sm13 24

________________________________________________________________________
PROBLEM (13.24) A continuous beam of constant flexural rigidity EI is loaded and supported as
shown in Fig. P13.24. Determine:
(a) The stiffness matrix for each element.
(b) The system stiffness matrix and the force-displacement relations
SOLUTION
(a) The three element stiffness are identical. Equation (13.22):
22
123
3
22
12 6 12 6
64 62
[] [] [] 12 6 12 6
62 64
LL
LL LL
EI
kkk LL
L
LL LL
−
⎡
⎤
⎢
⎥
−
⎢
⎥
===
⎢
⎥
−− −
⎢
⎥
−
⎣
⎦
(b) The beam stiffness matrix, [123
] [] [] []Kkk k++
4
, is assembled as
=
1122 33 4
vvvv
θ
θθθ
22
222
3
22
22
126 126 00 00
64 62 00 00
12 6 24 0 12 6 0 0
62 08 62 00
[] 0 0 12 6 24 0 12 6
00 62 08 62
00 00 126 126
00 00 62 64
LL
LL LL
LLL
LL L LL
EI
KLL L
L
LL L LL
LL
LL LL
−
⎡⎤
⎢⎥
−
⎢⎥
⎢⎥
−− −
⎢⎥
−
⎢⎥
=⎢⎥
−− −
⎢⎥
−
⎢⎥
⎢⎥
−− −
⎢⎥
−
⎢⎥
⎣⎦
2
Boundary conditions are :
2344
0vvv
θ
====
System governing relations, from . (13.21):
1
1
2
22
3
33
4
4
0
0
[]0
0
0
y
y
y
Pv
F
MK
F
M
F
M
θ
θ
θ
−
⎧⎫ ⎧⎫
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
⎪⎪
⎨⎬
⎪⎪
=⎨⎬
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
⎪⎪ ⎪⎪
⎩⎭
⎩⎭
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