Chapter 12 Elastic Stability of Columns Axial compressive loads can cause a sudden lateral deflection (Buckling) For columns made of elastic-perfectly plastic materials, P cr Depends primarily on E and I Independent of s yield and s ult For columns made of elastic strainhardening material, P cr Will also depend on the inelastic stress-strain behavior 1 Ideal column Perfectly straight Load lies exactly along central longitudinal axis Weightless Free of residual stresses Not subject to a bending moment or a lateral force 2 1
1. Introduce some basic concepts of column buckling 2. Physical description of the elastic buckling of columns a. For a range of lateral deflections b. For both ideal and imperfect slender columns 3. Derive Euler formula for a pin-pin column 4. Examine the effect of constraints 5. Investigate Local Buckling of thin-wall flanges of elastic columns with open cross sections 3 12.1 Introduction to the Concept of Column Buckling When an initially straight, slender column with pinned ends is subject to a compressive load P, failure occurs by elastic buckling when P = P cr P cr = π2 EI L 2 (12.1) When an ideal column has P < P cr, Column remains straight A lateral force will cause the beam to move laterally, but beam will return to straight position upon removal of the force Stable Equilibrium When an ideal column has P = P cr, Column can be freely moved laterally and remain displaced after removal of the lateral load Neutral Equilibrium When an ideal column has P > P cr Unstable 4 2
Magnitude of the buckling load is a function of the boundary conditions Buckling is governed by the SMALLEST area moment of inertia Real materials experience: plastic collapse or fracture (unrestrained lateral displacements) jamming in assembly (restrained lateral displacements) 5 12.2 Deflection Response of Columns to Compressive Loads 12.2.1 Elastic Buckling of an Ideal Slender Column Consider a straight slender pinned-end column made of a homogeneous material Load the column to P cr Lateral deflection is represented by Curve 0AB in Fig. 12.3a Fig. 12.3 Relation between load and lateral deflection for columns 6 3
P cr = π2 EI, σ L 2 cr = P cr A = π2 E L r where r is the radius of gyration (r 2 = I/A) L/r is the slenderness ratio For elastic behavior, s cr < s yield 2 (12.2) Fig. 12.3 Relation between load and lateral deflection for columns 7 Large Deflections Southwell (1941) showed that a very slender column can sustain a load greater than P cr in a bent position Provided the average s < s yield The load-deflection response is similar to curves BCD For a real column, the s yield is exceeded at some value C due to axial and bending stresses Fig. 12.3 Relation between load and lateral deflection for columns 8 4
By elementary beam theory: M x = EI R(x) (12.4) From calculus: 1 R = ± d2y dx 2 1+ dy dx 2 3/2 ± d2 y dx 2 (12.5) From Eqs. 12.4 and 12.5: M x By Eqs. 12.3 (M x by EI: where = ±EI d2 y dx 2 (12.6) = Py) and 12.6 after dividing d 2 y dx 2 + k2 y = 0 (12.7) k 2 = P EI (12.8) Fig. 12.4 Column with pinned ends 9 12.3 The Euler Formula for Columns with Pinned Ends Five methods: 1. Equilibrium 2. Imperfection 3. Energy 4. Snap through (more significant in buckling of shells than of beams) 5. Vibration (beyond scope of course) 12.3.1 The Equilibrium Method By equilibrium of moments about Point A: M A = 0 = M x + Py M x = Py (12.3) Eq. 12.3 represents a state of neutral equilibrium Fig. 12.4/5 Column with pinned 10 ends and FBD of lower portion 5
Fig. 12.6 Sign convention for internal moment. (a) Positive moment taken CW. (b) Positive moment taken CCW. 11 The b.c. s associated with Eq. 12.7 are: y = 0, for x = 0, L (12.9) For arbitrary values of k, Eq s. 12.7 and 12.9 admit only the trivial solution y = 0. However, nontrivial solutions exist for specific values (eigenvalues) of k. The general solutions to Eq. 12.7: y = A sin kx + B cos kx (12.10) where A and B are constants determined from the boundary conditions in Eq. 12.9. Thus, from Eq. 12.10: A sin kl = 0, B = 0 (12.11) For a nontrivial solution (A 0), Eq. 12.11 requires that kl = 0, 2 or: k = P EI = nπ, n = 1, 2, 3, L For each value of n, by Eq. 12.10, there exists a nontrivial solution (eigenfunction): y = A n sin nπx L (12.13) 12 6
