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:


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


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


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.


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.


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.


Boundary conditionsBoundary conditions

Displacement (essential) boundary conditions

Force (natural) boundary conditions


Different structural componentsDifferent structural components

Truss and beam structures

x

fy1

z

fy2 y

x

fx

z

Truss member Beam member


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


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


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



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

,

,

,

,



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


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

.



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

,

,


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



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



Hence, equilibrium equation in x direction

ufzyx xzxyxxx

Equilibrium equations in y and z directions

vfzyx yzyyyxy

wfzyx zzzyzxz


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


EQUATIONS FOR 2D SOLIDSEQUATIONS FOR 2D SOLIDS

Plane stressPlane strain

x

y

z

y

x



Txx yy zz yz xz xy

xx

yy

xy

xx

yy

xy

x

v

y

u

y

v

x

uxyyyxx

,,

(3D)




ε LU

where

0

0

x

y

y x

L, u

v

U



= 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)



ufzyx xzxyxxx

ufyx xyxxx

vfyx yyyxy

(3D)



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


EQUATIONS FOR TRUSS EQUATIONS FOR TRUSS MEMBERSMEMBERS

fx

y

x

z

xy

yy

xx

σ xx

x

ux



Hooke’s law in 1D

= E


ufx x

x

0

xx f

x

(Static)


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

x

fy1

y

fy2



Euler–Bernoulli theory

Centroidal axis

x

y



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



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


Moments and shear forcesMoments and shear forces

The stress and moment

M M

dx

y

x

xx


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



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



vAxfdx

dQy

Therefore,

yz fvAx

vEI

4

4

yz fx

vEI

4

4

(Static)


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

z, w

h

fzy, v

x, u



Thin plate theory or Classical Plate Theory (CPT)

Centroidal axis

x

y



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

,




= z Lw

where 2

2

2

2

2

x

y

x y

L



= 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



Stresses on isolated plate cellz

x

y

fz

h

xyxxxz

yx

yy

yz

O



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



= 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

,



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



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



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)


Mindlin plateMindlin plate

Neutral surface



yzu xzv ,

Therefore, in-plane strains = z L

where0

0

x

y

x y

L ,y

x



Transverse shear strains

y

wx

w

x

y

yz

xz

γ

Transverse shear stress

0[ ]

0xz

syz

G

G

D

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.

Documents

dynamic equations

d slide

quek chapter

d solids equations

constitutive equations

beams equations

structural mechanics

constitutive equations