Advanced Methods of Structural Analysis

148 6 Deflections of Elastic Structures

expression for displacement. This expression allows us to calculate slope, bend-
ing moments, and shear along the beam and is called the Universal equation of
elastic curve of a beam.
2.Universal equation of the elastic curve of a beam contains only two unknown
parameters foranynumber of portions.
A general case of a beam under different types of loads and the corresponding no-
tational convention is presented in Fig.6.2a. The origin is placed at the extreme left
end point of a beam, thex-axis is directed along the beam, andy-axis is directed
downward. SupportAis shown as fixed, however, it can be any type of support or
even free end. Loadqis distributed along the portionDE. Coordinates of points of
application of concentrated forceP, coupleM, and initial point of distributed load
qare denoted asawith corresponding subscriptsP,M,andq. This beam has five
portions (AB,BC,CD,DE,andEL), which leads to the ten constants of integrating
using the double integration method.

b

l

x

y

ak 1

ak 2

k 0

S T L

k

k 0

k-k 0

A

a x

3

x

y l

P

P

M q

aP

am a

q (^1) a
q 2
E
x 1
x 2
A
BD
L
C
Fig. 6.2 Initial parameters method notation
The initial parameter method requires the following rules to be entertained:
1.Abscisesxfor all portions should be reckoned from the origin; in this case
the bending moment expression foreach next portion contains all components
related to the previous portion.
2.Uniformly distributed load may start from any point but it must continue to the
very right point of the beam. If distributed load is interrupted (pointE,Fig.6.2a),
then this load should be continued till the very right point and action of the added
load must be compensated by the same but oppositely directed load, as shown in
Fig.6.2a. The same rule remains for load which is distributed by triangle law. If
load is located within the portionS-T(Fig.6.2b), it should be continued till the
very right pointLof the beam and action of the added load must be compen-
sated by the same but oppositely directed loads (uniformly distributed load with
intensityk 0 and load distributed by triangle law with maximum intensityk–k 0
at pointL). Both of these compensated loads start at pointTand do not interrupt
until the extremely right pointL.
3.All components of a bending moment within each portion should be presented
in unified form using the factor.x–a/in specified power, as shown in Table 6.1.
For example, the bending moments for the second and third portions (Fig.6.2a)
caused by the active loads only are