As previously stated, a strut-and-tie model relies on a simulation of the stress field in a structure in the ultimate limit state, when the concrete is cracked and the structure is close to reach collapse. Then one can consider that plastic redistribution has occurred in the structural member and that the chosen force distribution is possible to happen,
Identify and delineate D-regions (Section 1.5)
Determine the support reactions and the stress distribution at the boundaries of D-regions
Check bearing capacities at loads and supports
(Section 3.6.3)
Determine a strut-and-tie model
Determine the forces in the struts and ties
Design the required amount of reinforcement for each tie and the position of its axis
(Section 3.6.1)
Determine the provided area for each strut (Equation 3.9)
Check the nodal zones and the struts (Sections 3.6.2, 3.6.3)
Arrange the reinforcement
Refine the strut-and-tie model by modifying:
- geometry of structure - size of bearing areas - location of ties (as) Consider a structure to design and a certain
load combination
Modify dimensions or number of bearing areas
OK Not OK
Not OK OK
which does not mean that it is not much on the safe side. However, it is still important that the strut-and-tie model remains rather close to the linear elastic stress field for two reasons:
- to take into account the limited plastic redistribution in reinforced concrete - to provide acceptable performance in the service state
This is especially true in design of pile caps, which are reinforced concrete members with a low ability to plastic redistribution.
There are several ways to find an appropriate strut-and-tie model. A linear finite element analysis can be carried out to give an idea of the elastic stress field. The direction and intensity of principal stresses given by this analysis can provide good indications for the choice of the model. Some discretization methods using finite elements and optimisation criteria have also been developed in order to generate strut- and-tie models automatically, for instance by Kostic (2009) or for the software ForcePad. Besides, intuitive methods, such as the load path method (Schlaich et al.
1987) or the stress field method (Muttoni et al. 2008), can help the designer to position the struts and the ties by considering the resultants of the stress fields. These methods used together with the strut-and-tie method present the advantage to lead the designer to a better understanding of the mechanical behaviour of the structure.
Some other rules have to be followed when determining a strut-and-tie model for a structure, such as angle limitations (Section 3.5.1.1) and that the struts should not overlap or cross each other outside the node regions. Indeed, as struts are designed according to the concrete effective strength, it would lead to yielding in the overlapping area (Reineck 2002). On the other hand, ties can cross struts or other ties.
It is usually convenient to choose horizontal and vertical orthogonal ties, to obtain a need for reinforcement close to what is usually provided in practice. However, some other more advanced reinforcement layouts are sometimes used, and more efficient ways of reinforcing the member could be considered with the strut-and-tie models.
Another parameter affecting the choice of the model is the level of statical indeterminacy of the model. Indeterminate models increase the complexity of the procedure, as it is further discussed in Section 4.7.2. However, in some cases they can lead to more efficient models and a higher reliability in the service state. Indeed, in order to establish a statically determinate system the designer can be led to neglect solutions more complicated but closer to the elastic flow of forces. This could lead to severe cracking in some regions under service load.
3.5.1 Choice of the strut inclinations
3.5.1.1 Angle limitations
When building a strut-and-tie model, attention has to be paid to the choice of the inclination of the struts. Two kinds of problems could arise: on the one hand an inappropriate deviation angle at concentrated forces can lead to a too high need of plastic redistribution and strain compatibility problems between stressed and unstressed regions. On the other hand too small angles between struts and ties can also lead to strain compatibility problems. The recommendations of Schọfer in fib bulletin 3 (1999, cited in Engstrửm 2009) are given hereafter, using the notations of Figure 3.3.
Figure 3.3 Angle recommendations in a deep beam with stirrups, for deviation of concentrated forces and between struts and ties
a) Deviation of concentrated loads
α ≈ 30° and α < 45° (3.1)
Additionally, the stresses under concentrated loads should be directly spread out when entering the D-region.
b) Angles between struts and ties
θ1 ≈ 60° and θ1 > 45° (3.2)
θ2 ≈ 45° and θ2 > 30° (3.3)
θ3 ≈ 45° and θ3 > 30° (3.4)
The recommendations on minimum angles to use in a strut-and-tie model differ between different authors and codes. For instance, in the ACI Building Code the minimum angle between a strut and a tie joining at a node is set to 25°.
It should be noticed in Figure 3.3 that the angle limitation for the deviation of the concentrated force of the column applies to the inclined strut; evidently the horizontal strut is not concerned as it does not directly derive from the spreading of the concentrated force but it is needed for the equilibrium of the model.
When the concentrated force is transferred in the model by several inclined struts, the limitation should apply to the angle of the resultant of the forces in the struts.
However, this statement should go together with the appreciation of the designer, who should distinguish the cases where it can be accepted and where it could lead to any compatibility problem.
3.5.1.2 Optimal design
Several different strut-and-tie models can be chosen for a given problem. However some of them are more efficient than others. For instance, when the strut inclinations
F/2 F/2
α θ1
α
θ2
θ3
θ1
F/2
θ1 and θ2 decrease in Figure 3.3, the forces in the struts and the forces in the horizontal reinforcement increase, while less vertical ties may be needed, hence reducing the amount of shear reinforcement required.
Schlaich et al. (1987) defined the optimal strut-and-tie model, for a certain load case, as being the one with the lowest need for reinforcement. This model would also be the one for which the strain energy is minimum, because the strains in the reinforcement are more important than the ones in the concrete.