8. 'short' vs. 'long' columns
Buckling behaviour - 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
stiffness and is short)
Conversely, the buckling
resistance is low when the
member is long or “slender”.
9. • Traditional design - based on Euler analysis of ideal columns -
an upper bound 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 the use of Perry Robertsons curves.
• Modern codes advocate the use of multiple-column curves
for design.
• Although these design procedures are more accurate in
predicting the buckling load of practical columns,
Euler's theory helps in understanding the behaviour of
slender columns
Compression members
10. Euler analysis –
ELASTIC BUCKLING OF AN IDEAL COLUMN OR STRUT WITH
PINNED END
Assumptions.
• The material of which the strut is made is
homogeneous and linearly elastic (i.e. it obeys
Hooke’s Law).
• The strut is perfectly straight and there are no
imperfections.
• The loading is applied at the centroid of the
cross section at the ends.
11. Initially, the strut will remain straight for all values of P,
but at a particular value P = Pcr, it buckles.
Euler buckling analysis
13. Euler buckling analysis
Buckling load Vs. central (lateral) deflection relationship
While there are several buckling modes each corresponding to n = 1, 2, 3…
the lowest stable buckling mode corresponds to n = 1.
14. Euler buckling analysis
• The lowest value of the critical load is given by Pcr = π2EI / L2
• The strut can remain straight for any value of P. Under incremental
loading, when P reaches a value of Pcr = π2EI / L2 the strut can buckle in
the shape of a half-sine wave; the amplitude of this buckling deflection is
indeterminate.
• At higher values of the loads given by n2π2EI / L2 other sinusoidal buckled
shapes (n half waves) are possible. However, the column will be in
unstable equilibrium for all values of P > π2EI / L2 whether it be straight or
buckled. This means that the slightest disturbance will cause the column to
deflect away from its original position.
• Elastic Instability - a condition in which the structure has no tendency to
return to its initial position when slightly disturbed, even when the
material is assumed to have an infinitely large yield stress.
Thus Pcr = π2EI / L2 represents the maximum load that the strut can
usefully support.
17. 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 exceed fy , shown by A-A1
A column would fail by buckling at a stress given by (π2 E / λ2). 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 λc
19. DESIGN OF AXIALLY LOADED
COLUMNS
The behaviour 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.
20. Effect of initial out-of-straightness
The column will fail at a lower load Pf when the deflection becomes
large enough. (Pf < Pcr ) The corresponding stress is denoted as ff
21. Theoretical and actual load-deflection response
of a strut with initial imperfection
For very stocky members, the initial out of straightness has a very negligible effect
and the failure is at plastic squash load.
For a very slender member, the lower bound curve is close to the elastic critical
stress (fcr) curve.
At intermediate values of slenderness the effect of initial out of straightness is very
marked and the lower bound curve is significantly below the fy line and fcr line.
22. Effect of eccentricity of applied loading
Strength curves for eccentrically loaded columns
load carrying capacity is reduced (for stocky members) even for low
values of λ .
23. 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 tips reduces 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
24. 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.
26. MODIFICATION TO THE PERRY-
ROBERTSON APPROACH
- very stocky columns (e.g. stub columns) resisted loads in excess of their
squash load of fy.A
- column strength values are lower than fy. even in very low slenderness
cases.
- by modifying the slenderness, λ to (λ - λ0) a plateau to the design curve at
low slenderness values is introduced.
30. Stipulations of IS: 800
• For various types of column cross sections including Indian Standard rolled steel
sections (as against Universal Column sections), CHS, SHS, RHS and Heavily Welded
sections, IS: 800 recommends the classifications following the column buckling
curves a, b, c and d
• Apart from types of column cross-sections, the respective geometric dimensions of
individual structural elements and their corresponding limits also guide Buckling
Class. For example, if the ratio of overall height to the overall width of the flange of
rolled I section i.e. h/b is greater than 1.2 and the thickness of flange is less than
40 mm, the buckling class corresponding to axis z-z will be guided by Curve a. This
shows for same slenderness ratio, more reserve strength is available for the case
described above in comparison to built-up welded sections. For various types of
column cross-sections, Table 10 of IS: 800 defines the buckling classes with respect
to their respective limits of height to width ratio, thickness of flange and the axis
about which the buckling takes place (to be determined based on the column
buckling curves as indicated in fig. corresponding to both major and minor axis)
33. • Only very short columns can be loaded upto
yield stress – basic mech. of materials
• For long columns buckling occurs prior to
developing full material strength
• Stability theory is necessary for designing
compression members
• Square and circular tubes – ideal sections – r is
same in the two axes
Compression members
34. 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
35. Imperfections
• The material not being isotropic and
homogeneous
• Geometric variations of columns
• Eccentricity of load
36. Possible failure modes
• Local Buckling
• Squashing
• Overall flexural buckling
• Torsional and flexural- torsional buckling
37. Local
Buckling
- Failure occurs by buckling of one or more individual plate
elements
- Flange or web, with no overall deflection in the direction
normal to the applied load
- Prevented by selecting suitable width to thickness ratios of
component plates
38.
39. 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
40. Overall flexural buckling
• This mode of failure
normally controls the
design of most comp.
members
• Failure occurs by excessive
deflection in the plane of
the weaker principal axis
• Increase in length – results
in column resisting
progressively less loads
42. Torsional & Flexural buckling
• A combination of flexural - torsional
buckling is also possible
43. Open sections
Singly symmetric and for section that have no symmetry –
flexural- torsional buckling must be checked
Sections always rotate about shear centre
Shear centre lies on the axis of symmetry
45. 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
46. Open Sections
• Open sections that are doubly symmetric or
point symmetric are not subjected to flexural
torsional buckling because their Shear centre
and centroid coincide
48. Built-up sections
• Failure of a component
member may occur, if
joints between
members are sparsely
placed
• Codes specify rules to
prevent such failures
49. Compression Members
Short Intermediate Long
Failure stress = yield stress
No buckling occurs
L < 88.85 r
for fy = 250 MPa
No practical applications
Some fibres would have
yielded & some will still be
elastic
Failure by both yielding
and buckling
Behaviour is inelastic
Euler’s formula predicts
the strength
Buckling stress below
proportional limit
Elastic buckling
Behaviour of Compression members
60. Conclusions
Discussions regarding the following were made
• Elastic buckling of an ideally straight column
pin ended at both ends
• Factors affecting the column strengths
• column strength curves
• Elastic Torsional and Torsional-flexural
buckling
• Design of columns using multiple column
curves
