Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

5.10 The Reciprocal Theorem 153

P 2 followedbyP 1 ,wehave

U 2 =

P 2

2

(a 22 P 2 )+

P 1

2

(a 11 P 1 )+P 2 (a 21 P 1 ) (5.27)

Bytheprincipleofsuperposition,thestrainenergystoredisindependentoftheorderinwhichthe
loadsareapplied.Hence,

U 1 =U 2

anditfollowsthat

a 12 =a 21 (5.28)

Thus,initssimplestformthereciprocaltheoremstatesthat

Thedeflectionatapoint1inagivendirectionduetoaunitloadatapoint2inaseconddirectionis

equaltothedeflectionatthepoint2intheseconddirectionduetoaunitloadatthepoint1inthe

firstdirection.

Inasimilarmanner,wederivetherelationshipbetweenmomentsandrotations,thus

The rotation at a point 1 due to a unit moment at a point 2 is equal to the rotation at the point 2

producedbyaunitmomentatthepoint1.

Finally,wehave

Therotationatapoint1duetoaunitloadatapoint2isnumericallyequaltothedeflectionatthe

point2inthedirectionoftheunitloadduetoaunitmomentatthepoint1.

Example 5.12
Acantilever800mmlongwithaprop500mmfromthewalldeflectsinaccordancewiththefollowing
observationswhenapointloadof40Nisappliedtoitsend.

Distance(mm) 0 100 200 300 400 500 600 700 800
Deflection(mm) 0 −0.3 −1.4 −2.5 −1.9 0 2.3 4.8 10.6

Whatwillbetheangularrotationofthebeamatthepropduetoa30Nloadapplied200mmfromthe
wall,togetherwitha10Nloadapplied350mmfromthewall?

TheinitialdeflectedshapeofthecantileverisplottedasshowninFig.5.27(a)andthedeflectionsat
DandEproducedbythe40Nloaddetermined.Thesolutionthenproceedsasfollows.

DeflectionatDdueto40NloadatC=−1.4mm.

Hence,fromthereciprocaltheorem,thedeflectionatCduetoa40NloadatD=−1.4mm.
ItfollowsthatthedeflectionatCduetoa30NloadatD=−^34 ×1.4=−1.05mm.