eccentric tension and compression
TRANSCRIPT
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ECCENTRICTENSION
AND
COMPRESSION
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ECCENTRIC TENSION AND COMPRESSION
definition
side panels of a beam element are subjectedto a perpendicular forcenot passing
through their centroid
M M
N N
F F
+
=
y
xz
Mx
My
N
F
F = (N, Mx, My)
eccentric tension / compression
bending + simple tension / compression
deformation of a beam element: all points move axially, see uniform strains (centric
tension / compression) or assumptions of Bernoulli and Navier (classical beam theory)/!
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z
yMxNF
C
"uperposition of stress distributions according to our
former studies (assume uniaxial bending aboutx) if
the material resists bot tension !nd compression#
$
%
zN z
M$
z
+
"
#
(applies also for biaxial bending)
if the material resists compression
onl$# (e&g& concrete, soil)
F
$
%%
%
F
%
z
'/!
alculation of stresses:N/Aand any of the methods
discussed at bending (general formula, methods of
superposition, neutral axis, ulmann*s +ernel)
###
compressive and noncompressive
parts of the cross section, other
approach is needed
ECCENTRIC TENSION AND COMPRESSION
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SECTIONS O% TENSION&COMPRESSION MATERIA'S
/!
Reminder( e-uations of simple tension / compression and simple bending
dwC
C Cdx
dy
+
=
+.Fiz: z(x,y,z) dA=N (z)
.Mix: z(x,y,z)ydA=Mx(z)
.Miy: z(x,y,z)xdA=My(z)
A(z)
A(z)
A(z)
)EOMETRIC e-uationsSTATICA' e-uations
dwS(z)
dz
$ etc&
z(x,y,z) = $ y% xdy(z)
dz
dx(z)
dz
z(x,y,z) = zS(z)$ x(z)y% y(z)x
simple tension / compression
simple bending
effects ofN and (Mx, My)can be calculated
independently
N (Mx, My)
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/!
0etermination of stresses by superposition
Cx 1 2
y1
u
%
' stress distributions can be dra3nseparately:
for the angle of uit still holds:
= arctanMyIx
MxIy
z
Mx=M2
My=M
$%
$
%
neutral axis bet3eenMand the nd principalaxis but udoes not p!ss tro*g te centroid4
signs rather from
inspection again:
5fxandy
are principal:
N($)
zC
= N
EA6 x=
Mx
EIx6 y=
My
EI y
Nz=
N
A
Mx
Ixy
My
Iyx
N
$
$ z=
N
A
Mx
Ixy
My
Iyx
N
SECTIONS O% TENSION&COMPRESSION MATERIA'S
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7/!
8xample:
determine minimum and maximum normal stresses of the section in eccentric compression
M= 9!!,!' =,+Nm
I = 2'7cm)
I = 7)cm)
C = )9cm)
cm
7cm
' cm
Ix = I2 = 27! cm)
I y = I = )!cm)
! = 7,/;o =arctg!,/
y 1
x 1 2
!
it is already +no3n&&&
y 1
x 1 2
!
A
!
C
Mx= ,277+Nm
My
= 2,!;''+Nm
2,;99< ',2'!/
zA
=
9!
)
2),77
27! 2,;99
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%
$
$
%
C
M= ,+Nm
y 1
x 1 2
!
A
Mx= ,277+Nm
My= 2,!
;''+Nm
z 9,!![ +Ncm ]9,!!
7,!!!
;,!!;/!
%
$
$
$
SECTIONS O% TENSION&COMPRESSION MATERIA'S
biaxial bending part:
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%
$
$
%
S
M= ,+Nm
y 1
x 1 2
!
A
Mx= ,277+Nm
My= 2,!;''+Nm
z 9,!![ +Ncm ]9,!!
7,!!!
;,!!
'7,9;!
9/!
N= 9! +N(%)
%','''
%
%
$
$
$
SECTIONS O% TENSION&COMPRESSION MATERIA'S
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?
y 1
C
Fcentric (e
= !) >F1N:
uniform stress distribution,
(as if u3as at infinity)
Fat isnfinity (e> ?):
ifM=Fe is given,N=F> !
yieldsMonly> linear stress
distribution, upasses through C
M
generic case ###
NandM> linear
stress distribution
M
N(%)
z z z
SECTIONS O% TENSION&COMPRESSION MATERIA'S
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2!/!
x 1 2
%
u
F(%)
e
y 1
"pecial position:
there is#ustno stress 3ith
sign opposite toF
(uis just tangent to the cross
section, no intersection occurs)
@
e= # (point of appl& ofF#)M
N(%)
0efinition:
ACDNN*" E8FN8:
ocii of points of application ofF(axial) in the cross section, for 3hich the neutral axis u
is tangent to (not intersecting) the boundary of the cross section&
Fis located just at the
boundary of ulmann*s +ernel
assume thatFis compressive,
but could be (FG !) as 3ell
z
SECTIONS O% TENSION&COMPRESSION MATERIA'S
8ccentricity of the load and position of the neutral axis
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8-uivalent definition (3ithout proof):
ACDNN*" E8FN8:
ocii of points of application ofF(axial) in the cross section, for 3hich the neutral axis u
is tangent to (not intersecting) the boundary of the cross section&
ACDNN*" E8FN8:
5nner envelope of neutral axes pertaining to axial loadsFlocated at the convex hull of the
cross section (that is, ulmann*s +ernel is convex)&
corollary: ifFis located at the contour of the (convex hull of) the cross section, uis tangent to
ulmann*s +ernel !nd ,ice ,ers!
