Chap.03-1Thin Shells Introduction

8/13/2019 Chap.03-1Thin Shells Introduction

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Thin Shell ConcreteStructures

General Theory of Thin Shells

Universidad Catlica de Santa MariaOctubre, 2013


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INTRODUCTIONThe purpose of this chapter is first to introduce the design ofthin shells through 02 main aspects:

By describing the physical behavior of certain well-definedsystems.

By deriving the General Equations for the Elastic Analysis.

The design of concrete thin shells is more difficult to presentrationally than the design of most other common structuralsystems; the main reasons for this are basically two:

1.- Rigorous analysis is so extraordinarily complex that thedesigner must necessarily resort to simplifications, thisnaturally leads to extra caution and hence to conservativedesigning.


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INTRODUCTION

2.-The load-carrying capacity of thin curved structures often

far exceeds the prediction the prediction of even the mostrefined available analysis; hence thin shells are oftenconsiderably stronger then even the expert designer assume.

Based in this two aspects most of the studies on thin shellshave usually concentrated on one of two aspects:

a).- Analysis.

b).- Descriptions ofshell action.

In the aspect of design in consequence there are two distinctapproaches:


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Arch versus Dome

Let consider first the loads acting on the arch.

The uniform load on a parabolic arch producespractically no bending if the footings do not permithorizontal displacement. However, once the load is

changed to apartial load, substantial bending developsin the arch.


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Arch versus Dome In the case of the dome the uniform load is carried by

forces in the plane of the shell that are similar to theforces which carry the uniform load in the arch.However, in the dome additional forces (hoop forces)are set up at right angles to these arching or meridionalforces.

The existence of two sets of forces in two separatedirections within the shell makes this structuralsystems similar to plate structures.


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Arch versus Dome The hoop forces, however, do not enter into the

equations of vertical equilibrium; therefore the entire

vertical load must be transmitted by the meridionalforces.

Consider now the case of thedome with a partial load.As long as this load has a smooth variation, that is to

say as long as there is no strong discontinuity at thepoint where the load goes to zero, we can observe thatthe shell is able to carry this load almost entirely bythe same arching forces as before.


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Arch versus Dome This action is quite unlike the action of the arch, and

the reason for the difference is the existence of the

hoop forces.

The hoop forces restrain it (the bending) just as if stiffrings were wrapped around the structure.Thus

whereas the arch is best suited only for one type ofloading, the dome is well suited to almost any type of

loading within certain restr ictions.


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Arch versus Dome


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Arch versus Dome The boundary conditions or edge restraints also play a

fundamental aspect in the behavior of both systems.

If we give the arch a horizontal push at one side, thisforce must be held in equilibrium by an equal andopposite force acting at the other support. The only waythat the force can be transmitted through the system is up

through the arch, over the crown and down the otherside. The effect clearly is to produce high bendingmoments throughout the entire arch system, with amaximum moment at the crown.


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Arch versus Dome We see, therefore, that the arch is not only restricted in its

most efficient form by the nature of the loading but is

also sensitive to foundation displacements or edge forces.

Now let us give a corresponding horizontal force to thedome. Such a force may be considered as a uniform

horizontal thrust applied all around the circular edge ofthe dome. In cross section it would appear that thishorizontal force would create bending momentsthroughout the dome similar to those created in the arch.


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Arch versus Dome This is, however, not the case. As the horizontal force

tends to bend the shell, and thus to be carried up a

meridian,the hoop forces again come into play andcause a rapid damping of the bending so that at a

relatively short distance from the edge the bending

effect is no longer observable.


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Arch versus Dome Thus edge forces in equilibrium applied to an arch

propagate throughout the entire structural system and

createlarge bending moments, whereas similar forcesacting on a dome create bending moments in a verynarrow regionnear the edge andhave generally no effectthroughout a large portion of the structure.


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Beam versus Barrel Just as Arches are limited by functional requirements and

do not find as wide applications as beams, Domes are

also limited and do not find as wide application asbarrels.

To make a comparison, first consider the framing planshown (figure a) in the figure in which the loads are

essentially carried transversely by the slab andlongitudinally by the beams.


