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

All such moment increments that act on the cross section, along with the internal
bending moment M, must sum to zero in order to satisfy equilibrium about the z axis:
Σ Mz
= −∫ yσ
x dA − M = 0
A
If Equation (8.3) is substituted for σ x , then the bending moment M can be related to the
radius of curvature ρ:
E
M =− ∫ yσ
x dA =
A ρ
∫
A
2
y dA
(8.5)
Again from statics, recall that the integral term in Equation (8.5) is called the second
moment of area or, more commonly, the area moment of inertia:
I
z
=
∫
A
2
y dA
The subscript z indicates an area moment of inertia determined with respect to the z centroidal
axis (i.e., the axis about which the bending moment M acts). The integral term in
Equation (8.5) can be replaced by the moment of inertia I z , where
M
to give an expression relating the beam curvature to its internal bending moment:
=
EI z
ρ
1
κ = = M ρ EI
z
(8.6)
This relationship is called the moment–curvature equation, and it shows that the beam
curvature is directly related to the bending moment and inversely related to the quantity
EI z . In general, the term EI is known as the flexural rigidity, and it is a measure of the
bending resistance of a beam.
Flexure Formula
The relationship between the normal stress σ x and the curvature was developed in Equation
(8.3), and the relationship between the curvature and the bending moment M is given by
Equation (8.6). These two relationships can be combined, giving
σ
x
=− Eκy =−E ⎛ M ⎞
y
⎝ ⎜ EI ⎠
⎟
to define the stress produced in a beam by a bending moment:
z
243
NORMAL STRESSES IN bEAMS
In the context of mechanics of
materials, the area moment of
inertia is usually referred to as,
simply, the moment of inertia.
The radius of curvature ρ is
measured from the center of
curvature to the neutral surface
of the beam. (See Figure 8.5b.)
σ
x
=− My
I
z
(8.7)
Equation (8.7) is known as the elastic flexure formula or, simply, the flexure formula. As
developed here, a bending moment M that acts about the z axis produces normal stresses that
act in the x direction (i.e., the longitudinal direction) of the beam. The stresses vary linearly
in intensity over the depth of the cross section. The normal stresses produced in a beam by
a bending moment are commonly referred to as bending stresses or flexural stresses.
Examination of the flexure formula reveals that a positive bending moment causes
negative normal stresses (i.e., compression) for portions of the cross section above the
neutral axis (i.e., positive y values) and positive normal stresses (i.e., tension) for portions
below the neutral axis (i.e., negative y values). The opposite stresses occur for a negative
bending moment. The distributions of bending stresses for both positive and negative bending
moments are illustrated in Figure 8.6.