Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

208 CHAPTER 6 Matrix Methods

IfYoung’smodulusE=200000N/mm^2 andPoisson’sratioν=0.3,calculatethestressesatthecenter
oftheelement.

FromthefirstofEqs.(6.96),

u 1 =α 1 − 2 α 2 −α 3 + 2 α 4 =0.001 (i)

u 2 =α 1 + 2 α 2 −α 3 − 2 α 4 =0.003 (ii)

u 3 =α 1 + 2 α 2 +α 3 + 2 α 4 =−0.003 (iii)

u 4 =α 1 − 2 α 2 +α 3 − 2 α 4 =0(iv)

SubtractingEq.(ii)fromEq.(i),

α 2 −α 4 =0.0005 (v)

NowsubtractingEq.(iv)fromEq.(iii),

α 2 +α 4 =−0.00075 (vi)

ThensubtractingEq.(vi)fromEq.(v),

α 4 =−0.000625 (vii)

sofromeitherofEq.(v)orofEq.(vi)

α 2 =−0.000125 (viii)

AddingEqs.(i)and(ii),

α 1 −α 3 =0.002 (ix)

AddingEqs.(iii)and(iv),

α 1 +α 3 =−0.0015 (x)

ThenaddingEqs.(ix)and(x),

α 1 =0.00025 (xi)

andfromeitherofEq.(ix)orofEq.(x)

α 3 =−0.00175 (xii)

ThesecondofEqs.(6.96)isusedtodetermineα 5 ,α 6 ,α 7 ,α 8 inanidenticalmannertothepreceding.
Thus,

α 5 =−0.001

α 6 =0.00025

α 7 =0.002

α 8 =−0.00025