From Eq. 12.12, the corresponding Euler loads are: P = n2 π 2 EI L 2, n = 1, 2, 3, (12.14) The minimum P occurs for n = 1. This load is the smallest load for which a nontrivial solution is possible the critical load for the column. By 12.12/12.14, with n = 1: P = π2 EI L 2 = P cr (12.15) Euler formula for buckling of a column with pinned ends. The buckled shape of the column is determined from Eq. 12.13 with n = 1 : y = A 1 sin πx (12.16) L But, A 1 is indeterminate. The maximum amplitude of the buckled column cannot be determined by this approach. A 1 must be determined by the theory of elasticity. 13 12.3.2 Higher Buckling Loads; n >1 Higher buckling loads than P cr are possible if the lower modes are constrained By Eq. 12.14 for n=2 P = 4 π2 EI L 2 By Eq. 12.14 for n=2 P = 9 π2 EI L 2 In general: P = n 2 π2 EI L 2 = 4P cr, σ cr(2) = P A = 4 π2 E L r = 9P cr, σ cr(3) = P A = 9 π2 E L r = n 2 P cr, σ cr(n0) = P A = n2 π2 E 2 2 (12.17) (12.18) L r 2 (12.19) In practice, n=1 is the most significant. Fig. 12.7 Buckling modes: n=1, 2, 3 14 7
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19 12.2.2 Imperfect Slender Columns Real columns nearly always possess deviations from ideal conditions Unless a column is extremely slender, it will fail by yielding or fracture before failing by large lateral deflections An imperfect column may be considered as a perfect column with an eccentricity, e For small e, 0 B FG represents the Load-d curve (max load close to P cr ) For large e, 0 B IJ represents the Load-d curve (max load can be much lower than P cr ) Fig. 12.3 Relation between load and lateral deflection for columns 20 10
Failure of Columns of Intermediate Slenderness Ratio The load-d relations for columns of intermediate slender ratios are represented by the curves in Fig. 12.3c For such columns, a condition of instability is associated with Points B, F and I At these points, inelastic strain occurs and is followed, after only a small increase in load, by instability collapse at relatively small lateral deflections Fig. 12.3 Relation between load and lateral deflection for columns 21 Which Type of Failure Occurs? Two potential types of failures 1. Failure by excessive deflection before plastic collapse or fracture 2. Failure by plastic collapse or fracture Pure analytical approach is difficult Empirical methods are usually used in conjunction with analysis to develop workable design criteria Fig. 12.3 Relation between load and lateral deflection for columns 22 11
12.3.3 The Imperfection Method Acknowledge that a real column is usually loaded eccentrically, e Hence, the Imperfection Method is a generalization of the Equilibrium Method By equilibrium of moments about Point A M A = 0 = M x + Pe x L + Py M x = Pe x L Py Fig. 12.8 Eccentrically loaded pinned-end columns 23 M x = Pe x L Py (12.20) Recall, 12.6: M x = ±EI d2 y dx 2 Thus, M x = +EI d2 y = Pe x Py (12.20) dx 2 L Dividing by EI and recalling: k 2 = P EI gives: d 2 y + dx 2 k2 y = k2 ex L (12.21) The b.c. s are: y = 0 for x = 0, L (12.22) Giving the general solution of Eq. 12.21: y = A sin kx + B cos kx ex (12.23) L where A and B are constants determined by the boundary conditions. Hence, from Eq. s 12.22 and 12.23: y = e sin(kx) sin(kl) x L (12.24) 24 12
Rewriting 12.24: y = e sin(kx) sin(kl) x L (12.24) As the load P increases, the deflection of the column increases When sin(kl) = 0 for kl=np, n=1,2,3,, y The Imperfection Method gives the same result as the Equilibrium Method. 25 12.3.4 The Energy Method The Energy Method is based on the first law of thermodynamics The work that external forces perform on a system plus the heat energy that flows into the system equals the increase in internal energy of the system plus the increase in the kinetic energy of the system: δw + δh = δu + δk (12.25) For column buckling, assuming an adiabatic system δh = 0 If beam is disturbed laterally, then it may vibrate, but δk << δw Implies δw = δu (12.26) Can solve the problem using the Rayleigh method by reducing the problem to a single DOF, e.g. y(x) = A sin(p x / L) A more general form is to use a Fourier series: y x = a n sin nπx (12.27) L Eq. 12.27 satisfies the BCs y=0 @ x=0 and x=l 26 13
