13 Buckling of Columns CHAPTER OBJECTIVES Discuss the

13. 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. • Analyze the buckling with bending of a column. • Discuss inelastic buckling of a column. • Discuss methods used to design concentric and eccentric columns. 2005 Pearson Education South Asia Pte Ltd 1

13. Buckling of Columns CHAPTER OUTLINE 1. 2. 3. 4. 5. 6. 7. Critical Load Ideal Column with Pin Supports Columns Having Various Types of Supports *The Secant Formula *Inelastic Buckling *Design of Columns for Concentric Loading *Design of Columns for Eccentric Loading 2005 Pearson Education South Asia Pte Ltd 2

13. Buckling of Columns 13. 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 3

13. Buckling of Columns 13. 1 CRITICAL LOAD • Spring develops restoring force F = k , while applied load P develops two horizontal components, Px = P tan , which tends to push the pin further out of equilibrium. • Since is small, = (L/2) and tan ≈ . • Thus, restoring spring force becomes F = k L/2, and disturbing force is 2 Px = 2 P. 2005 Pearson Education South Asia Pte Ltd 4

13. Buckling of Columns 13. 1 CRITICAL LOAD • For k L/2 > 2 P , • For k L/2 < 2 P , • For k L/2 = 2 P , 2005 Pearson Education South Asia Pte Ltd 5

13. Buckling of Columns 13. 2 IDEAL COLUMN WITH PIN SUPPORTS • 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. 2005 Pearson Education South Asia Pte Ltd 6

13. Buckling of Columns 13. 2 IDEAL COLUMN WITH PIN SUPPORTS • In order to determine the critical load and buckled shape of column, we apply Eqn 12 -10, • 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. 2005 Pearson Education South Asia Pte Ltd 7

13. Buckling of Columns 13. 2 IDEAL COLUMN WITH PIN SUPPORTS • Summing moments, M = P , Eqn 13 -1 becomes • General solution is • Since = 0 at x = 0, then C 2 = 0. Since = 0 at x = L, then 2005 Pearson Education South Asia Pte Ltd 8

13. Buckling of Columns 13. 2 IDEAL COLUMN WITH PIN SUPPORTS • Disregarding trivial soln for C 1 = 0, we get • Which is satisfied if • or 2005 Pearson Education South Asia Pte Ltd 9

13. Buckling of Columns 13. 2 IDEAL COLUMN WITH PIN SUPPORTS • Smallest value of P is obtained for n = 1, so critical load for column is • This load is also referred to as the Euler load. The corresponding buckled shape is defined by • C 1 represents maximum deflection, max, which occurs at midpoint of the column. 2005 Pearson Education South Asia Pte Ltd 10

13. Buckling of Columns 13. 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 11

13. Buckling of Columns 13. 2 IDEAL COLUMN WITH PIN SUPPORTS • Buckling eqn for a pin-supported long slender column, 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. 2005 Pearson Education South Asia Pte Ltd 12

13. Buckling of Columns 13. 2 IDEAL COLUMN WITH PIN SUPPORTS • Expressing I = Ar 2 where A is x-sectional area of column and r is the radius of gyration of x-sectional area. cr = critical stress, an average stress in column just before the column buckles. This stress is an elastic stress and therefore cr Y 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 13

13. Buckling of Columns 13. 2 IDEAL COLUMN WITH PIN SUPPORTS • 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. 2005 Pearson Education South Asia Pte Ltd 14

13. Buckling of Columns 13. 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 15

13. Buckling of Columns 13. 2 IDEAL COLUMN WITH PIN SUPPORTS 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. 2005 Pearson Education South Asia Pte Ltd 16

13. Buckling of Columns EXAMPLE 13. 1 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. 2005 Pearson Education South Asia Pte Ltd 17

13. Buckling of Columns EXAMPLE 13. 1 (SOLN) Use Eqn 13 -5 to obtain critical load with Est = 200 GPa. 2005 Pearson Education South Asia Pte Ltd 18