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Beam versus Barrel Since the beam is relatively shallow compared to its span,

we know that the simple flexural theory is valid and that

the magnitude of stresses are directly proportional to thedepth and vary as a straight line from top to bottom.

Even in the slab the loads are essentially carried bybending moments; the sizes of which are functions of the

transverse span.


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Beam versus Barrel If then we imagine that the slab is given a curvature and

that the longitudinal beams are reduced in cross section

(Figure b), we would have a typical simply supportedbarrel shell spanning the same distance as the beams inthe previous figure. The Structural Action, however issignificantly Different.


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Beam versus Barrel Now that the slab iscurved, we might think that it would

act as an arch, but it does not because the slender edge

beam is unable to sustain horizontal forces. Thereforealthough the curved slab will try to act as an arch, thethrust contained at the springing lines will be very small.

Theprincipal structural actionof this system is found to

belongitudinal. In the previous case the beam

had to carry the entire loadwith perhaps some additional

flange help from the slab, butin this casethe enti re systemacts as a beam with curved

cross section to carry the load.


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Beam versus Barrel The principal action, therefore, of the barrel shell with

small flexible edge beams islongitudinal bending, but

the bending stresses are within the plane of the shellitself. Thus we may liken the entire shell-and-edge-beamsystem to a beam with the compression stresses near thecrown and with the tension stresses concentrated in theedge beam or either side.

Crown

Compression

Tension


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Beam versus Barrel The simple Flexural Theory, which leads to a straight-line

stress distribution, requires that all points within the cross

section of the member deflect exactly the same amount(the case of a solid rectangular beam). In a barrel shell thecross section may undergo substantial lateral distortion,and it is principally this distortion which causes thelongitudinal stresses to depart from the straight linedistribution of the beam theory.

Lateral

Distortion


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Beam versus Barrel It is true thatthelonger the span in comparison with the

transverse-chord width,the more the entire cross

section behaves as a beam and The shorter the spancompared with the chord width, the more the structurebehaves as an arch with a supported sloped deep beamnear the edge.


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Definitions and Assumptions

We shall proceed then to develop The General Elastic

Theory of Thin Shells firs formulated by A.E. H. Love in1888.

The following step after having studied the structural

behavior of the two most common thin shell structures isto develop the General Equation and then to reduce themfor specific systems in the following chapters.

A Thin Shell can be defined as a curved slab whosethickness h is relatively small compared with its otherdimensions and compared with its radii of curvature rxand ry.


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Definitions and Assumptions The surface that bisects the shell thickness is called the

middle surface, and by specifying the form of this surface

and the thickness h at every point, we completely definethe geometry of the shell.

The Analysis of Thin Shells consists first ofestabli shingthe Equilibrium of a differential element cut from theshell and second ofachieving Strain Compatibil i ty sothat each element remains continuous with each adjacent

element after deformation.


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Definitions and Assumptions

Stress resultants and Stress

Couples, defined as the totalforces and moments acting perunit length of middle surfaceare the integrals of stress overthe shell Thickness.


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Definitions and Assumptions In the following figure we can observe the stress

resultants and stress couples.

/ 2

/ 21

h

z xh

y

zN dz

r

/2

/ 21

h

y yh

x

zN dz

r

/2

/ 21

h

xy xyh

y

zN dz

r

/ 2

/21

h

yx yxh

x

zN dz

r

/ 2

/21

h

x xyh

y

zQ dz

r

/ 2

/21

h

y yxh

x

zQ dz

r

/2

/2 1h

x xh

y

zM z dzr

/2

/2 1

h

y yh

x

zM z dzr

/ 2

/21

h

xy xyh

y

zM z dz

r

/2

/21

h

yx yxh

x

zM z dz

r


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Definitions and Assumptions If the terms z/ry and z/rx are neglected when they appear

with unity, then from the previous equations considering

that we can establish that.

The stress is neglectedbecause it is very small

compared to and andhence has only a smalleffect on the strains and

xy yx

xy yxN N

xy yxM M

z

x y

x

y


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Definitions and AssumptionsSmall Deflection Theory: It is normally the basis for

analysis of thin shells, implies that the deflections under

loads are small enough so that changes in geometry of theshell will not alter the static equilibrium of the system.This assumptions is also used in analysis to apply theprinciple of Superposition.

