Finite Element Method by G. R. Liu and S. S. Quek 1 Finite Element Method INTRODUCTION TO MECHANICS FOR SOLIDS AND STRUCTURES for readers of all backgrounds.

1Finite Element Method by G. R. Liu and S. S. Quek

FFinite Element Methodinite Element Method

INTRODUCTION TO MECHANICS

FOR SOLIDS AND STRUCTURES

for readers of all backgroundsfor readers of all backgrounds

G. R. Liu and S. S. Quek

CHAPTER 2:

2Finite Element Method by G. R. Liu and S. S. Quek

CONTENTSCONTENTS

INTRODUCTION– Statics and dynamics– Elasticity and plasticity– Isotropy and anisotropy– Boundary conditions– Different structural components

EQUATIONS FOR THREE-DIMENSIONAL (3D) SOLIDS EQUATIONS FOR TWO-DIMENSIONAL (2D) SOLIDS EQUATIONS FOR TRUSS MEMBERS EQUATIONS FOR BEAMS EQUATIONS FOR PLATES

3Finite Element Method by G. R. Liu and S. S. Quek

INTRODUCTIONINTRODUCTION

Solids and structures are stressed when they are subjected to loads or forces.

The stresses are, in general, not uniform as the forces usually vary with coordinates.

The stresses lead to strains, which can be observed as a deformation or displacement.

Solid mechanics and structural mechanics

4Finite Element Method by G. R. Liu and S. S. Quek

Statics and dynamicsStatics and dynamics

Forces can be static and/or dynamic. Statics deals with the mechanics of solids and

structures subject to static loads. Dynamics deals with the mechanics of solids and

structures subject to dynamic loads. As statics is a special case of dynamics, the

equations for statics can be derived by simply dropping out the dynamic terms in the dynamic equations.

5Finite Element Method by G. R. Liu and S. S. Quek

Elasticity and Elasticity and pplasticitylasticity

Elastic: the deformation in the solids disappears fully if it is unloaded.

Plastic: the deformation in the solids cannot be fully recovered when it is unloaded.

Elasticity deals with solids and structures of elastic materials.

Plasticity deals with solids and structures of plastic materials.

6Finite Element Method by G. R. Liu and S. S. Quek

Isotropy and Isotropy and aanisotropynisotropy

Anisotropic: the material property varies with direction.

Composite materials: anisotropic, many material constants.

Isotropic material: property is not direction dependent, two independent material constants.

7Finite Element Method by G. R. Liu and S. S. Quek

Boundary conditionsBoundary conditions

Displacement (essential) boundary conditions

Force (natural) boundary conditions

8Finite Element Method by G. R. Liu and S. S. Quek

Different structural componentsDifferent structural components

Truss and beam structures

x

fy1

z

fy2 y

x

fx

z

Truss member Beam member

9Finite Element Method by G. R. Liu and S. S. Quek

Different structural componentsDifferent structural components

Plate and shell structures

Plate Neutral surface

z

y x

Neutral surface

z

y x

h

h Shell

Neutral surface

Neutral surface

10Finite Element Method by G. R. Liu and S. S. Quek

EQUATIONS FOR 3D SOLIDSEQUATIONS FOR 3D SOLIDSStress and strainConstitutive equationsDynamic and static equilibrium equations

y

x

z

V

Sd

fs1

fs2

fb2 fb1

Sf

Sf Sd

Sd

11Finite Element Method by G. R. Liu and S. S. Quek

Stress and strainStress and strain

Stresses at a point in a 3D solid:

zxxz

yzzy

yxxy

Txx yy zz yz xz xy

xy

xz

xx

yy

yz

yx

zy

zz

zx

y

x

z

yy

yz

yx

zy

zz

zx

xy

xz

xx

12Finite Element Method by G. R. Liu and S. S. Quek

Stress and strainStress and strain

Strains

Txx yy zz yz xz xy

y

w

z

v

x

w

z

u

x

v

y

u

z

w

y

v

x

u

yzxzxy

zzyyxx

,

,

,

,

13Finite Element Method by G. R. Liu and S. S. Quek

Stress and strainStress and strain

Strains in matrix form

LUwhere

0 0

0 0

0 0

0

0

0

x

y

z

z y

z x

y x

L

w

v

u

U

14Finite Element Method by G. R. Liu and S. S. Quek

Constitutive equationsConstitutive equations

= c or

xy

xz

yz

zz

yy

xx

xy

xz

yz

zz

yy

xx

c

ccsy

ccc

cccc

ccccc

cccccc

66

5655

464544

36353433

2625242322

161514131211

.

15Finite Element Method by G. R. Liu and S. S. Quek

Constitutive equationsConstitutive equations

For isotropic materials

2

02

.

