9. Buckling of Columns 9. Buckling of Columns CHAPTER OBJECTIVES • Discuss the behavior of columns. • Discuss the buckling of columns. • Determine the axial load needed to buckle an ideal column. Chapter 7: Buckling ©2005 Pearson Education South Asia Pte Ltd 1 ©2005 Pearson Education South Asia Pte Ltd 9. Buckling of Columns 9. Buckling of Columns CHAPTER OUTLINE Introduction 1. Critical Load 2. Ideal Column with Pin Supports 3. Columns Having Various Types of Supports 2 • Buckling is a failure mode characterized by a sudden failure of a structural member subjected to high compressive stresses, where the actual compressive stresses at failure are smaller than the ultimate compressive stresses that the material is capable of withstanding. • This mode of failure is also described as failure due to elastic instability. ©2005 Pearson Education South Asia Pte Ltd 3 ©2005 Pearson Education South Asia Pte Ltd 4 9. Buckling of Columns 9. Buckling of Columns 7.1 CRITICAL LOAD 7.1 CRITICAL LOAD • Long slender members subjected to axial compressive force are called columns. • The lateral deflection that occurs is called buckling. • The maximum axial load a column can support when it is on the verge of buckling is called the critical load, Pcr. ©2005 Pearson Education South Asia Pte Ltd • Spring develops restoring force F = k∆, while applied load P develops two horizontal components, Px = P tan θ, which tends to push the pin further out of equilibrium. • Since θ is small, ∆ = θ(L/2) and tan θ ≈ θ. • Thus, restoring spring force becomes F = kθL/2, and disturbing force is 2Px = 2Pθ. 5 ©2005 Pearson Education South Asia Pte Ltd 9. Buckling of Columns 9. Buckling of Columns 7.1 CRITICAL LOAD 7.2 IDEAL COLUMN WITH PIN SUPPORTS • For kθL/2 > 2Pθ, • An ideal column is perfectly straight before loading, made of homogeneous material, and upon which the load is applied through the centroid of the xsection. • We also assume that the material behaves in a linear-elastic manner and the column buckles or bends in a single plane. kL stable equilibrium 4 • For kθL/2 < 2Pθ, P< P> kL 4 6 unstable equilibrium • For kθL/2 = 2Pθ, kL Pcr = neutral equilibrium 4 ©2005 Pearson Education South Asia Pte Ltd 7 ©2005 Pearson Education South Asia Pte Ltd 8 9. Buckling of Columns 9. Buckling of Columns 7.2 IDEAL COLUMN WITH PIN SUPPORTS 7.2 IDEAL COLUMN WITH PIN SUPPORTS • Summing moments, M = −Pν, Eqn 13-1 becomes d 2υ ⎛ P ⎞ (13 - 2) + ⎟υ = 0 2 ⎜ • In order to determine the critical load and buckled shape of column, we apply Eqn 12-10, d 2υ (13 - 1) EI 2 = M dx • Recall that this eqn assume the slope of the elastic curve is small and deflections occur only in bending. We assume that the material behaves in a linear-elastic manner and the column buckles or bends in a single plane. dx • General solution is ⎛ P ⎞ ⎛ P ⎞ υ = C1 sin ⎜ x ⎟ + C2 cos⎜ x⎟ ⎝ EI ⎠ ⎝ EI ⎠ (13 - 3) • Since ν = 0 at x = 0, then C2 = 0. Since ν = 0 at x = L, then 9 ©2005 Pearson Education South Asia Pte Ltd ⎝ EI ⎠ ⎛ P ⎞ C1 sin ⎜ L⎟ = 0 ⎝ EI ⎠ ©2005 Pearson Education South Asia Pte Ltd 9. Buckling of Columns 9. Buckling of Columns 7.2 IDEAL COLUMN WITH PIN SUPPORTS 7.2 IDEAL COLUMN WITH PIN SUPPORTS • Smallest value of P is obtained for n = 1, so critical load for column is π 2 EI Pcr = 4L2 • This load is also referred to as the Euler load. The corresponding buckled shape is defined by πx υ = C1 sin L • Disregarding trivial soln for C1 = 0, we get ⎛ P ⎞ sin ⎜ L⎟ = 0 ⎝ EI ⎠ • Which is satisfied if P L = nπ EI • or P= ©2005 Pearson Education South Asia Pte Ltd n 2π 2 EI L2 10 n = 1,2,3,... • C1 represents maximum deflection, νmax, which occurs at midpoint of the column. 11 ©2005 Pearson Education South Asia Pte Ltd 12 9. Buckling of Columns 9. Buckling of Columns 7.2 IDEAL COLUMN WITH PIN SUPPORTS 7.2 IDEAL COLUMN WITH PIN SUPPORTS • A column will buckle about the principal axis of the x-section having the least moment of inertia (weakest axis). • For example, the meter stick shown will buckle about the a-a axis and not the b-b axis. • Thus, circular tubes made excellent columns, and square tube or those shapes having Ix ≈ Iy are selected for columns. ©2005 Pearson Education South Asia Pte Ltd • Buckling eqn for a pin-supported long slender column, π 2 EI (13 - 5) Pcr = 2 L Pcr = critical or maximum axial load on column just before it begins to buckle. This load must not cause the stress in column to exceed proportional limit. E = modulus of elasticity of material I = Least modulus of inertia for column’s x-sectional area. L = unsupported length of pinned-end columns. 