Patrick de Buhan

A Multiphase Model for Assessing the Overall Yield Strength of Soils Reinforced by Linear Inclusions

Conceived as an extended homogenization procedure, a multiphase approach for ascertaining the macroscopic behavior of reinforced soil structures has been developed in the last years. This contribution is dedicated to the evaluation of the yield strength properties of soils reinforced by linear inclusions by making use of a homogenization procedure, in which the reinforced soil is regarded as a periodic composite, as a first calculation, and using the multiphase model. It appears from such a calculation that only the multiphase model is able to capture scale and boundary effects, which may play an important role in the yield design of reinforced structures. The decisive element is the introduction of a parameter characterizing the strength of the interaction between two continuous media (“phases”) representing the soil and the reinforcing inclusions, respectively. A preliminary analysis suggests that such a parameter varies in direct proportion to the inverse of a scale factor.

The “Mixed Modeling” Approach to Reinforced Structures

Most civil engineering structures involving inclusion-reinforced materials, such as reinforced concrete beams and slabs or reinforced geotechnical structures, are commonly designed by resorting to an implicit mechanical model in which the material to be reinforced (concrete, soil, rock, etc.) is regarded as a three-dimensional ( 3D ) continuous medium , whereas the reinforcing inclusions (steel bars, piles or geomembranes) are treated as one-dimensional ( 1D ) (beams) or two-dimensional ( 2D ) (plates) structural elements . Hence, the denomination of “ mixed modeling ” to which this chapter is devoted.

The Homogenization Approach: Reinforced Materials as Macroscopically Homogeneous Anisotropic Media

While the “mixed modeling” approach provides a simplified, but suitable method for designing reinforced structures, as far as a limited number of inclusions is involved, it obviously becomes inappropriate in two frequently encountered situations. First in the case when the number of inclusions is very large (up to several hundreds); second when the stiffness or strength characteristics of the reinforcing material are not so much greater than those of the surrounding matrix material, so that, in spite of their geometrical slenderness, the inclusions can hardly be modeled as one-dimensional (1D) beam or two-dimensional (2D) membrane elements. As will be seen further on, a typical illustration of the latter situation is provided by the example of foundation soils improved by columnar inclusions.

Applications of the Multiphase Approach Part 2: Load-Bearing Capacity and Stability Analysis of Reinforced Structures

This chapter discusses some recent applications of the multiphase model to the design of reinforced structures, which are being loaded beyond their initial yield point up to failure.

Applications of the Multiphase Approach Part 1: Static and Dynamic Stiffness of Piled Raft Foundations

This chapter are devoted to the presentation of some illustrative applications of the multiphase approach to geotechnical engineering problems involving soils reinforced by linear inclusions. The present chapter is more specifically focused on evaluating the overall stiffness of piled raft foundations under either (quasi) static or truly dynamic conditions, in the context of linear elasticity .

Yield Design of Reinforced Concrete Beams, Plates and Shells

This chapter addresses different aspects of the yield design of composite structural elements such as beams, plates and shells. A typical example of such a situation is the case of reinforced concrete members. The aim of this chapter is to show how to derive macroscopic strength properties in terms of generalized stresses such as membrane forces and bending moments in order to perform the yield design at the structure level. This “up-scaling” procedure will make use of homogenization concepts such as those presented in the previous chapters. Since complete analytic solutions are rarely available, related numerical aspects will also be briefly discussed.

Mechanical Modeling of Reinforced Materials as Multiphase Systems

The concept of a multiphase model of heterogeneous media, such as reinforced materials, originates from the basic intuition that the macroscopic behavior of this kind of materials can be appropriately described as the superposition of several (and not only one as in the classical homogenization procedure) continua, called phases . It is therefore strongly reminiscent of that classically developed in the field of poromechanics where a fluid saturated porous solid (soil or rock) is modeled by two such mutually interacting fluid and solid phases. It should be noted that, in closer connection to our topic, this concept had been already advocated by Stern and Bedford and Bedford and Drumheller for modeling the dynamic behavior of elastic laminates.

Yield Design of Periodically Heterogeneous Plates

This work addresses the determination of the overall strength capacities of periodically heterogeneous plates within a yield design framework. Illustrative applications focus, notably, on reinforced concrete slabs in fire conditions. A homogenization procedure and related numerical tools are proposed to compute macroscopic strength criteria expressed in terms of generalized forces (membrane and bending solicitations). To this end, a yield design auxiliary problem is formulated on the representative three-dimensional unit cell and a numerical resolution by a static approach is presented, making use of simple 3D equilibrium finite elements. A particular emphasis is put on the link between the local strength criterion of steel and concrete and the resulting optimization problem, which can be formulated, either as a second-order cone programming (SOCP) problem or, more generally, as a semi-definite programming (SDP) problem. A first illustrative example of a concrete slab with a single array of steel bars will be used to validate the approach. Then, the influence of fire conditions on the strength capacities of reinforced concrete slabs will be investigated and numerical computations will be confronted to experimental results.