002

000

000

000

1211

1211

1211

11

1211

121211

cc

ccsy

ccc

cc

ccc

c

)1)(21(

)1(11

Ec

)1)(21(12

E

c

Gcc

21211

)1(2 E

G

,

,

16Finite Element Method by G. R. Liu and S. S. Quek

Dynamic equilibrium equationsDynamic equilibrium equations

Consider stresses on an infinitely small block

xy +d xy

xz +d xz

xx +d xx

yy +d yy

yz +d yz

yx +d yx

zy +d zy

zz +d zz zx +d zx

y x

z

yy yz

yx

zy zz

zx

xy

xz

xx

d x d y

d z

17Finite Element Method by G. R. Liu and S. S. Quek

Dynamic equilibrium equationsDynamic equilibrium equations

Equilibrium of forces in x direction including the inertia forces

Note: d d ,

d d ,

d d

xxxx

yxyx

zxzx

xx

yy

zz

xy+dxy

xz+dxz

xx+dxx

yy+dyy

yz+dyz

yx+dyx

zy+dzy

zz+dzz

zx+dzx

yy

yz

yx

zy

zz

zx

xy

xz

xx

dx dy

dz

external force inertial force

( d )d d d d ( d )d d d d

( d )d d d d d d d

xx xx xx yx yx yx

zx zx zx x

y z y z x z x z

x y x y f u x y z

18Finite Element Method by G. R. Liu and S. S. Quek

Dynamic equilibrium equationsDynamic equilibrium equations

Hence, equilibrium equation in x direction

ufzyx xzxyxxx

Equilibrium equations in y and z directions

vfzyx yzyyyxy

wfzyx zzzyzxz

19Finite Element Method by G. R. Liu and S. S. Quek

Dynamic and static equilibrium equationsDynamic and static equilibrium equations

In matrix formT

b L f Uor

Tb L cLU f U

For static case

0Tb L cLU f

Note:

x

b y

z

f

f

f

f

20Finite Element Method by G. R. Liu and S. S. Quek

EQUATIONS FOR 2D SOLIDSEQUATIONS FOR 2D SOLIDS

Plane stressPlane strain

x

y

z

y

x

21Finite Element Method by G. R. Liu and S. S. Quek

Stress and strainStress and strain

Txx yy zz yz xz xy

xx

yy

xy

xx

yy

xy

x

v

y

u

y

v

x

uxyyyxx

,,

(3D)

22Finite Element Method by G. R. Liu and S. S. Quek

Stress and strainStress and strain

Strains in matrix form

ε LU

where

0

0

x

y

y x

L, u

v

U

23Finite Element Method by G. R. Liu and S. S. Quek

Constitutive equationsConstitutive equations

= c

2

1 0

1 01

0 0 1 / 2

E

c (For plane stress)

1 01

(1 )1 0

(1 )(1 2 ) 11 2

0 02(1 )

E

c(For plane strain)

24Finite Element Method by G. R. Liu and S. S. Quek

Dynamic equilibrium equationsDynamic equilibrium equations

ufzyx xzxyxxx

ufyx xyxxx

vfyx yyyxy

(3D)

25Finite Element Method by G. R. Liu and S. S. Quek

Dynamic and static equilibrium equationsDynamic and static equilibrium equations

In matrix formT

b L f Uor

Tb L cLU f U

For static case

0Tb L cLU f

Note: x

by

f

f

f

26Finite Element Method by G. R. Liu and S. S. Quek

EQUATIONS FOR TRUSS EQUATIONS FOR TRUSS MEMBERSMEMBERS

fx

y

x

z

xy

yy

xx

σ xx

x

ux

27Finite Element Method by G. R. Liu and S. S. Quek

Constitutive equationsConstitutive equations

Hooke’s law in 1D

= E

Dynamic and static equilibrium equationsDynamic and static equilibrium equations

ufx x

x

0

xx f

x

(Static)

28Finite Element Method by G. R. Liu and S. S. Quek

EQUATIONS FOR BEAMSEQUATIONS FOR BEAMS Stress and strainConstitutive equationsMoments and shear forcesDynamic and static equilibrium equations

x

fy1

y

fy2

29Finite Element Method by G. R. Liu and S. S. Quek

Stress and strainStress and strain

Euler–Bernoulli theory

Centroidal axis

x

y

30Finite Element Method by G. R. Liu and S. S. Quek

Stress and strainStress and strain

0xy

yu

x

v

Assumption of thin beam

Sections remain normal

Slope of the deflection curve

yLvx

vy

x

uxx

2

2

2

2

x L

where

xx = E xx yELvxx

31Finite Element Method by G. R. Liu and S. S. Quek

Constitutive equationsConstitutive equations

xx = E xx

Moments and shear forcesMoments and shear forcesConsider isolated beam cell of length dx

dx

Mz

Mz + dMz

Q Q + dQ

(fy(x)- vA ) dx

x

y

32Finite Element Method by G. R. Liu and S. S. Quek

Moments and shear forcesMoments and shear forces

The stress and moment

M M

dx

y

x

xx

33Finite Element Method by G. R. Liu and S. S. Quek

Moments and shear Moments and shear forcesforces

Since yELvxx

Therefore,2

22

d ( d )z xx z z

A A

vM y A E y A Lv EI Lv EI

x

Where

2dz

A

I y A (Second moment of area about z axis – dependent on shape and dimensions of cross-section)

