4.1 The Basics of a Timoshenko Beam
91
Table 4.2 Different formulations of the partial differential equation for a Timoshenko beam in the
X -Z plane (X -axis: right facing; Z -axis: upward facing)
Configuration
Partial differential equation
I
k
G
A
E
,
, ,
,
s
Y
E I Y
d 2 φ Y
dX 2 − k s G A
du Z
dX + φ Y
= 0
k s G A
d 2 u Z
dX 2 +
dφ Y
dX
= 0
G
A
E
,
k s
X
E
,
,
,
( )
X
( )
X
( )
X
( )
X
( )
I Y
d
dX
E(X )I Y (X )
dφ Y
dX
−
k s (X )G(X )A(X )
du Z
dX + φ Y
= 0
d
dX
k s (X )G(X )A(X )
du Z
dX + φ Y
= 0
X
( )
q Z
E I Y
d 2 φ Y
dX 2 − k s G A
du Z
dX + φ Y
= 0
k s G A
d 2 u Z
dX 2 +
dφ Y
dX
= −q Z (X )
X
( )
m Y
E I Y
d 2 φ Y
dX 2 − k s G A
du Z
dX + φ Y
= −m Y (X )
k s G A
d 2 u Z
dX 2 +
dφ Y
dX
= 0
k(X)
E I Y
d 2 φ Y
dX 2 − k s G A
du Z
dX + φ Y
= 0
k s G A
d 2 u Z
dX 2 +
dφ Y Y
dX
= k(X )u Z
dφ Y
dX
= −
d
2 u Z
dX 2 −
q Z
k s G A
,
(4.1)
d
3
φ Y
dX 3 = −
d
4 u Z
dX 4 −
d
2 q Z
k s G AdX 2 .
(4.2)
One-time differentiation of the first PDE provided in Table 4.1 gives:
E I Y
d
3
φ Y
dX 3 − k s AG
d
2 u Z
dX 2 +
dφ Y
dX
= 0 .
(4.3)