PSI - Issue 6
L.M. Kagan-Rosenzweig / Procedia Structural Integrity 6 (2017) 216–223 L.M. Kagan-Rosenzweig / Structural Integrity Procedia 00 (2017) 000–000
222
7
l
1
0
0
=
−
.
( )
w x w y dy ( )] [ ( )
f x
l
x
0 b is
For this rod of constant cross-section, factor
l
l
2240 3
0
0
0
2
=
f x l x dx ) ( )( −
l x dx ) ( −
= −
.
b
EI
If c K K 0.9 = , the error in moment at support and the maximum span moment calculation is less than 1%.
4.3. Statically indeterminate rod of variable cross-section
The rigidity of the rod loaded according to Fig. 3, a, is given in the form
4
(1 EI EI = − α x
)
,
0
1 α < . In the absence of compressive load, the horizontal reaction in the upper support is
where parameter
l
l
3
2
− EI l x 2
− EI l x
(
)
(
)
0
0
0
= H q
,
dx
dx
and the bending moment and deflection are
x
2
0
l x −
ξ ( )
(
)
M
= ξ − (
0
0
0
M H q = −
ξ
,
.
)
w
x
d
ξ ( )
2
EI
0
0 0 = w
0 a ,
0 b are calculated by Eq. (15). Eq. (14) with
0
Parameters
gives:
/ P P P
0
0 0 0 w a b x + +
= +
.
(
)
M M
−
1
c
2 ql M
2
2 ql M
0.5 α =
0.8 α =
2
1
1
0
0
x / l
x / l
Fig. 4. Moments in the rod of variable cross-section
The critical force and the exact solution needed to evaluate the proposed results, both depend on parameter α
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