Advanced Methods of Structural Analysis

6.2 Initial Parameters Method 151

For positive bending moments atxdue to coupleM,forceP, and uniformly dis-

tributed loadq, the expanded equations for displacement and slope are

EIy.x/DEIy 0 CEI 0 x

M.xaM/^2

2Š



P.xaP/^3

3Š



q.xaq/^4

4Š

;

(6.5)

EI.x/DEI 0 M.xaM/

P.xaP/^2

2



q.xaq/^3

6

: (6.6)

Expressions for bending moment and shear force may be obtained by formula

M.x/DEIy^00 .x/; Q.x/DEIy^000 .x/: (6.7)

Advantages of the initial parameters method are as follows:

1.Initial parameters method allows to obtain theexpressionfor elastic curve of the
beam. The method is very effective in case of large number of portions of a beam.
2.Initial parameters method do not require to form the expressions for bending
moment at different portions of a beam and integration of differential equation
of elastic curve of a beam; a procedure of integration was once used for deriving
the Universal equation of a beam and then only simple algebraic procedures are
applied according to expression (6.3).
3.The number of unknown initial parameters is always equals two and does not
depend on the number of portions of a beam.
4.Initial parameters method may be effectively applied for beams with elastic sup-
ports and beams subjected to displacement of supports. Also, this method may
be applied for statically indeterminate beams.

Example 6.1.A simply supported beam is subjected to a uniformly distributed load

qover the span (Fig.6.3). The flexural stiffnessEIis constant. Derive the expres-

sions for elastic curve and slope of the beam. Determine the slope at the left and

right supports, and the maximum deflection.

Solution.The origin is placed at the left support, thex-axis is directed along the

beam and they-axis is directed downward. The forces that should be taken into

Fig. 6.3 Design diagram of

simply supported beam and

its deflected shape

l

x

y

RA=0.5ql

Initial

parameters:

y 0 =0, q 0 ≠ 0

q y(l)=0

x q 0 y(x)ymax