Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

216 CHAPTER 6 Matrix Methods

Fig. P.6.7

P.6.10 Itisrequiredtoformthestiffnessmatrixforarectangularelementofside2a× 2 bandthicknesstforuse
in“planestress”problems.
(a) Assumeasuitabledisplacementfield.
(b) Formthe[C]matrix.
(c) Obtain

∫

vol[C]

T[D][C]dV.

Notethatthestiffnessmatrixmaybeexpressedas

[Ke]=[A−^1 ]T

⎡

⎣

∫

vol

[C]T[D][C]dV

⎤

⎦[A−^1 ]

P.6.11 Asquareelement1234,whosecornershavecoordinatesx,y(inmeters)of(−1,−1),(1,−1),(1,1),and
(−1,1),respectively,wasusedinaplanestressfiniteelementanalysis.Thefollowingnodaldisplacements(mm)
wereobtained:

u 1 =0.1 u 2 =0.3 u 3 =0.6 u 4 =0.1

v 1 =0.1 v 2 =0.3 v 3 =0.7 v 4 =0.5

If Young’s modulusE=200 000 N/mm^2 and Poisson’s ratioν=0.3, calculate the stresses at the center of
theelement.

Ans. σx=51.65N/mm^2 ,σy=55.49N/mm^2 ,τxy=13.46N/mm^2.

P.6.12 A rectangular element used in plane stress analysis has corners whose coordinates in meters referred
to an Oxyaxes system are 1(−2,−1), 2(2,−1), 3(2, 1), and 4(−2, 1). The displacements of the corners (in
meters)are

u 1 =0.001 u 2 =0.003 u 3 =−0.003 u 4 = 0

v 1 =−0.004 v 2 =−0.002 v 3 =0.001 v 4 =0.001

IfYoung’smodulusis200000N/mm^2 andPoisson’sratiois0.3,calculatethestrainsatthecenteroftheelement.

Ans. εx=−0.000125,εy=0.002,γxy=−0.0015.