13. Buckling of Columns EXAMPLE 13. 1 (SOLN) This force creates an average compressive stress in the column of Since cr < Y = 250 MPa, application of Euler’s eqn is appropriate. 2005 Pearson Education South Asia Pte Ltd 19

13. Buckling of Columns EXAMPLE 13. 2 The A-36 steel W 200 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. 2005 Pearson Education South Asia Pte Ltd 20

13. Buckling of Columns EXAMPLE 13. 2 (SOLN) From table in Appendix B, column’s x-sectional area and moments of inertia are A = 5890 mm 2, Ix = 45. 5 106 mm 4, and Iy = 15. 3 106 mm 4. By inspection, buckling will occur about the y-y axis. Applying Eqn 13 -5, we have 2005 Pearson Education South Asia Pte Ltd 21

13. Buckling of Columns EXAMPLE 13. 2 (SOLN) When fully loaded, average compressive stress in column is Since this stress exceeds yield stress (250 N/mm 2), the load P is determined from simple compression: 2005 Pearson Education South Asia Pte Ltd 22

13. Buckling of Columns 13. 3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS • From free-body diagram, M = P( ). • Differential eqn for the deflection curve is • Solving by using boundary conditions and integration, we get 2005 Pearson Education South Asia Pte Ltd 23

13. Buckling of Columns 13. 3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS • Thus, smallest critical load occurs when n = 1, so that • 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. 2005 Pearson Education South Asia Pte Ltd 24

13. Buckling of Columns 13. 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 25

13. Buckling of Columns 13. 3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS Effective length 2005 Pearson Education South Asia Pte Ltd 26

13. Buckling of Columns 13. 3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS Effective length • Many design codes provide column formulae that use a dimensionless coefficient K, known as thee effective-length factor. • Thus, Euler’s formula can be expressed as 2005 Pearson Education South Asia Pte Ltd 27

13. Buckling of Columns 13. 3 COLUMNS HAVING VARIOUS TYPES OF SUPPORTS Effective length • Here (KL/r) is the column’s effective-slenderness ratio. 2005 Pearson Education South Asia Pte Ltd 28

13. Buckling of Columns EXAMPLE 13. 3 A W 150 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 sp that the column does not buckle nor material exceed the yield stress. Take Est = 200 GPa and Y = 410 MPa. 2005 Pearson Education South Asia Pte Ltd 29

13. Buckling of Columns EXAMPLE 13. 3 (SOLN) 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 mm 4 and Iy = 1. 83 106 mm 4 from Appendix B. 2005 Pearson Education South Asia Pte Ltd 30

13. Buckling of Columns EXAMPLE 13. 3 (SOLN) Applying Eqn 13 -11, By comparison, buckling will occur about the y-y axis. 2005 Pearson Education South Asia Pte Ltd 31

13. Buckling of Columns EXAMPLE 13. 3 (SOLN) Area of x-section is 3060 mm 2, so average compressive stress in column will be Since cr < Y = 410 MPa, buckling will occur before the material yields. 2005 Pearson Education South Asia Pte Ltd 32

13. Buckling of Columns EXAMPLE 13. 3 (SOLN) 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, 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 33

13. Buckling of Columns *13. 4 THE SECANT FORMULA • The actual criterion for load application on a column is limited to either a specified deflection of the column or by not allowing the maximum stress in the column exceed an allowable stress. • We apply a load P to column at a short eccentric distance e from centroid of x-section. • This is equivalent to applying a load P and moment M’ = Pe. 2005 Pearson Education South Asia Pte Ltd 34

13. Buckling of Columns *13. 4 THE SECANT FORMULA • From free-body diagram, internal moment in column is • Thus, the general solution for the differential eqn of the deflection curve is • Applying boundary conditions to determine the constants, deflection curve is written as 2005 Pearson Education South Asia Pte Ltd 35

13. Buckling of Columns *13. 4 THE SECANT FORMULA Maximum deflection • Due to symmetry of loading, both maximum deflection and maximum stress occur at column’s midpoint. Therefore, when x = L/2, = max, so 2005 Pearson Education South Asia Pte Ltd 36

