FEM: Nonlinear Beam Deflection Model (with Temperature)

FEM: Nonlinear Beam Deflection

Mohammad Tawfik #WikiCourses

http://WikiCourses.WikiSpaces.com

Introduction to the Finite

Element Method

Nonlinear Beam Deflection




Nonlinear Strain Displacement

Relation




Nonlinear Strain Displacement

Relation • The von Karman large deflection strain-

displacement relation for the deflections u,

and w can be written as follows

2

22

2

1

x

wz

x

w

x

ux

zm




Using the relations …

}{

xx1 32

aH

axw

w

}{

1

bH

bxu

u

bwbbw wNwTHw 1

mummu wNwTHu 1




For the derivatives …

bbbw wBwT

dx

dH

dx

dw

1

bbbbw wBwT

dx

Hd

dx

wd

1

2

2

2

2

mmmmu wBwT

dx

dH

dx

du

1




Giving the strain relation …

• Where:

• Is a scalar function of x

bbbmm

m

wzww

z

BBB

2

1

}{

x

w




Inplane Forces and Bending

Moments




Inplane Forces and Bending

Moments

• For homogeneous symmetric cross sections:

T

T

M

N

D

A

M

N

0

0

EAQhA EIQh

D 12

3

TEAdzzyxTQNh

hT

2/

2/),,(

0),,(2/

2/

h

hT zdzzyxTQM




Expanding equation

Tm

Tbmm

Tm

NNN

NwAwA

NAN

BB2

1

][

bb wDDM B][}]{[




The Potential Energy

Volume height

TT

ijij dzMNdVU }{}{

dz

wDw

NwAwA

ww

Uheight

bb

T

b

T

b

Tbmm

TTT

b

T

m

T

m

BB

BB

BB

][

2

1

*




Skipping the details …

• We get the element equation as:

m

b

mb

bmnm

TN

m

b

W

W

n

n

nn

kT

k

k

00

02

3

1

01

11

2

1

00

0

0

0

Tm

b

p

p 0

0




Where …

• The linear stiffness matreces

dzDkheight

b

T

bb BB ][][

dzAkheight

m

T

mm BB][




Where …

• The nonlinear stiffness matreces

dzAnnheight

T

m

T

bmmb BB]1[]1[

dzBNnheight

m

T

nm B1

dzAnheight

TT

BB2

3]2[




Where …

• The thermal effect terms

dzBNkheight

T

T

TN B][

dzNpheight

T

T

mTm B




In a compact form …

TTN PPWNNKK

2

3

11

2

1




Note

• For details for the derivation, follow the link

on the pages: • https://wikicourses.wikispaces.com/Topic+Nonlinear+Solid+Mechani

cs

• https://eau-esa.wikispaces.com/Topic+Nonlinear+Solid+Mechanics

https://eau-esa.wikispaces.com/Topic+Nonlinear+Solid+Mechanics

https://wikicourses.wikispaces.com/Topic+Nonlinear+Solid+Mechanics

https://wikicourses.wikispaces.com/Topic+Nonlinear+Solid+Mechanics










Solving the Equations for

Post-Buckling Deflection




When no external loading exists

• We may write the equations in the form:

• Introducing a new function:

TTN PWNNKK

2

3

11

2

1

023

11

2

1

TTN PWNNKKW




Using truncated Taylor

expansion • Linear approximation:

• Where:

WdW

WdWWW

tan21 KNNKK

dW

WdTN




Using Newton Algorithm:

Repeat, until:

TiiiTNi PWNNKKW

2

3

11

2

1

iiiWWK 1tan

ii WKWi

1

tan1

11 iii WWW

toliW 1max




Maximum deflection vs. Temp.

Education

FEM: Nonlinear Beam Deflection Model (with Temperature)