Geometry of Middle Surface

8/10/2019 Geometry of Middle Surface

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COORDINATE SYSTEM OF THE SURFACE

Prepared by: Prof. P G Patel Introduction to Shell 1

The parametric representation of the surfacecan be given, as follows:in vector form as r = r (,)

In scalar form as X = X(,); Y = Y(,) ;Z = Z(,) ;

Therefore these parameters, & can becalled curvilinear coordinates of a givensurface

If the & coordinate lines are mutually

perpendicular at all points on a surface (i.e., the angles between the tangents tothese lines are equal to 90 ), the curvilinearcoordinates are said to be orthogonal.


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The derivatives of the position vector r with respect to thecurvilinear coordinates & are given by the following:


r, & r, are the tangent vectors at any point of the surface tothe & coordinate lines, respectively (Fig. 11.1).Indeed, since & are scalar quantities, the directions of

vectors r, & r, coincide with directions of the vector dr.This vector dr points to the chords joining points M and N for r,

and M and N1 for r, .Since the latter are assumed to be orthogonal, then r, . r, = 0


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PRINCIPAL DIRECTIONS AND LINES OF CURVATURE

3Prepared by: Prof. P G Patel Introduction to Shell


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Consider one normal sections as a plane curve, as shown inFig. 11.4. The position of a point M on the curve isdetermined by a single arc length coordinate s, measuredfrom a suitable datum point. Let M and M1 be twoneighboring points defininga short arc of length ds. At Mand M1 the normals OM and OM1 are drawn to the curve

(there is only one normal at each point, since the curve issmooth) and they are inclined with respect to a suitabledatum direction at angles and +d, respectively.

Assume that the two normals intersect at a point O (thecenter of curvature of the arc). The center of curvature is

defined strictly in terms of a limiting process in which ds0. The length OM (i.e., the distance from the center ofcurvature to point M) is called the radius of curvature, , ofthe given curve at point M.



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We also introduce lines of the principal curvature or simplylines of curvature. A line of curvature is a curve on thesurface with the property that, at any point of the curve, ithas a common tangent with the principal directions. Fromthe above, it follows that the curvature takes on an extreme

value at that point of the lines of curvature. Hence, at anypoint of a smooth surface there is at least one set ofprincipal directions and two orthogonal lines of curvature.

The principal directions and lines of curvature may coincideor may not coincide, depending upon the surface geometry.

Figure 11.5a shows the surface of revolution. It is describedby rotating a plane curve (called a meridian) about the axisof rotation (the Z axis in Fig. 11.5).



7/29Prepared by: Prof. P G Patel Introduction to Shell 7

Figure 11.5a shows the surface ofrevolution. It is described byrotating a plane curve (called ameridian) about the axis of rotation(the Z axis in Fig. 11.5).

Thus, meridians at any point of thesurface are one of the lines ofcurvature and, at the same time, oneof the principal normal sections.

The radius of curvature of themeridian equals MO1 and it is thefirstprincipal radius of curvature ofthe surface of revolution, R1.


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From now on we assume that the coordinate lines & arethe lines of curvature. This system of the coordinate lineshas particularly simple properties, so that the equations ofthe theory of shells acquire a relatively simple form in thissystem. To be able to use the corresponding formulas of thegeneral theory of shells, knowledge of the lines ofcurvature is required, and the determination of thesecurves for a given surface is, in general, a fairly complicatedproblem. However, for many of types of shells used in

practice, the geometry of the middle surface is of a simplenature (e.g., surfaces of revolution, cylindrical surfaces,etc.), so that the lines of curvature are already known.



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THE FIRST AND SECOND QUADRATIC FORMS OF SURFACES

dr = r, d+ r, dDenoting

a11= r, .r, ;

a12

=r,

.r,;

a22= r, . r,

ds2= dr. dr = a11(d)2+ 2a12(d.d) + a22(d)

2

The expression on the right-hand side of this equation is called the firstquadratic form of the surfacedefinedby the vector r(and )



11/29Prepared by: Prof. P G Patel Introduction to Shell 11


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SHELLS IN ENGINEERING STRUCTURES



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SHELLS IN ENGINEERING STRUCTURES



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Advantages of SHELL



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FUNDAMENTALS OF SHELLS



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Owing to the curvature of the surface, shells aremore complicated than flat plates because theirbending cannot, in general, be separated fromtheir stretching.