13. Buckling of Columns *13. 4 THE SECANT FORMULA Maximum deflection • Therefore, to find Pcr, we require 2005 Pearson Education South Asia Pte Ltd 37

13. Buckling of Columns *13. 4 THE SECANT FORMULA The secant formula • Maximum stress in column occur when maximum moment occurs at the column’s midpoint. Using Eqns 13 -13 and 13 -16, • Maximum stress is compressive and 2005 Pearson Education South Asia Pte Ltd 38

13. Buckling of Columns *13. 4 THE SECANT FORMULA The secant formula • Since radius of gyration r 2 = I/A, • max = maximum elastic stress in column, at inner concave side of midpoint (compressive). • P = vertical load applied to the column. P < Pcr unless e = 0, then P = Pcr (Eqn 13 -5) • e = eccentricity of load P, measured from the neutral axis of column’s x-sectional area to line of action of P. 2005 Pearson Education South Asia Pte Ltd 39

13. Buckling of Columns *13. 4 THE SECANT FORMULA The secant formula • c = distance from neutral axis to outer fiber of column where maximum compressive stress max occurs. • A = x-sectional area of column • L = unsupported length of column in plane of bending. For non pin-supported columns, Le should be used. • E = modulus of elasticity of material. • r = radius of gyration, r = √(I/A), where I is computed about the neutral or bending axis. 2005 Pearson Education South Asia Pte Ltd 40

13. Buckling of Columns *13. 4 THE SECANT FORMULA Design • Once eccentricity ratio has been determined, column data can be substituted into Eqn 13 -19. • For max = Y, corresponding load PY is determined from a trial-and-error procedure, since eqn is transcendental and cannot be solved explicitly for P Y. • Note that PY will always be smaller than the critical load Pcr, since Euler’s formula assumes unrealistically that column is axially loaded without eccentricity. 2005 Pearson Education South Asia Pte Ltd 41

13. Buckling of Columns *13. 4 THE SECANT FORMULA IMPORTANT • Due to imperfections in manufacturing or application of the load, a column will never suddenly buckle, instead it begins to bend. • The load applied to a column is related to its deflections in a nonlinear manner, so the principle of superposition does not apply. • As the slenderness ratio increases, eccentrically loaded columns tend to fail at or near the Euler buckling load. 2005 Pearson Education South Asia Pte Ltd 42

13. Buckling of Columns EXAMPLE 13. 6 The W 200 59 A-36 steel column shown is fixed at its base and braced at the top so that it is fixed from displacement, yet free to rotate about the y-y axis. Also, it can sway to the side in the y-z plane. Determine the maximum eccentric load the column can support before it either begins to buckle or the steel yields. 2005 Pearson Education South Asia Pte Ltd 43

13. Buckling of Columns EXAMPLE 13. 6 (SOLN) From support conditions, about the y-y axis, the column behaves as if it was pinned at the top, fixed at the base and subjected to an axial load P. About the x-x axis, the column is free at the top and fixed at the base, and subjected to both axial load P and moment M = P(200 mm). 2005 Pearson Education South Asia Pte Ltd 44

13. Buckling of Columns EXAMPLE 13. 6 (SOLN) y-y axis buckling: Effective length factor is Ky = 0. 7, so (KL)y = 0. 7(4 m) = 2. 8 m = 2800 mm. Using table in Appendix B to determine Iy for the section and applying Eqn 13 -11, 2005 Pearson Education South Asia Pte Ltd 45

13. Buckling of Columns EXAMPLE 13. 6 (SOLN) x-x axis yielding: Kx = 2, so (KL)x = 2(4 m) = 8 m = 8000 mm. From table in Appendix B, A = 7580 mm 2, c = 210 mm/2 = 105 mm, and rx = 89. 9 mm, applying secant formula, 2005 Pearson Education South Asia Pte Ltd 46

13. Buckling of Columns EXAMPLE 13. 6 (SOLN) x-x axis yielding: Solving for Px by trial and error, noting that argument for secant is in radians, we get Since this value is less than (Pcr)y = 5136 k. N, failure will occur about the x-x axis. Also, = 419. 4 103 N / 7580 mm 2 = 55. 3 MPa < Y = 250 MPa. 2005 Pearson Education South Asia Pte Ltd 47

