Euler Buckling Formula

12 Key excerpts on "Euler Buckling Formula"

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  Mechanics of Materials, International Adaptation

  • Roy R. Craig, Jr., Eric M. Taleff(Authors)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  Let us examine some important implications of the Euler buckling-load for- mula, Eq. 10.16. We can express this in terms of critical (buckling) stress.   cr cr   P A E Ar AL 2 2 2 ( ) or   cr /  2 2 E L r ( ) Euler Buckling Stress (10.18) where σ cr = the critical (elastic buckling) stress. E = the modulus of elasticity. r I A = / = the radius of gyration. L = the length of the member between supports. The quantity L/r is called the slenderness ratio of the column. Curves of σ cr versus L/r for structural steel and for an aluminum alloy are plotted in Fig. 10.8. From Eq. 10.18 and from Fig. 10.8 we observe the following characteristics of elastic buckling of ideal columns: • The only material property that enters directly into either the elastic buckling load or the buckling stress is the modulus of elasticity, E, which represents the stiffness of the material. Therefore, one way to increase the elastic buckling load of a member would be to use a member that is made of material with a higher E value. • The elastic buckling load is inversely proportional to the square of the length of the column. Figure 10.8 illustrates this length effect. Figure 10.8 Graphs of Euler’s formula for structural steel and for an aluminum alloy. 483 Aluminum (E = 73 GPa) L/r Steel (E = 200 GPa) 414 345 275 207 138 69 0 50 42 89 100 150 200 0 σ Y = 414 MPa σ Y = 250 MPa σ cr (MPa) 10.2 The Ideal Pin-Ended Column; Euler Buckling Load 639 • Euler’s formula is valid only for “long” columns, that is, columns whose L/r ratio leads to a critical stress below the compressive proportional limit, σ PL . (Since σ PL is generally not available, the compressive yield stress σ Y is usually substituted for it.) The values of L/r marking the limit of valid- ity of Euler’s formula for steel and for an aluminum alloy are illustrated in Fig. 10.8. • The buckling load can be increased by increasing the value of the cross- sectional moment of inertia, I.

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  Mechanics of Materials

  • Timothy A. Philpot(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Consequently, the Euler buckling equations are reliable predictors for long columns. Long, slender columns, however, are not very efficient, since the Euler buckling stress for these columns is much less than the proportional limit for the steel. Several representative empirical design formulas for centrically loaded steel, aluminum, and wood columns will be presented to introduce basic aspects of column design. Structural Steel Columns Structural steel columns are designed in accordance with specifications published by the American Institute for Steel Construction (AISC). The AISC Allowable Stress Design 1 pro- cedure differentiates between short- and intermediate-length columns and long columns. The transition point between these two categories is defined by an effective-slenderness ratio of  KL r 4.71 E Y  This effective-slenderness ratio corresponds to an Euler buckling stress of 0.44 Y . For short and intermediate-length columns with effective-slendernes ratios less than or equal to 4.71 E/ Y , the AISC formula for the critical compression stress is  KL r 4.71 0.658 when E Y  cr   Y   Y  e (16.22) where  e is the elastic buckling stress (i.e., Euler stress) given by e    2 E KL r 2 (16.23) For long columns with effective-slenderness ratios greater than 4.71 E/ Y , the AISC formula simply multiplies the Euler buckling stress by a factor of 0.877 to account for initial column crookedness. This reduction accounts for the fact that no real column is per- fectly straight. The AISC formula for the critical compression stress of long columns is  KL r 4.71 when E Y  cr  0.877 e  (16.24) The AISC recommends that the effective-slenderness ratios of columns should not exceed 200. 1 Specification for Structural Steel Buildings, ANSI/AISC 360-10, American Institute of Steel Construction, Chicago, 2010. 713 EMPIRICAL COLUMN FORMULAS—CENTRIC LOADING

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  Fundamentals of Engineering Elasticity

  (Revised 2nd Printing)

  • S F Borg(Author)
  • 1990(Publication Date)
  • WSPC

    (Publisher)

