sm10 112 113

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________________________________________________________________________
PROBLEMS (10. 112 and 10.113) A cantilever beam supports a partial uniform load of intensity
w as shown in Figs. P10.112 and P10.113. Determine:
(a) The angle of rotation at free end B.
(b) The deflection at the free end B.
SOLUTION (10.112)
23
11
12
224 2 96 2 6 3
wL L wL L L L
Ax
EI EI
=− =− = + =
23
22
15
28 2 32 2 3 6
wL L wL L L L
Ax
EI EI
=− =− = + =
23
33
133
()
38 2 48 4 2 8
wL L wL L L
Ax
EI EI
=− =− = =
(a) 0
/123BA B A AAA
θθθ =− =++
33
(1 3 2)
96 16
BwL wL
EI EI
θ =− + + =
Continued on next slide
x
B
v
B
θ
M
A
Tangent at A
M
EI
x
28wL EI−3
x
B
Parabolic
Spandrel
A
w
A
3
3EI
2
8
wL
EI
−
B
EI
A
1
2
38wL−
224wL EI−
L/
2
L/
2
A
2
2
x
1
x

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(b) /BA B
tv=
333
253
96 3 32 6 48 8
wL L wL L wL L
EI EI EI
=− − −
or
4(8 30 9)
1152
BwL
vEI
=− + + 4
47
1152 wL
EI
=
↓
SOLUTION (10.113)
23
11
3326
bh wa wa
Aa
EI EI EI
==− =−
33
2
23
11
()
222
wa wa
AAawa
EI EI
=− =− =−
123
37 5 8
44 2 3
aa a a
xa x x
=+ = = =
(a) 0
/123BA B A AAA
θθθ =− =++
33
7
(266)
12 6
Bwa wa
EI EI
θ =− + + =
Continued on next slide
D
x
B
v
B
θ
a
A
Tangent at A
M
EI
22wa EI−
3
x
B
Spandrel
Parabolic
A
w
D
B
C
2
32wa EI−
a
a
1
x
A
3
A
2
A
1
2
x

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(b) /112233BA B
tvAxAxAx== + +
44
69
(7 30 32)
24 24
Bwa wa
vEI EI
=− + + =− 4
23
8wa
EI
=
↓