On the other hand, a plate may be considered as aspecial limiting case of a shell that has nocurvature;

Consequently, shells are sometimes referred to as

curved plates. This is the basis for the adoption ofmethods from the theory of plates, discussed inPart I, into the theory of shells.



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CLASSES OF SHELL

Thin shellsA shell is called thin if the maximum value of the ratio h/R

(where R is the radius of curvature of the middlesurface) can be neglected in comparison with unity

For an engineering accuracy, a shell may be regarded asthin if the following condition is satisfied:

Max h/R 20

Thick shells

Shells for which this inequality is violated are referred toas thick shells.



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BRIEF OUTLINE OF THE LINEAR SHELL THEORIES

Love was the first investigator to present a successfulapproximation shell theory based on classical linearelasticity. To simplify the straindisplacement relationshipsand, consequently, the constitutive relations, Love applied,to the shell theory, the Kirchhoff hypotheses developed

originally for the plate bending theory, together with thesmall deflection and thinness of the shell assumptions.This set of assumptions is commonly called the KirchhoffLove assumptions.



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The Love theory of thin elastic shells is also referred to as thefirst-order approximation shell theory. In spite of itspopularity and common character, Loves theory was notfree from some deficiencies, including its inconsistenttreatment of small terms, where some were retained andothers were rejected, although they were of the same order.This meant that, for certain shells, Loves differentialoperator matrix on the displacements, in the equations ofequilibrium, became unsymmetric. Obviously, this violated

Bettistheorem of reciprocity. Lovestheory also containedsome other inconsistencies. The need for a mathematicallyrigorous two-dimensional set of the shell equationsemploying the KirchhoffLove assumptions led to different

versions of the first-order approximation theories. 20Prepared by: Prof. P G Patel Introduction to Shell


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E. Reissner developed the linear theory of thin shells (alsothe first-order approximation theory) where someinadequacies of Lovestheory were eliminated. He derivedequations of equilibrium, straindisplacement relations,and stress resultants expressions for thin shells directly

from the three-dimensional theory of elasticity, byapplying the LoveKirchhoff hypotheses and neglectingsmall terms of order z/Ri (where Ri = 1; 2 are the radii ofthe curvature of the middle surface) compared with unity

in the corresponding expressions.



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Sanders also developed the first-order-approximation shelltheory from the principle of virtual work and by applyingthe KirchhoffLove assumptions. Sanders theory of thinshells has removed successfully the inconsistencies of the

Love theory. A version that retains terms of magnitudecompatible with those retained by E. Reissner and resolvesthe inconsistency in the expression for twist was developedby Koiter .

Timoshenkostheory of thin shellswas very close to the Lovetheory. General relations and equations were obtained byapplying the KirchhoffLove hypotheses and neglectingterms z/Ri in comparison with unity.

Naghdianalyzed the accuracy of the LoveKirchhoff theoryof thin elastic shells. 22Prepared by: Prof. P G Patel Introduction to Shell


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Lurye , Flugge , and Byrne independently developed thesecond-order approximation theory of shells. The generalrelations and equations of this theory are the direct resultof the application of the Kirchhoff hypotheses together

with the small-deflectionassumption to the correspondingequations of the three-dimensional theory of elasticity. Thesecond-order approximation theory attempts a morecareful discard of terms z/Ri. It retains these terms incomparison with unity in the straindisplacement relationsand stress resultant equations. Applications of this theoryhave generally been restricted to circular cylindrical shells.In addition, the general relations and equations of thissecond-order approximation shell theory are found to be

cumbersome for application. 23Prepared by: Prof. P G Patel Introduction to Shell


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LOADING-CARRYING MECHANISM OF SHELLSWe can say that shells fall into a class of plates as arches relate

to straight beams under the action of transverse loading.