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(12.33) 29 30 15
12.12. Determine the Euler load for the column shown in Figure 12.8c. See the discussion on the imperfection method in Section 12.3. Fig. 12.8c 31 32 16
33 12.4 Euler Buckling of Columns with Linear Elastic End Constraints Consider a straight elastic column with linear elastic end constraints Apply an axial force P The potential energy of the column-spring system is (12.34) The displaced equilibrium position of the column is given by the principle of stationary potential energy Fig. 12.10 Elastic column with linear elastic end constraints 34 17
By Eq. 12.34, set dv=0 Eq. 12.37 is the Euler equation for the column Eq. 12.38 are the BCs (Includes both the natural (e.g. y =0 implies no moment at a pin) and forced (specified, e.g. y=0 at ends) BCs) 35 (12.40) 36 18
If any of the end displacements (y 1,y 2 ) and the end slopes (y 1, y 2 ) of the column are forced (given), then they are not arbitrary and the associated variations must vanish These specified conditions are called forced BCs (also called geometric, kinematic, or essential BCs) e.g., for pinned ends y 1 =0 @ x=0 and y 2 =0 @ x=l Therefore, dy 1 =dy 2 =0 Then the last two of Eqs. 12.38 are identically satisfied The first two of Eqs. 12.38 yield the natural (unforced) BCs for the pinned ends Because y 1 and y 2 and hence dy 1 and dy 2 are arbitrary (i.e. nonzero) Also for the pinned ends K 1 =K 2 =0 Therefore, Eqs. 12.38 give the natural BCs (because EI>0) y 1 = y 2 = 0 (12.42) Eqs. 12.39, 12.41 and 12.42 yield B = C = D = 0 and A sin KL = 0, i.e. the result P cr =p 2 EI/L 2 37 For specific values of K 1, K 2, k 1 and k 2 that are neither zero nor infinity The buckling load is obtained by setting the determinate D of the coefficients A, B, C and D in Eq. 12.40 Usually must be solved numerically 38 19
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12.5 Local Buckling of Columns Consider a column that is formed with several thin-wall parts e.g., a channel, an angle or a wide-flange I-beam Depending on the relative cross-sectional dimensions of a flange or web Such a column may fail by local buckling of the flange or web, before it fails as an Euler column 45 Consider the example If the ratio t/b is relatively large, the column buckles as an Euler column (global buckling) If t/b is relatively small, the column fails by buckling or wrinkling, or more generally, Local Buckling 46 23
Local buckling of a compressed thin-wall column may not cause immediate collapse of the column. However, It alters the stress distribution in the system Reduces the compressive stiffness of the column Generally leads to collapse at loads lower than the Euler P cr In the design of columns in building structures using hot-rolled steel, local buckling is controlled by selecting cross sections with t/b ratios s.t. the critical stress for local buckling will exceed the s yield of the material Therefore, local buckling will not occur before the material yields Local buckling is controlled in cold-formed steel members by the use of effective widths of the various compression elements (i.e., leg of an angle or flange of a channel) which will account for the relatively small t/b ratio. These effective widths are then used to compute effective (reduced) crosssection properties, A, I and so forth. 47 48 24
Fig. 12.11 Buckling loads for local buckling and Euler buckling for columns made of 245 TR aluminum (E=74.5 GPa) 49 50 25
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