13. Buckling of Columns *13. 5 INELASTIC BUCKLING • Application of Euler’s equation requires that the stress in column remain BELOW the material’s yield point when column buckles. So it only applies to long slender columns. • In practice, most are intermediate columns, so we can study their behavior by modifying Euler’s equation to apply for inelastic buckling. • Consider a stress-strain diagram as shown. 2005 Pearson Education South Asia Pte Ltd 48

13. Buckling of Columns *13. 5 INELASTIC BUCKLING • Proportional limit is pl, and modulus of elasticity E is slope of line AB. • A plot of Euler’s hyperbola is shown having a slenderness ratio as small as (KL/r)pl, since at this pt, cr = pl. • When column about to buckle, change in strain that occurs is within a small range , so E for material can be taken as the tangent modulus Et. 2005 Pearson Education South Asia Pte Ltd 49

13. Buckling of Columns *13. 5 INELASTIC BUCKLING • Et is defined as the slope of stress-strain diagram at pt D. • At time of failure, column behaves as if it were made of a material having lower stiffness than when it behaves elastically, Et < E. • In general, as slenderness ratio increases, critical stress for a column continues to rise and tangent modulus for material decreases. 2005 Pearson Education South Asia Pte Ltd 50

13. Buckling of Columns *13. 5 INELASTIC BUCKLING • Thus, we substitute the material’s tangent modulus Et for E into Euler’s equation, • This is the tangent modulus or Engesser’s eqn. 2005 Pearson Education South Asia Pte Ltd 51

13. Buckling of Columns EXAMPLE 13. 7 A solid rod has a diameter of 30 mm and is 600 mm long. It is made of a material that can be modeled by the stress-strain diagram shown. If it is used as a pin -supported column, determine the critical load. 2005 Pearson Education South Asia Pte Ltd 52

13. Buckling of Columns EXAMPLE 13. 7 (SOLN) Radius of gyration and slenderness ratio are Applying Eqn 13 -20 yields 2005 Pearson Education South Asia Pte Ltd 53

13. Buckling of Columns EXAMPLE 13. 7 (SOLN) Assume that critical stress is elastic. From diagram, Thus, Eqn (1) becomes Since cr > pl = 150 MPa, inelastic buckling occurs. From second line segment of diagram, we have 2005 Pearson Education South Asia Pte Ltd 54

13. Buckling of Columns EXAMPLE 13. 7 (SOLN) Applying Eqn (1) yields Since value falls between 150 MPa and 270 MPa, it is indeed the critical stress. Critical load on the rod is therefore 2005 Pearson Education South Asia Pte Ltd 55

13. Buckling of Columns *13. 6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING • To account for behavior of different-length columns, design codes specify several formulae that will best fit the data within the short, intermediate, and long column range. Steel columns • Structural steel columns are designed on the basis of formulae proposed by the Structural Stability Research Council (SSRC). • Factors of safety are applied to the formulae and adopted as specs for building construction by the American Institute of Steel Construction (AISC). 2005 Pearson Education South Asia Pte Ltd 56

13. Buckling of Columns *13. 6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING Steel columns • For long columns, the Euler formula is used. A factor of safety F. S. = 23/12 ≈ 1. 92 is applied. Thus for design, • Value of slenderness ratio obtained by 2005 Pearson Education South Asia Pte Ltd 57

13. Buckling of Columns *13. 6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING Steel columns • For slenderness ratio lesser than (KL/r)c, the design eqn is 2005 Pearson Education South Asia Pte Ltd 58

13. Buckling of Columns *13. 6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING Steel columns 2005 Pearson Education South Asia Pte Ltd 59

13. Buckling of Columns *13. 6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING Aluminum columns • Design equations are specified by the Aluminum Association, applicable for specific range of slenderness ratios. • For a common alloy (2014 -T 6), we have 2005 Pearson Education South Asia Pte Ltd 60

13. Buckling of Columns *13. 6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING Aluminum columns 2005 Pearson Education South Asia Pte Ltd 61

