Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

Problems 61

Fig. P.2.2

asanAirystressfunctionanddeterminethecoefficientsA,B,andC.

Ans. A= 2 Pl/td^3 , B=− 2 P/td^3 , C= 3 P/ 2 td.

P.2.3 The cantilever beam shown in Fig. P.2.3 is in a state of plane strain and is rigidly supported atx=L.
Examinethefollowingstressfunctioninrelationtothisproblem:

φ=

w

20 h^3

(

15 h^2 x^2 y− 5 x^2 y^3 − 2 h^2 y^3 +y^5

)

Fig. P.2.3

Showthatthestressesactingontheboundariessatisfytheconditionsexceptforadistributeddirectstressatthe
freeendofthebeamwhichexertsnoresultantforceorbendingmoment.

Ans. Thestressfunctionsatisfiesthebiharmonicequation:

•Aty=h,σy=w,andτxy=0,boundaryconditionssatisfied.

•Aty=−h,σy=−w,andτxy=0,boundaryconditionssatisfied.

Directstressatfreeendofbeamisnotzero,andthereisnoresultantforceorbendingmomentatthefreeend.

P.2.4 Athinrectangularplateofunitthickness(Fig.P.2.4)isloadedalongtheedgey=+dbyalinearlyvarying
distributedloadofintensityw=pxwithcorrespondingequilibratingshearsalongtheverticaledgesatx=0andl.