Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

154 CHAPTER 5 Energy Methods

Fig.5.27

(a) Given deflected shape of propped cantilever; (b) determination of the deflection of C.

Similarly, the deflection at C due to a 10N load at E=−^14 ×2.4=−0.6mm. Therefore, the
total deflection at C, produced by the 30 and 10N loads acting simultaneously (Fig. 5.27(b)), is
−1.05−0.6=−1.65mmfromwhichtheangularrotationofthebeamatB,θB,isgivenby

θB=tan−^1

1.65

300

=tan−^1 0.0055

or

θB= 0 ◦ 19 ′

Example 5.13
AnelasticmemberispinnedtoadrawingboardatitsendsAandB.WhenamomentMisappliedat
A,ArotatesθA,BrotatesθB,andthecenterdeflectsδ 1 .ThesamemomentMappliedtoBrotatesB,
θCanddeflectsthecenterthroughδ 2 .FindthemomentinducedatAwhenaloadWisappliedtothe
centerinthedirectionofthemeasureddeflections,bothAandBbeingrestrainedagainstrotation.

The three load conditions and the relevant displacements are shown in Fig. 5.28. Thus, from
Fig. 5.28(a) and (b), the rotation at A due toMat B is, from the reciprocal theorem, equal to the
rotationatBduetoMatA.Hence,

θA(b)=θB

ItfollowsthattherotationatAduetoMBatBis

θA(c),1=

MB

M

θB (i)

Also,therotationatAduetounitloadatCisequaltothedeflectionatCduetounitmomentatA.
Therefore,

θA(c),2

W

=

δ 1

M