13 ©2005 Pearson Education South Asia Pte Ltd 9. Buckling of Columns 9. Buckling of Columns 7.2 IDEAL COLUMN WITH PIN SUPPORTS 7.2 IDEAL COLUMN WITH PIN SUPPORTS • Expressing I = Ar2 where A is x-sectional area of column and r is the radius of gyration of x-sectional π 2E area. (13 - 6) σ cr = 2 (L r ) σcr = critical stress, an average stress in column just before the column buckles. This stress is an elastic stress and therefore σcr ≤ σY E = modulus of elasticity of material L = unsupported length of pinned-end columns. r = smallest radius of gyration of column, determined from r = √(I/A), where I is least moment of inertia of column’s x-sectional area A. ©2005 Pearson Education South Asia Pte Ltd 14 • The geometric ratio L/r in Eqn 13-6 is known as the slenderness ratio. • It is a measure of the column’s flexibility and will be used to classify columns as long, intermediate or short. 15 ©2005 Pearson Education South Asia Pte Ltd 16 9. Buckling of Columns 9. Buckling of Columns 7.2 IDEAL COLUMN WITH PIN SUPPORTS 7.2 IDEAL COLUMN WITH PIN SUPPORTS IMPORTANT • Columns are long slender members that are subjected to axial loads. • Critical load is the maximum axial load that a column can support when it is on the verge of buckling. • This loading represents a case of neutral equilibrium. ©2005 Pearson Education South Asia Pte Ltd IMPORTANT • An ideal column is initially perfectly straight, made of homogeneous material, and the load is applied through the centroid of the x-section. • A pin-connected column will buckle about the principal axis of the x-section having the least moment of intertia. • The slenderness ratio L/r, where r is the smallest radius of gyration of x-section. Buckling will occur about the axis where this ratio gives the greatest value. 17 18 ©2005 Pearson Education South Asia Pte Ltd 9. Buckling of Columns 9. Buckling of Columns EXAMPLE 13.1 EXAMPLE 13.1 (SOLN) A 7.2-m long A-36 steel tube having the x-section shown is to be used a pin-ended column. Determine the maximum allowable axial load the column can support so that it does not buckle. Use Eqn 13-5 to obtain critical load with Est = 200 GPa. Pcr = π 2EI L2 = π 2 [200(106 ) kN/m2 ]⎢ π (75)4 − π (70)4 ⎥(1 m/1000mm)4 ⎡1 ⎣4 1 4 (7.2 m)2 ⎤ ⎦ = 228.2 kN ©2005 Pearson Education South Asia Pte Ltd 19 ©2005 Pearson Education South Asia Pte Ltd 20 9. Buckling of Columns 9. Buckling of Columns EXAMPLE 13.1 (SOLN) EXAMPLE 13.2 The A-36 steel W200×46 member shown is to be used as a pin-connected column. Determine the largest axial load it can support before it either begins to buckle or the steel yields. This force creates an average compressive stress in the column of σ cr = Pcr 228.2 kN(1000 N/kN ) = A π (75)2 − π (70)2 mm2 [ ] = 100.2 N/mm2 = 100 MPa Since σcr < σY = 250 MPa, application of Euler’s eqn is appropriate. 21 ©2005 Pearson Education South Asia Pte Ltd 22 ©2005 Pearson Education South Asia Pte Ltd 9. Buckling of Columns 9. Buckling of Columns EXAMPLE 13.2 (SOLN) EXAMPLE 13.2 (SOLN) From table in Appendix B, column’s x-sectional area and moments of inertia are A = 5890 mm2, Ix = 45.5×106 mm4,and Iy = 15.3×106 mm4. By inspection, buckling will occur about the y-y axis. Applying Eqn 13-5, we have Pcr = When fully loaded, average compressive stress in column is P 1887.6 kN(1000 N/kN ) σ cr = cr = A 5890 mm2 = 320.5 N/mm2 Since this stress exceeds yield stress (250 N/mm2), the load P is determined from simple compression: P 250 N/mm2 = 5890 mm2 P = 1472.5 kN π 2 EI L2 = π 2 [200(106 ) kN/m 2 ](15.3(10 4 ) mm 4 )(1 m / 1000 mm )4 (4 m )2 = 1887.6 kN ©2005 Pearson Education South Asia Pte Ltd 23 ©2005 Pearson Education South Asia Pte Ltd 24 9. Buckling of Columns 9. Buckling of Columns 7.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS 7.