Advanced Methods of Structural Analysis

158 6 Deflections of Elastic Structures

Fig. 6.8 Fixed-fixed beam
subjected to linear settlement
of support

RB

B

DB

RA MB

MA

l

A EI

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 initial parameters arey 0 D 0 and 0 D 0 then

EIy.x/D

RAx^3

6



MAx^2

2

: (a)

For right fixed support the displacement isB, therefore

EIy.l/D

RAl^3

6



MAl^2

2

DEIB (b)

Expression for slope isEIy^0 .x/D.RAx^2 =2/MAx. For right fixed support the
slope is zero, so

EIy^0 .l /D

RAl^2

2

MAlD0: (c)

Solution of (b) and (c) leads to the following results:

RAD

12 EI

l^3

B."/; MAD

6 EI

l^2

B: (d)

Substitution of (d) into (a) allows calculating the transversal displacement of any
point of the beam.
Other required reactions may be determined considering the equilibrium
equations:

RBD

12 EI

l^3

B.#/ and MBD

6 EI

l^2

B: (e)

The negative signs show that actual directions for reactions and moments do not
coincide with adopted direction in Fig.6.8.
Formulas (d) and (e) will be used for analysis of statically indeterminate frames
by the displacement method (Chap. 8 ).