- 1.1 Introduction
- 1.2 Scope of Treatment
- 1.3 Analysis and Design
- 1.4 Conditions of Equilibrium
- 1.5 Definition and Components of Stress
- 1.6 Internal Force-Resultant and Stress Relations
- 1.7 Stresses on Inclined Sections
- 1.8 Variation of Stress Within a Body
- 1.9 Plane-Stress Transformation
- 1.10 Principal Stresses and Maximum in-plane Shear Stress
- 1.11 Mohr's Circle for Two-Dimensional Stress
- 1.12 Three-Dimensional Stress Transformation
- 1.13 Principal Stresses in Three Dimensions
- 1.14 Normal and Shear Stresses on an Oblique Plane
- 1.15 Mohr's Circles in Three Dimensions
- 1.16 Boundary Conditions in Terms of Surface Forces
- 1.17 Indicial Notation
1.9 Plane-Stress Transformation
A two-dimensional state of stress exists when the stresses and body forces are independent of one of the coordinates, here taken as z. Such a state is described by stresses s x , s y , and t xy and the x and y body forces. Two-dimensional problems are of two classes: plane stress and plane strain. In the case of plane stress, as described in the previous section, the stresses s z , t xz , and t yz , and the z-directed body forces are assumed to be zero. The condition that occurs in a thin plate subjected to loading uniformly distributed over the thickness and parallel to the plane of the plate typifies the state of plane stress (Fig. 1.10). In the case of plane strain, the stresses t xz and t yz and the body force Fz are likewise taken to be zero, but s z does not vanish* and can be determined from stresses s x and sy .
Figure 1.10 Thin Plate in-plane loads.
We shall now determine the equations for transformation of the stress components s x , s y , and t xy at any point of a body represented by an infinitesimal element, isolated from the plate illustrated in Fig. 1.10. The z-directed normal stress s z , even if it is nonzero, need not be considered here. In the following derivations, the angle q locating the x' axis is assumed positive when measured from the x axis in a counterclockwise direction. Note that, according to our sign convention (see Sec. 1.5), the stresses are indicated as positive values.
Consider an infinitesimal wedge cut from the loaded body shown in Fig. 1.11a, b. It is required to determine the stresses s x' and t x'y' , which refer to axes x', y' making an angle q with axes x, y, as shown in the figure. Let side AB be normal to the x' axis. Note that in accordance with the sign convention, s x' and t x'y' are positive stresses, as shown in the figure. If the area of side AB is taken as unity, then sides QA and QB have area cos q and sin q, respectively.
Figure 1.11 Elements in plane stress.
Equilibrium of forces in the x and y directions requires that
where px and py are the components of stress resultant acting on AB in the x and y directions, respectively. The normal and shear stresses on the x' plane (AB plane) are obtained by projecting px and py in the x' and y' directions:
From the foregoing it is clear that . Upon substitution of the stress resultants from Eq. (1.16), Eqs. (a) become
Note that the normal stress sy' acting on the y' face of an inclined element (Fig. 1.11c) may readily be obtained by substituting q + p/2 for q in the expression for s x' . In so doing, we have
Equations (1.17) can be converted to a useful form by introducing the following trigonometric identities:
The transformation equations for plane stress now become
The foregoing expressions permit the computation of stresses acting on all possible planes AB (the state of stress at a point) provided that three stress components on a set of orthogonal faces are known.
Stress tensor. It is important to note that addition of Eqs. (1.17a) and (1.17c) gives the relationships
s x + s y = s x' + s y' = constant
In words then, the sum of the normal stresses on two perpendicular planes is invariant—that is, independent of q. This conclusion is also valid in the case of a three-dimensional state of stress, as shown in Section 1.13. In mathematical terms, the stress whose components transform in the preceding way by rotation of axes is termed tensor. Some examples of other quantities are strain and moment of inertia. The similarities between the transformation equations for these quantities are observed in Sections 2.5 and C.4. Mohr's circle (Sec. 1.11) is a graphical representation of a stress tensor transformation.
Polar Representations of State of Plane Stress
Consider, for example, the possible states of stress corresponding to s x = 14 MPa, s y = 4 MPa, and t xy = 10 MPa. Substituting these values into Eq. (1.18) and permitting q to vary from 0° to 360° yields the data upon which the curves shown in Fig. 1.12 are based. The plots shown, called stress trajectories, are polar representations: s x' versus q (Fig. 1.12a) and t x'y' versus q (Fig. 1.12b). It is observed that the direction of each maximum shear stress bisects the angle between the maximum and minimum normal stresses. Note that the normal stress is either a maximum or a minimum on planes at q = 31.66° and q = 31.66° + 90°, respectively, for which the shearing stress is zero. The conclusions drawn from this example are valid for any two-dimensional (or three-dimensional) state of stress and are observed in the sections to follow.
Figure 1.12 Polar representations of sx' and Tx'y' versus q (in megapascals) versus q.
Cartesian Representation of State of Plane Stress
Now let us examine a two-dimensional condition of stress at a point in a loaded machine component on an element illustrated in Fig. 1.13a. Introducing the given values into the first two of Eqs. (1.18), gives
= 4.5 + 2.5 cos 2q + 5 sin 2q
= –2.5 sin 2q + 5 cos 2q
Figure 1.13 Graph of normal stress sx' and shearing stress Tx'y' with angle q (for q 180°).
In the foregoing, permitting q to vary from 0° to 180° in increments of 15° leads to the data from which the graphs illustrated in Fig. 1.13b are obtained [Ref. 1.7]. This Cartesian representation demonstrates the variation of the normal and shearing stresses versus q 180°. Observe that the direction of maximum (and minimum) shear stress bisects the angle between the maximum and minimum normal stresses. Moreover, the normal stress is either a maximum or a minimum on planes
q = 31.7° and
q = 31.7° + 90°, respectively, for which the shear stress is zero. Note as a check that s
max + s
min = 9 MPa, as expected.
The conclusions drawn from the foregoing polar and Cartesian representations are valid for any state of stress, as will be seen in the next section. A more convenient approach to the graphical transformation for stress is considered in Sections 1.11 and 1.15. The manner in which the three-dimensional normal and shearing stresses vary is discussed in Sections 1.12 through 1.14.