The Strut-and-Tie Model (STM) is a tool for the analysis, design, and detailing of reinforced concrete members. It is essentially a truss analogy, based on the fact that concrete is strong in compression, and that steel is strong in tension. Truss members that are in compression are made up of concrete, while truss members that are in tension consist of steel reinforcement.
Chapter 23 (Appendix A in ACI 318-11 and prior), Strut-and-Tie Models, was introduced in ACI 318-02. The method presented in Chapter 23 provides a design approach, applicable to an array of design problems that do not have an explicit design solution in the body of the code. This method requires the designer to consciously select a realistic load path within the structural member in the form of an idealized truss. Rational detailing of the truss elements and compliance with equilibrium assures the safe transfer of loads to the supports or to other regions designed by conventional procedures. While solutions provided with this powerful analysis and design tool are not unique, they represent a conservative lower bound approach. As opposed to some of the prescriptive formulations in the body of ACI 318, the very visual, rational strut-and-tie model of Chapter 23 gives insight into detailing needs of irregular (load or geometric discontinuities) regions of concrete structures and promotes ductility at the strength limit stage. The only serviceability provisions in the current Chapter 23 are the crack control reinforcement for the struts.
The design methodology presented in Chapter 23 is largely based on the seminal articles on the subject by Schlaich et al., Collins and Mitchell, and Marti. Since publication of these papers, the strut-and-tie method has received increased attention by researchers and textbook writers (Collins and Mitchell, MacGregor and Wight). MacGregor described the background of STM provisions incorporated in ACI 318 Chapter 23 in ACI Special Publication SP-208.
1. B-regions represent portions of a member in which the “plane section” assumptions of the classical beam theory can be applied with a sectional design approach.
2. D-regions are all the zones outside the B-regions where cross-sectional planes do not remain plane upon loading. D-regions are typically assumed at portions of a member where discontinuities (or disturbances) of stress distribution occur due to concentrated forces (loads or reactions) or abrupt changes of geometry.
Based on St. Venant’s Principle, the normal stresses (due to axial load and bending) approach quasi-linear distribution at a distance approximately equal to the larger of the overall height (h) and width of the member, away from the location of the concentrated force or geometric irregularity. The following figure illustrates typical discontinuities, D-Regions (cross-hatched areas), and B-Regions.
Figure 2 - Load and Geometric Discontinuities
While B-regions can be designed with the traditional methods using applicable provisions from ACI 318, the strut and-tie model was primarily introduced to facilitate the design of D-regions, and can be extended to the B-regions as well. The strut-and-tie model depicts the D-region of the structural member with a truss system consisting of compression struts and tension ties connected at nodes as shown in the following figure. This truss system is designed to transfer the factored loads to the supports or to adjacent B-regions. At the same time, forces in the truss members should maintain equilibrium with the applied loads and reactions.
Figure 3 - Strut-and-Tie Model (STM)
3. Struts are the compression elements of the strut-and-tie model representing the resultants of a compression field. Both parallel and fan shaped compression fields can be modeled by their resultant compression struts as shown in the following figure.
Figure 4 - Strut-and-Tie Model
4. Ties consist of conventional deformed reinforcing steel, prestressing steel, or both, plus a portion of the surrounding concrete that is concentric with the axis of the tie. The surrounding concrete is not considered to resist axial force in the model. However, it reduces the elongation of the tie (tension stiffening), in particular, under service loads. It also defines the zone in which the forces in the struts and ties are to be anchored.
5. Nodes are the intersection points of the axes of the struts, ties and concentrated forces, representing the joints of a strut-and-tie model. To maintain equilibrium, at least three forces should act on a given node of the model. Nodes are classified depending on the sign of the forces acting upon them (e.g., a C-C-C node resists three compression forces, a C-T-T node resists one compression forces and two tensile forces, etc.) as shown in following figure.
Figure 5 - Classification of Nodes
6. A nodal zone is the volume of concrete that is assumed to transfer strut and tie forces through the node. The early strut-and-tie models used hydrostatic nodal zones, which were lately superseded by extended nodal zones.
a. The faces of a hydrostatic nodal zone are perpendicular to the axes of the struts and ties acting on the node, as depicted in the following figure. The term hydrostatic refers to the fact that the in-plane stresses are the same in all directions. (Note that in a true hydrostatic stress state the out-of-plane stresses should be also equal). Assuming identical stresses on all faces of a C-C-C nodal zone with three struts implies that the ratios of the lengths of the sides of the nodal zones (wn1 : wn2 : wn3) are proportional to the magnitude of the strut forces (C1 : C2 : C3). Note, that C denotes compression and T denotes tension.
Figure 6 - Hydrostatic Nodal Zone
b. The extended nodal zone is a portion of a member bounded by the intersection of the effective strut width, ws, and the effective tie width, wt. This is illustrated in following figure.
Figure 7 - Extended Nodal Zone
1.2. Strut-and-Tie Model Design Procedure
A design with the strut-and-tie model typically involves the following steps:
1. Define and isolate D-regions.
2. Compute resultant forces on each D-region boundary.
3. Devise a truss model to transfer the resultant forces across the D-region. The axes of the struts and ties, are oriented to approximately coincide with the axes of the compression and tension stress fields respectively.
4. Calculate forces in the truss members using hand calculations, analysis aid tables, or structural analysis software based on the complexity of the selected truss model - STM.
5. Determine the effective widths of the struts and nodal zones considering the forces from the previous steps and the effective concrete strengths (defined in 23.4.3 and 23.9.2).
6. Provide reinforcement for the ties considering the steel strengths defined in 23.7.2. The reinforcement must be detailed to provide proper anchorage either side of the critical sections. In addition to the strength limit states, represented by the strut-and-tie model, structural members should be checked for serviceability requirements. Traditional elastic analysis can be used for deflection checks. Crack control can be verified using provisions of 24.3.2, assuming that the tie is encased in a prism of concrete corresponding to the area of tie (R23.8.1).