L inear Elastic Behavior: It is also used in analysis ofthin shells, provides a direct relationship between stressand strain by which the equilibrium of the stressresultants and stress couples is related to the Strain

Compatibility Equations.

Besides the two general theories just describe we will alsotake into account two more assumptions:


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Definitions and Assumptions1.- Points on lines normal to the middle surface beforedeformation remain on lines normal to the middle surface

after deformation.2.- Deformations of the shell due to radial shears (Qx andQy) are neglected.

D fi iti d A ti


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Definitions and Assumptions The two previous assumptions have been used in

elementary beam theory as plane sections remain plane

after bending and deformations due to shears areneglected.

Based in the last two assumptions and definitions, we canformulate the general shell theory in five steps:

1.-Determining the Equilibrium of forceson the differentialelement; There are 05 (Five) available Equations with 08(Eight) Unknowns.

0X0Y

0Z

0xM 0yM 0zM


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D fi iti d A ti


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Definitions and Assumptions4.- Transforming the Force-Strain relationships into Force-Displacement equations (Still six equations with three

unknowns).

5.- Obtaining a Complete Formulation by combining theforce-displacement equations with the equil ibrium

equations(11 equations with 11 unknowns).

E ilib i


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Equilibrium The following two figures show the differential element,

the stress resultants and the vectors of stress couples.

Eq ilibri m


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Equilibrium From the six equations we are going to consider only 5

equations; the equation we will not consider is the sum ofmoments about the z axis.

Then the number of equations are reduced to 5 equations andthe number of unknowns (Initially 10) reduces to 08.

The analysis will begin by deriving the first equation of the setof equations for equilibrium.

Equilibrium


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Equilibrium The negative sides of the element (those sides which

intersects at the origin) have the curved lengths of :

x xa d y ya d The positive sides of the element have the increased curved

lengths of :

x

x y xy

a

a d d

y

y x yx

aa d d

We can visualized the terms x and y as radii of curvature

and dx and dy as small angles, but strictly speaking, thelatter terms are theparameters of the two famil ies of curves

which define the elemental sides and x and y arecoeff icients which convert the parameters to lengths:

In the previous figure we can visualize that the z direction isalways taken perpendicular to the middle surface and hence

along the lines of the radii of curvature rx and ry.

Equilibrium


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Equilibrium The small angles which are formed by the radii will be:

Where x = rx,the angle is simply dx.

The angles dx and dy are small so we can take cosdx = 1,sindx = tandx = dx. Each stress resultant or stress couple

will be multiplied by the length of the element side on which itacts so that forces and moments can be summed. Consider Nx:

The second term in the previous equation includes the changeboth in the stress resultant and in the length of the elementalside; if we expand the expression it results in:

xx

x

ad

r

x yy

ad

r

yxx y y x x y x y

x x

aNN a d N d a d d

a a

y yx xx x y y x y x x y

x x x x

a aN NN d d a d d d d d

Equilibrium


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Equilibrium In the previous equation the first two terms can be combined

as:

The third term in the previous equation can be neglectedbecause it represents a Second-Order effect. Then the firstexpression is reduced to an equation of only one term which

simply expresses the change in force:

x yx y

x

N a

d d

x y yN a d

With respect to x. The partial derivative is used becausethe force:

x y yN a d

may change also with respect toy.

Equilibrium


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Equilibrium We now consider Ny and the figure allow us to observe that

Ny does not act perpendicularly to the x axis but at some

angle:

y x x yx

x x

a d d

a d

.y y

x

x x

a d

a

So that in the x direction the component of the total forceNyxdx is:

.y y y

y x x y x y

x x x

a d aN a d N d d

a


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E ilib i


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Equilibrium The Shear-Stress resultant Nyx gives an expression equals to:

yx x

yx x x yx y x y xy y

N

N a d N d d d

E ilib i


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Equilibrium The last Equations can also be reduced to:

yx x

yx x x yx y x y xy y

N

N a d N d d d

yx xx y

y

N ad d

All of the stresses resultants acting in the plane of the shellhave been considered. The stress couples obviously give noforce components in the x, y or z directions, but the radial-shear-stress resultants must be considered in the analysis.

