Elastic Beam Model

8/10/2019 Elastic Beam Model

1/12

THE ELASTIC

BEAM MODEL


2/12

Mechanics of materials: beams / bars

2/12

Bar:body with an extension considerablelarger in oneof the three directions

it has a well-defined axisand a perpendicular cross section

Model of elastic beam: a series of differential! beam elements

! different from the bar support in tatics!

Beam element: ri"idpanels with distributed" elasticconnection

Basic assumptions:

# $lanar cross sections

# small dis$lacements deformations!

dz

z differential#length$: dz


3/12

The $rinci$le of $lanar cross sections

%/12

&f an undeformed" planar cross section undergoes deformation' it remains

# $lanar' and

# con"r%ent with its initial shape()ather#planar and ri"id&cross sections:no trans*erse contraction + ,"(

M M

'

'

Basic inds of deformation:

implification: it is accepted forall inds of deformations

( (

TT

stretching

bending

shearing

twisting


4/12

The $rinci$le of small dis$lacements

xx x

f x" +xf x" +x

f x" . 1

sinxxx

%

%!x

/

/!cosx1

x2

2 !

x0

0 ! tanxx

2x%

1/

0/12

cos 1 sin tan

aylor series of trigonometrical functions:

lcos

l

lsin ltan

l

l

l

l l


5/12

y

xz

State )ariables of a beam element

/12

3&435internal forces' stresses"

6&78M3&435deformations' strains"

Mx

y

xz

Vx

Vy

N

Myy

xz

y

xz

y

xz

y

xz

y

xz

T

y

z

y

x

x

z

dx

dy dz

dv

du dw

* Vx' Vy'N"29shear' 19normal"

+ Mx'My' T"29bending' 19twisting"

*

+

de

d,

% relati*e displacements"

d, dx' dy' dz"% relati*e rotations"

dz- dex' dey' dez"

dz

c o m p o n e n t s c o m p o n e n t s

de du' dv' dw"


6/12

/12

Def.:;AB= A B

this compared to this

C

;

BA

A

B

;AB

BA

;BA

C+ ,

relati)edispl(: difference of two absol%teones

absolute relati*e

displacements u' v' w ;u' ;v' ;wrotations x ' y ' z ;x';y';z

ex

' ey

' ez

;ex

'

;ey

'

;ezor

dz

zN z"

N


7/12

dzis drawn largerfor practical reasons

#elementary stripe$"

>/12

3&435internal forces' stresses"

6&78M3&435deformations' strains"

dz

z

w z"r z"

$z


8/12

E1%ations of the beam element

F/12

3&435 or 8G@&5&B)&@M set 6&78M3&435 or D8CM8)&435 set

2O*CESI(TE*(AL 2O*CES!

e(g(N"

H I

differentiation integration

H Ie(g( "

ST*ESSES

STATICALor

E34ILIB*I4MeJuations

5I(EMATICALor

6EOMET*ICALeJuations

DE2O*MATIO(S

e(g( u"

H I

differentiation integration

H Ie(g( !"

ST*AI(S

MATE*IAL or 7H8SICALeJuations

material constants:$' % + ,"


9/12

K/12

STATICAL/ E34ILIB*I4M eJuations 5I(EMATICAL / 6EOMET*ICAL eJuations

L&'x: Vx= zxx'y" dA

L&'y: Vy= zyx'y" dA

L&'z: N = zx'y" dA

LM'x: Mx= zx'y"ydA

LM'y: My= zx'y"xdA

LM'z: T= zxx'y"y

Documents

Elastic Beam Model