Bharath Columns ShearCentre

8/12/2019 Bharath Columns ShearCentre

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COMPRESSION MEMBERS


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KEY OBJECTIVES

History

Introduction-Compression members

Elastic buckling of an ideal column

Strength of practical column Concepts of effective lengths

Torsional and torsional-flexural buckling


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HISTORY

LEONARD EULER, the most prolific mathematician introduced the term

bucklingand derived the formula for it and popularly known as EulersBuckling Formula or EULERS FORMULA.

Later JOSEPH-LOUIS-LAGRANGE, mathematician developed a complete

set of buckling loads and the associated buckling modes.

Columns with eccentric loads and columns with initial curvatures were first

formulated and. studied by THOMAS YOUNG.

ANATOLE HENRI ERNEST LAMARLE, a French engineer, proposed

correctly that the Euler formula should be used below the proportional limit,

while experimentally determined formulas should be used for shorter columns.

F. ENGESSER, a German engineer, proposed the tangent modulus theory, in

which the elastic modulus is replaced by the tangent modulus of elasticity when

proportional stress is exceeded i.e. upto yield stress in which tangent modulus

is replaced by reduced modulus of elasticity or double modulus of elasticity.

The Euler buckling formula is still used from past three centuries, later for

column design and is still valid for long columns with pin-supported ends

Thats the power of Eulers Logical Thinking.


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INTRODUCTION

Compression Members

Compression members are a type of axially loaded member in which the

external forces are working to make the object shorter.

Applications are

Columns in Building Columns supports Compression Members in

Bridges


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INTRODUCTION

Compression Members in Trusses-Struts

Compression Members in Towers


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INTRODUCTION

A long column fails

by predominant buckling

A short column fails by

compression yield

Buckled shape

Fig 1: short vs long columns

Buckling behavior - large deformations

developed in a direction normal to that of the

loading that produces it.

The buckling resistance is high when the

member is short or stocky (i.e. the member

has a high bending sti ff ness and is shor t)

Conversely, the buckl ing resistance is low

when the member is long or slender.


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Traditional design - based on Euler analysis of ideal columns - an upperbound to the buckling load.

Practical columns are far from ideal & buckle at much lower loads.

The first significant step in the design procedures for such columns was theuse of Perry Robertsons curves.

Modern codes advocate the use of multiple-column curves for design.

Although these design procedures are more accurate in predicting thebuckling load of practical columns,

Euler 's theory helps in understanding the behaviour of slender columns

Only very short columns can be loaded upto yield stressbasic mech. ofmaterials

For long columns buckling occurs prior to developing full material strength Stability theory is necessary for designing compression members

Square and circular tubesideal sectionsradius r is same in the twoaxes

Compression members


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Initially, the strut will remain straight for all

values of P, but at a particular value P = Pcr, it

buckles.

Euler buckling analysis


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Euler buckling analysis


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Euler buckling

S h f i ll l d d i i i ll


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Strength curve for an axially loaded initially

straight pin-ended column

A strut under compression can resist only a max. force given by fy

.A, when plastic squashing

failure would occur by the plastic yielding of the entire cross section; this means that the

stress at failure of a column can never exceedfy, shown by A-A1

A column would fail by buckling at a stress given by (2E / 2). This is indicated by B-B1.

The changeover from yielding to buckling failure occurs at the point C, defined by a

slenderness ratio given by c


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DESIGNOF AXIALLY LOADED COLUMNS

The behavior of practical columns subjected to axial compressive loading: Very short columns subjected to axial compression fail by yielding. Very long

columns fail by buckling in the Euler mode.

Practical columns generally fail by inelastic buckling and do not conform to the

assumptions made in Euler theory. They do not normally remain linearly elastic

upto failure unless they are very slender Slenderness ratio (L/r ) and material yield stress (fy) are dominant factors

affecting the ultimate strengths of axially loaded columns.

The compressive strengths of practical columns are significantly affected by (i)

the initial imperfection (ii) eccentricity of loading

(iii) residual stresses and (iv) lack of defined yield point and strain hardening.


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Effect of initial out-of-straightness

The column will fail at a lower load Pfwhen the deflection becomes

large enough. (Pf < Pcr ) The corresponding stress is denoted as ff


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Effect of eccentricity of applied loading

Strength curves for eccentr ical ly loaded columns

Load carrying capacity is reduced (for stocky members) even for low values

of .


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Effect of residual stress

As a consequence of the differential heating and cooling in the rolling and forming

processes, there will always be inherent residual stresses.

Only in a very stocky column (i.e. one with a very low slenderness) the residual

stress causes premature yielding

For struts having intermediate slenderness, the premature yielding at the tipsreduces the effective bending stiffness of the column; in this case, the column will

buckle elastically at a load below the elastic critical load and the plastic squash

load.

Distribution of residual stresses


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Typical column design curve

Ultimate load tests on practical columns reveal a scatter band of results shown in

Fig. 1.

A lower bound curve of the type shown therein can be employed for design

purposes.


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Robertsons Design Curve


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MODIFICATION TO THE PERRY-

ROBERTSON APPROACH

- very stocky columns (e.g. stub columns) resisted loads in excess of their squash

load offy.A

- column strength values are lower than fy. even in very low slenderness cases.

-by modifying the slenderness, to ( -0) a plateau to the design curve at low

slenderness values is introduced.


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Buckling

class of

crosssections


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Different

column c/s

shapes

Simple

Compression

Members


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Different

column c/s

shapes

Built-up Columns


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Depends on

Material of the column

c/s configuration

Length of the column

Support conditions at the ends

Residual stresses

Imperfections

Strength of a column

Imperfections

The material not being isotropic and

homogeneous

Geometric variations of columns

Eccentricity of load

Possible failure modes Local Buckling

Squashing

Overall flexural buckling

Torsional and flexural- torsional

buckling


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Squashing

When length is small (stocky column)no local buckling

the column will be able to attain its full strength or squash load

Squash load = yield stress x area of c/s


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Overall flexural buckling

This mode of failure normally

controls the design of most

comp. members

Failure occurs by excessivedeflection in the plane of the

weaker principal axis

Increase in lengthresults in

column resisting progressivelyless loads


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Torsional buckling

Torsional buckling occurs

by twisting about the shearcentre in the longitudinal

axis


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Torsional & Flexural buckling

A combination of flexural - torsional

buckling is also possible


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Open sections

Singly symmetric and for section that have no symmetryflexural-

torsional buckling must be checked

Sections always rotate about shear centre

Shear centre lies on the axis of symmetry

Open sections that are doubly symmetric or point symmetric are not

subjected to flexural torsional buckling because their Shear centre and

centroid coincide


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Open sections


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Shear Centre

The shear

center(also

known as the

elastic axis or

torsional axis)is an imaginary

point on a

section, where

a shear force

can be applied

without

inducing any

torsion


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Compression MembersShort Intermediate Long

Failure stress = yield stress

No buckling occurs

L < 88.85 r

for fy = 250 MPa

No practical applications

Some fibers would have

yielded & some will still be

elastic

Failure by both yielding

and buckling

Behavior is inelastic

Eulers formula predicts

the strength

Buckling stress below

proportional limit

Elastic buckling

Behavior of Compression members


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Elastic bucklingElastic (Euler) Buckling


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Inelastic Buckling


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Slenderness Ratio


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Actual Length

Effecti e Length


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Effective Length


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Appropriate Radius of Gyration


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Design Compressive Stress and Strength

Bharath Columns ShearCentre

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