bharath columns shearcentre
TRANSCRIPT
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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