Civil Engineering Reference
In-Depth Information
where
C
1
and
C
2
are unknown constants and
µ
2
=
P
/
EI
. Substituting the boundary
conditions
v
=
0at
x
=
0 and
L
gives
w
µ
2
P
w
µ
2
P
sin
µ
L
(1
C
1
=
C
2
=
−
cos
µ
L
)
so that the deflection is determinate for any value of
w
and
P
and is given by
cos
µ
x
1
sin
µ
x
x
2
w
µ
2
P
−
cos
µ
L
sin
µ
L
w
2
P
2
µ
2
=
+
+
−
Lx
−
v
(21.48)
In beam columns, as in beams, we are primarily interested in maximum values of stress
and deflection. For this particular case the maximum deflection occurs at the centre
of the beam and is, after some transformation of Eq. (21.48)
sec
µ
L
1
wL
2
8
P
w
µ
2
P
v
max
=
2
−
−
(21.49)
The corresponding maximum bending moment is
wL
2
8
M
max
=−
P
v
max
−
or, from Eq. (21.49)
1
w
µ
2
sec
µ
L
2
M
max
=
−
(21.50)
x
P
wL
2
w
L
y
wL
y
2
F
IGURE
21.13
Bending of a uniformly loaded
beam-column
P