let us analyse the increasing eccentricity ofF:@
(there also exists a pure geometric definition H not considered here&&&)
SECTIONS O% TENSION&COMPRESSION MATERIA'S
8ccentricity of the load and position of the neutral axis
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Iracing the contour of ulmann*s +ernel
2'/!
x 1 2u
F
y 1
general case (using the method of superposition)
ey= -z=!=
N
A
Mx
Ixy
My
Iyx
Na
$
starting from the e-uation of u:
N
A=
Mx
Ixy
My
Iyx
N
N
A=
Mx
Ixy
My
Iyx
F Fey Fex
2=A e
y
Ixy
Aex
Iyx
e-uation of uas a line
3ith axial segments a% $(here a% $J !):
2= 2$
y2a
x
Mx=Fey
My=%Fex
ey=Ix
A$=
ix
$6 ex=
Iy
Aa=
iy
aix =
Ix
A6 iy=
Iy
A: radius of gyration
ex= -
SECTIONS O% TENSION&COMPRESSION MATERIA'S
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2/!
x 1 2
u
F
y 1
ey= -
5fFis at one of the principal axes (): >
MKK (2): uniaxial bending >
uis also parallel to (2)
8xample: determine contour points of ulmann*s +ernel alongyof a rectangular section
s
"
$
z
assume thatFG!:
(ifF3as compressive: $ L %)
z=!=N
A
Nex
Ixy
F Fey
ey = Ix
A"/ =
s"'
2s"
" =
"
7
brea+point > line,
curve > curve,
line > brea+point
SECTIONS O% TENSION&COMPRESSION MATERIA'S
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2/!
5mportance of ulmann*s +ernel:
in the analysis of no tension materials
ulmann*s +ernel for simple sections
a
$
a &
&/
a/a/'
$/'
SECTIONS O% TENSION&COMPRESSION MATERIA'S
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27/!
Ddditional assumptions:
% symmetric cross section and compressi,e forcelocated at the axis of symmetry% principle of plane cross sections still applies but Moo+e*s la3 holds just in part&&
IO PO""5B8 D"8":
a) the force is located inside ulmann*s +ernel, then only compressive stresses occur irrespective of the resistance to tension >
"AP8FPO"5I5ON "MOA0 "I5 B8 DPP580
b) the force is located inside the convex hull of the section but outside the
+ernel: compressive (zQ !) and noncompressive (z= !) parts >
CO05R580 8SA55BF5AC ON05I5ON" DF8 N88080
@
SECTIONS O% NO TENSION MATERIA'S
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u
2;/!
)EOMETRIAI e-uationsSTATICA' e-uations
dw(y') =y'dxy'
x'z
.Fiz: z(y'%z) dA=F (z) J !
.Mix': z(y',z)y'dA=yFF (z)
A(
A(
$ rig& c&s&:x=y=!
)xy=!
z(y',z) = y'dx(z)
dz
z(y',z) = x(z)y'
)zx
= !, )zy
= !
u
z(y',z) =Ez(y',z), if z(y',z) J !,
z(y',z) = !, if z(y',z) T !
MATERIA' e-uations! ! !
! ! !
! ! z
.=
! ! !
! ! !
! ! z
=
dxy'
F
F= (N, Mx
) x'at the neutral axis(not through C4)A(
*: +no3n
yF: sought for
yFFQMx
0EAM E'EMENTS O% NO TENSION MATERIA'
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29/!
)EOMETRIC e-uationsSTATICA' e-uations
$ etc&
Rirstly, "IDI&: .MixU .Fiz
yFz=A(
zy ' % zy 'dA
A
(
zy ' % zdAU CDI $ V8OC
yF
z =
xEA(
y 'dA
xEA
(
y 'dA=
Ix 'z
Sx 'z
.Fiz: z(y'%z) dA=F (z) J !
.Mix': z(y',z)y'dA=yFF (z)A(A(
z(y',z) = y'd
x(z)
dz
z(y',z) = x(z)y'
z(y',z) =Ez(y',z), if z(y',z) J !,z(y',z) = !, if z(y',z) T !
MATERIA' e-uations
moment e-uilibrium (line ofF 1 resultant of z(y',z))
implicite condition foryF
.Fizxz=
Fz
ESx 'zF
z+y% "ax
z= Fz
Sx 'zy '"ax z
F
(better 3ith absolute values&&&)
0EAM E'EMENTS O% NO TENSION MATERIA'
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Criteri!
'!&Bending and tension or compression& 8ccentric tension or compression&
'2&Bending and tension or compression for bars of tensioncompression materials&
ulmanYs (internal) +ernel& 0efinition of neutral axis&'&"tresses in cross section 3ith eccentric compressive load in axis of symmetry
3ith notension material&
Oter e1!m 2*estions
;2&0etermine the stresses for eccentric tension/compression by using the general
method (method of neutral axis) of biaxial bending4 Ihe material behaviour is the
same for tension and compression& 0ra3 the stress diagrams&
;&0etermine the stresses for eccentric tension/compression by using the
superposition method of s+e3 bending4 Ihe material behaviour is the same for
tension and compression&;'&
;&alculate the stresses in a cross section loaded symmetrically by an eccentric
compressive force, if the material cannot resist tension&!/!