified version of Method 2. By equilibrium, the horizontal component of the compressive strut force divided into the two flanges should be:
(Eqn. 7.60)
The simplified Method 2 fixes the angle θ in this equation such that its cotangent is equal to 2.0 (θ ≈ 26.6º). This reduces the equation for horizontal components in the flanges to:
(Eqn. 7.61)
In using equation 7.61 the following considerations should be applied:
• The absolute value of the factored moment, │Mu│, acting with the factored shear, Vu, should not be taken less than │Vu – Vp │multiplied by the shear depth, dv.
Chapter 7 – Longitudinal Analysis & Design 172 of 369
• The areas of reinforcing bars and prestressing steel not fully developed at the cross section under consideration should be reduced in proportion to their lack of full development.
• If the longitudinal strain, es, calculated by equation 7.61 is negative, it can either be taken equal to zero or recomputed by equation 7.61 where the denominator is replaced with (EsAs + EpAps + EcAc). Ac in this revised denominator is the area of concrete on the flexural side of the member. However, εs should not be taken less than -0.4 x 10-3.
• If the axial tension is large enough to crack the flexural compression face of the section, the value calculated by equation 7.61 should be doubled.
• β and θ may be calculated with values of εs larger than that predicted by equation 7.61.
However, es should not be taken greater than 6.0 x 10-3.
With the sectional forces resolved into the idealized cross section and the longitudinal strain computed, the shear resistance contributed by the concrete can be determined. LRFD Article 5.8.3.3 defines this contribution for the simplified form of Method 2:
(Eqn. 7.62)
The parameter β, for the simplified approach, is defined in LRFD Article 5.8.3.4.2. When the minimum required web (transverse) reinforcing is provided, β is defined as:
(Eqn. 7.63)
In most cases, the transverse requirements at the critical sections of post-tensioned concrete box girders will require more than the minimum shear reinforcing requirements. This reinforcing is often used throughout the span to simplify the tying of reinforcing steel. The spacing of this reinforcing could be increased where demand is lower, even to that less than the minimum requirements. In this case, the LRFD Specifications provides an equation for β when the minimum amount of transverse reinforcing is not provided. In all cases, the minimum transverse reinforcing spacing requirements must be met.
Note: It is interesting to be aware of the forms that AASHTO LRFD equations take as a result of using consistent units. In the case of equation 7.62 the units are ksi for the concrete strength and inches for bv and dv. The parameter β has no units. Historically, the concrete contribution to shear resistance was first expressed as a function of the concrete strength in psi. Later, this contribution was defined as a multiple of the square root of the concrete strength (again in psi), with that multiple typically ranging between a factor of 2 and 4. This multiple is now expressed as the β term, with the constant of 0.0316 added for consistent units of ksi. Though more cumbersome in appearance, it may be that equation is better understood from the historical perspective as:
The concrete strength term in the last expression of this equation is in psi. The engineer is encouraged to consider other LRFD equations of similar form in order to retain the historical development of the code.
( 0.0316 ') ' 1000 1
c c v v c v v
V =β f b d =β f b d
Chapter 7 – Longitudinal Analysis & Design 173 of 369 7.4.4.2 Method 3 (Historical Empirical)
Past editions of the AASHTO Standard Specifications for Bridge Design included an approach to shear design based on the nature of girder cracking initiation. Figure 7.36 shows a portion of a continuous concrete girder with zones that generally define two types of shear cracks and their typical general locations. At the ends of girders and near points of contraflexure girder bending is small and web cracking initiates by principle tensile stresses in the webs large enough to crack the concrete. In regions of significant flexure girders can crack vertically on the tension faces as a result of longitudinal flexural stresses greater than the tensile capacity of the concrete. If cracking continues into the webs, the effect of shear stresses in these regions can change the direction of these cracks in the webs to be inclined shear cracks.
Figure 7.36 – Types and Locations of Reinforced and Prestressed Girder Cracking
The contribution of the concrete in Method 3 is different depending on the type of cracking likely to be found. Two resistance expressions, one for each type of cracking, are defined in LRFD Article 5.8.3.4.3. This article requires that concrete resistance for each expression be evaluated, and the lesser of the two be used for design.
In regions of web shear cracking, the expression for the shear capacity of the concrete is determined from LRFD Equation 5.8.3.4.3-3:
(Eqn. 7.64)
Where: fpc = compressive stress at the center of gravity of the cross section after all losses (ksi)
The first term of equation 7.64 represents, in the form of a shear force, a principle tension in the web required to offset the axial compression from the post-tensioning and a conservative estimate of the tensile strength of the concrete. This combination of terms was empirically derived. The second term of equation 7.64 represents the vertical component of the post- tensioning at the section. When equation 7.64 governs the evaluation of the concrete resistance, Vp in equation 7.52 (LRFD Equation 5.8.3.3-1) is set equal to zero.
In regions of flexure-shear cracking, the expression for the shear capacity of the concrete is determined from LRFD Equation 5.8.3.4.3-1:
(Eqn. 7.65)
Chapter 7 – Longitudinal Analysis & Design 174 of 369 Where: Vd = shear force at section do to the unfactored combination of DC and DW (kips)
Vi = factored shear force at section due to externally applied loads occurring simultaneously with Mmax (kips)
Mcre = moment causing flexural cracking moment at section due to externally applied loads (kip-in)
Mmax = moment causing flexural cracking moment at section due to externally applied loads (kip-in)
The second and third terms of equation 7.65 represent a shear force consistent with a bending moment that causes cracking at the section. The first term of equation 7.65 is an empirically derived expression that accounts for an increase concrete resistance developed as the flexure crack transforms into a shear crack.