Buckling of Compression Members

• Members in compression have the potential to buckle: to suddenly lose the ability to carry load my moving laterally with respect to the load.

  

• The load which causes a member to buckle depends on the following member properties:
  • The unbraced length: L
  • The cross section size and "spread-outness": I
  • The material stiffness: E

• The end conditions are also important, since they change the effective length of the member. This is accounted for by an "effective length factor", denoted by k.

• The following table shows effective length factors.

  

• The following is a form of the Euler Buckling Formula, which predictes the Critical Buckling Load P_cr, based on these factors.

  

  For the purposes of member design, it is useful to use an allowable stress to assess buckling resistance, rather than a failure load, since allowable stresses are commonly used in other types of member design, such as bending.

• Expressing buckling resistance to an allowable stress requires two additional terms:
  • To express the resistance as a stress, divide the load by the cross sectional area A, to define the Critical Buckling Stress, f_cr.

    

    Where:

    

    r is the "radius of gyration" which measures the "spread-outness" of a cross section with respect to an axis.

    kL/r is the Slenderness Ratio, a unitless ratio that accounts for the effects of cross section (r), unbraced length (L), and end conditions (k).

  • To express the resistance as an allowable stress, it is necessary to include a safety factor (F_s) to define an Allowable Buckling Stress, F_cr:

    

• In some cases, a member may buckle in the Strong Axis direction (based on a strong axis bending shape), or a Weak Axis direction, as shown below. The allowable stress for each is calculated using appropriate k, L, and r properties for the direction.

  Strong Axis	Weak Axis

  Summary

• Compression members have (at least) three potential failure modes:

  • Crushing (inititated by material failure)

    F_c

  • Strong Axis Buckling (inititated by instability)

    F_crx

  • Weak Axis Buckling (inititated by instability)

    F_cry

• The allowable compression stress for the member is the smallest of these three.