Chap. 7 Analysis of 2d Solid Mechanics Problems CAE 기본개념 … · 2016-02-27 · 1 1 Chap.7 Structural Analysis CAE 기본개념 소개 Chap. 7 Analysis of 2d Solid Mechanics

1

1 Chap.7 Structural Analysis

CAE 기본개념 소개

Chap. 7 Analysis of 2d Solid Mechanics Problems


Torsion Member (Arbitrary Cross-Section)

[Review] Torsion of a Circular Shaft

The shear stress distribution

The angle of twist

The maximum shear stress (Radius: R)

T: the applied torque

r: the radial distance

J: the polar moment of inertia

Tr

J

TL

JG

L: the length of the member

G: the shear modulus

34max

2

21 R

T

R

TR

2



Torsion of a Bar with a Rectangular Cross Section

The angle of twist

The maximum shear stress (Radius: R)

For high aspect ratio (w/h > 10)



Prandtl Formulation for Torsion Formulations

The governing differential equation

The shear stress components

The applied torque

where : the angle of twist per unit length

φ : the stress function

3


Heat Transfer Problems

[Review] 2D Heat Transfer Formulation (Rectangular elements)

Galerkin formulation as a generalized form

Elementary conductance matrix & Thermal load vector

2 2

1 2 32 20

T T T

A A A

T TC dA C dA C dA

x x

S S S

1 2 3, , x yC k C k C q

2112

1221

1221

2112

6

2211

2211

1122

1122

6

][ )(

w

lk

l

wkeK

1

1

1

1

4

)( Aqe F

C1 C2 C3



Torsion vs. 2d Heat Transfer Problems

The governing differential equation

Elementary stiffness matrix & load vector

2 2

2 22 0G

x y

Torsion eq.

Heat diffusion eq. 02

2

2

2

q

y

Tk

x

Tk

4


Beams and Frames

Beam

A structural member whose cross-sectional dimensions are relatively smaller than length

Commonly subjected to transverse loading: Bending

Engineering applications: buildings, bridges, automobiles, and airplane structures

Truss vs. Beam

Truss: Loads are applied only at joints

Beam: Loads are applied at any locations

Approximations

Normal planes to the neutral axis are maintained to the normal planes after deflection

The effects of the shear stresses are neglected


Beams and Frames

Beam Formulation (Euler Beam Theory)

The flexure formula

The deflection of the neutral axis (v)

M: the applied bending moment

y: distance from the neutral axis

I: the second moment of inertia

My

I

2

2

dEI M x

dx

3

3

dM xdEI V x

dx dx

4

4

dV xdEI w x

dx dx

5


Beams and Frames

FE Formulation – Elementary Stiffness Matrix

The strain energy for an arbitrary beam element (e)

Recognizing the integral is the second moment of inertia (I)

Approximation of the deflection (v) with

the 3rd order polynomial

2

Ay dA

2 3

1 2 3 4c c x c x c x


Node Node i (x = 0) Node j (x = L)

Displacement

Slope

Beams and Frames


The element’s end conditions (nodal values)

4 equations with four unknowns: solve for

1 1ic U

2 2

0

i

x

dc U

dx

2 3

1 2 3 4 1jc c L c L c L U

2

2 3 4 22 3 j

x L

dc c L c L U

dx

1 2 3 4, , ,c c c c

2 3

1 2 3 4c c x c x c x

6


Beams and Frames


Expression using the shape function

End conditions: Verification

1 1 2 2 1 1 2 2i i i i j j j jS U S U S U S U


Beams and Frames


Governing equation in terms of the shape functions

The 2nd derivatives of the shape functions

2

11 2 2 3

6 12ii

d S xD

dx L L

2

22 2 2

4 6ii

d S xD

dx L L

2

1

1 2 2 3

6 12j

j

d S xD

dx L L

2

2

2 2 2

2 6j

j

d S xD

dx L L

2

2

dEI M x

dx

7


Beams and Frames


The strain energy for an arbitrary beam element

Minimum total potential energy principle

22

( )

202

Le EI

dxx

2

2

2

T T

x

D U D U U D D U

eFU

0 =1, 2, 3, 4

e

k k k

FU kU U U

since


Beams and Frames


The strain energy term

The elementary stiffness matrix

2 2

3

2 2

12 6 12 6

6 4 6 2

12 6 12 6

6 2 6 4

e

L L

L L L LEI

L LL

L L L L

K

8


Beams and Frames

FE Formulation – Elementary Load Vector

Calculation of the

equivalent nodal loads

Reverse the signs

(pp. 326 ~ 327)


