Craig's Soil Mechanics 7th Edition
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followed. In analysis an appropriate idealization of the stress–strain relationship is employed. One such idealization is shown by the dotted lines in Figure 5.2(a), linearly elastic behaviour being assumed between O and Y0 (the assumed yield point) followed by unrestricted plastic strain (or flow) Y0P at constant stress. This idealization, which is shown separately in Figure 5.2(b), is known as the elastic– perfectly plastic model of material behaviour. If only the collapse condition in a practical problem is of interest then the elastic phase can be omitted and the rigid– perfectly plastic model, shown in Figure 5.2(c), may be used. A third idealization is the elastic–strain hardening plastic model, shown in Figure 5.2(d), in which plastic strain beyond the yield point necessitates further stress increase. If unloading and reloading were to take place subsequent to yielding in the strain hardening model, Figure 5.1 Two-dimensional state of stress in an element. Elasticity and plasticity 137 as shown by the dotted line Y00U in Figure 5.2(d), there would be a new yield point Y00 at a stress level higher than that at Y0. An increase in yield stress is a characteristic of strain hardening. No such increase takes place in the case of perfectly plastic (i.e. non-hardening) behaviour, the stress at Y00 being equal to that at Y0 as shown in Figures 5.2(b) and (c). A further idealization is the elastic–strain softening plastic model, represented by OY0P0 in Figure 5.2(d), in which the plastic strain beyond the yield point is accompanied by stress decrease. In plasticity theory  the characteristics of yielding, hardening and flow are considered; these are described by a yield function, a hardening law and a flow rule, respectively. The yield function is written in terms of stress components or principal stresses. The Mohr–Coulomb criterion (Equation 4.6) is one possible yield function if perfectly plastic behaviour is assumed. Alternatively, the yield function could be expressed in terms of critical-state parameters (Section 4.5). The hardening law represents the relationship between the increase in yield stress and the corresponding plastic strain components. The flow rule specifies the relative (i.e. not absolute) magnitudes of the plastic strain components during yielding under a particular state of stress. The hardening law and the flow rule can also be expressed in terms of critical- state parameters. In practice the most widely used solutions are those for the vertical stress at a point below a loaded area on the surface of a soil mass. It has been shown  that the vertical stress increment at a given point below the surface due to foundation loading is insensitive to a relatively wide range of soil characteristics such as heterogeneity, Figure 5.2 (a) Typical stress–strain relationship, (b) elastic–perfectly plastic model, (c) rigid– perfectly plastic model, and (d) elastic–strain hardening and softening plastic models. 138 Stresses and displacements anisotropy and non-linearity of the stress–strain relationship. Accordingly, solutions from linear elastic theory, in which the soil is assumed to be homogeneous and isotropic, are sufficiently accurate for use in most cases. The main exceptions are loose sands and soft clays, particularly where they are overlain by a relatively dense or stiff stratum. It should be noted, however, that increments of horizontal stress and of shear stress are relatively sensitive to soil characteristics. The insensitivity of the vertical stress increment depends on the assumption of a uniform pressure distribution, as would be the case for a flexible foundation. In the case of a stiff foundation the contact pressure is non-uniform, the exact distribution depending on the soil characteristics. A comprehensive collection of solutions from the theory of elasticity has been published by Poulos and Davis . The finite elementmethod is nowwidely used for the calculation of stresses and displace- ments. For example, the finite element program CRISP  enables the soil to be treated as either a non-homogeneous or an anisotropic elastic material; alternatively, elastoplastic behaviour can be modelled by means of critical-state parameters, enabling soil displace- ments up to failure to be determined. A comprehensive treatment of finite element analysis in geotechnical engineering has been published by Potts and Zdravkovic [14, 15]. Displacement solutions from elastic theory can be used at relatively low stress levels. These solutions require a knowledge of the values of Young’s modulus (E ) and Poisson’s ratio (�) for the soil, either for undrained conditions or in terms of effective stress. Poisson’s ratio is required for certain stress solutions. It should be noted that the shear modulus (G), where G ¼ E 2ð1þ �Þ ð5:3Þ is independent of the drainage conditions, assuming that the soil is isotropic. The volumetric strain of an element of linearly elastic material under three principal stresses is given by �V V ¼ 1� 2� E ð��1 þ��2 þ��3Þ If this expression is applied to soils over the initial part of the stress–strain curve, then for undrained conditions �V/V ¼ 0, hence � ¼ 0:5. The undrained value of Young’s modulus is then related to the shear modulus by the expression Eu ¼ 3G. If consolidation takes place then �V/V > 0 and � < 0:5 for drained or partially drained conditions. In principle, the value of E can be estimated from the curve relating principal stress difference and axial strain in an appropriate triaxial test. The value is usually deter- mined as the secant modulus between the origin and one-third of the peak stress, or over the actual stress range in the particular problem. However, because of the effects of sampling disturbance, it is preferable to determine E (or G) from the results of in-situ tests. One such method is to apply load increments to a test plate, either in a shallow pit or at the bottom of a large-diameter borehole, and to measure the resulting vertical displacements. The value of E is then calculated using the relevant displace- ment solution, an appropriate value of � being assumed. Elasticity and plasticity 139 The pressuremeter The shear modulus (G) can be determined in situ by means of the pressuremeter. The original pressuremeter was developed in the 1950s by Menard in an attempt to overcome the problem of sampling disturbance and to ensure that the macro-fabric of the soil is adequately represented. Menard’s original design, illustrated in Figure 5.3(a), consists of three cylindrical rubber cells of equal diameter arranged coaxially. The device is lowered into a (slightly oversize) borehole to the required depth and the central measuring cell is expanded against the borehole wall by means of water pressure, measurements of the applied pressure and the corresponding increase in volume of the cell being recorded. Pressure is applied to the water by compressed gas (usually nitrogen) in a control cylinder at the surface. The increase in volume of the measuring cell is determined from the movement of the gas–water interface in the control cylinder, readings normally being taken at times of 15, 30, 60 and 120 s after a pressure increment has been applied. The pressure is corrected for (a) the head difference between the water level in the cylinder and the test level in the borehole, (b) the pressure required to stretch the rubber cell and (c) the expansion of the control cylinder and tubing under pressure. The two outer guard cells are expanded under the same pressure as in the measuring cell but using compressed gas; the increase in volume of the guard cells is not measured. The function of the guard cells is to eliminate end effects, ensuring a state of plane strain adjacent to the measuring cell. Figure 5.3 Basic features of (a) Menard pressuremeter and (b) self-boring pressuremeter.