# In-plane loading

## Walls and beams

### Stress fields

Using stress fields, both the multidimensional load-bearing behaviour of concrete and the contribution of distributed reinforcement layers can be considered with any desired level of approximation.

Plane discontinuous stress fields are suitable for practical applications and are composed of several individual basic elements connected through discontinuities. These models are formed by a group of principal stress trajectories subjected to uniaxial or biaxial stresses (in the case of nodal regions with biaxial uniform compressive stress states – often inaccurately referred to as “hydrostatic” – the group of stress trajectories degenerates). These trajectories can be parallel to each other (struts and ties) or divergent (fans and arches).

### Stress fields with prestressing

The maximum spans of reinforced concrete structures are limited due to crack formation and long-term deformations (creep), since the deformations become unacceptably large in the case of large slenderness, as required for long-span structures.

With prestressing, these problems can be solved by compensating the stresses caused by permanent loads – or part of them – by the anchorage and deviation forces of the prestressing. The prestressing is thus an essential basis for long-span, slender and economical concrete structures. It was essential for the breakthrough and success of concrete structures, especially in bridge construction.

This chapter deals with the treatment of prestressing as anchorage, deviation and friction forces and the design and verification using strut-and-tie models or stress fields.

### Compatibility and deformation capacity

The application of the static and kinematic methods of the theory of plasticity allows bracketing the ultimate load of a structure. Those methods implicitly assume that the deformation capacity is sufficient, since the ultimate load is determined based on the assumption of a perfectly plastic behaviour. Due to the limited ductility of reinforced concrete, the question arises whether the state of equilibrium on which the design is based can actually occur or if premature (brittle) failures might occur.

No clear solutions can be found for this question in principle. A basic problem is that the real loading and restraint history of a structure is not sufficiently known. Usually, it is assumed that the initial (load-free) state is free of stresses and deformations. In reality, however, there are always more or less considerable initial stresses and deformations due to previous loads or shrinkage. A second problem arises from the fact that the calculation of deformations requires a large number of material and system parameters. These are often either insufficiently known or subjected to large scatter.

For these reasons, in practical applications, merely some reliable estimates of the relevant deformation parameters are used. Detailed calculations are rather complex and only justified for special problems. However, one should always be aware of the achievable accuracy and the significance of the obtained results.

This chapter first reviews models for concrete in compression and bonded reinforcement. Then it shows how the deformation capacity of beams can be investigated and how the plastic deformation capacity can be checked.

## Membrane elements

### Equilibrium and yield conditions

This chapter discusses equilibrium and yield conditions for membrane elements.

In the first part, as a repetition of Stahlbeton I, the equilibrium conditions are established and the yield conditions for orthogonally reinforced membrane elements are derived. Subsequently, yield conditions for inclined reinforcement are shown. Finally, the yield conditions for orthogonally reinforced membrane elements – which assume a constant concrete compressive strength – are extended to account for the influence of the strain state on the concrete compressive strength.

### Compatibility and deformation capacity

This chapter examines the load-deformation behaviour of reinforced concrete membrane elements using the example of orthogonally reinforced membrane elements under uniform in-plane loading. In particular, the Cracked Membrane Model (CMM) developed at ETH Zurich will be introduced. This mechanically consistent model enables a realistic investigation of the load-deformation behaviour. The load-bearing and deformation behaviour of reinforced concrete membrane elements is generally quite complex. First, the basic features of the behaviour of membrane elements (not predominantly stressed by compression) are described. Afterwards, possible calculation models and solution methods for the behaviour of orthogonally reinforced membrane elements in the cracked state are discussed.

## Numerical modelling

In this chapter, we discuss how the mechanical models that have been introduced in the previous chapters can be used in numerical approaches that allow for a more efficient structural design. A profound knowledge of the underlying methods and their limits of applicability is essential for a safe and correct application of numerical models. The engineer should always keep the control over the design and understand the behaviour of the structure while using any kind of software for structural analysis or design.