Thin plates and shells: theory, analysis, and applications

672 Pages·2001·3.347 MB·English

by  Theodor Krauthammer, Eduard Ventsel

Part I
Thin Plates
1
Introduction
1.1 GENERAL
Thin plates are initially flat structural members bounded by two parallel planes,
called faces, and a cylindrical surface, called an edge or boundary. The generators
ofthe cylindrical surfaceareperpendiculartotheplanefaces.The distancebetween
theplanefacesiscalledthethickness(h)oftheplate.Itwillbeassumedthattheplate
thicknessissmallcomparedwithothercharacteristicdimensionsofthefaces(length,
width,diameter,etc.).Geometrically,platesareboundedeitherbystraightorcurved
boundaries (Fig. 1.1). The static or dynamic loads carried by plates are predomi-
nantly perpendicular to the plate faces.
The load-carrying action of a plate is similar, to a certain extent, to that of
beams or cables; thus, plates can be approximated by a gridwork of an infinite
number of beams or by a network of an infinite number of cables, depending on
the flexural rigidity of the structures. This two-dimensional structural action of
plates results in lighter structures, and therefore offers numerous economic advan-
tages. The plate, being originally flat, develops shear forces, bending and twisting
moments to resist transverse loads. Because the loads are generally carried in both
directions and because the twisting rigidity in isotropic plates is quite significant, a
plate isconsiderablystifferthanabeamofcomparablespanandthickness.So,thin
plates combine light weight and a form efficiency with high load-carrying capacity,
economy, and technological effectiveness.
Because of the distinct advantages discussed above, thin plates are extensively
used in all fields of engineering. Plates are used in architectural structures, bridges,
hydraulic structures, pavements, containers, airplanes, missiles, ships, instruments,
machine parts, etc. (Fig. 1.2).
Weconsideraplate,forwhichitiscommontodividethethicknesshintoequal
halvesbyaplaneparalleltoitsfaces.Thisplaneiscalledthemiddleplane(orsimply,
1
2 Chapter1
Fig. 1.1
Fig. 1.2
Introduction 3
the midplane)of the plate (Fig. 1.3). Being subjected to transverse loads,an initially
flat plate deforms and the midplane passes into some curvilinear surface, which is
referred to as the middle surface. With the exception of Secs 3.8 and 4.8, we will
consider only plates of constant thickness. For such plates, the shape of a plate is
adequatelydefinedbydescribingthegeometryofitsmiddleplane.Dependingonthe
shapeofthismidplane,wewilldistinguishbetweenrectangular,circular,elliptic,etc.,
plates.
A plate resists transverse loads by means of bending, exclusively. The flexural
properties of a plate depend greatly upon its thickness in comparison with other
dimensions. Plates may be classified into three groups according to the ratio a=h,
wherea isatypicaldimensionof aplate in aplane andh isaplate thickness. These
groups are
1. Thefirstgroupispresentedbythickplateshavingratiosa=h(cid:1)8...10.The
analysisofsuchbodiesincludesallthecomponentsofstresses,strains,anddisplace-
mentsasforsolidbodiesusingthegeneralequationsofthree-dimensionalelasticity.
2. Thesecondgroupreferstoplateswithratiosa=h(cid:2)80...100.Theseplates
are referred to as membranes and they are devoid of flexural rigidity. Membranes
carry thelateralloadsbyaxialtensileforces N(andshearforces)actingintheplate
middle surface as shown in Fig. 1.7. These forces are called membrane forces; they
produce projection on a vertical axis and thus balance a lateral load applied to the
plate-membrane.
3. The most extensive group represents an intermediate type of plate, so-
called thin plate with 8...10 (cid:1)a=h(cid:1)80...100. Depending on the value of the
ratio w=h, the ratio of the maximum deflection of the plate to its thickness, the
part of flexural and membrane forces here may be different. Therefore, this group,
in turn, may also be subdivided into two different classes.
a. Stiffplates.Aplatecanbeclassifiedasastiffplateifw=h(cid:1)0:2.Stiffplates
areflexurallyrigidthin plates. Theycarry loads twodimensionally,mostlyby inter-
nalbendingandtwistingmomentsandbytransverseshearforces.Themiddleplane
deformations and the membrane forces are negligible. In engineering practice, the
termplateisunderstoodtomeanastiffplate,unlessotherwisespecified.Theconcept
of stiff plates introduces serious simplifications that are discussed later. A funda-
Fig. 1.3
4 Chapter1
Fig. 1.4
mental feature of stiff plates is that the equations of static equilibrium for a plate
element may be set up for an original (undeformed) configuration of the plate.
b. Flexible plates. If the plate deflections are beyond a certain level,
w=h(cid:2)0:3, then, the lateral deflections will be accompanied by stretching of the
middle surface. Such plates are referred to as flexible plates. These plates repre-
sent a combination of stiff plates and membranes and carry external loads by the
combined action of internal moments, shear forces, and membrane (axial) forces.
Such plates, because of their favorable weight-to-load ratio, are widely used by
the aerospace industry. When the magnitude of the maximum deflection is con-