13. Buckling of Columns *13. 6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING Timber columns • Timber design formulae published by the National Forest Products Association (NFPA) or American Institute of Timber Construction (AITC). 2005 Pearson Education South Asia Pte Ltd 62

13. Buckling of Columns *13. 6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING Timber columns • NFPA’s formulae for short, intermediate and long columns having a rectangular x-section of dimensions b and d (smallest dimension), 2005 Pearson Education South Asia Pte Ltd 63

13. Buckling of Columns *13. 6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING Procedure for analysis Column analysis • When using any formula to analyze a column, or to find its allowable load, it is necessary to calculate the slenderness ratio in order to determine which column formula applies. • Once the average allowable stress has been computed, the allowable load in the column is determined from P = allow. A. 2005 Pearson Education South Asia Pte Ltd 64

13. Buckling of Columns *13. 6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING Procedure for analysis Column design • If a formula is used to design a column, or to determine the column’s x-sectional area for a given loading and effective length, then a trial-and-check procedure generally must be followed if the column has a composite shape, such as a wide-flange section. • One way is to assume the column’s x-sectional area, A’, and calculate the corresponding stress ‘ = P/A’. 2005 Pearson Education South Asia Pte Ltd 65

13. Buckling of Columns *13. 6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING Procedure for analysis Column design • Also, with A’ use an appropriate design formula to determine the allowable stress allow. • From this, calculate the required column area Areq’d = P/ allow. • If A’ > Areq’d, the design is safe. When making comparison, it is practical to require A’ to be close to but greater than Areq’d, usually within 2 -3%. A redesign is necessary if A’ > Areq’d. 2005 Pearson Education South Asia Pte Ltd 66

13. Buckling of Columns *13. 6 DESIGN OF COLUMNS FOR CONCENTRIC LOADING Procedure for analysis Column design • Whenever a trial-and-check procedure is repeated, the choice of an area is determined by the previously calculated required area. • In engineering practice, this method for design is usually shortened through the use of computer software or published tables and graphs. 2005 Pearson Education South Asia Pte Ltd 67

13. Buckling of Columns EXAMPLE 13. 8 An A-26 steel W 250 149 member is used as a pinsupported column. Using AISC column design formulae, determine the largest load that it can safely support. Est = 200(103) MPa, Y = 250 MPa. 2005 Pearson Education South Asia Pte Ltd 68

13. Buckling of Columns EXAMPLE 13. 8 (SOLN) From Appendix B, A = 19000 mm 2; rx = 117 mm; ry= 67. 4 mm. Since K = 1 for both x and y axes buckling, slenderness ratio is largest if ry is used. Thus From Eqn 13 -22, 2005 Pearson Education South Asia Pte Ltd 69

13. Buckling of Columns EXAMPLE 13. 8 (SOLN) Here 0 < KL/r < (KL/r)c, so Eqn 13 -23 applies Allowable load P on column is 2005 Pearson Education South Asia Pte Ltd 70

13. Buckling of Columns EXAMPLE 13. 10 A bar having a length of 750 mm is used to support an axial compressive load of 60 k. N. It is pinsupported at its ends and made from a 2014 -T 6 aluminum alloy. Determine the dimensions of its x-sectional area if its width is to be twice its thickness. 2005 Pearson Education South Asia Pte Ltd 71

13. Buckling of Columns EXAMPLE 13. 10 (SOLN) Since KL = 750 mm is the same for x-x and y-y axes buckling, largest slenderness ratio is determined using smallest radius of gyration, using Imin = Iy: Since we do not know the slenderness ratio, we apply Eqn 13 -24 first, 2005 Pearson Education South Asia Pte Ltd 72

13. Buckling of Columns EXAMPLE 13. 10 (SOLN) Checking slenderness ratio, Try Eqn 13 -26, which is valid for KL/r ≥ 55; 2005 Pearson Education South Asia Pte Ltd 73

13. Buckling of Columns EXAMPLE 13. 10 (SOLN) From Eqn (1), Note: It would be satisfactory to choose the x-section with dimensions 27 mm by 54 mm. 2005 Pearson Education South Asia Pte Ltd 74

