Proper modeling involves free-body diagrams and equations
of equilibrium. However, it is important to remember that the
equilibrium equations of statics are valid only for forces or for
moments of forces, and not for stresses.

Stress

The intensity of a force is called stress and is de?ned as the
force acting on an in?nitesimal area. A normal stress ? is de?ned as

where dF is a differential normal force acting on a differential area
dA. It is often useful to calculate the average normal stress ? = P/A,
where P is the resultant force on an area A. A shear stress ? caused
by a shearing force V is de?ned likewise,

An average shear stress is obtained from V/A.
It is helpful to consider the general cases of stresses using rectangular
elements in two and three dimensions, while ignoring the deformations
caused by the stresses.

Plane Stress

There are relatively simple cases where all stress vectors lie in the
same plane. This is represented by a two-dimensional element in
Figure 1.5.1, where ?x and/or ?y may be either tensile (pulling on the
element as shown) or compressive (pushing on the element; not shown).
Normal stresses are easy to visualize and set up correctly.

FIGURE 1.5.1 Generalized plane stress.

Shear stresses need to be discussed here in a little detail. The notation
means that ?xy, for example, is a shear stress acting in the y direction,
on a face that is perpendicular to the x axis. It follows that ?yx is
acting in the x direction, on a face that is perpendicular to the y axis.
The four shear stress vectors are pointed as they are because of the
requirement that the element be in equilibrium: the net forces and
moments of forces on it must be zero. Thus, reversing the direction
of all four ?s in Figure 1.5.1 is possible, but reversing less than
four is not realistic.

Three-Dimensional State of Stress

The concept of plane stress can be generalized for a three-dimensional
element as shown in Figure 1.5.2, working with the three primary faces
of the cube and not showing stresses on the hidden faces, for clarity.

FIGURE 1.5.2 Three-dimensional general state of stress.

Again, the normal stresses are easy to set up, while the shear stresses
may require considerable attention. The complex cases of stresses
result from multiaxial loading, such as combined axial, bending,
and torsional loading. Note that even in complex situations simpli?cations
are possible. For example, if the right  face in Figure 1.5.2 is a free
surface, ?_x = ?_xz = ?_xy = 0. This leaves a plane stress state with
?_y, ?_z, and ?_yz, at most.

Stress Transformation

A free-body element with known stresses on it allows the calculation
of stresses in directions other than the given xyz coordinates.
This is useful when potentially critical welded or glued joints, or ?bers
of a composite, are along other axes. The stress transformations are
simplest in the case of plane stress and can be done in several ways.
In any case, at a given point in a material there is only one state
of stress at a particular instant. At the same time, the components
of the stresses depend on the orientation of the chosen coordinate
system.

The stress transformation equations depend on the chosen coordinate
system and the sign convention adopted. A common arrangement is
shown in Figure 1.5.3, where (a) is the known set of stresses and
(b) is the unknown set, denoted by primes.

FIGURE 1.5.3 Elements for stress transformation.

In the present sign convention an outward normal stress is positive,
and an upward shear stress on the right-hand face of the element is
positive. The transformation equations are

If a result is negative, it means that the actual direction of the stress
is opposite to the assumed direction.

Principal Stresses

It is often important to determine the maximum and minimum values
of the stress at a point and the orientations of the planes of these stresses.
For plane stress, the maximum and minimum normal stresses,
called principal stresses, are obtained from

There is no shear stress acting on the principal planes on which the
principal stresses are acting. However, there are shear stresses on
other planes. The maximum shear stress is calculated from

This stress acts on planes oriented 45? from the planes of
principal stress. There is a normal stress on
these planes of ?max, the average of ?x and ?y,

Mohrs Circle for Plane Stress

The equations for plane stress transformation have a graphical
solution, called Mohrs circle, which is convenient to use in engineering
practice, including back-of-the-envelope calculations. Mohrs circle
is plotted on a ? ? coordinate system as in Figure 1.5.4, with the center C
of the circle always on the ? axis at ?ave = (?x + ?y)/2
and its radius The positive ? axis is downward
for convenience, to make ? on the element and the corresponding 2?
on the circle agree in sense (both counterclockwise here).
The following aspects of Mohrs circle should be noted:
1. The center C of the circle is always on the ? axis, but it may move
left and right in a dynamic loading situation. This should be considered
in failure prevention.
2. The radius R of the circle is ?max, and it may change, even pulsate,
in dynamic loading. This is also relevant in failure prevention.
3. Working back and forth between the rectangular element and the
circle should be done carefully and consistently. An angle ? on the element
should be represented by 2? in the corresponding circle. If ? is positive
downward for the circle, the sense of rotation is identical in the element
and the circle.

FIGURE 1.5.4 Mohrs circle.

4. The principal stresses ?1 and ?2 are on the ? axis (? = 0).
5. The planes on which ?1 and ?2 act are oriented at 2?p from the
planes of ?x and ?y (respectively) in the circle and at ?p in the element.
6. The stresses on an arbitrary plane can be determined by their ?
and ? coordinates from the circle. These coordinates give magnitudes
and signs of the stresses. The physical meaning of +? vs. ?
regarding material response is normally not as distinct as +? vs. ?
(tension vs. compression).
7. To plot the circle, either use the calculated center C coordinate and
the radius R, or directly plot
the stress coordinates for two mutually perpendicular planes and draw
the circle through the two points (A and B in Figure 1.5.4) which must
be diametrically opposite on the circle. Special Cases of Mohrs
Circles for Plane Stress See Figures 1.5.5 to 1.5.9

FIGURE 1.5.5 Uniaxial tension.

FIGURE 1.5.6 Uniaxial compression.

FIGURE 1.5.7 Biaxial tension: ?x = ?y
(and similarly for biaxial compression: ?x = ?y ).

FIGURE 1.5.8 Pure shear.

FIGURE 1.5.9 Biaxial tension-compression: ?x = ?y
(similar to the case of pure shear).
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