Chapter 4. Example 1: Five-Column Bent Cap of a Skewed Bridge
4.4.8 Step 8: Perform Shear Serviceability Check
The estimated diagonal cracking strength of the concrete can be compared to the unfactored service level shear to determine the likelihood of the formation of service cracks.
Identifying the critical region for the serviceability check depends on the service shear, effective depth, web width, and shear span at a given point. The serviceability check allows designers to estimate the likelihood of diagonal cracking due to highly stressed diagonal struts. The diagonal cracking strength, Vcr, can be estimated by the following expression (refer to Section 2.7):
3.58”
Nodal Zone Extended
Nodal Zone
75.96°
60.41°
23.0”
L Bearing Area C
Available Length
26.1”
Critical Section
NODE V
Assume Prismatic Strut
2” min.
≈ 10”
Angle Considering the Subdivision of
Node NN
[ ( )] √
but not greater than √ nor less than √ where:
a = shear span (in.)
d = effective depth of the member (in.)
f’c = specified compressive strength of concrete (psi) bw = width o m m ’ w i n.)
Applying the AASHTO LRFD (2010) Service I load case to the bent cap and analyzing it as a continuous beam reveals that the region near Column 4 is critical (recall that the load case maximizes shear near Column 4). The highest service shear occurs between the support reaction at Column 4 and the load at Node Q. A portion of the loaded bearing area, however, is directly above the column reaction. Therefore, the shear span between the load at Node Q and Column 4 is not critical. Loads will flow directly from the location of the applied load to the support.
Although the serviceability behavior of the short shear span between Node Q and the column does not need to be checked, the possibility of diagonal crack formation within the shear span between Column 4 and the load at Node R should be considered. Within this region, the magnitude of the service shear is 255.7 kips (the shear between Nodes Q and R). The shear span, or the distance between the load at Node R and the reaction of Column 4, is 57.9 inches.
The shear span-to-depth ratio, a/d, is calculated to be 1.51 (a/d = 57.9 in./38.4 in.). Please recall from Section 2.7 that the upper and lower limits of the diagonal cracking load equation occur at a/d ratios of 0.5 and 1.5, respectively. Therefore, the magnitude of Vcr for the region right of Column 4 (i.e., between Node R and Column 4) is:
√ √ ( )( )
- Expect diagonal cracks
The data point of the normalized service shear for this region is plotted in Figure 4.32 (labeled
“Right o Column 4”). Fu th di cu i on g ding thi plot can be found in Section 2.7.
The shear serviceability check reveals the risk of the formation of diagonal cracks in the region right of the column when the full service loads act on the bent cap. The required crack control reinforcement should help to minimize the widths of the cracks that may form and alleviates the cause for concern regarding significant diagonal crack formation in this region.
Moreover, the shear force measured at first diagonal cracking exhibits significant scatter (refer to the experimental data of Figure 4.32 relative to the data point for the region under consideration).
Lastly, the expression for Vcr presented above estimates the diagonal cracking load with a reasonable amount of conservatism. For these reasons, significant serviceability problems are not expected within the region right of Column 4 given the current service load case.
(4.3)
Figure 4.32: Diagonal cracking load equation with experimental data and the normalized service shear for two regions of the bent cap (adapted from Birrcher et al., 2009)
The region between the load at Node P and Column 4 is checked next. Here, the shear force due to service loads is 330.3 kips. Although the magnitude of this shear force is greater than the magnitude of the shear force in the region right of Column 4 (255.7 kips), it is less critical due to the shorter shear span. For the region left of Column 4, the estimated diagonal cracking strength is:
[ (
)] (√ )( )( ) - Diagonal cracking is not expected
This value is within the √ and √ limits. Since the estimated diagonal cracking load, 353 kips, is greater than the service shear, 330.3 kips, diagonal cracks are not expected to form in this region for the particular service load case being considered (refer to corresponding data point in Figure 4.32).
Please recall that the bent cap that exists in the field has the same geometry as the bent cap of this design example but has a specified concrete compressive strength of 3.6 ksi. Using this value in the calculations above would slightly lower the magnitudes of the estimated diagonal cracking strengths.
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Vcrack/ √f 'c bwd
Shear Span to Depth Ratio (a/d)
Other Studies Current Project
Evaluation Database N = 59
Vcr
√f ′cbwd = 6.5 – 3 (a/d)
TxDOT Project 0-5253 Other Studies TxDOT Project 0-5253 Other Studies Evaluation Database
N = 59
Right of Column 4
Left of Column 4