Civil Engineering Reference
In-Depth Information
y
Centre of curvature
Neutral plane
R
Slope
d
y
d
x
F
IGURE
13.1
Deflection and
curvature of a beam
due to bending
y
x
O
the curvature of the beam also varies along its length; Eq. (9.11) therefore gives the
curvature at a particular section of a beam.
Consider a beam having a vertical plane of symmetry and loaded such that at a section
of the beam the deflection of the neutral plane, referred to arbitrary axes O
xy
,is
v
and
the slope of the tangent to the neutral plane at this section is d
v
/d
x
(Fig. 13.1). Also,
if the applied loads produce a positive, i.e. sagging, bending moment at this section,
then the upper surface of the beam is concave and the centre of curvature lies above
the beam as shown. For the system of axes shown in Fig. 13.1, the sign convention
usually adopted in mathematical theory gives a positive value for this curvature, i.e.
d
2
v/
d
x
2
1
R
=
(13.1)
1
+
(
d
v/
d
x
)
2
3
/
2
For small deflections d
v
/d
x
is small so that (d
v
/d
x
)
2
is negligibly small compared with
unity. Equation (13.1) then reduces to
d
2
v
d
x
2
1
R
=
(13.2)
whence, from Eq. (9.11)
d
2
v
d
x
2
M
EI
=
(13.3)
Double integration of Eq. (13.3) then yields the equation of the deflection curve of
the neutral plane of the beam.
In the majority of problems concerned with beam deflections the bending moment
varies along the length of a beam and therefore
M
in Eq. (13.3) must be expressed as
a function of
x
before integration can commence. Alternatively, it may be convenient
in cases where the load is a known function of
x
to use the relationships of Eq. (3.8).
Thus
d
3
v
d
x
3
S
EI
=−
(13.4)
