Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

5.4Application to the Solution of Statically Indeterminate Systems 137

orassuminglinearelasticity
∫

half-frame

M

EI

∂M

∂SA

ds=

∫

half-frame

M

EI

∂M

∂SD

ds=0(iii)

InAB,

M=−SArsinθ and

∂M

∂SA

=−rsinθ,

∂M

∂SD

= 0

InDB,

M=SDx and

∂M

∂SA

=0,

∂M

∂SD

=x

InCB,

M=SCrsinφ=

(

M 0

r

−SA−SD

)

rsinφ

Thus,

∂M

∂SA

=−rsinφ and

∂M

∂SD

=−rsinφ

SubstitutingtheseexpressionsinEq.(iii)andintegrating,wehave

3.365SA+SC=M 0 /r (iv)

SA+2.178SC=M 0 /r (v)

which,withEq.(ii),enableSA,SD,andSCtobefound.Inmatrixform,theseequationsarewrittenas
⎧
⎨
⎩

M 0 /r

M 0 /r

M 0 /r

⎫

⎬

⎭

=

⎡

⎣

111

3.356 0 1

1 0 2.178

⎤

⎦

⎧

⎨

⎩

SA

SD

SC

⎫

⎬

⎭

(vi)

fromwhichweobtain
⎧
⎨
⎩

SA

SD

SC

⎫

⎬

⎭

=

⎡

⎣

0 0.345 −0.159

1 −0.187 −0.373

0 −0.159 0.532

⎤

⎦

⎧

⎨

⎩

M 0 /r

M 0 /r

M 0 /r

⎫

⎬

⎭

(vii)

whichgive

SA=0.187M 0 /rSD=0.44M 0 /rSC=0.373M 0 /r

Again the square matrix of Eq. (vi) has been inverted to produce Eq. (vii). The bending moment
distributionwithdirectionsofbendingmomentisshowninFig.5.18.
Sofarinthischapter,wehaveconsideredtheapplicationoftheprincipleofthestationaryvalueof
thetotalcomplementaryenergyofelasticsystemsintheanalysisofvarioustypesofstructure.Although
themajorityoftheexamplesusedtoillustratethemethodareoflinearlyelasticsystems,itwaspointed
outthatgenerallytheymaybeusedwithequalfacilityforthesolutionofnonlinearsystems.