  After this we will discuss the rather arbitrary modifications made to the Euler formula by various specifications which enable engineers to design steel columns and also columns of other materials. fThere are other instabilities associated with beam action than the one considered in this chapter. The present discussion is confined to compression, or column, instability. An equally important instability may occur in members subjected to torsional loads which is characterized by a sudden twisting type of failure. We shall not discuss this instability here. The interested student will find a rather complete treatment of this topic in Ref. 20. BENDING INSTAMUTY — DESIGN OF COLUMNS 195 Finally we shall indicate how critical buckling loads may be obtained by the use of finite differences. 11-2 The Euler Solution for a Hinged-Hinged Column Consider a column hinged at both ends and loaded as shown in Fig. 11.3a. This is a hinged-hinged column. EI is constant P p CR, (a) (b) Fig. 11.3 In the Euler analysis, the critical buckling load, P C R, (or PEuier, as we shall occasionally call it) is taken as the load which will just cause the bar to bend. Stated in another manner, the load Pck will keep the bar in a slightly deflected position, as shown by Fig. 11.3b. In this figure, if the load is reduced below P C R, the bar will straighten; it the load is increased above Pen, the bar will fail. Hence P C R is the critical value of the load. At the center of the beam, 8 is a small, arbitrary deflection. The Bernoulli-Euler bending relation, which holds for small deflections and for bars in which stress is proportional to strain, is (see Eq. 6.46)

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  Mechanics of Materials

  An Integrated Learning System

  • Timothy A. Philpot, Jeffery S. Thomas(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  For a given problem, the slenderness ratio must be computed to determine whether or not the Euler equation is valid. 16.5 Empirical Column Formulas—Centric Loading The Euler Buckling Formulas for the critical buckling load [Equation (16.16)] and critical buckling stress [Equation (16.17)] were derived for ideal columns. In con- sidering ideal columns, it was assumed that the column was perfectly straight, that the compression load was applied exactly at the centroid of the cross section, and that the column material remained below its proportional limit during buck- ling. Practical columns, however, rarely satisfy all of the conditions assumed for ideal columns. Although the Euler equations give reasonable predictions for the strength of long, slender columns, early researchers soon found that the strength of short and intermediate-length columns were not well predicted by these formulas. A representative graph of the results from numerous column load tests plotted as a function of slenderness ratio is shown in Figure 16.11. The graph shows a scat- tered range of values that transition from the yield stress for the very shortest col- umns to the Euler buckling stress for the very longest columns. In the broad range of slenderness ratios between these two extremes, neither the yield stress nor the Euler buckling stress is a good predictor of the strength of the column. Further- more, most practical columns fall within this intermediate range of slenderness ratios. Consequently, practical column design is based primarily on empirical for- mulas that have been developed to represent the best fit of test results for a range FIGURE 16.11 Repre- sentative column test data for a range of slenderness ratios. σ σ σ  = = Y cr cr cr 16.5 Empirical Column Formulas—Centric Loading 593 of realistic full-size columns. These empirical formulas incorporate appropriate fac- tors of safety, effective-length factors, and other modifying factors.

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  Mechanics of Materials

  • William F. Riley, Leroy D. Sturges, Don H. Morris(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Buckling of a column is caused by deterioration of what was a stable state of equilibrium to an unstable one, not by failure of the material of which the column is composed. For long, slender columns, the maximum load for which the column is in stable equilibrium (the critical buckling load) occurs at stress levels much less than the proportional limit for the material. For a straight, slender, pin-ended column that is centrically loaded by axial compressive forces P at the ends and that has experienced a small lateral deflection, the differential equation for the elastic curve is EI d 2 v dx 2 = M r = − Pv which has the solution v = A sin px + B cos px The minimum value of load P for a nontrivial solution is P cr = π 2 EI L 2 (9-1) REVIEW PROBLEMS 611 The value given by Eq. 9-1 is called the critical buckling load or the Euler load. The second moment of the cross-sectional area I in Eq. 9-1 refers to the axis about which bending occurs. When I is replaced by Ar 2 , where r is the radius of gyration about the axis of bending, Eq. 9-1 becomes P cr A = π 2 E ( L/ r ) 2 = σ cr (9-2) The quantity L/r is called the slenderness ratio and is determined for the axis about which bending tends to occur. For a pin-ended, centrically loaded column, bending occurs about the axis of minimum second moment of area (minimum radius of gyration). Equation 9-1 agrees well with experiment if the slenderness ratio is large (L/r > 140 for steel columns). Short compression members (L/r < 40 for steel columns) can be treated as compression blocks where yielding occurs before buck- ling. Columns that lie between these extremes are analyzed by using empirical formulas (column design codes). REVIEW PROBLEMS 9-85* A 20-ft-long timber (E = 1200 ksi and σ e = 2.4 ksi) col- umn has the cross section shown in Fig. P9-85. The timbers are nailed together so that they act as a unit.

