1、 Example 1: Bridge Pier Hammerhead Bent Cap Robin G. Tuchscherer Michael D. Brown Oguzhan Bayrak Synopsis The design of a hammerhead bent cap is presented. The design is performed according to the ACI 318 Appendix A Strut-and-Tie Modeling (STM) provisions. The objective of this paper is to present a
2、 design example for designers who do not have a large amount of experience with strut-and-tie modeling. Strut-and-tie modeling is relatively new to ACI 318 as provi- sions were first included in the 2002 edition. The procedure may be used for the design of both reinforced and prestressed concrete st
3、ructures. The following example is intended to supplement the basic example problems available in textbooks. For additional background information, a review of textbook-level material ACI SP 208 (2002), Schlaich et al. (1987) is recommended. Robin G. Tuchscherer is a Project Engineer at Datum Engine
4、ers, Inc.in Austin, Texas. He received his PhD and MSE from the University of Texas at Austin (2008 and 2006 respec- tively); and his BSCE from the University of Wisconsin-Milwaukee (1999). Michael D. Brown is a Staff Engineer at Whitlock, Dalrymple, Poston, and Associates, Aus- tin, Texas. He recei
5、ved his PhD from the University of Texas at Austin in 2005. His BSCE (2000) and MSE (2002) are also from the University of Texas at Austin. He is an associate member of ACI-ASCE Committee 445 Shear and Torsion. Oguzhan Bayrak, FACI, is an Associate Professor in the Department of Civil, Environmen- t
6、al, and Architectural Engineering and holds the Charles Elmer Rowe Fellowship in Engineering at the University of Texas at Austin, where he serves as Director of the Phil M. Ferguson Structural Engineering Laboratory. He is a member of ACI Committees 341, Earth- quake-Resistant Concrete Bridges, and
7、 E803, Faculty Network Coordinating Committee, and Joint ACI-ASCE Committees 441, Reinforced Concrete Columns, and 445, Shear and Tor- sion. 1- 2 Example 1: Hammerhead Bent Cap 1 Description of design task 1.1 Geometry and Loads The external geometry of the hammerhead bent cap used for this example
8、is shown in Fig. 1-1. In general, the bent cap dimensions are controlled by the size of the bearing plates. For this ex- ample, the bent cap in loaded asymmetrically in order to account for the critical unbalanced load case of live load on one side of the bridge. The loads acting on the cap include
9、the factored dead load, factored lane load, and include an allowance for impact. The factored self-weight of the bent cap was divided equally, and is also included in the applied loads. Fig. 1-1: Geometry and loads for hammerhead bent cap 1.2 Materials Concrete: f c = 5,000 psi (34.5 MPa) Reinforcem
10、ent: f y= 60,000 psi (414 MPa) Example 1: Hammerhead Bent Cap 1-3 1.3 Statement of design problem A deep beam design must be treated differently than a sectional design (or slender beam design) because the assumptions used to derive the sectional theory are no longer valid. A deep beam is a member w
11、hose shear span-to-depth ratio, a/d, is relatively small such that nonlinear shearing strains dominate the behavior. According to MacGregor (1997), a deep beam can be defined as follows: a beam in which a significant amount of load is carried to the supports by a compres- sion thrust joining the loa
12、d and the reaction. This occurs if a concentrated load acts closer than about 2d to the support, or for uniformly loaded beams with a span-to-depth ratio, l n /d, less than about 4 to 5. Strut-and-tie modeling (STM) is a relatively simple and effective method that can be used for the design of compl
13、ex regions of discontinuity. A strut-and-tie model idealizes the complex flow of stresses in a structural member as axial elements in a truss member. Concrete struts resist the compressive stress fields and reinforcing steel ties resist the tensile stress fields. Struts and ties intersect at regions
14、 called nodes. Struts, ties, and nodes are the three elements that comprise a STM and they must be proportioned to resist the applied forces. Failure modes of a STM reflect the failure modes of a deep beam region and include the: (i) crushing of concrete in a strut or at the face of a node; (ii) yie
15、lding of a tie; (iii) or anchorage failure of a tie. Further examples illustrating the case of a load near a support are evaluated and presented in ACI Special Publication 208 (2002). The purpose of the example presented herein is to provide fur- ther guidance for the design of a cantilever deep bea
16、m structure. 2 Design procedure The shear span-to-depth ratio of the hammerhead bent cap in this example is less than two. As such, the structure may be classified as a deep beam region and may be designed according to the strut-and-tie provisions in Appendix A. For this example, the structure is fi
