Cantilever Beam Deflection: Problem & Solution

________________________________________________________________________
PROBLEM (13.18) A cantilever of constant flexural rigidity EI supports a concentrated load P at
its free end (Fig. P13.18). Determine:
(a) The deflection and angle of rotation
1
v1
θ
at the free end.
(b) The reactions R2 and M2 at the fixed end.
SOLUTION
Equation (13.22):
22
3
22
12 6 12 6
64 62
[] 12 6 12 6
62 64
LL
LL LL
EI
KLL
L
LL LL
−
⎡⎤
⎢⎥
−
⎢⎥
=⎢⎥
−− −
⎢⎥
−
⎣⎦
(a) Boundary conditions are 22
0v
θ
=
=. Equation (13.21a) becomes
1
2
21
12 6
064
v
PL
EI LL
L
θ
−
⎧
⎫
⎧⎫ ⎡ ⎤
=
⎨⎬ ⎨
⎢⎥
⎩⎭ ⎣ ⎦
⎩⎭
⎬
Solving,
23
11
23
PL PL
v
EI EI
θ ==−
(b) Equation (13.21a):
3
122
2
13
222
2
12 6 12 6 3
64 62 2
12 6 12 6 0
62 64 0
y
y
FLL
PL EI
MLL LLPL EI
EI
FLL
L
MLL LL
−
⎧
⎫
−
⎧⎫ ⎡⎤
⎪
⎪
⎪⎪ ⎢⎥
−
⎪⎪
⎨⎬
⎪ ⎪
⎢⎥
=⎨ ⎬
⎢⎥
−− −
⎪⎪ ⎪ ⎪
⎢⎥
⎪⎪ ⎪ ⎪
−
⎣⎦
⎩⎭ ⎩⎭
⎪ ⎪
=⎨ ⎬
⎪⎪⎪ ⎪
⎪⎪⎪ ⎪
−
⎩⎭
⎩⎭ P
Solving
⎪⎪
⎨⎬
1
1
2
2
0
y
y
FP
M
FP
MPL
−
⎧⎫
⎧⎫
⎪⎪
⎪⎪
where .
22y
FR==
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