12.4 Flexural Vibration of Beams
453
The stress–strain relation of the beam element is nothing but the bending moment–
curvature relationship.
M = − E I
d
2
w
dx 2
(12.49)
Equation (12.49) written in matrix form for generality
{σ } = [D] {∈}
(12.50)
where
{σ } = M, [D] = E I and {∈} = −
d
2
w
dx 2
(12.51)
Differentiating both sides of Eq. (12.48) twice with respect to x, we get
{∈} =
6
L 2 −
12x
L 3
4
L
−
6x
L 2
−
6
L 2 +
12x
L 3
2
L
−
6x
L 2
{X } e
(12.52)
or
{∈} = [B] {X } e
(12.53)
Combining Eqs. (12.50) and (12.53), we get
{σ } = [D] [B] {X } e
(12.54)
Based on the minimum potential energy principle shown in the previous section,
the stiffness matrix of the element is given by
[K ] e =
L
0
[B]
T
[D] [B] dx
(12.55)
or
[K ] e =
E I
L 3
⎡
⎢
⎢
⎣
12
Symmetrical
6L 4L
2
− 12 − 6L 12
6L 2L
2
− 6L
4L
2
⎤
⎥
⎥
⎦
(12.56)