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KUL-49.4250 MODELS FOR BEAM, PLATE

AND SHELL STRUCTURES

Spring-2012

https://noppa.tkk.fi/noppa/kurssi/kul-49.4250/etusivu

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BAR

A thin body in 2 directions.

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STRING

A thin body in 2 directions. Curved version of the bar model!

teachers.sduhsd.k12.ca.us/.../GatewayArch.jpgwww.math.udel.edu/.../Chain/Demo%20015.jpg

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STRING MODEL: THE CURVATURE EFFECT

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MEMBRANE

A thin body in one direction. Membrane is a curved version of a thin slab!

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CURVED BEAM

A thin body in two directions. Curved version of the beam model.

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SHELL

Para

A thin body in one direction. Curved version of the plate model.

www.modot.org/newsroom/images/Planetarium.JPG www.scottspeck.com/.../north_point/DSCN3526a.jpg

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LEARNING OUTCOMES OF THE COURSE

Student is able to represent the scalar, vector and dyad (tensor) quantities and operators

of continuum mechanics in Cartesian and non-Cartesian coordinate systems,

knows the kinematic and kinetic assumptions of the beam, plate, and shell models,

is able to derive boundary value problems for beams, plates, and shells by using the

principle of virtual work, and

is able to solve the boundary value problems in simple cases either analytically or

approximately with a continuous approximation and the principle of virtual work.

Prerequisites are Kul-49.3200 Mechanics of Materials II, linear algebra and boundary

value problems.

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LECTURE 1/11 : COORDINATE SYSTEMS

1 Quantities of mechanics: scalars, vectors, tensors

2 Dyad and tensor algebra

3 Orthogonal curvilinear material coordinates

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LEARNING OUTCOMES

Student knows the summation convention, the basic concepts of index notation, and

the meanings of the delta and permutation symbols,

is able to represent the directed quantities (vectors and dyads) of continuum

mechanics by using the index notation and manipulate and simplify expressions, and

knows how to derive the basis vector derivatives of polar, cylindrical, spherical and

intrinsic coordinate systems.

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QUANTITIES OF MECHANICS

The common quantities of mechanics can be classified into scalars a magnitude, vectors

a

magnitude & direction and dyads a

magnitude & direction & direction.

Scalar a

Vectory y X Y Z

a a i a j a k a I a J a K

Dyad xx xy zy zz XX XY ZY ZZa a ii a ij a kj a kk a II a IJ a KJ a KK

Quantities are invariant with respect to coordinate system, but components depend on the

basis!

componentbase vector

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FUNDAMENTAL AND DERIVED QUANTITIES

Fundamental quantities are chosen to be length L [m], mass m [kg] and force F [N] (time

is not important in Kul-49.4250). Derived quantities have definitions in terms of the

fundamental quantities

Density: /m V

3[ ] kgm

Position vector: r xi yj zk

[ ] mr

Cauchy stress:

xx xy xz

yx yy yz

zx zy zz

i

i j k j

k

/F A 2[ ] Nm

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COMPONENT REPRESENTATION

A square matrixcan be taken as the componentrepresentation of a dyad and a column or

row matrix that of a vector. The component notation is convenient in a Cartesian

coordinate system.

Invariant y z y

z

i a

a a a a j i j k a

k a

& xx xy xz

yx yy yz

zx zy zz

a a a i

a i j k a a a j

a a a k

Component

x

y

z

a

a

a

a &

xx xy xz

yx yy yz

zx zy zz

a a a

a a a

a a a

a first index rowsecond index column

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SUMMATION CONVENTION

A once repeated (dummy) index in a term means summation over all the values of the

index set I. The index set depends on the setting (usually implicit). A non-repeated (free)

index takes all the values of the index set.

