Civil Engineering Reference
In-Depth Information
y
M
A
M
B
F
IGURE
21.15
Beam-column
supporting end
moments
A
B
P
P
x
by using the principle of superposition and the results of the previous case. First we
imagine that
M
B
acts alone with the axial load,
P
. If we assume that the point load,
W
,
moves towards Band simultaneously increases so that the product
Wa
M
B
then, in the limit as
a
tends to zero, we have themoment
M
B
applied at B. The deflection
curve is then obtained from Eq. (21.56) by substituting
µ
a
for sin
µ
a
(since
µ
a
is now
very small) and
M
B
for
Wa
. Thus
=
constant
=
sin
µ
x
sin
µ
L
−
M
B
P
x
L
v
=
(21.58)
We find the deflection curve corresponding to
M
A
acting alone in a similar way. Sup-
pose that
W
moves towards A such that the product
W
(
L
−
a
)
=
constant
=
M
A
. Then
as (
L
−
a
) tends to zero we have sin
µ
(
L
−
a
)
=
µ
(
L
−
a
) and Eq. (21.57) becomes
sin
µ
(
L
M
A
P
−
x
)
(
L
−
x
)
v
=
−
(21.59)
sin
µ
L
L
The effect of the two moments acting simultaneously is obtained by superposition of
the results of Eqs (21.58) and (21.59). Hence, for the beam-column of Fig. 21.15
sin
µ
x
sin
µ
L
−
sin
µ
(
L
M
B
P
x
L
M
A
P
−
x
)
(
L
−
x
)
v
=
+
−
(21.60)
sin
µ
L
L
Equation (21.60) is also the deflected form of a beam-column supporting eccentrically
applied end loads at A and B. For example, if
e
A
and
e
B
are the eccentricities of
P
at
the ends A and B, respectively, then
M
A
=
Pe
A
,
M
B
=
Pe
B
, giving a deflected form of
e
B
sin
µ
x
e
A
sin
µ
(
L
x
L
−
x
)
(
L
−
x
)
v
=
sin
µ
L
−
+
−
(21.61)
sin
µ
L
L
Other beam-column configurations featuring a variety of end conditions and loading
regimes may be analysed by a similar procedure.
21.6 E
NERGY
M
ETHOD FOR
T
HE
C
ALCULATION OF
B
UCKLING
L
OADS IN
C
OLUMNS
(R
AYLEIGH
-R
ITZ
M
ETHOD)
The fact that the total potential energy of an elastic body possesses a stationary value
in an equilibrium state (see Section 15.3) may be used to investigate the neutral equi-
librium of a buckled column. In particular the energy method is extremely useful when
the deflected form of the buckled column is unknown and has to be 'guessed'.