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS • From free-body diagram, M = P(δ − ν). • Differential eqn for the deflection curve is d 2υ • Thus, smallest critical load occurs when n = 1, so that π 2EI (13-9) Pcr = 2 L • By comparing with Eqn 13-5, a column fixedsupported at its base will carry only one-fourth the critical load applied to a pin-supported column. P P (13 - 7 ) = υ δ EI dx 2 EI • Solving by using boundary conditions and integration, we get + ⎡ P ⎞⎤ x ⎟⎥ ⎝ EI ⎠⎦ ⎛ υ = δ ⎢1 − cos⎜ ⎣ (13 - 8) ©2005 Pearson Education South Asia Pte Ltd 25 ©2005 Pearson Education South Asia Pte Ltd 9. Buckling of Columns 9. Buckling of Columns 7.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS 7.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS Effective length • If a column is not supported by pinned-ends, then Euler’s formula can also be used to determine the critical load. • “L” must then represent the distance between the zero-moment points. • This distance is called the columns’ effective length, Le. ©2005 Pearson Education South Asia Pte Ltd 26 Effective length 27 ©2005 Pearson Education South Asia Pte Ltd 28 9. Buckling of Columns 9. Buckling of Columns 7.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS 7.3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS Effective length • Many design codes provide column formulae that use a dimensionless coefficient K, known as the effective-length factor. (13 - 10) Le = KL Effective length • Here (KL/r) is the column’s effective-slenderness ratio. Factor of safety ; • Thus, Euler’s formula can be expressed as Pcr = σ cr = π 2 EI ( KL )2 π 2E (KL r )2 (13 - 11) • Pcr = critical or maximum axial load on column just before it begins to buckle. • P = largest allowable axial load on column (13 - 12) ©2005 Pearson Education South Asia Pte Ltd 29 ©2005 Pearson Education South Asia Pte Ltd 9. Buckling of Columns 9. Buckling of Columns EXAMPLE 13.3 EXAMPLE 13.3 (SOLN) A W150×24 steel column is 8 m long and is fixed at its ends as shown. Its load-carrying capacity is increased by bracing it about the y-y axis using struts that are assumed to be pin-connected to its mid-height. Determine the load it can support that the column does not buckle or material exceed the yield stress. Take Est = 200 GPa and σY = 410 MPa. ©2005 Pearson Education South Asia Pte Ltd 30 Buckling behavior is different about the x and y axes due to bracing. Buckled shape for each case is shown. The effective length for buckling about the x-x axis is (KL)x = 0.5(8 m) = 4 m. For buckling about the y-y axis, (KL)y = 0.7(8 m/2) = 2.8 m. We get Ix = 13.4×106 mm4 and Iy = 1.83×106 mm4 from Appendix B. 31 ©2005 Pearson Education South Asia Pte Ltd 32 9. Buckling of Columns 9. Buckling of Columns EXAMPLE 13.3 (SOLN) EXAMPLE 13.3 (SOLN) Applying Eqn 13-11, (Pcr )x = π 2 EI x ( KL )2x = (Pcr ) y = ( KL )2y = ] ( ) π 2 200 106 kN/m 2 13.4 10−6 m 4 (Pcr )x = 1653.2 kN π 2 EI y [ ( ) Area of x-section is 3060 mm2, so average compressive stress in column will be [ ( ) (4 m )2 σ cr = ] ( ) π 2 200 106 kN/m 2 1.83 10−6 m 4 ( ) Pcr 460.8 103 N = = 150.6 N/mm2 2 A 3060 m Since σcr < σY = 410 MPa, buckling will occur before the material yields. (2.8 m ) 2 (Pcr ) y = 460.8 kN By comparison, buckling will occur about the y-y axis ©2005 Pearson Education South Asia Pte Ltd 33 ©2005 Pearson Education South Asia Pte Ltd 9. Buckling of Columns 9. Buckling of Columns EXAMPLE 13.3 (SOLN) CHAPTER REVIEW • Buckling is the sudden instability that occurs in columns or members that support an axial load. • The maximum axial load that a member can support just before buckling occurs is called the critical load Pcr. • The critical load for an ideal column is determined from the Euler eqn, Pcr = π2EI/(KL)2, where K = 1 for pin supports, K = 0.5 for fixed supports, K = 0.7 for a pin and a fixed support, and K = 2 for a fixed support and a free end. NOTE: From Eqn 13-11, we see that buckling always occur about the column axis having the largest slenderness ratio. Thus using data for the radius of gyration from table in Appendix B, ⎛ KL ⎞ = 4 m(1000 mm/m) = 60.4 ⎜ ⎟ 66.2 mm ⎝ r ⎠x ⎛ KL ⎞ = 2.8 m(1000 mm/m ) = 114.3 ⎜ ⎟ 24.5 mm ⎝ r ⎠y Hence, y-y axis buckling will occur, which is the same conclusion reached by comparing Eqns 13-11 for both axes. ©2005 Pearson Education South Asia Pte Ltd 34 35 ©2005 Pearson Education South Asia Pte Ltd 36
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