Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

Problems 217

P.6.13 Aconstantstraintriangularelementhascorners1(0,0),2(4,0),and3(2,2)andis1unitthick.Iftheelasticity
matrix[D]haselementsD 11 =D 22 =a,D 12 =D 1 =b,D 13 =D 23 =D 31 =D 32 =0,andD 33 =c,derivethestiffness
matrixfortheelement.

Ans.

[Ke]=

1

4

⎡

⎢⎢

⎢

⎢⎢

⎢⎢

⎢

⎢⎣

a+c

b+ca+c

−a+c −b+ca+c

b−ca−c −b−ca+c

− 2 c − 2 c − 2 c 2 c 4 c

− 2 b − 2 a 2 b − 2 a 04 a

⎤

⎥⎥

⎥

⎥⎥

⎥⎥

⎥

⎥⎦

P.6.14 Thefollowinginterpolationformulaissuggestedasadisplacementfunctionforderivingthestiffnessofa
planestressrectangularelementofuniformthicknesstshowninFig.P.6.14.

u=

1

4 ab

[(a−x)(b−y)u 1 +(a+x)(b−y)u 2 +(a+x)(b+y)u 3

+(a−x)(b+y)u 1 ]
FormthestrainmatrixandobtainthestiffnesscoefficientsK 11 andK 12 intermsofthematerialconstantsc,d,
andedefinedinthefollowing.

Fig. P.6.14

Intheelasticitymatrix[D]

D 11 =D 22 =cD 12 =dD 33 =e and D 13 =D 23 = 0

Ans. K 11 =t( 4 c+e)/6,K 12 =t(d+e)/4.