Chapter 3 Prestressing with Post-Tensioning
7.4 Strength Limit Verification—Shear
7.4.3 Sectional Model Nominal Shear Resistance
Shear design of reinforced and prestressed concrete girders in the AASHTO LRFD specifications in flexural regions has long been based on a truss analogy. Figure 7.29 shows a typical parallel chord truss for a constant depth beam. The beam is loaded with a concentrated vertical load at mid-span. The vertical force is carried to the supports through successive inclined compression struts. Vertical tension members lift the vertical component at the bottom of a diagonal strut to the top of the adjacent inclined strut. The horizontal component of the diagonal strut is held in equilibrium by the compressive forces in top chord members and tensile forces in the bottom chord members.
Figure 7.29 – Web Width based on Horizontal Widths
Using truss analogies for shear design of reinforced concrete beams has its origins in the beginning of the last century. Early applications used shear reinforcing to carry the entire shear forces predicted in the vertical tension members of the truss. Orientation of the inclined compressive struts (θ) for these early solutions was typically 45 degrees from horizontal.
Later testing of concrete beams reinforced using truss analogies revealed shear strengths greater than that provided by the reinforcing alone (Vs in LRFD). The additional shear resistance was observed to be a somewhat complex combination of behaviors that were a function of the magnitude and nature of beam cracking under ultimate loads. Tests of this nature led to the empirical development of a shear resisting mechanism associated with the girder concrete (Vc in LRFD).
Still later work in the 1980’s advanced the estimation of the concrete contribution from a purely empirically based solution to a more analytical approach using the Modified Compression Field Theory (MCFT). The MCFT methods still have an empirical component, but one based only on the nature of reinforced webs to resist in-plane forces. The LRFD Specifications provide both empirically based and MCFT based methods for determining the component of shear resistance provided by the girder concrete.
The nominal shear resistance of a reinforced or prestressed concrete member with transverse reinforcing is presented as the lesser of that predicted by LRFD Equations 5.8.3.3-1 and 5.8.3.3-2:
(Eqn. 7.52)
Chapter 7 – Longitudinal Analysis & Design 166 of 369 (Eqn. 7.53)
Where, Vc = shear resistance provided by the cross section concrete Vs = shear resistance provided by transverse reinforcing
Vp = shear resistance provided by the component of the effective prestressing force in the direction of the applied shear
bv = the effective web width (LRFD Article 5.8.2.9) dv = the effective shear depth (LRFD Article 5.8.2.9)
The three components of resistance will be discussed in the following sections. This section continues by providing more detail with regard to girder dimensions to be used for shear design.
7.4.3.1 Effective Web Width
LRFD Article 5.8.2.9 defines the effective web width as being measured parallel to the neutral axis. A literal interpretation of this article is depicted in figure 7.30 for the cross section of design example 1. The widths of all webs in this cross section, vertical and inclined, are 12 inches perpendicular to their axes. A horizontal cut along the neutral axis would cross the vertical webs with their 12-inch widths. The horizontal cut through the inclined webs would see horizontal widths, bh, equal to 13.42 inches for a web slope of 2:1. The total web width along the horizontal cut is then 62.84 inches.
Figure 7.30 – Web Width based on Horizontal Widths
Unfortunately, the web width computed above is inconsistent with the internal equilibrium of shear flow around the cross section with regard to the inclined webs. Consider the cross section of the single cell box girder with inclined webs shown in figure 7.31. A shear force, V, is applied to the cross section. The inclination of the webs requires that the sum of the shear flow in the inclined webs is:
(Eqn. 7.54)
The vector summation of this force and a horizontal force in the top slab resolves to one half of the applied shear force.
Chapter 7 – Longitudinal Analysis & Design 167 of 369 Figure 7.31 – Shear Flow in Single Cell Box Girder
The resulting shear stress acting on the web is then:
(Eqn. 7.55)
Noting that:
(Eqn. 7.56)
Reduces Equation 7.55 to:
(Eqn. 7.57)
Thus, vertical shear forces are related to web widths perpendicular to their inclined axes.
Extending these considerations to the cross section shown in figure 7.30, it can be argued that the appropriate web width should be 60 inches as opposed to the 62.84 inches of width along a horizontal cut through the webs. Using the horizontal width would lead to limit state verifications that are approximately 5 percent unconservative. By comparison, the verifications of the single cell box girder, using a similar web slope, would be unconservative by 11 percent. The engineer should give appropriate consideration to the web width used for each cross section.
In addition to the previous discussion, the effective web width must include reductions in width to account for ducts embedded in the webs. Figure 7.32 shows one web of the box girder of design example 1. Also shown is the shear stress distribution for a given vertical shear force and a detail of the flow of shear around ducts. To accommodate the shear flow concentrations, LRFD Article 5.8.2.9 requires a subtraction of one half of the duct diameter for loads applied to the bridge when the duct is ungrouted and a subtraction of one fourth of the diameter for loads applied after the duct is grouted.
Chapter 7 – Longitudinal Analysis & Design 168 of 369 Figure 7.32 – Shear Stress and Shear Flow Around Ducts
7.4.3.2 Effective Shear Depth
The effective depth for shear, dv, is required to be at least equal to the distance between the centroids of compression in the concrete and tension in the tensile elements. Figure 7.33 shows one of the webs of the cross section of the box girder of design example 1 at one of the interior supports. The centroid of the compression in the concrete was determined as the center of gravity of the stress block at nominal loading. The corresponding tension force passes through the center of gravity of the post-tensioning tendons.
Figure 7.33 –Effective Depth for Shear Calculations
Chapter 7 – Longitudinal Analysis & Design 169 of 369 The minimum effective shear is depth can be found as the nominal moment capacity of the section divided by either the internal compressive or tensile force:
(Eqn. 7.58)
The effective shear depth need not be taken less than 90 percent of the depth measured from the extreme compression fiber to the centroid of the prestressing steel, de, or 72 percent of the overall depth of the box girder.