Strut-and-Tie Model (STM)

The STM is based on the lower-bound theory of limit analysis. In the STM, the complex flow of internal forces in the D-Region under consideration is idealized as a truss carrying the imposed loading through the region to its supports. This truss is called strut-and-tie model and is a statically admissible stress field  in lower-bound (static) solutions. Like a real truss, a strut-and-tie model consists of struts and ties interconnected at nodes (also referred to as nodal zones or nodal regions). A selection of strut-and-tie models for a few typical 2-D D-Regions is illustrated in Figure 1. As shown in the figure, struts are usually symbolized using broken lines, and ties are usually denoted using solid lines.

Figure 1   Examples of Strut-and-Tie Models for Common Structural Concrete Members

Strut-and-Tie Model Components
Struts are the compression members of a strut-and-tie model and represent concrete stress fields whose principal compressive stresses are predominantly along the centerline of the strut. The idealized shape of concrete stress field surrounding a strut in a plane (2-D) member, however, can be prismatic (Figure 2(a)), bottle-shaped (Figure 2(b)), or fan-shaped (Figure 2(c)) [1]. Struts can be strengthened by steel reinforcement, and if so, they are termed reinforced struts.

Figure 2   Basic Type of Struts in a 2-D Member: (a) Prismatic (b) Bottle-Shaped (c) Fan-Shaped

Ties are the tension members of a strut-and-tie model. Ties mostly represent reinforcing steel, but they can occasionally represent prestressing steel or concrete stress fields with principal tension predominant in the tie direction.
Nodes are analogous to joints in a truss and are where forces are transferred between struts and ties. As a result, these regions are subject to a multidirectional state of stress. Nodes are classified by the types of forces being connected. Figure 3 shows basic types of nodes in a 2-D member; in the figure, C is used to denote compression and T is used to denote tension.

Figure 3   Basic Type of Nodes: (a) CCC  (b) CCT  (c) CTT  (d) TTT

Uniqueness of Strut-and-Tie Models
As a statically admissible stress field, a strut-and-tie model has to be in equilibrium externally with the applied loading and reactions (the boundary forces) and internally at each Node. In addition, reinforcing or prestressing steel is selected to serve as the ties, the effective width of each strut is selected, and the shape of each nodal zone is constructed such that the strength is sufficient. Therefore, only equilibrium and yield criterion need to be fulfilled for an admissible strut-and-tie model. The third requirement in solid mechanics framework, namely the strain compatibility, is not considered.
As a result of these relaxed requirements, there is no unique strut-and-tie model for a given problem. In other words, more than one admissible strut-and-tie model may be developed for each load case as long as the selected truss is in equilibrium with the boundary forces and the stresses in the struts, ties, and nodes are within the acceptable limits. The lower-bound theorem guarantees that the capacity obtained from all statically admissible stress fields is lower than or equal to the actual collapse load. However, as a result of limited ductility in the structural concrete, there are only a small number of viable solutions for each design region. Figure 4 illustrates an example in which one solution is preferable to another. Due to the point load at the tip of the cantilever portion, the upper part of the beam is likely to develop horizontal tensile stresses along the beam. Therefore, the model with the upper horizontal tie (Figure 4(a)) is preferable to that shown in Figure 4(b). The latter only effectively resists the tension in the upper region near the middle support.

Figure 4   Two statically admissible strut-and-tie models for a cantilevered deep beam under vertical loading: (a) Workable truss (b) Less favorable truss due to excessive ductility demands

Reference:
1) Schlaich, J. and Schäfer, K., "Design and Detailing of Structural Concrete Using Strut-and-Tie Models," The Structural Engineer, Vol. 69, No. 6, March 1991, pp. 113-125.

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