STATE OF STRESS CAUSED BY COMBINED LOADINGS

Here it is required that the material be homogeneous and behave in a
linear elastic manner. Also, Saint-Venant’s principle requires that the
stress be determined at a point far removed from any discontinuities in
the cross section or points of applied load.
Internal Loading.
• Section the member perpendicular to its axis at the point
where the stress is to be determined; and use the equations of
equilibrium to obtain the resultant internal normal and shear
force components, and the bending and torsional moment
components.
• The force components should act through the centroid of the
cross section, and the moment components should be
calculated about centroidal axes, which represent the principal
axes of inertia for the cross section.
Stress Components.
• Determine the stress component associated with each internal
loading.
Normal Force.
• The normal force is related to a uniform normal-stress
distribution determined from s = N>A.
Shear Force.
• The shear force is related to a shear-stress distribution
determined from the shear formula, t = VQ>It.
Bending Moment.
• For straight members the bending moment is related to a
normal-stress distribution that varies linearly from zero at
the neutral axis to a maximum at the outer boundary of the
member. This stress distribution is determined from the
flexure formula, s = -My>I. If the member is curved, the
stress distribution is nonlinear and is determined from
s = My>[Ae(R - y)].
Torsional Moment.
• For circular shafts and tubes the torsional moment is
related to a shear-stress distribution that varies linearly
from zero at the center of the shaft to a maximum at the
shaft’s outer boundary. This stress distribution is
determined from the torsion formula, t = Tr>J.
Thin-Walled Pressure Vessels.
• If the vessel is a thin-walled cylinder, the internal pressure
p will cause a biaxial state of stress in the material such
that the hoop or circumferential stress component is
s1 = pr>t, and the longitudinal stress component is
s2 = pr>2t. If the vessel is a thin-walled sphere, then the
biaxial state of stress is represented by two equivalent
components, each having a magnitude of s2 = pr>2t.
Superposition.
• Once the normal and shear stress components for each loading
have been calculated, use the principle of superposition and
determine the resultant normal and shear stress components.
• Represent the results on an element of material located at a
point, or show the results as a distribution of stress acting over
the member’s cross-sectional area.