Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

on the x, y, and z planes. Stresses in the x′–y′–z′
coordinate system, however, are related to those in
the x–y–z coordinate system, and through a mathematical
process called stress transformation,
stresses can be converted from one coordinate
system to another. If the normal and shear stresses
on the x, y, and z planes at point Q are known (Figures
12.3, 12.4, and 12.5), then the normal and
shear stresses on any plane passing through point Q
can be determined. For this reason, the stresses on
these planes are called the state of stress at a
point. The state of stress can be uniquely defined
by three stress components acting on each of three
mutually perpendicular planes.
The state of stress at a point (such as point Q in the preceding figures) is conveniently
represented by stress components acting on an infinitesimally small cubic element of material
known as a stress element (Figure 12.6). The stress element is a graphical symbol that
represents a point of interest in an object (such as a shaft or a beam). The six faces of the
cubic element are each identified by the outward normal to the face. For example, the
positive x face is the face whose outward normal is in the direction of the positive x axis.
The coordinate axes x, y, and z are arranged as a right-hand coordinate system.
The stress components σ x , σ y , and σ z are normal stresses that act on the faces that
are perpendicular to the x, y, and z coordinate axes, respectively. There are six shear
stress components acting on the cubic element: τ xy , τ xz , τ yx , τ yz , τ zx , and τ zy . However,
only three of these shear stresses are independent, as will be demonstrated subsequently.
Specific values associated with stress components are dependent upon the orientation of
the coordinate axes. The state of stress shown in Figure 12.6 would be represented by a
different set of stress components if the coordinate axes were rotated.
Stress Sign conventions
FIGURE 12.3 Stresses acting on
an x plane at point Q in the body.
Normal stresses are indicated by the symbol σ and a single subscript that indicates the
plane on which the stress acts. The normal stress acting on a face of the stress element is
positive if it points in the outward normal direction. In other words, normal stresses are
positive if they cause tension in the material. Compressive normal stresses are negative.
Shear stresses are denoted by the symbol τ followed by two subscripts. The first
subscript designates the plane on which the shear stress acts. The second subscript indicates
the direction in which the stress acts. For example, τ xz is shear stress on an x face
acting in the z direction. The distinction between a positive and a negative shear stress
depends on two considerations: (1) the face of the stress element upon which the shear
stress acts and (2) the direction in which the stress acts.
z
P 3
P 4
y
P 2
τ xz
τ xy
σ x
Q
P 5
x
z
z
y
σ z
P 2
Q
τ zy
τ zx
P 1
FIGURE 12.5 Stresses acting on a
z plane at point Q in the body.
z
σ
z
P 4
y
σ y
Q τ yx
τ yz
τ yz
y
P 5
FIGURE 12.4 Stresses acting on a
y plane at point Q in the body.
σ
y
τ yx
τ zy τ xy
τ zx
τ x
xz
σ
FIGURE 12.6 Stress element
representing the state of stress
at a point.
x
x
x
A shear stress is positive if it
• acts in the positive coordinate direction on a positive face of the stress element, or
• acts in the negative coordinate direction on a negative face of the stress element.
For example, a shear stress on a positive x face that acts in a positive z direction is a positive
shear stress. Similarly, a shear stress that acts in a negative x direction on a negative y
face is also considered positive. The stresses shown on the stress element in Figure 12.6
are all positive.
mecmovies 12.5 presents
an animated discussion of
terminology used in stress
transformations.
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