M M

dx

y

x

xx

34Finite Element Method by G. R. Liu and S. S. Quek

Dynamic and static equilibrium equationsDynamic and static equilibrium equations

Forces in the x direction

0)( dxvAxfdQ y

vAxfdx

dQy

Moments about point A

0))(2

1 2 x(dvA-fxdQdM yz

dx

M z

M z + dM z

Q Q + dQ

(fy(x)- vA ) dx

A

Qdx

dM z 3

3

x

vEIQ z

35Finite Element Method by G. R. Liu and S. S. Quek

Dynamic and static equilibrium equationsDynamic and static equilibrium equations

vAxfdx

dQy

Therefore,

yz fvAx

vEI

4

4

yz fx

vEI

4

4

(Static)

36Finite Element Method by G. R. Liu and S. S. Quek

EQUATIONS FOR PLATESEQUATIONS FOR PLATES Stress and strainConstitutive equationsMoments and shear forcesDynamic and static equilibrium equationsMindlin plate

z, w

h

fzy, v

x, u

37Finite Element Method by G. R. Liu and S. S. Quek

Stress and strainStress and strain

Thin plate theory or Classical Plate Theory (CPT)

Centroidal axis

x

y

38Finite Element Method by G. R. Liu and S. S. Quek

Stress and strainStress and strain

Assumes that xz = 0, yz = 0

x

wzu

y

wzv

,

Therefore,

2

2

x

wz

x

uxx

2

2

y

wz

y

vyy

y x

wz

x

v

y

uxy

2

2

,

39Finite Element Method by G. R. Liu and S. S. Quek

Stress and strainStress and strain

Strains in matrix form

= z Lw

where 2

2

2

2

2

x

y

x y

L

40Finite Element Method by G. R. Liu and S. S. Quek

Constitutive equationsConstitutive equations

= c where c has the same form for the plane stress case of 2D solids

2

1 0

1 01

0 0 1 / 2

E

c

41Finite Element Method by G. R. Liu and S. S. Quek

Moments and shear forcesMoments and shear forces

Stresses on isolated plate cellz

x

y

fz

h

xyxxxz

yx

yy

yz

O

42Finite Element Method by G. R. Liu and S. S. Quek

Moments and shear forcesMoments and shear forces

Moments and shear forces on a plate cell dx x dy

z

x

yO

dx

dy

Qy

MyMyx

Qy+dQy

Myx+dMyx

My+dMy

Qx

MxMxy

Qx+dQx

Mxy+dMxyMx+dMx

43Finite Element Method by G. R. Liu and S. S. Quek

Moments and shear forcesMoments and shear forces

= c = c z Lw

Like beams,

32d ( d )

12

x

p y

A Axy

Mh

M z z z z w w

M

M c L cL

Note thatxd

x

QQd x

x

ydy

QQd y

y

,

44Finite Element Method by G. R. Liu and S. S. Quek

Moments and shear forcesMoments and shear forces

Therefore, equilibrium of forces in z direction

0)()()(

ydxdwhfxdydy

Qydxd

x

Qz

yx

or

whfy

Q

x

Qz

yx

Qy

Qx

Mx+dMx

Qx+dQx My+dMy

My

Myx+dMyx

Mxy+dMxy

Qy+dQy Myx o

x

y

x

dy

dx

A A Moments about A-A

y

M

x

MQ xyx

x

45Finite Element Method by G. R. Liu and S. S. Quek

Dynamic and static equilibrium equationsDynamic and static equilibrium equations

y

M

x

MQ xyx

x

whfy

Q

x

Qz

yx

w

yx

y

xh

M

M

M

xy

y

x

12

2

2

2

2

2

3

c

46Finite Element Method by G. R. Liu and S. S. Quek

Dynamic and static equilibrium equationsDynamic and static equilibrium equations

zfwhy

w

yx

w

x

wD

)2(

4

4

22

4

4

4

zfy

w

yx

w

x

wD

)2(4

4

22

4

4

4

where

)1(12 2

3

EhD

(Static)

47Finite Element Method by G. R. Liu and S. S. Quek

Mindlin plateMindlin plate

Neutral surface

48Finite Element Method by G. R. Liu and S. S. Quek

Mindlin plateMindlin plate

yzu xzv ,

Therefore, in-plane strains = z L

where0

0

x

y

x y

L ,y

x

49Finite Element Method by G. R. Liu and S. S. Quek

Mindlin plateMindlin plate

Transverse shear strains

y

wx

w

x

y

yz

xz

γ

Transverse shear stress

0[ ]

0xz

syz

G

G

D