CE 221: MECHANICS OF SOLIDS I CHAPTER 10: BUCKLING OF COLUMNS · CE 221: MECHANICS OF SOLIDS I CHAPTER 10: BUCKLING OF COLUMNS By Dr. Krisada Chaiyasarn Department of Civil Engineering,

CE 221: MECHANICS OF SOLIDS I CHAPTER 10: BUCKLING OF COLUMNS By Dr. Krisada Chaiyasarn Department of Civil Engineering, Faculty of Engineering Thammasat university

Outline • Critical load •  Ideal column with pin supports • Columns having various types of supports

©2005 Pearson Education South Asia Pte Ltd

Critical Load • A long slender members subjected to an axial

compressive forces are called columns •  The lateral deflection is called buckling • Buckling can lead to a sudden and dramatic failure •  https://www.youtube.com/watch?v=jNwvub87l8o

Critical Load •  The critical load Pcr is the maximum axial load when a column is on the verge

of buckling •  If the load is greater than Pcr, then the column will deform laterally •  To understand, the spring mechanism is used •  Spring with stiffness k, small vertical force P, displace by Δ, hence the spring

produces a restoring force F = kΔ, and the horizontal force Px = Ptanθ

Critical Load

Critical Load •  Pcr is independent of the displacement θ •  When the mechanism is in neutral position, the load

is less than Pcr •  In summary, there are three states,

•  P = Pcr, bifurcation point, mechanism is in equilibrium for small θ, this point the mechanism will not return to original, nor it will move further out

•  P > Pcr, buckle •  P < Pcr, Stable

•  Pcr is not the largest load the column can support, but the load greater than this value causes the column to deflect even larger

•  However, in engineering design, this is considered the largest as a large deflection is not tolerable in the design

Ideal Column with Pin Supports •  Ideal column is one that is

•  Perfectly straight before loading •  The load is applied through a centroid •  Assume linear-elastic behaviour •  Column buckles or bends in a single

plane •  Generally, the above assumptions are

never accomplished •  From the figure below, P can be increased

until failure, but it may reach Pcr first •  Once reaching Pcr, a small lateral force F

will cause the column to remain in the deflected position, if the axial load P reduces, the column will straighten out, and any increase will cause further lateral deflection

Ideal Column with Pin Supports •  A column will remain stable or become unstable depends on its resistance to

bending •  The proof below will result in a homogeneous, second-order linear differential

equations

Ideal Column with Pin Supports

Ideal Column with Pin Supports

C1 cannot be obtained, since the exact deflection form is unknown once it has buckled

Ideal Column with Pin Supports •  The critical load depends only on E and and column dimensions, I and L •  Hence, a column made of high strength steel offers no advantage over the

lower strength steel as E is the same •  I increase Pcr as well, hence efficient column have far cross-sectional area

from the centroid, i.e. hollow section •  The column will buckle about the principal axis with the least moment of

inertia

Ideal Column with Pin Supports •  For the design purpose, we use the radius of gyration, •  The geometric ratio L/r or the slenderness ratio is a

measure of column’s flexibility, classifying columns as long, intermediate or short

Example

Columns having various types of supports •  The Euler load is derived for the pin connected supports. •  For other supports, such as a fixed support, the load displace δ at x the

displacement is v.

Columns having various types of supports

Effective Length •  L in the equation represents the unsupported

distance between the points of zero moment, this is called the effective length, Le

•  Many design codes provide the column’s effective length, a dimensionless coefficient, K called effective-length factor

•  KL/r is the effective-slenderness ratio

Effective Length

Example

Example