Complex Stress and Strain - Structural and Stress Analysis - page 373

Chapter 14 / Complex Stress and
Strain
In Chapters 7, 9, 10 and 11 we determined stress distributions produced separately
by axial load, bending moment, shear force and torsion. However, in many practical
situations some or all of these force systems act simultaneously so that the various
stresses are combined to form complex systems which may include both direct and
shear stresses. In such cases it is no longer a simple matter to predict the mode of
failure of a structural member, particularly since, as we shall see, the direct and shear
stresses at a point due to, say, bending and torsion combined are not necessarily the
maximum values of direct and shear stress at that point.
Therefore as a preliminary to the investigation of the theories of elastic failure in
Section 14.10 we shall examine states of stress and strain at points in structural mem-
bers subjected to complex loading systems.
14.1 R EPRESENTATION OF S TRESS AT A P OINT
We have seen that, generally, stress distributions in structural members vary through-
out the member. For example the direct stress in a cantilever beam carrying a point
load at its free end varies along the length of the beam and throughout its depth. Sup-
pose that we are interested in the state of stress at a point lying in the vertical plane
of symmetry and on the upper surface of the beam mid-way along its span. The direct
stress at this point on planes perpendicular to the axis of the beam can be calculated
using Eq. (9.9). This stress may be imagined to be acting on two opposite sides of a
very small thin element ABCD in the surface of the beam at the point (Fig. 14.1).
Since the element is thin we can ignore any variation in direct stress across its thickness.
Similarly, since the sides of the element are extremely small we can assume that σ has
the same value on each opposite side BC and AD of the element and that σ is constant
along these sides (in this particular case σ is constant across the width of the beam but
the argument would apply if it were not). We are therefore representing the stress at
a point in a structural member by a stress system acting on the sides and in the plane
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