CE 221: MECHANICS OF SOLIDS I CHAPTER 10: BUCKLING OF COLUMNS · CE 221: MECHANICS OF SOLIDS I...
Transcript of CE 221: MECHANICS OF SOLIDS I CHAPTER 10: BUCKLING OF COLUMNS · CE 221: MECHANICS OF SOLIDS I...
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CE 221: MECHANICS OF SOLIDS I CHAPTER 10: BUCKLING OF COLUMNS By Dr. Krisada Chaiyasarn Department of Civil Engineering, Faculty of Engineering Thammasat university
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Outline • Critical load • Ideal column with pin supports • Columns having various types of supports
©2005 Pearson Education South Asia Pte Ltd
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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
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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
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Critical Load
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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
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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
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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
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Ideal Column with Pin Supports
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Ideal Column with Pin Supports
C1 cannot be obtained, since the exact deflection form is unknown once it has buckled
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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
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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
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Example
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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.
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Columns having various types of supports
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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
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Effective Length
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Example
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Example
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