17、rst designed using a “simple” strut-and-tie model, which is subsequently referred to as Model 1. Next, the analysis is further developed using a “refined“strut-and-tie model, which is subsequently referred to as Model 2. The following steps are used: Step 1: Determine appropriate truss model Step 2:
18、 Proportion tie Step 3: Determine node geometry and check strength of nodes Step 4: Check strength of struts Step 5: Proportion crack control reinforcement Step 6: Check tie anchorage 1- 4 Example 1: Hammerhead Bent Cap 3 Design Calculations: Simple Model 1 3.1 Step 1: Determine appropriate truss mo
19、del The design of a hammerhead bent cap using a simple truss is presented in Fig. 1-2. The strut- and-tie model is initially approximated by aligning the struts and ties with the centroid of the compression zone and main tensile reinforcement, respectively. The loading pattern shown is asymmetric in
20、 order to account for an unbalanced live load. An unbalanced live load can occur at either side of the structure depending which traffic lanes are considered. Thus, struts and ties on both sides of the bent will both be designed for the critical stresses that occur at the side with the higher loadin
21、g. In addition, the pier must also be designed to resist the bending moment and axial load. This can be done using a conventional beam-column design and is not presented here. Typically in structures of this type, the column is slightly narrower than the bent cap. This detail ensures that the longit
22、udinal bars that extend from the column will fit inside the reinforcing cage in the bent cap. For the interest of simplicity, this example problem will contain a bent cap and column of the same width. Fig. 1-2: Simple truss model 1 (force in kips and kN; dimensions in in. and mm) According to the lo
23、wer bound theory of plasticity, the capacity of a STM is always less than the actual capacity of the structure provided the following requirements are met: (i) the model is in equilibrium, (ii) sufficient deformation capacity exists to distribute forces according to the as- sumed model, and (iii) th
24、e stresses applied to the elements do not exceed their yield or plastic flow capacity. The choice of the model is left to the discretion of the designer. However, if the orientation of the model varies significantly from the actual stress field, then the structure must undergo substantial deformatio
25、n in order to develop the poorly assumed model. Thus, it is good practice for the STM to agree well with the dominant mechanism of force transfer in the struc- ture. Large deviations from 45 of the angle between a strut and tie will demand excessive strains in the reinforcement together with extreme
26、ly wide crack openings at failure. Accordingly, the ACI 318-08 STM provisions limit the angle between a strut and tie to 25. Example 1: Hammerhead Bent Cap 1-5 3.2 Step 2: Proportion tie Tie ABCD is positioned 6.5 in. (165 mm) from the top of the section. The large tensile force in the tie results i
27、n two layers of longitudinal bars. The force in tie ABCD is 1083 kip (4817 kN). According to Section A.4 of ACI 318-08, the required area of reinforcement is: y ts nt f A F Solving for A ts , 2 2 , 500 , 15 . 1 . 24 60 75 . 0 1083 mm in ksi kip f F A y u req ts Therefore, provide 18 No. 11 bars 2 .
28、1 . 28 in A ts 18 36M bars ( 2 100 , 18 mm A ts ) It is of interest to note that the strength reduction factor used to determine the main longitudin- al reinforcement is 0.75 compared with 0.9 used for a conventional flexural analysis. This is at- tributed to the fact that the strength reduction fac
29、tors in ACI 318-08 coincide with member behavior rather than material. 3.3 Step 3: Determine node geometry and check strength of nodes 3.3.1 Nodes A and D (CCT Node) Now that the tie reinforcement has been chosen, proportions of the nodal region can be deter- mined. Nodal geometry is based on the co
30、ver dimensions, bearing dimensions, and strut angles. The design of Nodes A and D is based on the concentration of critical stresses at Node A. The node anchors one tie and is therefore considered to be a CCT node. The geometry of Node A is determined as shown in Fig. 1-3. The placement of Tie ABCD
31、and the dimension of the adjacent bearing pad determine the geometry of Node A. The width of the node and strut are assumed to be equal to the width of the bearing pad. The height of a CCT node is typically taken as twice the clear cover distance to the centroid of the reinforcement; or 13 in. (330
32、mm). Fig. 1-3: Elevation view of node A 1- 6 Example 1: Hammerhead Bent Cap The angle of the strut is estimated based on the global model. In this case: 88 . 32 . 99 . 64 tan 1 in in According to Section A.5 of ACI 318-08, the allowable stress on any face of a CCT node is: c cu f f 80 . 0 85 . 0 The
