sm13 23

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PROBLEM (13.23) A propped cantilever beam with flexural rigidities EI and EI/4 for the parts 1-2
and 2-3. respectively, supports concentrated load P and moment 3PL at point 2 (Fig. P13.23). What are
the displacements ,
2
v2
θ
, and 3
θ
?
Given: L = 1.2 m, P = 20 kN, E = 207 GPa, I = 10 x 106 mm4
SOLUTION
We have
13 ()
EI EI
LL
= and 233
4
() 4
EI EI EI
LL
==
L
2
[]kk
So, the stiffness materices []
1
+
and [ are the same as obtained in ]K
Example 13.5.
Boundary conditions are v11
0, 0,= and v30.
=
Equation (g) of Example 13.5
θ
=
becomes
22
2
3
28 18 30
18 51 39 23
276 30 39 111 0
vL
LLLP
EI LL
θ
θ
−−
⎧⎫ ⎡ ⎤⎧ ⎫
P
L
⎪ ⎪
⎢⎥
=−
⎨ ⎬
⎢⎥
⎪⎪ ⎪ ⎪
⎢⎥
−−
⎩⎭ ⎣ ⎦⎩ ⎭
⎪⎪
⎨⎬
33
82
84 21
276 69
48 12
PL PL
LL
EI EI
LL
−
⎧⎫ ⎧
⎪⎪ ⎪
==
⎨⎬ ⎨
⎪⎪ ⎪
−−
⎩⎭ ⎩
⎫
⎪
⎬
⎪
⎭
Substituting the given data:
233
23
3
8 1.936
(20 10 )(1.2) 168 0.041
69(207 10 )(10) 96 0.023
vm
rad
rad
θ
θ
−−
⎧⎫ ⎧ ⎫⎧ ⎫
×m
⎪ ⎪⎪ ⎪
==
⎪⎪
⎨⎬ ⎨ ⎬⎨ ⎬
×
⎪⎪ ⎪ ⎪⎪ ⎪
−−
⎩⎭ ⎩ ⎭⎩ ⎭
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