The following figure shows the Qx stress resultants:

Equilibrium


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Equilibrium The horizontal component

of the total force on the

sloping side is:

yx xx x y x y x

x x x

aQ aQ d a d d d

r

In which dx sindx.

The previous equation willcontain terms with bothQx and dQx and both yand dy.

Equilibrium


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Equilibrium As in the previous cases, it is consistent to neglect all the

higher terms and thus retain only the Qx term as follows:

TheQy stress resul tants wil l contr ibute to the equil ibrium inthe x dir ection i f ax and ay are not pr incipal radi i of

curvature. The following figure shows the element withcurvatures which result in a total distance of zx+zy+zxy at thecorner farthest from the origin.

. . . xx y y xx

aQ a d d r

Equilibrium


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Equilibrium Zx and zy are due to the following slopes:

Because the change in the slope we can find that:

x

z

y

z

21 x yxy x y x y x y

x y x y xy

a azz a a d d d d

a a r

21 1

xy x y x y

z

r a a

Equilibrium


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Equilibrium

This term is called the twist of the surface with respect to the xand y axes. Thus the change in slope of Qy is:

21 1

xy x y x y

z

r a a

xy y y

x x xy

z a d

a d r

The total component of Qy in X direction is:

y y

y x x

xy

a dQ a d

r

Where The element is bounded by lines of principalcurvatures rx and ry, zxy=0, the twist rxy= and the lastexpression vanishes.

Equilibrium


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Equilibrium The loads are included in the form of pressure components

(px,py,pz) acting on the differential element of an area

axdxaydy:

Finally the complete Equilibrium Equation X=0 is obtainedcombining the following equations:

x x x y yp a d a d

x yx y

x

N ad d

.

y y y

y x x y x y

x x x

a d aN a d N d d

a

.x x x

xy y y xy y x

y y y

a d aN a d N d da

yx x

x y

y

N ad d

. . . xx y y xx

aQ a d d

r y y

y x x

xy

a dQ a d

r

x x x y yp a d a d

Equilibrium


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Equilibrium Summing the terms in the equation it yields:

The second Equilibrium Equation Y=0 has the followingform:

x y yx xy xy xyx x y y

N a N aa aN N

0x y x

y x y x x y

xy x

a a aQ Q a p a a

r r

y x xy yyxx yxy y x x

N a N aaaN N

0y x y

x y x y y x

yx y

a a aQ Q a p a a

r r

Equilibrium


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Equilibrium For the third Equilibrium Equation Z=0, que Qx stress

resultants give:

This equation can be reduced to:

yxx y y x y y x y

x x

aQQ a d Q d a d d

x yx y

x

Q ad d

For the third Equilibrium Equation Z=0, que Qy stress

resultants give:

y xx y

y

Q ad d

Equilibrium


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Equilibrium In the analysis of the thin shell the z direction is always

defined here as perpendicular to the middle surface; hence it is

always in the direction of the principal radii of curvature.

The components of the total forces due to Nx, Ny, Nxy andNyx in the z direction are:

xx y y x

x

aN a d d

r

y

y x x y

y

aN a d d

r

xxy y y x

xy

aN a d d

r

y

yx x x y

xy

aN a d d

r

The force due to loadings can be defined as:

z x y x yp a a d d

Equilibrium


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Equilibrium Combining the previous equations we can establish the

equilibrium in the Z direction.

Z=0

x y y x x xx y xy y

x y x xy

Q a Q a a a

N a N ar r

0y y

yx x y x x x y

xy y

a aN a N a p a a

r r

Equilibrium


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Equilibrium The next step to finish the set of equilibrium equations is to

balance the moments about the x axis. The My moments give:

y xy x x y y x y x

y y

M aM a d M d a d d

The previous expression can be reduced to:

y xy x

y

M ad d


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Equilibrium


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Equilibrium

Finally the moment Mxy has a total contribution of:

y y y

yx x x yx x y

x x x

a d aM a d M d d

a a a

xy y

xy y y xy x y x y

x x

M aM a d M d a d d

xy y x yx

M a d d


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Equilibrium


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Equilibrium We can also establish the equilibrium equations about the y

axis:

