kul-49_4250_lecture_01
TRANSCRIPT
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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