Chap.01-1General Theory of Plates

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Elements of Plate-Bending

theory

Universidad Catlica de Santa Maria

August 20, 2013

Introduction

Hipdromo de Zarzuela en Madrid


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Introduction

Idea Bsica del Teatro de la Opera en Sydney

Introduction

Vista General del Teatro de la Opera en Sydney


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Introduction

Vista General del Teatro de la Opera en Sydney

Introduction

Vista Posterior del Teatro de la Opera en Sydney


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Introduction

Vista en Elevacin del Teatro de la Opera en Sydney

Introduction

Detalle de la cubierta del Teatro de la Opera en Sydney


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Introduction

El mecanismo resistente de las lminas en estado

de membrana es mucho mas efectivo que elmecanismo de flexin.

Introduction Platesand Shellsare initiallyflat and curved

structural elements respectively.

Plates Flat Structural Elements

Shells Curved Structural Elements

The thicknesses in both cases are muchsmaller than the other dimensions.

Street manhole covers

Side panels and roof of buildings

Many practical engineering problems fall intocategories Plates in bending or Shells inbending.


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Introduction

In plates it is common to divide the thickness tinto

equal halves by a plane parallel to the faces; thisplane is called the midplane of the plate.

The plate thickness is measured in a directionnormal to the midplane.

The flexural properties of a plate depend greatlyupon its thickness in comparison with its otherdimensions.

Plates may be classified into three groups:

- Thin plates with small deflections.

- Thin plates with large deflections.

- Thick plates.

To define a thin plate for purposes of technicalcalculations, the ratio of the thickness to the

smaller span length should be less than 1/20.

The Plate and Shell materials are homogeneousand isotropic.

A homogeneous material displays identicalproperties throughout; when the properties arethe same in all directions, the material is calledisotropic.

1

t 1

l 20

Introduction


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Comportamiento General de los

Elementos Plate (Losas) Consideremos primero un elemento tipo plate libre de carga en

el cual los ejes x e y coinciden con el plano medio y enconsecuencia la deflexin en el eje z es cero como lo muestra lafigura.

Definiremos luego los componentes de desplazamiento en undeterminado punto, tanto en la direccin del eje x, y y en el eje z:

Para la direccin X u

Para la direccin Y v

Para la direccin Z w

Luego cuando debido a carga lateral,se presenten deformaciones, la

superficie media en un punto (xa, ya)

presentar una deflexin w.

The Fundamental assumptions of the small-deflection theory ofbending or so-called classical theory for isotropic, homogeneous,elastic, thin plates is based on the Geometry of Deformations:

1.- The deflection of the midsurface is small compared with thethickness of the plate. The slope of the deflected surface istherefore very small and the square of the slope is anegligible quantity in comparison with unity.

General Behavior of Plates


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The above assumptions, known as theKirchhoff hypothesesare analogous to those associated with thesimple Bending

Theory of beams.

Because of the resulting decrease in complexity, a three-dimensional plate problem reduces to one involving onlytwo dimensions.

When the deflections are not small, the bending of plates isaccompanied by strain in the midplane, and assumptions (1)and (2) are inapplicable.

In thick plates, the shearing stresses are important, as inshort, deep beams. Such plate are treated by means of amore general theory owing to the fact that assumptions (3)and (4) are no longer appropriate.

General Behavior of Plates

Strain-Curvature Relations As a consequence of the assumption (3) of the

foregoing section, the strain-displacement relationsreduce to:

u

xx

v

yy

u

yxy

v

x

w0

zz

0xzw u

x z

0yz

w v

y z

Note that these expressions are also referred to as theKinematic Relations, treating the geometry of strainrather than the matter of cause and effect.


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Strain-Curvature Relations Integrating the 4thequation we obtain:

w 0z

z

Indicating that the lateral deflection does not varyover the plate thickness. In a like manner,

integration of the expressions for the vertical shearstrains we have:

w . zz

w . zz w ( , )w x y

Strain-Curvature Relations Integrating the 5thand 6thequation we obtain:

0xzw u

x z

xz

u w

z x

. .xzw

u z zx

. .xzw

u z zx

. .xzw

u z zx

0. ( , )w

u z u x yx

0yzw v

y z

yz

v w

z y

. .yzw

v z zy

. .yzw

v z zy

. .yzw

v z zy

0. ( , )w

v z v x yy


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It is clear that uo(x,y) and vo(x,y) represent,

respectively, the values of u and v on the midplane.Based on assumption (2) we conclude thatuo=vo=0. Thus:

. w

u zx

.

wv z

y

Strain-Curvature Relations

We see that the above equations are consistent withthe assumption (3). Substitution of the aboveequations into the first 6 equations yields:

u

xx

For the First Equation:

. w

u zx

2

2.

du wz

x x

2

2.x

du wz

x x

2

2.x

wz

x



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1

v

yy

For the Second Equation:

. w

v zy

2

2.

dv wz

y y

2

2.y

dv wz

y y

2

2.y

wz

y

For the Third Equation:

uy

xyv

x

. w

u zx

.

wv z

y


u

yxy

v

x

. w

u zx

2

.du w

zy x y

. w

v zy

2

.dv w

zx y x

2 2

. .xyw w

z zx y y x

2

2 .xyw

zx y



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1

2

2 .xyw

z

x y

2

2

.x

wz

x

2

2.y

wz

y

These formulas provide the strains at any point in theplate.

The curvature of a plane curve is defined as the rate ofchange of the slope angle of the curve with respect todistance along the curve. Because of assumption (1), thesquare of a slope may be regarded as negligible and thepartial derivatives above represent the curvatures of the

plate. The curvatures k (kappa) at the midsurface in planes

parallel to xz, yz and xy planes are, respectively:


1x

x

wkr x x

1y

y

wk

r y y

1xy

xy

wk

r x y

Where kxy=kyx

Clearly these equations are therate at which the slopes vary over

the plate. The last of these

expressions is also referred to as

the Twist of the midplane with

respect to x and y axes.



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The local twist of a plate element is shown in the

following figure:


The strain-curvature relations considering the lastequations may be expressed in the following form:

2

2.x

wz

x

x

wk

x x

.

x xz k

2

2.yw

z y

y

w

k y y

.y yz k

2

2 .xyw

zx y

xy

wk

x y

2 .

xy xyz k



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An examination of the last equations shows that a circleof curvature can be constructed similarly to Mohrs

Circle of strains. The following figure shows a plateelement and a circle of curvature in which n and trepresent perpendicular directions at a point on themidsurface. The principal or maximum and minimumcurvatures are indicated by k1 and k2. The planesassociated with these curvatures are called the principalplanes of curvature.


The curvature and the twist of a surface vary with the angle, measured in the clockwise direction from the set of axes

xy to the xyset. It is seen that when the two principal curvatures are the same

Mohrs circle shrinks to a point. This means that thecurvature is the same in all directions; there is no twist in anydirection. The surface is purely spherical at that point. Fromthe circle and equations of curvature we have:

' 'x y x yk k k k



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The sum of the two curvatures in perpendiculardirections at a point, called the average curvature,is thus invariant with respect to rotation of the

coordinate axes. This assertion is valid at anylocation on the midsurface.


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Chap.01-1General Theory of Plates