sm13 20

________________________________________________________________________
PROBLEM (13.20) A beam supported by a pin, a spring of stiffness k, and a roller at points 1, 2, and
3, respectively, is subjected to a concentrated load P at point 2 (Fig. P13.20). Determine:
(a) The nodal displacements.
(b) The nodal forces and spring force.
Given: L = 4 m, P = 6 kN, EI = 14 MN
⋅
m2 , k = 200 kN/m
SOLUTION
Due to symmetry, only one-half of the beam need be considered.
22
13
22
12 6 12 6
64 62
[] 12 6 12 6
62 64
LL
LL LL
EI
kLL
L
LL LL
−
⎡
⎤
⎢
⎥
−
⎢
⎥
=
⎢
⎥
−− −
⎢
⎥
−
⎣
⎦
3
23
0000
000
[] 0000
0000
kL EI
EI
kL
⎡
⎤
⎢
⎥
⎢
⎥
=
⎢
⎥
⎢
⎥
⎣
⎦
1
2
L
P
/
k/2
2
2
Therefore
22
3
3
22
12 6 12 6
64 62
[] 12 6 12 6
62 64
LL
LL LL
EI
KkL
LLL
EI
LL LL
−
⎡⎤
⎢⎥
−
⎢⎥
=⎢⎥
−− + −
⎢⎥
⎢⎥
−
⎢⎥
⎣⎦
(a) Boundary conditions are 10v
=
and 20.
θ
=
Equation (13.21a)
with
22 y
FP=− and 10:M
=
2
1
3
32
46
0
2612
LL
EI kL v
PLLEI
θ
⎡⎤
−
⎧
⎫
⎧⎫⎢⎥
=
⎨⎬ ⎨
⎢⎥
−−+
⎩⎭ ⎩⎭
⎢⎥
⎣⎦
⎬
m rad
Introduce the data and solve:
vm
21
3.9669 1.4876
θ
=− =−
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Continued on next slide

(b)
122
3
3
3
2
22
2
12 6 12 6 0
64 62
01.4876 (10 )
3.9669
12 6 12 6 0
62 64
y
y
LL
FLL LL
EI kL
FLLL
EI
MLL LL
−
−
⎡⎤
⎧⎫ ⎧⎫
⎢⎥
−
⎪⎪ ⎪⎪
⎢⎥
−
⎪⎪ ⎪ ⎪
=⎢⎥
⎨⎬ ⎨ ⎬
−
−− + −
⎢⎥
⎪⎪ ⎪ ⎪
⎢⎥
⎪⎪ ⎪ ⎪
⎩⎭
⎩⎭ −
⎢⎥
⎣⎦
or
1
2
2
2.60
3.4
10.42
y
y
Fk
Fk
N
N
M
kN m
⎧⎫⎧
⎪⎪⎪
=−
⎨⎬⎨
⎪⎪⎪
−⋅
⎩⎭⎩
⎫
⎪
⎬
⎪
⎭
C
kN== ↑
200(3.9669) 793 ( )
spring
FN==
From symmetry:
FF
13
2.6
yy
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
reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner is unlawful.