Position: i i x x y y z zr r e r e r e r e

( { , , }I x y z )

Stress: ij i j xx x x xy x y yz y z zz z ze e e e e e e e e e

Elastic material: ijkl i j k l xxxx x x x x xxxy x x x y zzzz z z z zE E e e e e E e e e e E e e e e E e e e e

The order of the basis vectors is important (vector product is notcommutative)!

in the order of indices!index set

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IMPORTANT DEFINITIONS

Summation convention: 1 1 2 2i Ii i i i n na b a b a b a b a b

Comma notation: ,/i j i ja x a

Delta symbol: {0,1}ij i je e

( i j ije e

)

Permutation symbol: ( ) { 1,0,1}ijk i j k e e e

( i j ijk k e e e

)

Identity ( ): ijk imn jm kn jn km

Determinant: Det( )ijk lmn il jm kna a a a

Sign of ijk changes in each permutation of the indices. The definitions are useful in a

Cartesian coordinate system and in a curvilinear orthogonal system!

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RULES TO BE REMEMBERED

A once repeated (dummy) index in a term means summation over all the values of the

index set I (summation convention). A dummy index can be changed to some other

symbol not already appearing in the term.

A non-repeated (free) index takes all the values of the index set.

Index set may depend on the problem dimension and it is not necessarily repeated inevery occasion of the use.

Never use the same dummy symbol in the sums of a term.

A delta symbol eats indices

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EXAMPLE (J.N.Reddy 1.1).Prove the following properties of ij and ijk (assume that

{1,2,3}I ) (a) 3ij ij (b) 6ijk jki , (c) 0ijk ijF whenever ij jiF

Definition of delta-symbol 1ij when i j and 0ij whenever i j :

11 22 33 3ij ij ii

Identity ( ) ijk imn jm kn jn km gives

( ) ( ) (3 9) 6ijk jki ijk ikj jk kj jj kk jj jj kk

Identity 2a a a and property ijk jik give

1 1 1( ) ( ) ( ) 0

2 2 2ijk ij ijk ij jik ji ijk ij ijk ji ijk ij jiF F F F F F F

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INDEX REPRESENTATION

Index notation is very useful e.g. in vector manipulations and proving vector identities

involving the gradient operator. As examples, in a Cartesian coordinate system i j ije e

&i j ijk k

e e e

& ( )ijk i j k

e e e

& /i i

e x

:

Inner product: ( )i i j j i j i j i j ij i ia b a e b e a b e e a b a b

Outer product: ( )i i j j i j i j ijk i j k a b a e b e a b e e a b e

Divergence: ( )i j j i j j j j i j j ii i i i ia e a e e e a a e e e a ax x x x x

Curl:j

i j j i j j k kij j ijk k i i i i

aa e a e e e a e a e

x x x x

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EXAMPLE.Show that the following identities hold:

(a) ( )v v v

,

(b) ( ) ( ) ( )a b c a c b a b c

Use the index notation in a Cartesian coordinate system.

( ) ( ) div( ) grad( )i j j i ii i i

v e e v v v v v

x x x

( ) ( ) ( )i i j j k i j k i j k i j k l lin njk a b c a e e b e a b c e e e a b c e

( ) ( )i j k l nil njk i j k l ij lk ik lj i j i j i i j ja b c a b c e a b c e a b c e a b c e

( ) ( ) ( )a b c b a c c a b

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EXAMPLE. Use the index notation in a Cartesian coordinate system to prove the identity

1( ) ( )2

a a a a a a .

Let us start from the left hand side

( ) ( ) ( )k ki i j k k i i l ljk nil ljk i nj j j

a aa a a e e a e a e e a e

x x x

( ) ( )jk k i

lin ljk i n in jk ik jn i n i j i ij j j j

aa a aa a a e a e a e a e

x x x x

21( )

2a a a a a

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STRUCTURAL SYSTEM AND MATERIAL SYSTEM

In a particle model, particles are identified by natural numbers (or labels). In a continuum

model, particles are identified by coordinates ( , , )x y z

of the material coordinate system

which moves and deforms with the body i.e. closed system of particles.

