Technology & Engineering
Stress Transformation Equations
Stress transformation equations are used in engineering to analyze how stress changes when a material is subjected to different loading conditions. These equations help engineers understand how forces applied to a material affect its internal stress distribution. By using stress transformation equations, engineers can predict how a material will respond to various types of loading, which is crucial for designing safe and efficient structures.
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4 Key excerpts on "Stress Transformation Equations"
eBook - PDF- Timothy A. Philpot(Author)
- 2014(Publication Date)
- Wiley(Publisher)
522 STRESS TRANSFORMATIONS Equations (12.3), (12.4), (12.5), and (12.6) are called the plane Stress Transformation Equations. They provide a means for determining normal and shear stresses on any plane whose outward normal is (a) perpendicular to the z axis (i.e., the out-of-plane axis), and (b) oriented at an angle with respect to the reference x axis. Since the transformation equations were derived solely from equilibrium considerations, they are applicable to stresses in any kind of material, whether it is linear or nonlinear, elastic or inelastic. Stress Invariance The normal stress acting on the n face of the stress element shown in Figure 12.11 can be determined from Equation (12.5). The normal stress acting on the t face can also be ob- tained from Equation (12.5) by substituting 90° in place of , giving the following equation: t x y x y xy 2 2 2 2 cos sin (12.7) If the expressions for n and t [Equations (12.5) and (12.7)] are added, the following relationship is obtained: n t x y (12.8) This equation shows that the sum of the normal stresses acting on any two orthogonal faces of a plane stress element is a constant value, independent of the angle . This mathematical characteristic of stress is termed stress invariance. Stress is expressed with reference to specific coordinate systems. The stress transfor- mation equations show that the n–t components of stress are different from the x–y compo- nents, even though both are representations of the same stress state. However, certain func- tions of stress components are not dependent on the orientation of the coordinate system. These functions, called stress invariants, have the same value regardless of which coordi- nate system is used.
- Amar Khennane(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
Such relations are called constitutive equations, since they describe the macroscopic behavior resulting from the internal constitution of the material. Materials, however, exhibit different behaviors over their entire range of deformations. As such, it is not possible to write one set of mathematical equations to describe these behaviors. Yet, for many engineering applications, the theory of linear elasticity offers a useful and reliable model for analysis. 5.2 STRESS TENSOR 5.2.1 D EFINITION Let us consider a body in equilibrium under external forces as represented in Figure 5.1. Let us take a cut through the body, as represented by the plane , and denote by dA an infinitesimal element of the internal cross section. A force d F is exerted on this small area. It represents the influence of the right section on the left section of the body. The vector d F can be expressed in terms of its normal and tangential components, d F n and d F t , to the surface dA . The stresses acting on the surface are then given as σ n = lim dA → 0 d F n dA (5.1) σ t = lim dA → 0 d F t dA (5.2) It can be seen that d F t has also two components on the plane of the surface dA . In total, therefore, there are three stress components: one normal and two tangential. However, as the infinitesimal element dA shrinks to a point, there will be an infinite number of planes passing through that point. It would be impossible therefore to consider all of them. However, if we choose three mutually perpendicular planes, as represented in Figure 5.2, the stresses can be written for all of 135 136 Introduction to Finite Element Analysis Using MATLAB and Abaqus dA dA dF t 1 dF t dF t 2 dF dF n ∑ FIGURE 5.1 Internal force components. 3 σ 31 σ 33 σ 32 σ 23 σ 22 σ 21 σ 12 σ 11 σ 13 2 1 FIGURE 5.2 Stress components at a point.
eBook - PDF- J. N. Reddy(Author)
- 2007(Publication Date)
- Cambridge University Press(Publisher)
4 Stress Measures Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone. Albert Einstein 4.1 Introduction In the beginning of Chapter 3, we have briefly discussed the need to study defor- mation and stresses in material systems that we may design for engineering applica- tions. All materials have certain threshold to withstand forces, beyond which they “fail” to perform their intended function. The force per unit area, called stress, is a measure of the capacity of the material to carry loads, and all designs are based on the criterion that the materials used have the capacity to carry the working loads of the system. Thus, it is necessary to determine the state of stress in a material. In the present chapter, we study the concept of stress and its various measures. For instance, stress can be measured per unit deformed area or undeformed area. As we shall see shortly, stress at a point in a three-dimensional continuum can be measured in terms of nine quantities, three per plane, on three mutually perpendic- ular planes at the point. These nine quantities may be viewed as the components of a second-order tensor, called stress tensor. Coordinate transformations and principal values associated with the stress tensor and stress equilibrium equations will also be discussed. 4.2 Cauchy Stress Tensor and Cauchy’s Formula First we introduce the true stress, that is, stress in the deformed configuration κ that is measured per unit area of the deformed configuration κ . The surface force acting on a small element of area in a continuous medium depends not only on the magnitude of the area but also upon the orientation of the area. It is customary to denote the direction of a plane area by means of a unit vector drawn normal to that plane, as discussed in Section 2.2.3.
eBook - PDF- J. N. Reddy(Author)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
151 4 STRESS MEASURES Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone. —– Albert Einstein (1879–1955) 4.1 Introduction In the beginning of Chapter 3, we briefly discussed the need for studying de-formations and stresses in material systems that we may design for engineering applications. All materials have certain thresholds to withstand forces, beyond which they “fail” to perform their intended function. The force per unit area, called stress , is a measure of the capacity of the material to carry loads, and all designs are based on the criterion that the materials used have the capacity to carry the working loads of the system. Thus, it is necessary to determine the state of stress in a material. In this chapter we study the concept of stress and its various measures. For instance, stress can be measured per unit deformed area or undeformed area. As we shall see shortly, stress at a point in a three-dimensional continuum can be measured in terms of nine quantities, three per plane, on three mutually perpendicular planes at the point. These nine quantities may be viewed as the components of a second-order tensor, called a stress tensor . Coordinate transformations and principal values associated with the stress tensor and stress equilibrium equations are also discussed. 4.2 Cauchy Stress Tensor and Cauchy’s Formula 4.2.1 Stress Vector First we introduce the true stress, that is, the stress in the deformed configu-ration κ that is measured per unit area of the deformed configuration κ . The surface force acting on a small element of (surface) area in a continuous medium depends not only on the magnitude of the area but also on the orientation of the area. It is customary to denote the direction of a plane area by means of a unit vector drawn normal to that plane.
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