It is known that the efficiency of the arch form lies primarilyin resisting the transverse load with a thrust N, thusminimizing the shear force V and bending moment M.

It is possible to specify the arch shape and the manner of itsloading in such a way that the arch does not experiencebending at all. In this case, the arch is in the so-calledmoment-less state of stress. For example, for a parabolic

arch, bending will not be induced by a vertical loaduniformly distributed over its chord.

Thus, the ability of arches to support certain transverse loadswithout bending is the reason for their structural

advantageover straight beams. 24Prepared by: Prof. P G Patel Introduction to Shell


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LOADING-CARRYING MECHANISM OF SHELLS

A shell mainly balances an applied transverse load, much like an arch, bymeans of tensile and compressive stresses, referred to as the membraneor direct stresses. These stresses are uniformly distributed over theshell thickness. Such a state of stress is called the momentless ormembrane state of stress. Although the shear force and bending andtwisting moments are still present in the general case of loading, the

efficiency of the shell form rests with the presence of the membranestresses, as the primary means of resistance with the bending stressresultants and couples are minimized.

Thus, shells, like arches over beams, possess an analogous advantage overplates; however, with the following essential difference

while an arch of a given form will support only one completelydetermined load without bending, a shell of a given shape has,provided its edges are suitably supported, as a rule, the sameproperty for a wide range of loads which satisfy only very generalrequirements



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The membrane stress condition is an ideal state at whicha designer should aim. It should be noted thatstructural materials are generally far more efficient inan extensional rather in a flexuralmodebecause:

1. Strength properties of all materials can be usedcompletely in tension (or compression), since all fibersover the cross section are equally strained and load-carrying capacity may simultaneously reach the limitfor the whole section of the component.

2. The membrane stresses are always less than thecorresponding bending stresses for thin shells underthe same loading conditions.



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The highest efficiency of a shell, as a structural member, isassociated with its curvature and thinness. Owing to theshell curvature, the projections of the direct forces on thenormal to the middle surface develop an analog of anelasticfoundationunder the shell. So, it can be said that ashell resists an applied transverse loading as a flat plate

resting on an elastic foundation. This phenomenon canexplain an essential increase in strength and stiffness of ashell compared with a plate.

Thus, as a result of the curvature of the surface, a shell

acquires a spatial stiffness that gives to it a larger load-carrying capacity and develops the direct stresses.

Owing to its thinness, a shell may balance an appliedtransverse loading at the expense of the membrane stresses

mainly, with bending actions minimized. 27Prepared by: Prof. P G Patel Introduction to Shell


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Note that the pure bending conditions have no advantages

and should be avoided because shells, in view of their smallthickness, possess a low strength for this deformation.

However, sometimes bending conditions cannot be avoided.It turns out that strong severe bending conditions arelocalized only in a small domain near some discontinuities

in loading and geometrical conditions, as well as nearsupports, etc. As we move away from such a disturbancezone, the bending stresses will diminish rapidly and aconsiderable part of the shell will be in the moment lessstress condition.

Therewith, the thinner the shell the faster this decrease ofbending stresses.Shell thinness demonstrates the high efficiency of shells. It is

associated with the shells low weight and simultaneouslyits high strength.



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LOADING-CARRYING MECHANISM OF SHELLSHowever, a shells thinness is, at the same time, a weak point

because all the advantages mentioned earlier hold for a tensile

state of stress. In this case, a shell material is stretched and itsstrength properties are used completely.

On the other hand, a thinness of shells manifests itself incompression. External forces, as before, are effectivelytransformed in the constant membrane stresses over the shell

thickness.However, the trouble is that the level of the critical stresses at

buckling be sufficiently low. This level is just determined by theshell thickness. The thinner the shell, the lower is the level of the

critical stresses. The latter can be many factors smaller than theproportional limit of the shell material.

In this case, the efficiency of thin shells can be reducedconsiderably. To avoid the possibility of buckling, a shellstructure should be designed in such a way that a dominant part

of the structure is in tension 29P d b P f P G P t l I t d ti t Sh ll

Geometry of Middle Surface

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