Eccentric Tension and Compression

8/10/2019 Eccentric Tension and Compression

1/20

ECCENTRICTENSION

AND

COMPRESSION


2/20

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)/!


3/20

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)


5/20

/!

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



6/20

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


7/20

%

$

$

%

C

M= ,+Nm

y 1

x 1 2

!

A

Mx= ,277+Nm

My= 2,!

;''+Nm

z 9,!![ +Ncm ]9,!!

7,!!!

;,!!;/!

%

$

$

$


biaxial bending part:


8/20

%

$

$

%

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(%)

%','''

%

%

$

$

$



9/20

?

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



10/20

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


8ccentricity of the load and position of the neutral axis


11/20

22/!

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&&&)


8ccentricity of the load and position of the neutral axis


12/20


13/20

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= -



14/20

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


""/7

s/7


15/20

2/!

5mportance of ulmann*s +ernel:

in the analysis of no tension materials

ulmann*s +ernel for simple sections

a

$

a &

&/

a/a/'

$/'



16/20

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


17/20

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'


18/20

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'

-

-


19/20

2


20/20

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&!/!

Eccentric Tension and Compression

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