Structural( , , )X Y Z

coordinate system is needed in the description of geometry.

y

A x4

OX z1

2

YZ

Pi

structuralmaterial

body

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CARTESIAN COORDINATES , ,x y z

In a Cartesian coordinate system, a particle is identified by its distances , ,x y z from the

planes 0x , 0y and 0z , respectively.

Mapping: i ir r e xi yj zk

Basis: , ,/i i i ie r r e

& 1ih

Derivatives: , 0j j ii

e er

Cartesian coordinate system is useful as a reference system as basis vectors of the system

are constants!

reference system! P

O

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BASIS VECTORS; RECIPE

A generic mapping between two coordinate systems consists of relationships between

coordinates and basis vectors. Here the basis vectors are defined as normalized partial

derivatives of the position vector with respect to the coordinates (just a convenient

choice).

(a) Start with the position vector ( , , )i ir r e

in a Cartesian system

(c) Take derivatives , ,i ih r r e

(d) Scaling coefficients h h h h

(e) Normalize to getthe basis /e h h

Scaling coefficients are needed later e.g. in connection with the

-operator!

constants!

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POLAR COORDINATES ,r

In a curvilinear rectangular polar coordinate system, a particle is identified by its distance

rfrom the origin and angle from a chosen line.

Mapping: cos sinrr re r i r j

Basis:

c s

s c

re i

e j

&

1rh

h r

Derivatives:

r

r

e e

e e

and 0

re

r e

r

P

O

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The derivatives can be obtained in the following manner:

cos sin

sin cos

re i

e j

and

cos sin

sin cos

rei

ej

cos sin sin cos

( ) )sin cos cos sin

re i i

e j j

sin cos cos sin 0 1

cos sin sin cos 1 0

r r r

r

e e e e

e e e e

constants!

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DERIVATIVES OF BASIS VECTORS; RECIPE

(a) Start with the relationship [ ]

e i

e F j

e k

1[ ]

i e

j F e

k e

(b) Take derivative on both sides ( [ ])

e i

e F js se k

{ , , }s

(c) Retain the basis1

( [ ])[ ]

e e

e F F es s

e e

constants!

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CYLINDRICAL COORDINATES , ,r z

A particle is identified by its distancerfrom thez-axis origin, angle from thex-axis and

distancez from thexy-plane ( 0z ):

Mapping: cos sinr r i r j zk

Basis:

c s 0

s c 0

0 0 1

r

z

e i

e j

e k

&

1

1

r

z

h

h r

h

Derivatives:

0 1 0

1 0 0

0 0 0

r r

z z

e e

e e

e e

, otherwise zeros

r

P

O

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SPHERICAL COORDINATES , ,r

A particle is identified by its distance r from the origin, angle from the x-axis in xy-

plane ( 0z ) and angle from thez-axis:

Mapping: (s c s s c )r r i j k

Basis:

s c s s c

s c 0

c c c s s

re i

e j

e k

&

1

s

r

z

h

h r

h r

Derivatives:

0

0

0

re

er

e

,

s

s c

c

r

r

e e

e e e

e e

, 0r

r

e e

e

e e

,

r

P

O

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EXAMPLE. Derive the derivative expressions of the basis vectors re

, e

and e

with

respect to , ,r in terms of the basis vectors re

, e

and e

. Use the general recipe and the

relationship

s c s s c

s c 0

c c c s s

re i

e j

e k

Answer:

r

r

e s e

e s e c e

e c e

, 0

r

r

e e

e

e e

,

0

0

0

re

e

r e

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The generic recipegives

1

0

( ) 0

0

r re e

e F F er r

e e

1

0 s 0

( ) s 0 c

0 c 0

r r re e e

e F F e e

e e e

1

0 0 1

( ) 0 0 0

1 0 0

r r re e e

e F F e e

e e e