Structural Plasticity

Plasticity is one of the underlying principles in the design of structures, especially metal and reinforced concrete structures. Numerous textbooks and monographs on structural plasticity and plastic design have been published since the 1950s (Baker et al., 1956; Baker, 1956; Neal, 1956; Heyman, 1958; Heyman, 1971; Hodge, 1959; Hodge, 1963; Horne, 1964; Horne, 1978; Baker and Heyman, 1969; Save and Massonnet, 1972; Chen, 1975; Chen, 1982; Morris and Randall, 1979; Horne and Morris, 1981; Zyczkowski, 1981; König and Maier, 1981; König, 1987; Mrazik et al., 1987; Save et al., 1997; Nielsen, 1999). The European Recommendations for the design of steelwork and reinforced concrete structures apply widely the plastic behavior of materials (Horne and Morris, 1981).

The mechanical behavior at a point of a solid can be represented by stress and strain components in three-dimensional space. Consider a generic point O of an elementary parallelepiped of a continuum referred to by orthogonal Cartesian axes x, y, z as shown in Fig. 2.1. Each of the three faces in the reference planes is in general subjected to one normal stress and two shear stresses. The state of stress at O is thus characterized by these nine stress components.

A yield condition or yield criterion describes a material failure in structural plasticity. It defines the threshold state of a material between elastic and plastic or brittle failure deformations. It is very important to a dopt a proper yield criterion in the design of a structure. The estimated load-bearing capacity of structures may be significantly affected by the choice of different yield criteria. Great efforts have been devoted to the formulation of yield criteria. Many different yield criteria have been proposed during the past 100 years (Pisarenko and Lebedev, 1976; Zyczkowski, 1981; Chen, 1982; 1998; Yu, 2002b; 2004). Amongst them, the Tresca criterion (Tresca, 1864), the Huber-von Mises criterion (Huber, 1904; von Mises, 1913), and the twin-shear yield criterion (Yu, 1961; 1983) are three representative criteria which can be used for materials that have identical strength in tension and compression (non-SD materials), and the shear strengths of τ _Y =0.5σ _Y , τ _y =0.577σ _y and τ _y =0.677σ _y , respectively, where σ _y is the uniaxial yield strength. The Drucker-Prager criterion is an extension of the Huber-von Mises criterion. Although the Drucker-Prager criterion has been widely applied for non-metallic materials, it contradicts some experimental results for geomaterials. The Mohr-Coulomb strength criterion (Mohr, 1900) and the twin-shear strength criterion (Yu, 1985) give the lower and upper bounds of the convex yield curves. Generally, a yield criterion is suitable for a certain type of material only.

To understand plastic limit analysis it is helpful to review the behavior of an elastic-plastic solid or structure subjected to mechanical loading. An inelastic solid will yield at a specific magnitude of the applied load. The corresponding load is called the elastic limit of the structure. If the external load exceeds the elastic limit, a plastic region starts to spread through the structure. With further expansion of the yield area, the displacement of the structure progressively increases. At another critical load, the plastic region becomes so large as not to resist the unconstrained plastic flow in the solid. The load cannot be increased beyond this point. The collapse load is called the plastic limit of the structure. Plastic limit analysis involves an associated flow rule of the adopted yield criterion. The plastic limit load is also registered as the load-bearing capacity of the structure.

The circular plate has been used widely as an important structural element in many branches of engineering. Reliable prediction of the load-bearing capacity of circular plates is crucial for optimum structural design. The load-bearing capacity of circular plates by using the Tresca yield criterion and Huber-von Mises criterion has been given by (1954), (1954), and (1994), et al. The design of circular plates based on the plastic limit load was discussed by (1960). Nine cases including a simply supported circular plate, clamped circular plate, annular plate, a built-in at inner edge and simply supported along the outer edge plate, shearing force along the outer edge and built-in at the inner edge, etc. were studied (Hu, 1960). A systematical summary was given by (1960), (1959); (1963), (1972), (1981), (1985) and (1997).

The unified solutions of plastic limit analyses for a simply supported circular plate have been described in last chapter. Clamped circular plates are one of the typical structural elements in many branches of engineering. The limit analyses for a clamped circular plate were studied by using the Tresca, Hubervon Mises and Mohr-Coulomb criteria (Wang and Hopkins, 1954; Hu, 1960; Hodge, 1963; Zyczkowski, 1981; Nielsen, 1999).

An annular plate is a very common structural component in many branches of engineering, such as mechanical, aeronautic, and civil fields. Limit analysis of the plate is important in revealing its structural behavior and load-bearing capacity. The load-bearing capacity of circular plates has been given in terms of the Huber-von Mises criterion (Hopkins and Prager, 1953) and the Tresca yield criterion (Hopkins and Wang, 1954). The results are applicable for the materials which has τ _s =0.5σ _s and τ _s =0.577σ _s , respectively. The load-bearing capacity of annular plates was studied by (1960), (Hodge 1959, Hodge 1963), (1972), (1981), (1997), et al.

