Advanced Methods of Structural Analysis

200 6 Deflections of Elastic Structures

6.12.A uniform circular bar is clamped at pointBand carrying the coupleM 0 ,the
forcesPandHat free endA(Fig.P6.12). Compute the following displacements at
the free endA('D 0 ): (a) vertical; (b) horizontal; (c) angle of rotation. Check the
reciprocal unit displacement theorem.

B

H

x

P

M 0

x

y

R

Rj

dj

A ds =Rd

Fig. P6.12

Ans.

Displacement caused by

P H M 0

EIvert P

R^3

4

H

R^3

2

M 0 R^2

EIhor PR

3

2

HR^3



3

4 ^2



M 0 R^2



2 ^1



EI PR^2 HR^2



2

 1



M 0

R

2

6.13.Three-hinged uniform semicircular arch of radiusRcarrying the concentrated
forcePat pointC(Fig.P6.13). The flexural stiffness of the arch isEI.Calculate
the vertical displacement of the hingeC.

R R

x

R

y

A B

P

DC

C

Fig. P6.13

Ans.CD

PR^3

2 EI

. 3/.