0y xx y y xy yx x x x yx x y y

a aM a M M M a Q a a

Equilibrium


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Equilibrium Then we will establish the Five Equilibrium Equations for the

Differential Element of the Thin Shell:

0Fx y x yxx y y xy yx x y

x x y y xy

a a aaN a N N N a Q

r

0x y

x x x y

x

a a

Q p a ar

0Fy

y x yx

y x x yx xy y xy y x x xy

a a aaN a N N N a Q

r

0x y

y y x y

y

a aQ p a a

r

Equilibrium


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Equilibrium

0Mx

0My

0Fz x y x yx y y x xx y x xy

a a a aQ a Q a N N

r r

0x y x y

yx y z x y

xy y

a a a aN N p a ar r

0yxy x x yx xy y y x yy y x x

aaM a M M M a Q a a

0y xx y y xy yx x x x yx x y y

a aM a M M M a Q a a

Strain-Displacement Relationships


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Strain Displacement Relationships The following figure shows the differential element after

deformation. The components of displacement in the x, y and

z directions are taken as u, v and w respectively.



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Strain Displacement Relationships Consider now, the linear element xdx which undergoes a

change in length in the x direction of:

xx y x

x x

aud v w d

r

Which over the arc length xdx results in a strain of themiddle surface:

0

1 xx

x x x y y x

au v w

a a r

The first term represents the extension component along the x

axis (cosy = 1). The second term represents the extension bylateral movement of the element bacause of the increase inlength of the positive side. The third term expresses thecontraction due to a decrease in radius of curvature associatedwith inward radial displacement. Since z is always taken

normal to the middle surface, w is also normal to that surface.



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Strain Displacement Relationships

The same procedure can be considered to determine anexpression for strain in y direction.

The shearing strain xyo represents the total angular changebetween ydy and xdx. The middle surface strains arethus:

0

1 xx

x x x y y x

au v w

a a a r

0

1 yy

y y y x x y

av u w

a a a r

0 1 1 2yxxyx x y y x y y x y x xy

aav u u w

a a a a a a r

The first two terms in xyo represent the angle change as inthe case of a plate.



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Strain Displacement Relationships

The third and fourth terms express the angle change caused bythe increase length of the positive sides. The last termrepresents the effect of the twist of the surface.

0

1 1 2yxxy

x x y y x y y x y x xy

aav u u w

a a a a a a r

The rotation of the negative side of the element in the figurecan be expressed in terms of the displacements as:

x

x x x xy

u w

r a r

y

y y y xy

w u

r a r



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Strain Displacement Relationships The change in curvature is the change in rotation d per arc

length (d) or:

The sides of the element are not parallel, because of thechanging values of x and y. Hence a rotation y in the ydirection will produce a component ni the x direction:

1 1x

x x x x x x x xy

u wa a r a r

1 1y

y y y y y y y xy

w u

a a r a r

x xy y

y y y y y xy

a w u

a r a r

1y y x

x x y x y y y y xy

a w u

a a a r a r

Or a change in curvature per arc length of:



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Strain Displacement Relationships The complete expressions for change in curvature become:

Xxy is the change in twist, where the first two terms representthe total value if the sides of the element are parallel. Theremaining two terms account for the changing values of ayand ax.

1 yx xx

x x x y y

a

a a a

1 y yx

y

y y y x x

a

a a a

1 12

y y yx x xxy

y y x x x y y y x x

aa

a a a a a a

Stress-Strain Relationships


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Stress Strain Relationships The strain x at a distance z from the middle surface is

composed of the strain xo due to extension of the middle

surface and the strain due to bending.2 1

1

x

L L

L

1 1

x

zL ds

r

2 1 1

'

xo

x

zL ds

r

1 1 1'

1

xox x

x

x

z zr r

zr

1 1 1' '

1

xox x x

x

x

z z zr r r

zr



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Stress Strain Relationships By dropping the small terms z/r x and z/rx when they appear

with unity, we obtain.

1 1'

x xo

x x

zr r

And with X called the change in

curvature we can define:

x xo xz

y yo yz

2xy xyo xyz

xy is the Shearing Strain.



