sm13 29

________________________________________________________________________
PROBLEM (13.29) A stepped beam with fixed ends supports a concentrated center load P that
produces a vertical deflection Δ at the midpoint 2 as depicted in Fig. P13.29. Determine, in terms of EI,
L, and Δ , as needed:
(a) The load P.
(b) The slope at point 2.
SOLUTION
Boundary conditions are
1133 2
vv v
θ
θ === =−Δ (given)
Equation (13.22):
22 2
12
33
22 2
24 12 24 12 12 6 12 6
12 8 12 4 64 62
[] []
24 12 24 12 12 6 12 6
12 4 12 8 62 64
LL L
LL LL LLLL
EI EI
kk
LL L
LL
LL LL LLLL
−−
⎡⎤⎡
⎢⎥⎢
−−
⎢⎥⎢
==
⎢⎥⎢
−− − −− −
⎢⎥⎢
−−
⎣⎦⎣
2
2
L
L
⎤
⎥
⎥
⎥
⎥
⎦
After assembling [ and considering the boundary conditions, the pertinent ]K
equations are found as
22
32
24 12 12 6
012684
PL
EI LLL L
L
L
θ
−
Δ
−+−
⎧
⎫
⎧⎫ ⎡ ⎤
=
⎨⎬ ⎨⎬
⎢⎥
−+
⎩⎭ ⎣ ⎦
⎩⎭
Multiplying,
2
3(36 6 )
EI
P
LL
θ
−= −Δ+ (1)
22
3
0(612
EI LL
L)
θ
=−Δ+ (2)
(a) Equation (1) is then
3
33 EI PL
Δ
=↓
(b) Equation (2) gives
22L
θ
Δ
=
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