siderably greater than the plate thickness, the membrane action predominates. So,
if w=h>5, the flexural stress can be neglected compared with the membrane
stress. Consequently, the load-carrying mechanism of such plates becomes of
the membrane type, i.e., the stress is uniformly distributed over the plate
thickness.
The above classification is, of course, conditional because the reference of the
plate to one or another group depends on the accuracy of analysis, type of loading,
boundary conditions, etc.
WiththeexceptionofSec.7.4,weconsideronlysmalldeflectionsofthinplates,
a simplification consistent with the magnitude of deformation commonly found in
plate structures.
1.2 HISTORY OF PLATE THEORY DEVELOPMENT
Thefirstimpetustoamathematicalstatementofplateproblems,wasprobablydone
by Euler, who in 1776 performed a free vibration analysis of plate problems [1].
Chladni, a German physicist, discovered the various modes of free vibrations
[2]. In experiments on horizontal plates, he used evenly distributed powder, which
formedregularpatternsafterinductionofvibration.Thepowderaccumulatedalong
the nodallines,where noverticaldisplacementsoccurred. J.Bernoulli [3]attempted
to justify theoretically the results of these acoustic experiments. Bernoulli’s solution
was based on the previous work resulting in the Euler–D.Bernoulli’s bending beam
theory.J.Bernoullipresentedaplateasasystemofmutuallyperpendicularstripsat
right angles to one another, each strip regarded as functioning as a beam. But the
governingdifferentialequation,asdistinctfromcurrentapproaches,didnotcontain
the middle term.
Introduction 5
The French mathematician Germain developed a plate differential equation
thatlackedthewarpingterm[4];bytheway,shewasawardedaprizebytheParisian
Academy in 1816 for this work. Lagrange, being one of the reviewers of this work,
corrected Germain’s results (1813) by adding the missing term [5]; thus, he was the
first person to present the general plate equation properly.
Cauchy[6]andPoisson[7]werefirsttoformulatetheproblemofplatebending
onthebasisofgeneralequationsoftheoryofelasticity.Expandingallthecharacter-
isticquantitiesintoseriesinpowersofdistancefromamiddlesurface,theyretained
onlytermsofthefirstorderofsmallness.Insuchawaytheyobtainedthegoverning
differential equation for deflections that coincides completely with the well-known
Germain–Lagrange equation. In 1829 Poisson expanded successfully the Germain–
Lagrange plate equation to the solution of a plate under static loading. In this
solution, however, the plate flexural rigidity D was set equal to a constant term.
Poissonalso suggested setting up three boundary conditionsfor any point on a free
boundary. The boundary conditions derived by Poisson and a question about the
number and nature of these conditions had been the subject of much controversy
and were the subject of further investigations.
The first satisfactory theory of bending of plates is associated with Navier [8],
who considered the plate thickness in the general plate equation as a function of
rigidityD.Healsointroducedan‘‘exact’’methodwhichtransformedthedifferential
equation into algebraic expressions by use of Fourier trigonometric series.
In 1850 Kirchhoff published an important thesis on the theory of thin plates
[9]. In this thesis, Kirchhoffstated two independentbasic assumptions that are now
widelyacceptedintheplate-bendingtheoryandareknownas‘‘Kirchhoff’shypoth-
eses.’’ Using these assumptions, Kirchhoff simplified the energy functional of 3D
elasticity theory for bent plates. By requiring that it be stationary he obtained the
Germain-Lagrange equation as the Euler equation. He also pointed out that there
exist only two boundary conditions on a plate edge. Kirchhoff’s other significant
contributions are the discovery of the frequency equation of plates and the intro-
duction of virtual displacement methods in the solution of plate problems.
Kirchhoff’s theory contributed to the physical clarity of the plate bending theory
and promoted its widespread use in practice.
LordKelvin(Thomson)andTait[10]providedanadditionalinsightrelativeto
theconditionofboundaryequationsbyconvertingtwistingmomentsalongtheedge
of a plate into shearing forces. Thus, the edges are subject to only two forces: shear
and moment.
Kirchhoff’s book was translated by Clebsh [11]. That translation contains
numerous valuable comments by de Saint-Venant: the most important being the
extension of Kirchhoff’s differential equation of thin plates, which considered, in a
mathematically correct manner, the combined action of bending and stretching.
Saint-Venant also pointed out that the series proposed by Cauchy and Poissons as
a rule, are divergent.
The solution of rectangular plates, with two parallel simple supports and the
other two supports arbitrary, was successfully solved by Levy [12] in the late 19th
century.
At the end of the 19th and the beginning of the 20th centuries, shipbuilders
changed their construction methods by replacing wood with structural steel. This
change in structural materials was extremely fruitful in the development of various