Advanced Methods of Structural Analysis

6.2 Initial Parameters Method 157

Fig. 6.7 Fixed-rolled beam

subjected to the angular

settlementof support

q 0

l

y

MA RA qx^0 y(x)

A

B

Initial

parameters:

y 0 = 0, q 0

q (l )

Solution.According to (6.5), the expression for elastic curve is

EIy.x/DEIy 0 CEI 0 x



RA.x0/^3

3Š

C

MA.x0/^2

2Š

:

Since the initial parametery 0 D 0 ,then

EIy.x/DEI 0 x

RAx^3

6



MAx^2

2

: (a)

This equation contains two unknownsMAandRA. For their calculation we have

two additional equations.

1.The transverse displacement at the right support equals zero

EIy.l/DEI 0 l

MAl^2

2



RAl^3

6

D0: (b)

2.Bending moment at supportBequals zero. The expression for bending moment
may be obtained by twice differentiating (a)

M.x/DEIy^00 .x/DMACRAx: (c)

AtxDlwe have

M.l/DMACRAlD0: (d)

Solving (b) and (d) with respect toMAandRAleads to the following expressions

for reactions

MAD

3 EI

l

 0 ;RAD

3 EI

l^2

 0 : (e)

The formulas (e) are presented in TableA.3; they are necessary for analysis of

statically indeterminate frames by the displacement method (Chap. 8 ).

Example 6.6.The beamABis clamped at the left and right ends. Derive equation

of the elastic curve for the beam if the vertical relative displacement of supports is

B(Fig.6.8). Calculate corresponding reactions.