sm13 21

________________________________________________________________________
PROBLEM (13.21) A cantilever beam 1-2 of length L and constant flexural rigidity EI carries a
concentrated load P at the midspan as shown in Fig. P13.21. Determine the vertical deflection and
angle of rotation
2
v
2
θ
at the free end by replacing the applied load with the equivalent nodal loads acting at
each end of the beam (see Table B.15).
SOLUTION
Boundary conditions are
11
00v
θ
=
=
We have
21
8
PL
MM=− =
12 2
yy
P
FF
=
=−
Equation (13.21a) reduces to
1
2
32
2126
864
v
PL
EI
PL L L
L
θ
−−
⎧
⎫
⎧⎫ ⎡ ⎤
=
⎨⎬ ⎨
⎢⎥
−
⎩⎭ ⎣ ⎦
⎩⎭
⎬
Inverting
2
2
2
2
38
636
vP
LL
LPL
EI L
θ
−
⎡⎤
⎧⎫
⎧
⎫
=
⎨⎬ ⎨
⎢⎥
⎩⎭
⎩⎭ ⎣⎦
⎬
3
2
548
8
PL EI
PL EI
⎧
⎫
−
=
⎨
⎬
−
⎩⎭
P
/2
P
L2/8
P
L2/8
1 2
P
/4
L
Figure (a). Equivalent
Nod
al f
o
r
ces.
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