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30.1 EULER S FORMULA / 30.2 30.2 EFFECTIVE LENGTH / 30.4 30.3 GENERALIZATION OF THE PROBLEM / 30.6 30.4 MODIFIED BUCKLING FORMULAS / 30.7 30.5 STRESS-LIMITING CRITERION / 30.8 30.6 BEAM-COLUMN ANALYSIS / 30.12 30.7 APPROXIMATE METHOD / 30.13 30.8 INSTABILITY OF BEAMS / 30.14 REFERENCES / 30.18
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A B(n) c(n) c(y), c(z) e E E(t) F(x) G h H I I(y), I(z) J k2 K K(0) Area of cross section Arbitrary constants Coefficients in series Distance from y and z axis, respectively, to outermost compressive fiber Eccentricity of axial load P Modulus of elasticity of material Tangent modulus for buckling outside of elastic range A function of x Shear modulus of material Height of cross section Horizontal (transverse) force on column Moment of inertia of cross section Moment of inertia with respect to y and z axis, respectively Torsion constant; polar moment of inertia P/EI Effective-length coefficient Spring constant for constraining spring at origin
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30.1 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
INSTABILITIES IN BEAMS AND COLUMNS 30.2
LOAD CAPABILITY CONSIDERATIONS
K(T, 0), K(T, L) l L Leff M, M M(0), M(L), Mmid M(0)cr Mtr M(y), M(z) n P Pcr r R s t T x y, z Y Ytr
Torsional spring constants at x = 0, L, respectively Developed length of cross section Length of column or beam Effective length of column Bending moments Bending moments at x = 0, L, and midpoint, respectively Critical moment for buckling of beam Moment due to transverse load Moment about y and z axis, respectively Integer; running index Axial load on column Critical axial load for buckling of column Radius of gyration Radius of cross section Running coordinate, measured from one end Thickness of cross section Torque about x axis Axial coordinate of column or beam Transverse coordinates and deflections Initial deflection (crookedness) of column Deflection of beam-column due to transverse load Factor of safety Stress Angle of twist
As the terms beam and column imply, this chapter deals with members whose crosssectional dimensions are small in comparison with their lengths. Particularly, we are concerned with the stability of beams and columns whose axes in the undeformed state are substantially straight. Classically, instability is associated with a state in which the deformation of an idealized, perfectly straight member can become arbitrarily large. However, some of the criteria for stable design which we will develop will take into account the influences of imperfections such as the eccentricity of the axial load and the crookedness of the centroidal axis of the column. The magnitudes of these imperfections are generally not known, but they can be estimated from manufacturing tolerances. For axially loaded columns, the onset of instability is related to the moment of inertia of the column cross section about its minor principal axis. For beams, stability design requires, in addition to the moment of inertia, the consideration of the torsional stiffness.
30.1 EULER S FORMULA
We will begin with the familiar Euler column-buckling problem. The column is idealized as shown in Fig. 30.1. The top and bottom ends are pinned; that is, the moments
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INSTABILITIES IN BEAMS AND COLUMNS 30.3
INSTABILITIES IN BEAMS AND COLUMNS
at the ends are zero. The bottom pin is fixed against translation; the top pin is free to move in the vertical direction only; and the force P acts along the x axis, which coincides with the centroidal axis in the undeformed state. It is important to keep in mind that the analysis which follows applies only to columns with cross sections and loads that are symmetrical about the xy plane in Fig. 30.1 and satisfy the usual assumptions of linear beam theory. It is particularly important in this connection to keep in mind that this analysis is valid only when the deformation is such that the square of the slope of the tangent at any point on the deflection curve is negligibly small compared to unity (fortunately, this is generally true in design applications). In such a case, the familiar differential equation for the bending of a beam is applicable. Thus, EI For the column in Fig. 30.1, M = Py We take E and I as constant, and let P = k2 EI Then we get, from Eqs. (30.1), (30.2), and (30.3), d2y + k2 y = 0 dx2 (30.4) (30.3) (30.2) d2y =M dx2 (30.1)
FIGURE 30.1 Deflection of a simply supported column. (a) Ideal simply supported column; (b) column-deflection curve; (c) free-body diagram of deflected segment.
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