Plate structures are widely used in aerospace, shipping, civil, and mechanical engineering. Plastic limit analyses of flat plates with different geometries can approximately estimate the load-bearing capacities of the plates. A lot of analytical solutions for flat plates have been reported by (1961), (1963), (1972), (1997), (1996), (1998). Their solutions are mainly based on the Tresca yield criterion, the Huber-von Mises yield criterion, or the Mohr-Coulomb strength criterion. The maximum principal stress criterion has also been applied for simplicity.

Thin-walled vessels and thick-walled cylinders are applied widely in industry as pressure vessels, pipes, gun barrels, cylinders of rockets, etc. The limit analyses of thick-walled hollow spheres and cylinders under internal pressure were discussed in detail by (1950), and (1962). Further studies on this subject were reported by (1958), (1962), (1965), and (1981). The Tresca yield criterion or the Huber-von Mises yield criterion is usually applied for the design of thin-walled pressure vessels. The result using the Huber-von Mises yield criterion for a spherical vessel is similar to that using the Tresca yield criterion. These solutions are applicable only for non-SD materials. It can be seen in the textbook of plasticity.

The dynamic elastic response of plates and beams has been studied thoroughly. On the other hand, it is complicated to derive a dynamic plastic response for structures because of the plastic constitutive model. The first solution to the dynamic response of a rigid and simply supported circular plate was derived by Hopkins and Prager in 1954. Over the past fifty years very many research efforts have been concentrated on this subject by considering various boundary, loading conditions, and plastic flow assumptions (Florence, 1977; Jones and Oliveira, 1980; Jones, 1968; 1971; Stronge and Yu, 1993). Membrane mode solutions for impulsively loaded circular plates were derived by (1979). The dynamic response and failure of fully clamped circular plates under impulsive loading was studied by (1993).

A rotating disc and cylinder, as shown in Fig. 11.1, are often used as vane wheel and rotating axle of a propeller in many branches of engineering. When disc and cylinder rotate at an angular speed ω about their central axis, the stress and displacement caused by the centrifugal force are axisymmetric, i.e., σ _r , σ _θ and radial displacement u _r are related to radius r only. The rotating disc is in the generalized plane stress state, and rotating cylinder is in the generalized plane strain state.

A lot of research work has been conducted on impact and penetration analysis. The penetration studies include various lab and field tests, analytical derivations and numerical simulations. Early works were mainly experimental studies. In the last three decades, analytical and numerical tools have been used increasingly as a substitute for costly experiments. The critical issue in an analytical penetration model is to formulate properly the resultant penetration resistance force applied on the missile by the target medium. The most well-known resistance function is based on the so-called dynamic cavity expansion theory. The theory was pioneered by (1945), who developed the equations for the quasi-static expansion of cylindrical and spherical cavities and estimated forces on conical nose punches pushed slowly into metal targets. Later (1950) and (1960) derived and discussed the dynamic and spherically symmetric cavity-expansion equations for an incompressible target material.

Many plates used in practical engineering are strengthened by stiffeners to achieve high strength and low structural weight. Stiffeners are usually placed along the orthogonal directions. Thus a plate strengthened with stiffeners will exhibit structural plane orthotropy with two orthogonal axes of symmetry (Tsai, 1968; Daniel and Ishai, 1994).

A wellbore structure is usually used for underground engineering. The wellbore should be kept stable as it is subjected to earth stress in the mining engineering. A petroleum wellbore sustains the earth stress around the rock as well as the internal pressure of the oil. The stability of the wellbore is of great importance for the successful drilling.

Correct prediction of the load-bearing capacity of structures is a crucial task in the analysis and design of engineering structures. The plastic limit load of structures from limit analysis or slip-line analysis is usually used as an index of the load-bearing capacity of the structure, subjected to a monotonic loading. When the loading is a repeated loading, the structures fail at a load which is lower than the plastic limit load. This is due to gradual deterioration caused by the alternating plasticity or by the incremental plasticity instead of sudden collapse.

The static shakedown theorem (Melan’s theorem, the first shakedown theorem, or the lower bound shakedown theorem, Melan, 1936) and the dynamic shakedown theorem (Koiter’s theorem, the second shakedown theorem, or the upper bound shakedown theorem, Koiter, 1953; 1956; 1960) and the unified solution of shakedown limit for a thick-walled cylinder have been described in Chapter 15. In this chapter we will deal with the shakedown analysis for a simply supported circular plate and a clamped circular plate. The unified solutions are given for non-SD materials.

Rotating cylinders are a commonly-used component in engineering. When they rotate at a constant velocity the centrifugal force is the main loading applied on the cylinders. Elasto-plastic analyses are needed for the cylinders under static loadings and the shakedown of cylinders should be taken into account when subjected to cyclic-variation loading. The shakedown theorem and analysis of structures were described in the literature (Kachanov, 1971; Martin, 1975; Zyczkowski, 1981; König, 1987; Mroz et al., 1995; Weichert et al., 2000). Limits to shakedown loads were discussed by (2001), (1994) and others.