Beams and Frames


9


Beams and Frames

Frames

Structural members rigidly connected with welded or bolted joints

Displacements (DOFs): [rotation and lateral displacement] + axial deformation

1 1,i ju u

2 2,i ju u

3 3,i ju u

DOFs of a beam


Beams and Frames

FE Formulation for a Frame

Coordinate transformation – use of the transformation matrix

u = T U

Local DOF Global DOF

10


Beams and Frames


Elementary stiffness matrix

1 2 3 1 2 3

1

2

2 23

31

2

2 23

0 0 0 0 0 0

0 12 6 0 12 6

0 6 4 0 6 2

0 0 0 0 0 0

0 12 6 0 12 6

0 6 2 0 6 4

i i i j j j

i

i

e i

xyj

j

j

u u u u u u

u

uL L

uL L L LEI

uL

uL L

uL L L L

K

1 2 3 1 2 3

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

i i i j j j

e

axial

u u u u u u

AE AE

L L

AE AE

L L

K

1iu

3ju

2ju

2iu

1ju

3iu


Beams and Frames


Elementary stiffness matrix in local coordinate system

Elementary stiffness matrix in local coordinate system

3 2 3 2

2 2

3 2 3 2

2 2

0 0 0 0

12 6 12 60 0

6 4 6 20 0

0 0 0 0

12 6 12 60 0

6 2 6 40 0

e

xy

AE AE

L L

EI EI EI EI

L L L L

EI EI EI EI

L L L L

AE AE

L L

EI EI EI EI

L L L L

EI EI EI EI

L L L L

K

11


Beams and Frames

Examples

Ex 8.1 Beam (pp. 328 ~ 331) Ex 8.2 Frame (pp. 334 ~ 338)


Plane Stress Formulation

[Review] 2D Structural Problems

Plans Stress

Plane Strain

T

xyyyxx

T

zxzxzz

0

2

1 0

1 01

10 0

2

xx xx

yy yy

xy xy

E

1 0

1 01 1 2

10 0

2

xx xx

yy yy

xy xy

E

T

xyyyxx

T

zxzxzz

0

12



[Review] Plane Stress Problems

T

xyyyxx

T

zxzxzz

0

2

1 0

1 01

10 0

2

xx xx

yy yy

xy xy

E

σ ν ε

T

XX YY ZZ XY YZ XZ

xx

u

x

yy

v

y

xy

u v

y x




The strain energy for an arbitrary element under biaxial loading

The displacement variable in terms of linear triangular shape functions

(Hooke’s law)

1

2

e

xx xx yy yy xy xyV

dV

1

2

Te

VdV σ ε

i ix j jx k kxu SU S U S U

i iy j jy k kyv SU S U S U

0 0 0

0 0 0

ix

iy

i j k jx

i j k jy

kx

ky

U

U

S S S Uu

S S S Uv

U

U

13




The strain-displacement relation

1

2xx i ix j jx k kx i ix j jx k kx

uS U S U S U U U U

x x A

1

2yy i iy j jy k ky i iy j jy k ky

vS U S U S U U U U

y y A

1

2xy i ix i iy j jx j jy k kx k ky

u vU U U U U U

y x A

ε B U




The strain energy equation

Differentiating wrt the nodal displacement


1

2

eT T

Vk k

dVU U

U B ν B U 1, 2,...,6k

14




The work done by concentrated loads

The work done by a distributed load with px and py components

The work done by the distributed load in a triangular element

Te

W U Q

e

x yA

W up vp dA




A concentrated load vector for a triangular element

The differentiation of the work done by the distributed load

Te

W U Q

ix

iy

e jx

jy

kx

ky

Q

Q

Q

Q

Q

Q

F

differentiating

where

15




Integrating along the ki-edge (Sj = 0)

e T

AdA F S p

2

0

0

x

y

e ij x

y

p

p

tL p

p

F

0

0

2

e xjk

y

x

y

ptL

p

p

p

Falong the

ij-edge

along the

jk-edge



Isoparametric Formulation – Q4 Element

Displacement field within an element

Position within the element

i ix j jx m mx n nxu SU S U S U S U

i iy j jy m my n nyv SU S U S U S U

i i j j m m n nx S x S x S x S x

i i j j m m n ny S y S y S y S y

16




Using the Jacobian of the coord. transformation

x y

x y

J

i i j j m m n n i i j j m m n n

i i j j m m n n i i j j m m n n

S x S x S x S x S y S y S y S y

S x S x S x S x S y S y S y S y




Jacobian matrix for a Q4 element

1 1 1 11

4 1 1 1 1

i j m n

i j m n

x x x x

x x x x

J

11 12

21 22

1 1 1 1

1 1 1 1

i j m n

i j m n

y y y y J J

J Jy y y y

1 22 12 22 12

21 11 21 1111 22 12 21

1 1

det

J J J J

J J J JJ J J J

J

J

17




The strain energy of an element

The strain-displacement relation

1 1

2 2

T Te

eV A

dV t dA ε ν ε ε ν ε (te: element thickness)




The strain-displacement relation (cont’d)

22 12

21 11

1

det

uu

J Jx

u J J u

y

J

22 12

21 11

1

det

vv

J Jx

v J J v

y

J

18




The strain-displacement relation (cont’d)

1 0 1 0 1 0 1 0

1 0 1 0 1 0 1 01

0 1 0 1 0 1 0 14

0 1 0 1 0 1 0 1

u

u

v

v

D

ix

iy

jx

jy

mx

my

nx

ny

U

U

U

U

U

U

U

U

U




The strain energy of an element (in natural coordinate)

The strain-displacement relation (matrix form)


1 1

1 1

1 1det

2 2

T Te

e eA

dA

t dA t d d

ε ν ε ε ν ε J

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Chap. 7 Analysis of 2d Solid Mechanics Problems CAE 기본개념 … · 2016-02-27 · 1 1 Chap.7 Structural Analysis CAE 기본개념 소개 Chap. 7 Analysis of 2d Solid Mechanics