elastic beam model
TRANSCRIPT
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THE ELASTIC
BEAM MODEL
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Mechanics of materials: beams / bars
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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
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The $rinci$le of $lanar cross sections
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&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
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The $rinci$le of small dis$lacements
xx x
f x" +xf x" +x
f x" . 1
sinxxx
%
%!x
/
/!cosx1
x2
2 !
x0
0 ! tanxx
2x%
1/
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cos 1 sin tan
aylor series of trigonometrical functions:
lcos
l
lsin ltan
l
l
l
l l
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y
xz
State )ariables of a beam element
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3&435internal forces' stresses"
6&78M3&435deformations' strains"
Mx
y
xz
Vx
Vy
N
Myy
xz
y
xz
y
xz
y
xz
y
xz
T
y
z
y
x
x
z
dx
dy dz
dv
du dw
* Vx' Vy'N"29shear' 19normal"
+ Mx'My' T"29bending' 19twisting"
*
+
de
d,
% relati*e displacements"
d, dx' dy' dz"% relati*e rotations"
dz- dex' dey' dez"
dz
c o m p o n e n t s c o m p o n e n t s
de du' dv' dw"
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Def.:;AB= A B
this compared to this
C
;
BA
A
B
;AB
BA
;BA
C+ ,
relati)edispl(: difference of two absol%teones
absolute relati*e
displacements u' v' w ;u' ;v' ;wrotations x ' y ' z ;x';y';z
ex
' ey
' ez
;ex
'
;ey
'
;ezor
dz
zN z"
N
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dzis drawn largerfor practical reasons
#elementary stripe$"
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3&435internal forces' stresses"
6&78M3&435deformations' strains"
dz
z
w z"r z"
$z
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E1%ations of the beam element
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3&435 or 8G@&5&B)&@M set 6&78M3&435 or D8CM8)&435 set
2O*CESI(TE*(AL 2O*CES!
e(g(N"
H I
differentiation integration
H Ie(g( "
ST*ESSES
STATICALor
E34ILIB*I4MeJuations
5I(EMATICALor
6EOMET*ICALeJuations
DE2O*MATIO(S
e(g( u"
H I
differentiation integration
H Ie(g( !"
ST*AI(S
MATE*IAL or 7H8SICALeJuations
material constants:$' % + ,"
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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