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

242
bENdINg
produce a resultant force dF given by σ x dA. (Recall that force can be thought of as the
product of stress and area.) In order to satisfy horizontal equilibrium, all forces dF for the
beam in Figure 8.5a must sum to zero, or, as expressed in terms of calculus,
Σ Fx
= ∫dF = ∫ σ x dA = 0
Substitution of Equation (8.3) for σ x yields
A
Σ = ∫ σ =− E
∫
ρ
=− E
Fx
x dA ydA
ρ∫
ydA = 0
A
A
A
(8.4)
In Equation (8.4), the elastic modulus E cannot be zero for a solid material. The radius of
curvature ρ could equal infinity; however, this would imply that the beam does not bend at
all. Consequently, horizontal equilibrium of the normal stresses can be satisfied only if
∫ ydA = 0
A
(a)
This equation states that the first moment of area of the cross section with respect to the z
axis must equal zero. From statics, recall that the definition of the centroid of an area with
respect to a horizontal axis also includes the first-moment-of-area term:
Keep in mind that this
conclusion assumes pure
bending of an elastic material.
If an axial force exists in the
flexural member or if the
material is inelastic, the neutral
surface will not pass through
the centroid of the
cross-sectional area.
y =
∫A
∫
Substituting Equation (a) into Equation (b) shows that equilibrium can be satisfied only if
y = 0; in other words, the distance y measured from the neutral surface to the centroid of
the cross-sectional area must be zero. Thus, for pure bending, the neutral axis must pass
through the centroid of the cross-sectional area.
As discussed in Section 8.1, the study of bending presented here applies to beams that
have a longitudinal plane of symmetry. Consequently, the y axis must pass through the
centroid. The origin O of the beam coordinate system (see Figure 8.5b) is located at the
centroid of the cross-sectional area. The x axis lies in the plane of the neutral surface and is
coincident with the longitudinal axis of the member. The y axis lies in the longitudinal
plane of symmetry, originates at the centroid of the cross section, and is directed vertically
upward (for a horizontal beam). The z axis also originates at the centroid and acts in the
direction that produces a right-hand x–y–z coordinate system.
A
ydA
dA
(b)
A moment is composed of a
force term and a distance term.
The distance term is often called
a moment arm. On area dA, the
force is σ x dA. The moment arm
is y, which is the distance from
the neutral surface to dA.
moment–curvature Relationship
The second equilibrium equation to be satisfied requires that the sum of moments must
equal zero. Consider again the area element dA and the normal stress that acts upon it
(Figure 8.5b). Since the resultant force dF acting on dA is located a distance y from the
z axis, it produces a moment dM about the z axis. The resultant force can be expressed as
dF = σ x dA. A positive normal stress σ x (i.e., a tensile normal stress) acting on area dA,
which is located a positive distance y above the neutral axis, produces a moment dM that
rotates in a negative right-hand rule sense about the z axis; therefore, the incremental
moment dM is expressed as dM = −yσ x dA.