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Stress Strain Relationships All the defined expressions up to this point are independent of

the material properties, however these properties will bedefined in order to establish the stress-strain relationships.

Nearly all analysis of thin shells is based on the assumption ofa material which is:

1.- Linearly Elastic.

2.- Isotropic.

3.- Homogeneous.

These assumptions will be used in the following derivations.

The strain in the x direction, for example, can be written interms of the stresses as:

1

x x y zE



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Stress Strain Relationships Following the assumptions of a linear elastic and isotropic

material we take the modulus of Elasticity E and Poissonsratio v as constants in all directions.

Following the assumption of an homogeneous material we donot include the transformed area of the reinforcement nor dowe consider the fact that the concrete may be chacked

Because we have already assumed z=0 the previous

expression simplifies to:

1x x y

E

1y y x

E



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Stress-Strain Relationships Both stresses can be expressed in terms of the strains in x and

y directions.

21x y x y y x

E

21x y x x

E

21xx yE

Then we can obtain three expressions for the stresses.

21x x y

E

2

1

y y x

E

.xy xyG



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Stress-Strain Relationships The Shear Modulus can be expressed in terms of the modulus

of Elasticity.

We may now write the general equations relating stress andstrain by substituting the previous equations:

2 1EG

. 2xy xyo xyG z

21x x y

E

2 ( )

1

x xo yo x y

Ez

2 ( )

1y yo xo y x

Ez



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p These stresses may now be substituted into the following

equations and integrated over h to obtain the stress resultantsand stress couples.

/ 2

/ 21

h

x xh

y

zN dz

r

( )x xo yoN K

/2

/21

h

y yh

x

zN dz

r

/ 2

/21

h

xy xyh

y

zN dz

r

/2

/2

1h

x xh y

zM z dz

r

/2

/21

h

y yh

x

zM z dz

r

/2

/ 21

h

xy xyh

y

zM z dz

r

( )y yo xoN K

yx yx xyoN N Gh

.x x yM D

.y y xM D 3

(1 )

6xy yx xy xy

GhM M D



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Stress Strain Relationships K is called the Extensial Rigidity, G is the Modulus of

Rigidity and D is the Flexural Rigidity for the Shell.

2 1E

G

21

EhK

3

212 1

EhD

It is observed that once the effect of curvature is neglected inthe first equations (Nx, Ny) the values of the resultantsbecome clearly divided into 2 groups:

- The Stress resultants Nx, Ny and Nxy=Nyx in which onlythe extensial strains of the middle surface appear.

- The Stress couples Mx, My and Mxy=-Myx in which onlythe bending strains or curvature changes of the middle surfaceappear.



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Stress Strain Relationships The Elastic Constants K and D correspond to comparable

values in one-dimensional stress analysis:

21

EhK

3

212 1

Eh

D

K and D are bigger than the corresponding one-dimensionalconstants by the factor 1-n2 which represents the increase in

rigidity caused by the restriction in lateral strains.

EA

EI

.A b h

3.

12

b hI


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Force-Displacement Equations


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Force Displacement Equations1 yx x

x

x x x y y

a

a a a

1 y yxy

y y y x x

a

a a a

1 12

y y yx x xxy

y y x x x y y y x x

aa

a a a a a a

1 1.

y y yx x xx

x x x y y y y y x x

aaM D

a a a a a a

1 1.y y yx x xyy y y x x x x x y y

a aM Da a a a a a

3 (1 ) 1 1

6 2

y y yx x xxy yx xy

y y x x x y y y x x

aaGh DM M

a a a a a a

Force-Displacement Equations


73/73

Force Displacement Equations It is adequate to remember that totation are expressed in terms

of displacement:

Then we can state that the06 previous equationscontain09

unknowns, 03 stress resultants, 03 stress couples and 03displacements.The 05 equations (Fx, Fy, Fz, Mx,My) contain08 unknowns (05 Stress Resultants and 03Stress Couples). Taking both groups of equations together wehave 11 equations and 11 unknowns then a Complete Solutio

is possible.

x

x x x xy

u w

r a r

yy y y xy

w u

r a r

Chap.03-1Thin Shells Introduction

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