Civil Engineering Reference
In-Depth Information
Substitution of Equations 8.48a and 8.48b into Equation 8.33 yields
d
2
y
d
z
2
+
Qz(L
−
a)
k
2
y
=−
for 0
≤
z
≤
a
,
(8.49a)
EIL
d
2
y
d
z
2
+
Qa(L
−
z)
k
2
y
=−
for
a
≤
z
≤
L
.
(8.49b)
EIL
The general solutions to Equations 8.49a and 8.49b are
Qz(L
−
a)
y
=
A
sin
kz
+
B
cos
kz
−
for 0
≤
z
≤
a
,
(8.50a)
EILk
2
Qa(L
−
z)
y
=
C
sin
kz
+
D
cos
kz
−
for
a
≤
z
≤
L
.
(8.50b)
EILk
2
DifferentiatingEquations8.50aand8.50bwithboundaryconditionsof
y(
0
)
=
y(L)
=
0 and noting that displacement,
y(a)
, and slope, d
y(a)/
d
z
, are continuous at
z
=
a
provides
M
Z
=−
EI
d
2
y
d
z
2
Q
k
sin
k(L
−
a)
sin
kL
=−
sin
kz
for 0
≤
z
≤
a
,
(8.51a)
sin
kz
tan
kL
−
cos
kz
M
Z
=−
EI
d
2
y
d
z
2
Q
sin
ka
k
=
for
a
≤
z
≤
L
.
(8.51b)
The maximum moment at the center span (by substitution of
z
=
L/
2 into
Equations 8.51a or 8.51b) is
2 tan
kL/
2
kL
.
QL
4
M
z
=
L
/
2
=
(8.52)
The tangent function in Equation 8.52 can be expanded in a power series as (Beyer,
1984)
kL
2
3
kL
2
5
kL
2
7
tan
kL
kL
2
+
1
3
2
15
17
315
2
=
+
+
+···
,
(8.53)
which when substituted into Equation 8.52, and after making further simplifications
similar to those outlined in
Section 8.4.2.1.1,
yields
1
,
QL
4
−
0.2
(P/P
e
)
M
z
=
L
/
2
≈
(8.54)
1
−
(P/P
e
)
where
[
(
1
−
0.2
(P/P
e
))/(
1
−
(P/P
e
))
]
is an approximate moment magnification
factor appropriate for use in design.
