Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

50 CHAPTER 2 Two-Dimensional Problems in Elasticity

Fig.2.2

(a) Required loading conditions on rectangular sheet in Example 2.2 forA=B=C= 0 ; (b) as in (a) but
A=C=D= 0.

thestateofpurebendingshowninFig.2.2(a).Alternatively,Fig.2.2(b)showstheloadingconditions
correspondingtoA=C=D=0inwhichσx=0,σy=By,andτxy=−Bx.
By assuming polynomials of the second or third degree for the stress function, we ensure that
the compatibility equation is identically satisfied for any values of the coefficients. For polynomials
of higher degrees, compatibility is satisfied only if the coefficients are related in a certain way. For
example,forastressfunctionintheformofapolynomialofthefourthdegree

φ=

Ax^4

12

+

Bx^3 y

6

+

Cx^2 y^2

2

+

Dxy^3

6

+

Ey^4

12

and

∂^4 φ

∂x^4

= 2 A 2

∂^4 φ

∂x^2 ∂y^2

= 4 C

∂^4 φ

∂y^4

= 2 E

SubstitutingthesevaluesinEq.(2.9)wehave

E=−( 2 C+A)

Thestresscomponentsarethen

σx=

∂^2 φ

∂y^2

=Cx^2 +Dxy−( 2 C+A)y^2

σy=

∂^2 φ

∂x^2

=Ax^2 +Bxy+Cy^2

τxy=−

∂^2 φ

∂x∂y

=−

Bx^2

2

− 2 Cxy−

Dy^2

2

ThecoefficientsA,B,C,andDarearbitraryandmaybechosentoproducevariousloadingconditions
asinthepreviousexamples.