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  Advanced Mechanics of Materials

  • Arthur P. Boresi, Richard J. Schmidt(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  These inelastic strains, however, may not be great enough to cause significant damage to the column. In fact, the maximum inelastic strain in the column, at impending buckling, is often of the same order of magnitude as that of the elastic strain at the proportional limit of the material. Usually, it is much less than the strain correspond- ing to the yield stress based on a 0.2% offset. By permitting this small amount of inelastic strain, a larger design load may be justified than if only strains within the elastic range were permitted. However, when the buckling load is finally reached, the deflection of the column may increase suddenly and result in a catastrophic collapse. 12.6.2 Two Formulas for Inelastic Buckling of an Ideal Column The buckling load for a column is the axial load that holds the column in a slightly deflected position. Since an ideal column will not bend under axial load, a small lateral force must be applied to produce the initial deflection. This loading procedure may be carried out in any number of ways. For instance, the loading history might be one of the following: 1. The lateral force may be applied first, the axial load required to hold the column in the slightly bent position may be applied next, and the lateral force is removed. 2. The unknown critical load may be applied first, the lateral force may be applied next to cause a lateral deflection, and then the lateral force is removed. For elastic behavior of the column, the solution for the Euler buckling load is the same for the two procedures, since the physical process is conservative (reversible) and, hence, does not depend on the strain history. For a system in which inelastic behavior occurs, the physical process is irreversible. Hence, the loading history (the order of force application) influences the resulting value of the buckling load.

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  Mechanics of Solids and Structures

  • Roger T. Fenner, J.N. Reddy(Authors)
  • 2012(Publication Date)
  • CRC Press

    (Publisher)

  In practice, Figure 8.27(a) provides a very idealized description of failure, particularly in the transition region from yielding to buckling at intermediate slenderness ratios. Figure 8.27(b) provides a more typical empirical representation of the behavior of real struts and columns, based on experimental data. This takes into account both imperfections and the effects of combined yielding and buckling. For short columns, it is reasonable to assume that failure occurs by yielding in uniform compression. While long columns fail by elastic buckling, it is advisable to use the Euler formula only in conjunction with substantial safety factors. For columns of intermediate slenderness ratio, empirical formulae appropriate for different classes of materials, and which incorporate appropriate safety factors, are available in design codes and national standards. Figure 8.27 8.3 STRUTS AND COLUMNS WITH OTHER END CONDITIONS 455 We are also now in a position to generalize the results of our analysis of eccen-trically loaded pin-ended struts. Replacing I by Ar 2 , and using Equation (8.51) to define P c , we can express Equation (8.19) in the form P A = | σ max | 1 + ce r 2 sec L e 2 r r P EA ! (8.53) This is often referred to as the secant formula . It defines the force per unit area which causes a particular maximum compressive stress, σ max , in a column of effective slenderness ratio L e /r , for a given value of the eccentricity ratio ce/r 2 , e being the eccentricity of the axial force P , and c the distance of the greatest compressive stress from the neutral axis. Setting σ max equal to the yield stress, σ Y , Figure 8.28 shows typical curves of P /A plotted against slenderness ratio for various values of the eccentricity ratio ce/r 2 . These curves define the onset of yielding rather than buckling. For long slender columns, the effect of eccentric loading is relatively small, but for short columns it is much greater.

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  Stability of Structures

  Principles and Applications

  • Chai H Yoo, Sung Lee(Authors)
  • 2011(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  1.13. Inelastic Buckling of Straight Column