13. Buckling of Columns EXAMPLE 13. 11 A board having x-sectional dimensions of 150 mm by 40 mm is used to support an axial load of 20 k. N. If the board is assumed to be pin-supported at its top and base, determine its greatest allowable length L as specified by the NFPA. 2005 Pearson Education South Asia Pte Ltd 75

13. Buckling of Columns EXAMPLE 13. 11 (SOLN) By inspection, board will buckle about the y axis. In the NFPA eqns, d = 40 mm. Assuming that Eqn 13 -29 applies, we have 2005 Pearson Education South Asia Pte Ltd 76

13. Buckling of Columns EXAMPLE 13. 11 (SOLN) Here Since 26 < KL/d 50, the solution is valid. 2005 Pearson Education South Asia Pte Ltd 77

13. Buckling of Columns *13. 7 DESIGN OF COLUMNS FOR ECCENTRIC LOADING • A column may be required to support a load acting at its edge or on an angle bracket attached to its side. • The bending moment M = Pe, caused by eccentric loading, must be accounted for when column is designed. Use of available column formulae • Stress distribution acting over x-sectional area of column shown is determined from both axial force P and bending moment M = Pe. 2005 Pearson Education South Asia Pte Ltd 78

13. Buckling of Columns *13. 7 DESIGN OF COLUMNS FOR ECCENTRIC LOADING Use of available column formulae • Maximum compressive stress is • A typical stress profile is also shown here. • If we assume entire x-section is subjected to uniform stress max, then we can compare it with allow, which is determined from formulae given in chapter 13. 6. • If max ≤ allow, then column carry the specified load. 2005 Pearson Education South Asia Pte Ltd 79

13. Buckling of Columns *13. 7 DESIGN OF COLUMNS FOR ECCENTRIC LOADING Use of available column formulae • Otherwise, the column’s area A is increased and a new max and allow are calculated. • This method of design is rather simple to apply and works well for columns that are short or intermediate length. 2005 Pearson Education South Asia Pte Ltd 80

13. Buckling of Columns *13. 7 DESIGN OF COLUMNS FOR ECCENTRIC LOADING Interaction formula • It is sometimes desirable to see how the bending and axial loads interact when designing an eccentrically loaded column. • If allowable stress for axial load is ( a)allow, then required area for the column needed to support the load P is 2005 Pearson Education South Asia Pte Ltd 81

13. Buckling of Columns *13. 7 DESIGN OF COLUMNS FOR ECCENTRIC LOADING Interaction formula • Similarly, if allowable bending stress is ( b)allow, then since I = Ar 2, required area of column needed to resist eccentric moment is determined from flexure formula, 2005 Pearson Education South Asia Pte Ltd 82

13. Buckling of Columns *13. 7 DESIGN OF COLUMNS FOR ECCENTRIC LOADING Interaction formula • Thus, total area A for the column needed to resist both axial force and bending moment requires that 2005 Pearson Education South Asia Pte Ltd 83

13. Buckling of Columns *13. 7 DESIGN OF COLUMNS FOR ECCENTRIC LOADING Interaction formula a = axial stress caused by force P and determined from a = P/A, where A is the x-sectional area of the column. b = bending stress caused by an eccentric load or applied moment M; b is found from b = Mc/I, where I is the moment of inertia of x-sectional area computed about the bending or neutral axis. 2005 Pearson Education South Asia Pte Ltd 84

13. Buckling of Columns *13. 7 DESIGN OF COLUMNS FOR ECCENTRIC LOADING Interaction formula ( a)allow = allowable axial stress as defined by formulae given in chapter 13. 6 or by design code specs. Use the largest slenderness ratio for the column, regardless of which axis it experiences bending. ( b)allow = allowable bending stress as defined by code specifications. 2005 Pearson Education South Asia Pte Ltd 85

