Civil Engineering Reference
In-Depth Information
6.12.2 Deformations of beams with initial crookedness
and twist
The deformations of a simply supported beam with initial crookedness and twist
caused by equal and opposite end moments
M
can be analysed by considering the
minor axis bending and torsion equations
EI
z
d
2
v
d
x
2
=−
M
(φ
+
φ
0
)
,
(6.86)
d
v
d
x
−
EI
w
d
3
φ
GI
t
d
φ
d
x
−
d
v
0
d
x
3
=
M
,
(6.87)
d
x
whichareobtainedfromequations6.81and6.82byaddingtheadditionalmoment
M
φ
0
and torque
M
d
v
0
/
d
x
induced by the initial twist and crookedness.
If the initial crookedness and twist rotation are such that
v
0
δ
0
=
φ
0
θ
0
=
sin
π
x
L
,
(6.14)
in which the central initial crookedness
δ
0
and twist rotation
θ
0
are related by
δ
0
θ
0
=
M
zx
π
2
EI
z
/
L
2
,
(6.15)
thenthesolutionofequations6.86and6.87whichsatisfiestheboundaryconditions
(equations 6.83-6.85) is given by
v
δ
=
φ
θ
=
sin
π
x
L
,
(6.16)
in which
δ
0
=
θ
δ
M
/
M
zx
1
−
M
/
M
zx
.
θ
0
=
(6.17)
The maximum longitudinal stress in the beam is the sum of the stresses due to
major axis bending, minor axis bending, and warping, and is equal to
d
2
(
v
+
d
f
φ/
2
)
d
x
2
W
el
,
y
−
EI
z
M
σ
max
=
.
W
el
,
z
L
/
2
If the elastic limit is taken as the yield stress
f
y
, then the limiting nominal stress
σ
L
for which this elastic analysis is valid is given by
σ
L
=
f
y
−
δ
0
N
cr
,
z
M
zx
1
+
d
f
2
N
cr
,
z
M
zx
1
W
el
,
z
M
L
1
−
M
L
/
M
zx
,
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