33、 design strength of all the faces of a node must be greater than the strength required from the strut-and-tie model. Based on the allowable stress, the strengths of each nodal face is checked as follows: For the top face of the node: kN kN F kip kip in in ksi A f F nn nz c n nn 110 , 3 250 , 12 700
34、754 , 2 . 0 . 36 . 0 . 30 5 80 . 0 85 . 0 75 . 0 85 . 0 For the left face of the node: kN kN F kip kip in in ksi A f F nn nz c n nn 817 , 4 307 , 5 083 , 1 193 , 1 . 0 . 36 . 0 . 13 5 80 . 0 85 . 0 75 . 0 85 . 0 The strength of the inclined face of the node (strut-to-node interface) is controlled by
35、 the allowa- ble stress of a strut and is given in Section 3.4.1. 3.3.2 Node E (CCC Node) The geometry of node E is determined as shown in Fig. 1-4. The height of the node, h a , is assumed to be equivalent to the depth of the compressive stress block determined from a conven- tional flexural analys
36、is. For a rectangular section, the depth of the compressive stress block can be determined as follows: w c s s a b f f A h 85 . 0For this example, the height of node E is estimated to be approximately 9 in. (230 mm). Similarly, the width of node E can be determined from a sectional analysis of the c
37、olumn consi- dering the interaction between flexure and compression. Aligning the strut with the centroid of the compressive stress block obtained from an ultimate analysis is not necessary for such a large cross-section. Rather, the height of node E can be estimated based on an approximation of the
38、 compression zone depth as it varies between the linear-elastic and ultimate stress diagram. For this example, the compression zone is estimated to be 18 in. (910 mm). Thus, the centroid of the strut framing into node F is located 9 in. (460 mm) from the face of the pier. In order to check node E us
39、ing strut-and-tie provisions, it is necessary to subdivide the node based on the proportion of compressive force that frames into the region from each direction. In addition, the multiple struts framing into the node from the right side can be resolved into a sin- gle force based on summation of vec
40、tors. The resulting configuration of node E is illustrated in Fig. 1-4. Example 1: Hammerhead Bent Cap 1-7 Fig. 1-4: Elevation view of node E According to Section A.5 of ACI 318-08, the allowable stress on any face of a CCC node is: c cu f f 0 . 1 85 . 0 The design strength of all the faces of a nod
41、e must be greater than the strength required from the strut-and-tie model. Based on the allowable stress, the strengths of the nodal faces are checked as follows: For the bottom face of the node: kN kN F kip kip in in ksi A f F nn nz c n nn 114 , 3 276 , 6 700 411 , 1 . 0 . 54 . 2 . 8 5 0 . 1 85 . 0
42、 75 . 0 85 . 0 For the right face of the node: kN kN F kip kip in in ksi A f F nn nz c n nn 817 , 4 890 , 6 083 , 1 549 , 1 . 54 . 9 5 0 . 1 85 . 0 75 . 0 85 . 0 The strength of the inclined face of the node (strut-to-node interface) is controlled by the allowa- ble stress of a strut and is given in
43、 Section 3.4.1. 3.3.3 Node F By inspection, the critical design of node H would be as a CCC node when the load case in a lane loading on the other side of the bent similar to the analysis presented in Section 3.3.2 for node E. The tensile force at node F is accounted for with the design of the longi
44、tudinal column reinforcement. 3.3.4 Node C Node C is a CTT node framed by tie CF, tie BC, and strut EC. Tie CF is required to maintain equilibrium caused by the unbalanced moment on the pier. A similar condition would exist at the other side of the pier if the loading were unbalanced to the other si
45、de of the structure. With a 1- 8 Example 1: Hammerhead Bent Cap conventional design, the unbalanced moment is transferred into the pier by extending the longi- tudinal column reinforcement vertically into the bent and lapping the hooked ends with the longi- tudinal reinforcement in the cap. Addition
46、ally, column ties and stirrups are provided in order to confine the joint region and develop the flexural strength of each member. A strut-and-tie model may be used to determine the column reinforcement in this region but such analysis is not included in this example. 3.4 Step 4: Check strength of s
47、truts Struts AE and DF connect the exterior applied loads to the support reaction. The design of both of these struts is based on the critically stressed Strut AE. Each interface of strut AE is checked as follows. According to Section A.3 of ACI 318-08, the effective compressive strength of concrete
48、 in a strut shall be taken as: c s ce f f 85 . 0 The efficiency factor, s , is based on the shape of the strut, its location, and the amount of trans- verse reinforcement crossing its path. The compressive stress field of strut AE will spread later- ally such that the width of the midsection will be wider than the width at the node interface (bot- tle-shaped). Provided that adequate crack control reinforcement is provided, the strength of the strut-to-node interface at Node A is checked
