Chapter 7. Example 3b: Inverted-T Straddle Bent Cap (Simply Supported)
7.4.1 Step 1: Develop Global Strut-and-Tie Model
The global STM for the simply supported inverted-T bent cap is shown in Figure 7.4.
The connection between each column and the bent cap is assumed to transfer vertical and horizontal forces only. In the absence of lateral forces, only a vertical reaction force exists at the centerline of each column. Bearing forces at the cap-to-column connection are therefore resisted by a single node, located above each column along the column centerline.
186
5.00’
2.75’ 6.25’
18.8 k 8.5 k
465.8 k 8.5 k
13.6 k
522.9 k
1767.2 k
542.5 k
18.4 k
1550.8 k 2597.1 k
2602.7 k
5.00’
L Column A
C CL Column B
7.6”
6.35”
3.84’
2572.8 k
899.5 k 2602.7 k 2071.1 k
-2071.1 k
-899.5 k -1767.2 k -2597.1 k -2572.8 k -1550.8 k
570.7 k 947.3 k
9.8 k
9.8 k
9.9 k
9.9 k
10.5 k 8.5 k 10.7 k
507.5 k 426.6 k 443.5 k
8.5 k 8.5 k
6.25’ 6.25’ 4.21’ 4.21’ 4.21’ 4.21’ 6.41’ 2.75’
507.5 k 448.7 k 917.8 k
A B C
H P
D E F G
I J K L M N
O
Figure 7.4: Global strut-and-tie model for the inverted-T bent cap (simply supported case)
Through a series of design iterations, the bottom chord of the STM is placed at the centroid of the longitudinal steel along the tension face of the bent cap. The maximum positive bending moment due to the applied loads is larger for the simply supported case than for the moment frame case (Example 3a). A greater amount of bottom chord reinforcement is therefore necessary, and the corresponding centroid of the reinforcement is farther from the bottom surface of the bent cap (in relation to Example 3a). As shown in Figure 7.4, the final location of the bottom chord is 7.6 inches from the bottom face of the cap.
The top chord of the global STM consists entirely of struts (positive moment exists along the length of the cap). For this reason, its position is not necessarily determined by the centroid of the longitudinal reinforcement along the top of the cap (refer to Example 3a). To achieve efficient use of the bent cap depth, the distance between the top and bottom chords of the STM (analogous to the moment arm, jd) and the width of the top chord struts (analogous to the rectangular compression stress block) should be optimized (Tjhin and Kuchma, 2002). In other words, the factored force acting on the back face of the most critical node located along the top chord should be nearly equal to its design strength (refer to Section 2.10.4 and Figure 2.18).
To optimize the STM, the critical section for flexure (i.e., the section with the largest force in the top chord) is first identified by analyzing the simply supported member. Applying the factored superstructure loads and the factored distributed self-weight to the bent cap reveals that the maximum positive moment occurs at Beam Line 1 (Mmax = 9972 kip-ft, refer to Figure 7.5). Although the STM geometry has not yet been defined, the designer can know that all the nodes along the top chord will be CCT nodes (only one vertical tie joins at each node). To strengthen the back faces of the nodes along the top chord, 20-#11 bars are provided as compression reinforcement along the length of the bent cap. The centroid of the 20 bars will be located a distance d’s of 4.9 inches from the top surface of the cap.
Figure 7.5: Determining the location of the top chord of the global STM
7.6”
a/2
Mmax= 9972 k-ft d = 52.4”
Take moment about this point
CCT Node (Node C) CCT Node (Node C)
The equation below is used to determine the optimal position of the top chord, where a is the width of the top chord struts (i.e., a/2 is the distance from the top surface of the cap to the top chord of the STM).
[ ( ) ]
[ ( ) ] Solving: ⁄
The AASHTO LRFD (2010) resistance factor, , is taken as 0.7 for compression in strut-and-tie models. The concrete efficiency factor, ν, of 0.7 corresponds to the back face of the CCT node at Beam Line 1 (Node C in Figure 7.4). The minimum width of the horizontal strut necessary to resist the top chord forces is 12.70 inches. The distance from the top surface of the bent cap to the top chord of the STM (a/2) is therefore 6.35 inches (see Figure 7.4).
Once the locations of the top and bottom chords are determined, vertical ties representing the hanger reinforcement within the stem of the bent are placed at the locations of the applied superstructure loads (Ties CK, EM, and GO in Figure 7.4). The proposed STM specifications of Chapter 3 state that the angle between a strut and a tie entering the same node should not be less than 25 degrees. To satisfy this requirement, additional vertical ties are necessary in four locations (Ties AI, BJ, DL, and FN). Diagonal struts are then positioned in each truss panel of the STM.
The final reinforcement layout within the stem of the bent cap is shown in Figure 7.16.
Several iterations were necessary to ensure that (1) the centroids of the longitudinal reinforcement correspond with the locations assumed for the main tension and compression steel during the STM development and (2) the amount of compression reinforcement allows the nodal strength checks of the top chord to be satisfied. Engineering judgment should always be exercised in determining the necessity of additional design iterations.
The statically determinate truss, simply supported at Nodes H and P, is analyzed under the action of the beam loads (at Nodes K, M, and O) and the tributary self-weight (at each node).
The truss analysis results in the internal member forces and external column reactions shown in Figure 7.4. Considering that the system is statically determinate, the column reactions obtained from the truss analysis are the same as those that would result from an analysis of the simply supported bent cap.