Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

104 CHAPTER 4 Virtual Work and Energy Methods

Fig.4.11

Determination of the rotation of a simply supported beam at a support using the unit load method.

Theinternalvirtualbendingmomentis

Mv= 1 −

1

L

x 0 ≤x≤L

Theexternalvirtualworkdoneis1θA(thevirtualsupportreactionsdonoworkasthereisnovertical
displacementofthebeamatthesupports),andtheinternalvirtualworkdoneisgivenbyEq.(4.21).
Hence,

1 θA=

1

EI

⎡

⎢

⎣

∫L/^2

0

W

2

x

(

1 −

x

L

)

dx+

∫L

L/ 2

W

2

(L−x)

(

1 −

x

L

)

dx

⎤

⎥

⎦ (i)

SimplifyingEq.(i),wehave

θA=

W

2 EIL

⎡

⎢

⎣

∫L/^2

0

(Lx−x^2 )dx+

∫L

L/ 2

(L−x)^2 dx

⎤

⎥

⎦ (ii)