sm13 27

________________________________________________________________________
PROBLEM (13.27) A propped cantilever beam with an overhang supports a concentrated load P as
shown in Fig. P13.27. The beam has a constant flexural rigidity EI. Determine:
(a) The stiffness matrix for each element.
(b) The system stiffness matrix
(c) The nodal displacements.
(d) The forces and moments at the ends of each member.
(e) The shear and moment diagrams.
SOLUTION
(a) Use Eq.(13.21a):
2211
θ
θ
vv
22
13
22
12 6 12 6
64 62
[] 12 6 12 6
62 64
LL
LL LL
EI
kLL
L
LL LL
−
⎡⎤
⎢⎥
−
⎢⎥
=⎢⎥
−− −
⎢⎥
−
⎣⎦
22 3
vv
3
θ
θ
22
23
22
12 6 12 6
64 62
[] 12 6 12 6
62 64
LL
LL LL
EI
kLL
L
LL LL
−
⎡⎤
⎢⎥
−
⎢⎥
=⎢⎥
−− −
⎢⎥
−
⎣⎦
(b) Assemble the global stiffness matrix of the beam: [12
] [] []Kkk+. Then,
=
1 1
22
1 1
2 2
222
3
2 2
3 2
22
3 3
12 6 12 6 0 0
64 62 0 0
12 6 24 0 12 6
62 08 62
00126126
0062 64
y
R
v
LL
MLL LL
R
v
LL
EI
MLL L LL
L
Fv
LL
MLL LL
θ
θ
θ
−
⎧⎫ ⎧
⎡⎤
⎪⎪ ⎪
⎢⎥
−
⎪⎪ ⎪
⎢⎥
⎪⎪ ⎪
⎢⎥
−− −
⎪⎪ ⎪
=
⎨⎬ ⎨
⎢⎥
−
⎪⎪ ⎪
⎢⎥
⎪⎪ ⎪
⎢⎥
−− −
⎪⎪ ⎪
⎢⎥
−
⎪⎪ ⎪
⎣⎦
⎩⎭ ⎩
⎫
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎭
(1)
(c) The boundary conditions are 11
0, 0,v
θ
=
= and 20v
=
, Hence
3
22
3
322
2
12 6 6
0642
0628
PLL
EI LL L
LLL L
v
θ
θ
−
⎧⎫ ⎡ ⎤⎧
⎪⎪ ⎪
⎢⎥
=
⎨⎬ ⎨
⎢⎥
⎪⎪ ⎪
⎢⎥
⎩⎭ ⎣ ⎦⎩
⎫
⎪
⎬
⎪
⎭
Continued on next slide
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Solving
32
333
73
,,
1244
PL PL PL
vEI EI EI
θθ =− = = 2
(d) Introducing these equations into Eq. (1), after multiplying :
332
5
,0,
2
y
FPM R=− = = P
21 1
31
,,
22
M
PL R P M PL=− =− =
y
M
-3
P
/2
x
x
P
1 2
3
P
L/2
L
P
L
P
L
+
+
(e)
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or
instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
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