Advanced Methods of Structural Analysis

156 6 Deflections of Elastic Structures

This equation contains the unknown initial parameter 0 and two unknown reactions

RAandRB. For their calculation we have three additional conditions. They are

EIy.l/D 0 ,EIy.2l/D 0 ,M.2l/D 0.

1.Displacement atxDl(supportB):

EIy.l/DEI 0 l

RAl^3

6

D0: (b)

2.Displacement atxD2l(supportC):

EIy.2l/DEI 0 2l

RA.2l /^3

6



RB.2ll/^3

6

C

q.2ll/^4

24

D 0 or

EI 0 2l

8RAl^3

6



RBl^3

6

C

ql^4

24

D0: (c)

3.Bending moment equation may be presented as

M.x/DEIy^00 .x/DRAxCRB.xl/

q.xl/^2

2

:

The bending moment at the supportCis

M.2l/DRA2lCRB.2ll/

q.2ll/^2

2

D 0 or RA2lCRBl

ql^2

2

D0:(d)

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

RAD

1

16

ql; RBD

5

8

ql; EI 0 D

ql^3

96

: (e)

The negative sign of initial slope shows that the angle of rotation at pointAis in the

counterclockwise.

Substitution expression (e) in the general expression (a) leads to the following

equation for elastic curve

EIy.x/D

ql^3

96

xC

1

16

ql

x^3

6



5

8

ql

.xl/^3

6

C

q.xl/^4

24

:

The following terms should be taken into account: for the first span – the first and

second terms only and for the second span – all terms of the last equation.

Example 6.5.The beamABis clamped at the left end and pinned at right end. The

beam is subjected to the angular displacement 0 at the left end as shown in Fig.6.7.

Derive the equation of the elastic curve and compute the reactions of supports.