  In the discussions presented heretofore, the assumption has been made that the material obeys Hooke's law. For this assumption to be valid, the stresses in the column must be below the proportional limit of the material. The linear elastic analysis is correct for slender columns. On the other hand, the axial stress in a shot column will exceed the proportional limit. Consequently, the elastic analysis is not valid for short columns, and the limiting load for short columns must be determined by taking inelastic behavior into account. Before proceeding to consider the development of the theory of inelastic column behavior, it would be informative to review its historic perspective. The Euler hyperbola was derived by Leonhard Euler in 1744 . It was believed at the time that the formula applied to all columns, short and slender. It was soon discovered that the formula was grossly unconservative for short columns; the Euler formula was considered to be completely erroneous and was discarded for a lengthy period of time, approximately 150 years. An anecdotal story reveals that people ridiculed Euler when he could not adequately explain why a coin (a compression member with an extremely small slenderness ratio) on an anvil smashed by a hammer yielded (flattened) instead of carrying an infinitely large stress. It is of interest to note that the concept of flexural rigidity, EI, was not clearly defined at the time, and the modulus of elasticity of steel was determined by Thomas Young in 1807. 10

  10 S.P. Timoshenko, History of Strength of Materials (New York: Dover Edition, 1983), p. 92.

  However, Theodore von Kármán developed the double-modulus theory in 1910 in his doctoral dissertation at Göttingen University under Ludwig Prandtl direction. It gained widespread acceptance and the validity of Euler's work reestablished if the constant modulus E is replaced by an effective modulus for short columns. Later in 1947, F.R. Shanley11

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  Statics and Mechanics of Materials, SI Edition

  • Barry Goodno, James Gere(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 1060 Chapter 14 Columns as shown in Fig. 14-9b. The constant 1 C represents the deflection at the midpoint of the column and may have any small value, either positive or negative. There- fore, the part of the load–deflection diagram corresponding to cr P is a horizontal straight line (Fig. 14-8). Thus, the deflection at the critical load is undefined, although it must remain small for our equations to be valid. Above the bifurca- tion point B, the equilibrium is unstable, and below point B it is stable. Buckling of a pinned-end column in the first mode is called the fundamental case of column buckling. The type of buckling described in this section is called Euler buckling, and the critical load for an ideal elastic column is often called the Euler load. The famous mathematician Leonhard Euler (1707–1783), generally recognized as the greatest mathematician of all time, was the first person to investigate the buckling of a slender column and determine its critical load (Euler published his results in 1744). Taking higher values of the index n in Eqs. (14-13) and (14-14) leads to an infinite number of critical loads and corresponding mode shapes. The mode shape for 2 n 5 has two half-waves, as pictured in Fig. 14-9c. The correspond- ing critical load is four times larger than the critical load for the fundamental case. The magnitudes of the critical loads are proportional to the square of n, and the number of half-waves in the buckled shape is equal to n.

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  Exact Solutions for Buckling of Structural Members

  • C.M. Wang, C.Y. Wang(Authors)
  • 2004(Publication Date)
  • CRC Press

    (Publisher)

  He assumed a flexural rigidity and a distributed load (force per length) of the form EI ( x ) = α 1 − x L a , ρ ( x ) = β 1 − x L b (2 . 8 . 24) where α, β, a, and b are positive constants. Notice these shapes terminate into a point at the top at x = L , and no top load is allowed. 58 EXACT SOLUTIONS FOR BUCKLING OF STRUCTURAL MEMBERS A moment balance as in Fig. 2.13 yields d dx EI ( x ) dθ dx + L x ρ ( x ) dx = 0 (2 . 8 . 25) Eqs. (2.8.24) and (2.8.25) become d dz z a dθ dz + µz b +1 θ = 0 (2 . 8 . 26) where z = 1 − x L , µ = βL 3 α ( b + 1) (2 . 8 . 27) The solution is θ = C 1 z (1 − a ) / 2 J ν ( u ) + C 2 z (1 − a ) / 2 J ν ( u ) √ z (1 − a ) / 2 Y ν ( u ) (2 . 8 . 28) where u = √ µ | κ | z κ , ν = a − 1 b + 3 − a , κ = b + 3 − a 2 (2 . 8 . 29) As discussed before, since the displacement cannot be integrated in a closed form, the boundary conditions that yield exact characteristic equations are either zero inclination (fixed or sliding) or zero moment (free). Lastly, we comment on some exact solutions of very specific forms in the literature which are not included here. The first type is the inverse solution, obtained from assuming a deflection form and adjusting the rigidity and/or axial load distribution to satisfy the differential equation (Elishakoff, 2000). The second type concerns an end load which is completely dependent on the given axial load distribution (e.g., dependent on self weight) (Li et al., 1995). 2.9 Timoshenko Columns 2.9.1 Columns under End Axial Load When the column is stocky, or built up (latticed or battened) or of a composite-type construction, the application of Euler (classical) beam theory will overestimate the buckling loads. This is due to the neglect of transverse shear deformation in the Euler beam theory. A more BUCKLING OF COLUMNS 59 refined beam theory, known as the first-order shear deformation theory or Timoshenko beam theory, that incorporates the shear deformation effect was proposed by Engesser (1891) and Timoshenko (1921).

