Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

250 CHAPTER 7 Bending of Thin Plates

In this chapter, we dealt exclusively with small deflections of thin plates. For a plate subjected
to large deflections, the middle plane will be stretched due tobendingso that Eq. (7.33) requires
modification. The relevant theory is outside the scope of this book but may be found in a variety of
references.

References

[1] Jaeger,J.C.,ElementaryTheoryofElasticPlates,PergamonPress,1964.
[2] Timoshenko,S.P.,andWoinowsky-Krieger,S.,TheoryofPlatesandShells,2ndedition,McGraw-Hill,1959.
[3] Timoshenko,S.P.,andGere,J.M.,TheoryofElasticStability,2ndedition,McGraw-Hill,1961.
[4] Wang,Chi-Teh,AppliedElasticity,McGraw-Hill,1953.

Problems..............................................................................................

P.7.1 A10-mmthickplateissubjectedtobendingmomentsMxequalto10Nm/mmandMyequalto5Nm/mm.
Calculatethemaximumdirectstressesintheplate.

Ans. σx,max=±600N/mm^2 , σy,max=±300N/mm^2.

P.7.2 FortheplateandloadingofproblemP.7.1,findthemaximumtwistingmomentperunitlengthintheplate
andthedirectionoftheplanesonwhichthisoccurs.

Ans. 2.5Nm/mmat45◦tothexandyaxes.

P.7.3 Theplateoftheprevioustwoproblemsissubjectedtoatwistingmomentof5Nm/mmalongeachedge,
inadditiontothebendingmomentsofMx=10Nm/mmandMy=5Nm/mm.Determinetheprincipalmomentsin
theplate,theplanesonwhichtheyact,andthecorrespondingprincipalstresses.

Ans. 13.1Nm/mm, 1.9Nm/mm, α=−31.7◦, α=+58.3◦, ±786N/mm^2 , ±114N/mm^2.

P.7.4 Athinrectangularplateoflengthaandwidth2aissimplysupportedalongtheedgesx=0,x=a,y=−a,
andy=+a. The plate has a flexural rigidityD, a Poisson’s ratio of 0,3 and carries a load distribution given by
q(x,y)=q 0 sin(πx/a).Ifthedeflectionoftheplatemayberepresentedbytheexpression

w=

qa^4

Dπ^4

(

1 +Acosh

πy

a

+B

πy

a

sinh

πy

a

)

sin

πx

a

,

determinethevaluesoftheconstantsAandB.

Ans. A=−0.2213,B=0.0431.

P.7.5 A thin, elastic square plate of sideais simply supported on all four sides and supports a uniformly
distributedloadq.Iftheoriginofaxescoincideswiththecenteroftheplate,showthatthedeflectionoftheplate
canberepresentedbytheexpression

w=

q

96 ( 1 −ν)D

[2(x^4 +y^4 )− 3 a^2 ( 1 −ν)(x^2 +y^2 )− 12 νx^2 y^2 +A],