2.3.1 Fundamentals of Strut-and-Tie Modeling
The principles that form the basis of strut-and-tie modeling ensure that the resulting structural design is conservative (i.e., is a lower-bound design). An STM design adheres to these principles if (1) the truss model is in equilibrium with external forces and (2) the concrete element has enough deformation capacity to accommodate the assumed distribution of forces (Schlaich et al., 1987). Proper anchorage of the reinforcement is an implicit requirement of the latter condition. Additionally, the compressive forces in the concrete as indicated by an analysis of the strut-and-tie model must not exceed the factored concrete strengths, and the tensile forces within the STM must not exceed the factored tie capacities. If all of the requirements above are satisfied, application of the STM procedure will result in a conservative design (i.e., lower-bound design).
Every STM consists of three components: struts, ties, and nodes. A basic STM representing the flow of forces through a simply supported beam is depicted in Figure 2.3. After calculating the external reactions and defining the geometry of the STM, the member forces of the truss model are calculated from statics. The compression members are referred to as struts, and the tension members are referred to as ties. Strut and ties are denoted by dashed lines and solid lines, respectively, in Figure 2.3 and throughout this guidebook. The struts and ties intersect at regions referred to as nodes. Due to the concentration of stresses from intersecting truss members, the nodes are the most highly stressed regions of a structural member.
Figure 2.3: Struts, ties, and nodes within a strut-and-tie model
When developing an STM, the locations of the struts and ties should ideally be based upon the flow of forces indicated by an elastic analysis. Placing the struts and ties in accordance with the elastic flow of forces ensures a safe design with minimal cracking at service load levels (Bergmeister et al., 1993). Further discussion concerning the placement of struts and ties is provided in Section 2.8.
Node
Tie
Strut
A strut-and-tie model can ultimately be tailored to any geometry and stress distribution that may be encountered in the design of D-regions. This versatility is simultaneously viewed as a primary advantage as well as a major challenge of the application of STM. The flexibility with which strut-and-tie modeling can be applied often leads to uncertainties and confusion for the designer: no one “correct” STM exists for any particular structure. If the principles required to achieve a lower-bound solution are satisfied, however, the engineer can be assured that a safe design will result. The desire to minimize uncertainties and formulate consistent STM design procedures within a design office is, nevertheless, understood. This guidebook is therefore meant to assist engineers with developing such procedures that can be applied to the design of structural components of highway bridges.
For further explanation of the theoretical background of STM, the reader is encouraged to reference the TxDOT Project 0-5253 report (Birrcher et al., 2009).
2.3.2Prismatic and Bottle-Shaped Struts
Struts can either be defined as prismatic or bottle-shaped depending on the uniformity of the stress fields in which they are located. As illustrated in Figure 2.4, prismatic struts are concentrated in regions where stresses are fairly uniform, such as the region at the top of a member in positive bending. Bottle-shaped struts are located in regions where the compressive stresses are able to spread laterally. Diagonal struts within a beam are bottle-shaped. The spreading of the compressive stresses produces tensile stresses transverse to the strut, causing diagonal cracks to form within the member. These tensile stresses reduce the efficiency of the concrete that comprises the strut. Orthogonal reinforcement is provided in the vicinity of bottle- shaped struts to carry the tensile forces, strengthen the strut, and control the bursting cracks that tend to develop. Although bottle-shaped struts are often idealized as prismatic struts, as illustrated in Figure 2.4, the effects of the transverse tensile stresses must not be overlooked.
Figure 2.4: Prismatic and bottle-shape struts within a strut-and-tie model (adapted from Birrcher et al., 2009)
Prismatic Strut
Idealized Prismatic Strut Bottle-Shaped
Strut
Tension Develops
2.3.3 Strut-and-Tie Model Design Procedure
A list of the steps typically followed when designing a deep structural component using the STM procedure is provided below. The procedure is based on the application of the STM specifications that were developed as a part of TxDOT Project 0-5253 and the current implementation project (5-5253-01). The proposed specifications as well as a brief overview of the work completed during project 0-5253 are presented in Chapter 3. The STM procedure provided below is generally followed in the design examples of Chapters 4 through 8 but is adapted to the particular design scenarios as necessary. While each step of the procedure will be independently described in the sections that follow, the designer should note that the steps are sometimes performed simultaneously. The STM procedure is presented in a flow-chart format in Figure 2.5.
1. Separate B- and D-regions – Determine which regions of the structural component are expected to exhibit deep beam behavior or if the entire component should be designed using STM.
2. Define load case – Calculate the factored loads acting on the structural component, and if necessary, make simplifying assumptions to develop a load case that can be applied to a reasonable STM.
3. Analyze structural component – Solve for the structural component’s support reactions assuming linear elastic behavior.
4. Size structural component using the shear serviceability check – Determine the initial geometry of the structural component by using the shear serviceability check introduced in the TxDOT Project 0-5253 report.
5. Develop strut-and-tie model – Position struts and ties to represent the actual flow of forces within the structural component, and determine the forces in the struts and ties.
6. Proportion ties – Specify the reinforcement needed to carry the force in each tie.
7. Perform nodal strength checks – Define the geometries of the critical nodes, and ensure the strength of each face is adequate to resist the applied forces determined from the analysis of the STM.
8. Proportion crack control reinforcement – Specify the required crack control reinforcement to restrain diagonal cracks formed by the transverse tensile stresses of bottle-shaped struts.
9. Provide necessary anchorage for ties – Ensure reinforcement is properly anchored at the nodal regions.
Figure 2.5: Strut-and-tie model design procedure
Separate B- and D-Regions
(Section 2.4)
Define Load Case
(Section 2.5)
Analyze Structural Component
(Section 2.6)
Size Structural Component
(Section 2.7)
Develop Strut-and-Tie Model
(Section 2.8)
Proportion Ties
(Section 2.9)
Perform Nodal Strength Checks
(Section 2.10)
Proportion Crack Control Reinforcement
(Section 2.11)
Provide Necessary Anchorage for Ties
(Section 2.12)