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  Computational Solid Mechanics

  Variational Formulation and High Order Approximation

  • Marco L. Bittencourt(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  For a small θ , sin θ ≈ θ , and equation (5.94) reduces to the following linear stability problem: ( k t − PL ) θ = 0 . (5.96) The solutions of the previous equation are θ = 0 and PL k t = 1 for any value of θ , both shown in Figure 5.47(b). The P cr = P = k t L relation is called the critical buckling load and represents the lowest compressive force that causes instability of the column. These concepts will be considered in the next section for the Euler column. For more general structures, the above equation represents an eigenvalue problem, with ( P , θ ) an eigenpair comprising the value of critical force and the respective buckling mode. 5.12.1 EULER COLUMN Consider a straight Euler column illustrated in Figure 5.48(a) with pinned supports at both ends. We wish to determine the lower compression force P , called Euler critical load, such that buckling occurs in the column. The buckling modes depend not only on the geometric properties of the column but also on the intensity of the compressive force. It is assumed that the considered columns have only global buckling modes due to bending. For cross-sections with two axes of symmetry (for instance, circular, rectangular, and an I-shaped), buckling can occur laterally in any direction. But for cross-sections with only one axis of symmetry, buckling occurs about the axis with the smallest moment of area. Thus, I min is denoted as the smallest moment of area of the cross-section in the formulation presented below. L P (a) Simply sup-ported column. y M y u P P x x z (b) FBD. Figure 5.48 Euler column. FORMULATION AND APPROXIMATION OF BEAMS 259 The deformed geometry of the Euler column is illustrated in Figure 5.48(b). The bending moment in a generic section x of the beam is given by M z ( x )= − Pu y ( x ) . Assuming that the column has the same material and constant cross-sections, we obtain the following differential equation substituting M z ( x ) into expression (5.32): d 2 u y ( x ) dx 2 = − P EI min u y ( x ) .

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  Strength of Materials

  A New Unified Theory for the 21st Century

  • Surya Patnaik, Dale Hopkins(Authors)
  • 2003(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  11.3 Perturbation Equation for Column Buckling Consider a perfectly straight column under a specified compressive load P , shown in Fig. 11-4 a . Consider the origin of the coordinate system at B and the x -and y -axes as marked. The column is pinned at A and B . However, the support at B is allowed to move or slide along the x axis. A perturbed configuration of the centerline is shown in Fig. 11-4 b . It has two displacement components: an axial displacement u ( x ) and a transverse displacement v ( x ). Both displacement components are small. The forces acting along the column centerline consist of an axial force F and a bending moment M . The two equilibrium equations are as follows: Axial equilibrium : F P ˆ 0 Rotational EE about B : M ‡ vF ˆ 0 Both EE are combined to obtain M ‡ Pv ˆ 0 11-1 † Equation (11-1) contains both a force variable ( M ) and a displacement variable ( v ). The moment curvature relationship is used to express the equation in either moments or displacements. In this relation the sign associated with the curvature ( M ˆ EI d 2 v dx 2 ) is ignored because the buckled form with equal probability can develop along the positive or negative y (or v) coordinate direction. Column Buckling 479 M ˆ EI d 2 v dx 2 11-2 † where E is the Young's modulus and I is the moment of inertia of the column section. Equation (11-1) can be expressed in either moment or in displacement. Buckling analysis uses the Young's modulus of the material and the moment of inertia of the column cross-section as well as the length of the column Equilibrium Equation Expressed in Moment This is obtained by differentiating Eq. (11-1) twice with respect to the x -coordinate and then eliminating displacement in favor of moment. d 2 M dx 2 ‡ P EI M ˆ 0 11-3 a † Equilibrium Equation Expressed in Displacement This is obtained by eliminating moment in favor of displacement. d 2 v dx 2 ‡ P EI v ˆ 0 11-3 b † Equations (11-3 a ) and (11-3 b ) are homogeneous equations.

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