2.3. IDEALIZED STRUCTURAL RESPONSES 17
applied to yield the well-known formula to predict the bending stress at a given distance from
the neutral axis:
where M refers to the resultant moment, I represents the cross-sectional property known as the
area moment of inertia about an axis passing through the centroid of the cross section, and y
represents the distance from the neutral axis toward the outer edge of the cross section where the
stress is being evaluated. e displacement of a beam due to a transverse loading can be determined
Figure 2.4: Stress distribution due to bending loads varies linearly through the cross section.
by integrating the fourth-order diﬀerential equation
where E is the modulus of elasticity and w is the load per unit length applied transversely to the
beam. is basic theory of beam bending is often referred to as Euler-Bernoulli beam theory.
2.3.4 TORSIONAL RESPONSE
Shear stresses can also develop when a torque is applied to a shaft. If the shaft is circular or
annular in cross section, it can be assumed that cross sections remain parallel and circular. From
this assumption, the shear stress due to torsion can be predicted at a point at a given radial distance,
, away from the center by the well-known formula
where T is the total torque carried by the section and J is the polar moment of inertia of the cross
section. ese stress components are illustrated in Fig. 2.5. Under these conditions, the axial twist
(sometimes referred to as angular displacement) along such a shaft of length L can be calculated
from the formula
where G is the modulus of rigidity.