6.2 Beam Bending Examples
        Ex. 6.2.1 | Ex. 6.2.2

Example 6.2.1

Given: The rectangular beam, built in at the left end, having length, L, and cross-section of width, b, and height, h, is acted upon by a point load, P, at its free end.

Req'd: Determine the moment of inertia, I, and maximum bending stress, sx, in the beam.

Sol'n: For a rectangular beam with width, b, and height, h, the Moment of Inertia for bending about the z-axis is:

A maximum moment of Mmax = -P·L occurs at x = 0, and the maximum bending stress occurs at the top and bottom surfaces, y = ±h/2; then:
NOTE:
ymax is often called c - the distance from the neutral axis to the outermost material point, or "furthest fiber" from wood engineering.

Example 6.2.2

Given: A simply supported solid circular rod with radius r = 1.2 in. and length L = 50 in. is subjected to a uniform distributed load of q(x) = 24 lbf/in.

Req'd:
(a)Determine the maximum Moment, Mmax in the rod.
(b)Calculate the Moment of Inertia, Iz of the rod.
(c)Determine the maximum bending stress smax in the rod.

Sol'n: (a) By summing the forces in the y-direction, the resultant forces can be shown to be R1 = R2 = qL/2.

Taking a "cut" in the rod at an arbitrary distance from the end (x), and treating the section as a FBD, the shear force and moment in the beam are:

V(x) = qx - R1 = qx -qL/2

V(x) = q
x -
L

2

and

M(x) = R1x - qx2/2 = qLx/2 - qx2/2

M(x) =
- q

2
x2 - Lx

The maximum moment occurs where V(x) = 0, or by symmetry when x = L/2:

Mmax = M(L/2) = - q

L2

8

-

L2

4

Mmax =

qL2

8


Shear and moment diagrams

(b) The moment of inertia about the z-axis of the circular rod is given by:
Iz =


y2 dA =
2p

0
r

0
(r sinq)2 r dr dq
Iz =
2p

0
r4 sin2q)

4
dq =
r4

8
q -
sin 2q

2
2p


0
Iz =
p r4

4

(c) The bending stress is given by
sx =
M·y

I
   
smax =
M(L/2)·r

I
smax =
(qL2/8)·r

p r4/4
=
qL2

2pr3
smax = 5.53 ksi