Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

62 CHAPTER 2 Two-Dimensional Problems in Elasticity

Fig. P.2.4

AsasolutiontothestressanalysisproblemanAirystressfunctionφisproposed,where

φ=

p

120 d^3

[5(x^3 −l^2 x)(y+d)^2 (y− 2 d)− 3 yx(y^2 −d^2 )^2 ]

Show thatφsatisfies the internal compatibility conditions and obtain the distribution of stresses within the
plate.Determinealsotheextenttowhichthestaticboundaryconditionsaresatisfied.

Ans. σx= px

20 d^3

[5y(x^2 −l^2 )− 10 y^3 + 6 d^2 y]

σy=

px

4 d^3

(y^3 − 3 yd^2 − 2 d^3 )

τxy=

−p

40 d^3

[5( 3 x^2 −l^2 )(y^2 −d^2 )− 5 y^4 + 6 y^2 d^2 −d^4 ].

The boundary stress function values ofτxydo not agree with the assumed constant equilibrating shears atx= 0
andl.

P.2.5 The cantilever beam shown in Fig. P.2.5 is rigidly fixed atx=Land carries loading such that the Airy
stressfunctionrelatingtotheproblemis

φ=

w

40 bc^3

(

− 10 c^3 x^2 − 15 c^2 x^2 y+ 2 c^2 y^3 + 5 x^2 y^3 −y^5

)

Findtheloadingpatterncorrespondingtothefunctionandcheckitsvaliditywithrespecttotheboundaryconditions.

Fig. P.2.5