30 Mechanics of Materials 2 Webb's approximation for the Smith-Southwell formula -+(引 Laterally loaded struts (a)Central concentrated load W nL n Maximum deflection tan 2nP 2 2 maximum bending moment (B.M.)= 2n tan 2 (b)Uniformly distributed load Maximum deflection nL maximum B.M. n2 Introduction Structural members which carry compressive loads may be divided into two broad categories depending on their relative lengths and cross-sectional dimensions.Short,thick members are generally termed columns and these usually fail by crushing when the yield stress of the material in compression is exceeded.Long,slender columns or struts,however, fail by buckling some time before the yield stress in compression is reached.The buckling occurs owing to one or more of the following reasons: (a)the strut may not be perfectly straight initially; (b)the load may not be applied exactly along the axis of the strut; (c)one part of the material may yield in compression more readily than others owing to some lack of uniformity in the material properties throughout the strut. At values of load below the buckling load a strut will be in stable eqilibrium where the displacement caused by any lateral disturbance will be totally recovered when the disturbance is removed.At the buckling load the strut is said to be in a state of neutral equilibrium,and theoretically it should then be possible to gently deflect the strut into a simple sine wave provided that the amplitude of the wave is kept small.This can be demonstrated quite simply using long thin strips of metal,e.g.a metal rule,and gentle application of compressive loads. Theoretically,it is possible for struts to achieve a condition of unstable equilibrium with loads exceeding the buckling load,any slight lateral disturbance then causing failure by buckling;this condition is never achieved in practice under static load conditions.Buckling occurs immediately at the point where the buckling load is reached owing to the reasons stated earlier.30 Mechanics of Materials 2 Webb's approximation for the Smith-Southwell formula P, + 0.26P omax=! [I+$( p,-p )] A Laterally loaded struts (a) Central concentrated load Maximum deflection = 2n P W nL maximum bending moment (B.M.) = - tan - 2n 2 (6) Uniformly distributed load Maximum deflection = maximum B.M. = sec - - 1 n "( ) Introduction Structural members which carry compressive loads may be divided into two broad categories depending on their relative lengths and cross-sectional dimensions. Short, thick members are generally termed columns and these usually fail by crushing when the yield stress of the material in compression is exceeded. Long, slender columns or struts, however, fail by buckling some time before the yield stress in compression is reached. The buckling occurs owing to one or more of the following reasons: (a) the strut may not be perfectly straight initially; (b) the load may not be applied exactly along the axis of the strut; (c) one part of the material may yield in compression more readily than others owing to some lack of uniformity in the material properties throughout the strut. At values of load below the buckling load a strut will be in stable eqilibrium where the displacement caused by any lateral disturbance will be totally recovered when the disturbance is removed. At the buckling load the strut is said to be in a state of neutral equilibrium, and theoretically it should then be possible to gently deflect the strut into a simple sine wave provided that the amplitude of the wave is kept small. This can be demonstrated quite simply using long thin strips of metal, e.g. a metal rule, and gentle application of compressive loads. Theoretically, it is possible for struts to achieve a condition of unstable equilibrium with loads exceeding the buckling load, any slight lateral disturbance then causing failure by buckling; this condition is never achieved in practice under static load conditions. Buckling occurs immediately at the point where the buckling load is reached owing to the reasons stated earlier