13. Buckling of Columns *13. 7 DESIGN OF COLUMNS FOR ECCENTRIC LOADING Interaction formula • Eqn 13 -31 is sometimes referred to as the interaction formula. • This approach requires a trial-and-check procedure. • Designer needs to choose an available column and check to see if the inequality is satisfied. • If not, a larger section is picked and the process repeated. • American Institute of Steel Construction specifies the use of Eqn 13 -31 only when the axial-stress ratio a/( a)allow ≤ 0. 15. 2005 Pearson Education South Asia Pte Ltd 86

13. Buckling of Columns EXAMPLE 13. 12 Column is made of 2014 -T 6 aluminum alloy and is used to support an eccentric load P. Determine the magnitude of P that can be supported if column is fixed at its base and free at its top. Use Eqn 13 -30. 2005 Pearson Education South Asia Pte Ltd 87

13. Buckling of Columns EXAMPLE 13. 12 (SOLN) K = 2. Largest slenderness ratio for column is By inspection, Eqn 13 -26 must be used (277. 1 > 55). 2005 Pearson Education South Asia Pte Ltd 88

13. Buckling of Columns EXAMPLE 13. 12 (SOLN) Actual maximum compressive stress in the column is determined from the combination of axial load and bending. Assuming that this stress is uniform over the xsection, instead of just at the outer boundary, 2005 Pearson Education South Asia Pte Ltd 89

13. Buckling of Columns EXAMPLE 13. 13 The A-36 steel W 150 30 column is pin-connected at its ends and subjected to eccentric load P. Determine the maximum allowable value of P using the interaction method if allowable bending stress is ( b)allow = 160 MPa, E = 200 GPa, and Y = 250 MPa. 2005 Pearson Education South Asia Pte Ltd 90

13. Buckling of Columns EXAMPLE 13. 13 (SOLN) K = 1. The geometric properties for the W 150 30 are taken from the table in Appendix B. We consider ry as it lead to largest value of the slenderness ratio. Ix is needed since bending occurs about the x axis (c = 157 mm/2 = 78. 5 mm). To determine the allowable bending compressive stress, we have 2005 Pearson Education South Asia Pte Ltd 91

13. Buckling of Columns EXAMPLE 13. 13 (SOLN) Then KL/r < (KL/r)c and so Eqn 13 -23 must be used. Assuming that this stress is uniform over the xsection, instead of just at the outer boundary, 2005 Pearson Education South Asia Pte Ltd 92

13. Buckling of Columns EXAMPLE 13. 13 (SOLN) Applying the interaction formula Eqn 13 -31 yields Checking application of interaction method for steel section, we require 2005 Pearson Education South Asia Pte Ltd 93

13. Buckling of Columns EXAMPLE 13. 14 Timber column is made from two boards nailed together so the xsection has the dimensions shown. If column is fixed at its base and free at its top, use Eqn 13 -30 to determine the eccentric load P that can be supported. 2005 Pearson Education South Asia Pte Ltd 94

13. Buckling of Columns EXAMPLE 13. 14 (SOLN) K = 2. Here, we calculate KL/d to determine which eqn to use. Since allow is determined using the largest slenderness ratio, we choose d = 60 mm. This is done to make the ratio as large as possible, and thus yield the lowest possible allowable axial stress. This is done even though bending due to P is about the x axis. 2005 Pearson Education South Asia Pte Ltd 95

13. Buckling of Columns EXAMPLE 13. 14 (SOLN) Allowable axial stress is determined using Eqn 13 -29 since 26 < KL/d < 50. Thus Applying Eqn 13 -30 with allow = max, we have 2005 Pearson Education South Asia Pte Ltd 96

13. Buckling of Columns 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 = 2 EI/(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. 2005 Pearson Education South Asia Pte Ltd 97

13. Buckling of Columns CHAPTER REVIEW • If axial loading is applied eccentrically to the column, then the secant formula must be used to determine the maximum stress in the column. • When the axial load tends to cause yielding of the column, then the tangent modulus should be used with Euler’s eqn to determine the buckling load. This is referred to as Engesser’s eqn. • Empirical formulae based upon experimental data have been developed for use in the design of steel, aluminum, and timber columns. 2005 Pearson Education South Asia Pte Ltd 98