Rankine Gordon Formula

4 Key excerpts on "Rankine Gordon Formula"

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  eBook - ePub

  Structural Members and Frames

  • Theodore V. Galambos(Author)
  • 2016(Publication Date)
  • Dover Publications

    (Publisher)

  1.29 )

  Rankine-Gordon formula

  Straight-line formula (Tetmaier)

  Parabolic formula {Johnson parabola)

  In these equations σac is the allowable stress, F.S. is the factor of safety, and K1 , K2 , and K3 are numerical constants.

  Examples for each of these formulas are

  AISC specification, 1949 (secondary members)

  Chicago building code

  (1.29 )

  AASHO specification (columns with riveted ends)

  (3.55 )

  These column equations apply for A7 steel (σY = 33 ksi), and σac is in units of ksi.

  The CRC basic column strength formula,

  (3.10 )

  which was used in Chapter 3 and which was alluded to several times in this chapter, is also an empirical curve in the form of a parabola. In its most general form it is equal to

  where arc is the maximum compressive residual stress and KL/r is the effective slenderness ratio. In the CRC guide this formula, with σrc = 0.5σY , was suggested as a compromise to having two separate equations for the strong and weak axis buckling strength of rolled steel wide-flange columns

  and it is shown in Fig. 4.34 . Over most of the inelastic range we can see that the CRC curve lies between the strong-axis and weak-axis solution. It provides a reasonably good estimate of the strength of rolled wide-flange shapes,

  (1.36 )

  but it is not conservative for welded steel columns (Fig. 9.22, Ref. 1.34 ) and for initially crooked solid round columns (Fig. 4.37 ).

  The CRC equation [in the form of Eq. (4.203) ] is the basis for column design in the 1963 AISC specifications

  (1.20 )

  for steel building construction and the 1962 AISI specifications

  (3.52 )

  for light-gage cold-formed steel construction. In these specifications σ

  ac

  = σ

  cr

  /F.S., where σcr is determined from Eq. (4.203)

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  eBook - ePub

  Stability Design of Steel Frames

  • Wai-Kai Chen(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  An eigenvalue analysis of a perfectly straight column allows us to determine the load at which bifurcation of equilibrium takes place when the initial straight configuration of the column ceases to be stable and an adjacent (stable) equilibrium configuration that corresponds to a slightly bent configuration is preferred. The effect of inelasticity and residual stresses can be allowed for by using the tangent modulus concept (Engesser, 1895) by replacing the elastic modulus E by the tangent modulus E t in the column buckling equation (Eq. (2.5.11)). Fig. 2.38 AISC and SSRC column design curves A load–deflection analysis of a geometrically imperfect column allows us to trace the load–deflection response of the member from the start of loading to failure by some numerical means. The load-carrying capacity of the column is obtained from the peak point of the load–deflection curve. The effect of geometrical and material nonlinearities can be explicitly accounted for in the numerical model. It is generally believed that the load–deflection approach provides a more accurate assessment of column strength. A more thorough discussion of these two approaches, including some historical highlights, was given by Chen and Lui (1985). In the following, we discuss several expressions used by the Structural Stability Research Council (SSRC), AISC and the European Convention of Constructional Steelworks (ECCS) for representing the column strength curves for the design of columns in steel frames. 1

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  eBook - ePub

  Introduction to Engineering Mechanics

  A Continuum Approach, Second Edition

  • Jenn Stroud Rossmann, Clive L. Dym, Lori Bassman(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  cr . We divide Euler’s formula by the column’s cross-sectional area:

  σ cr = P cr A = π 2 E I L e 2 A .

  (11.8)

  Next, we set the second moment of area I = Ar2 , where r is the cross-sectional area’s radius of gyration. We obtain the radius of gyration of various shapes using its definition, r = (I/A)1/2 :

  σ cr = π 2 E A r 2 L e 2 A = π 2 E r 2 L e 2 = π 2 E ( L e / r ) 2 .

  (11.9)

  The quantity Le /r is known as the column’s slenderness ratio. The critical stress is proportional to the elastic modulus of the material used and is inversely proportional to the square of this ratio. For sufficiently slender columns, σcr can be much lower than the material’s yield stress, and the column almost certainly fails due to buckling. If this critical buckling stress is greater than the material’s yield stress, the column in question likely yields in compression before it has the opportunity to buckle—this is often true for short, stubby columns.

  In practice, loads are rarely applied as we have modeled our P—a perfectly aligned axial load. To more realistically assess the likelihood of buckling, we must develop a model that includes the effects of load eccentricity.

  11.2  Effect of Eccentricity

  The lines of action of applied forces P are generally not through the cross section’s centroid, as we had optimistically modeled them in the previous section. We now analyze the potential for buckling when an eccentric, or off-center, load is applied, again beginning with a beam/column that is free to rotate at pinned ends. We see that this off-center load P applies a moment to the column even when it is straight. As illustrated in Figure 11.7 , the force P has a moment arm equal to its eccentricity e. We can thus replace the off-center P by a centric load, also with magnitude P, and a moment M = Pe, as shown in Figure 11.7 . No matter how small either P or e is, this moment M will cause some bending of the column. In a sense, we are calculating not how to make the column stay straight but how much bending is permissible to maintain a normal stress σ < σcr and a tolerable deflection wmax . As expected from the previous section, the solution to this question comes from a nonlinear expression. This means that we cannot use superposition and add the effect of M to our understanding of the solution for P

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  eBook - ePub

  Stability of Structures

  Principles and Applications

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

    (Publisher)

  Chapter 1. Buckling of Columns

  Contents

  1.1. Introduction1

  1.2. Neutral Equilibrium3

  1.3. Euler Load4

  1.4. Differential Equations of Beam-Columns8

  1.5. Effects of Boundary Conditions on the Column Strength15

  1.6. Introduction to Calculus of Variations18

  1.7. Derivation of Beam-Column GDE Using Finite Strain24

  1.8. Galerkin Method27

  1.9. Continuous Beam-Columns Resting on Elastic Supports29

  1.9.1. One Span29

  1.9.2. Two Span31

  1.9.3. Three Span32

  1.9.4. Four Span34

  1.10. Elastic Buckling of Columns Subjected to Distributed Axial Loads39

  1.11. Large Deflection Theory (The Elastica)45

  1.12. Eccentrically Loaded Columns—Secant Formula54

  1.13. Inelastic Buckling of Straight Column57

  1.13.1. Double-Modulus (Reduced Modulus) Theory58

  1.13.2. Tangent-Modulus Theory62

  1.14. Metric System of Units69

  General References69

  References71

  Problems73

  1.1. Introduction

  A physical phenomenon of a reasonably straight, slender member (or body) bending laterally (usually abruptly) from its longitudinal position due to compression is referred to as buckling. The term buckling

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