Design of Compression Members According to Australian Standards

Compression Members

Compression members are structural elements that are pushed together or carry a load, more technically they are subjected only to axial compression forces. 

Types of Compression Members 

• Stocky Member - members which are short and have low slenderness ratio.
• Slender Member - Members which are short and have high slenderness ratio. 
• Members with intermediate slenderness ratio (λ).

Compression Members can fail in:

• Yielding -  the applied forces will cause a compression strain, which results in the shortening of the strut in the direction of the applied forces. Under incremental loading, this shortening continues until the column yields or "squashes".  The load under which a very stocky member yields is called squash load (Ny). Generally, very stocky members undergo yielding.
• Buckling - axial shortening is observed only at the initial stages of incremental loading. Thereafter, as the applied forces are increased in magnitude, the strut becomes “unstable” and develops a deformation in a direction normal to the loading axis and its axis is no longer straight. The strut is said to have “buckled”.  The load at which a straight member bends is called elastic buckling load (No).  
• Yielding and Buckling - This is the most common type of failure. Members with intermediate slenderness ratio undergo both yielding and buckling. 

Buckling of Straight Members 

For a member with friction less hinges at each ends, its lower end fixed in position while its upper end free to move only in vertical direction:

• Stable equilibrium - if N<No, then member deflects  laterally by a small amount 'u'.  The member returns to its original shape after the load is removed. 
• Neutral equilibrium -  if N=No, then the member is deflected but doesn't return to its original shape after removal of the load. It is slightly deflected after removal of the load.
• Unstable equilibrium - If N>No, then the member is unstable and buckles. 

The load (No) at which a member buckles is given by:

No = π^2 E*I/L^2

    No = π^2 E*A/λ^2  

where λ is called slenderness ratio which is equal to L/r. 'r' is radius of gyration. 

Deflection along the member is given by:

u = 𝛿 sin(π z/L)

where,  𝛿 is the central deflection and z is height along the member.

Concept of Effective length 

The buckling load in the above section is specific to the hinged boundary condition at both ends.

Buckling load for different boundary conditions can is given by:

• Both ends fixed - No = (2.04 π^2 E*I) /L^2 
• One end fixed and the other end pinned - No = (1.384 π^2 E*I) / L^2 
• One end fixed and the other end free - No = (π^2 E*I)/(4.84 L^2 ) ​

Using the column, pinned at both ends as the basis of comparison, the critical load in all the above cases can be obtained by employing the concept of “effective length”, Le.

Effective length corresponds to the distance between the points of inflection in the buckled mode. The inflection points in the deflection shape of the column are the points at which the curvature of the column change sign and are also the points at which the internal bending moments are zero. The effective column length can be defined as the length of an equivalent pin-ended column having the same load-carrying capacity as the member under consideration. The smaller the effective length of a particular column, the smaller its danger of lateral buckling and the greater its load carrying capacity.

• Both ends fixed - No = (2.04 π^2 E*I) /L^2           (le = 0.7L, )
• One end fixed and the other end pinned - No = (1.384 π^2 E*I) / L^2              (le = 0.85 L)
• One end fixed and the other end free - No = (π^2 E*I)/(4.84 L^2 )                     (le = 2.2 L )​

Elastic buckling load of a member with effective length le is given by:

No = ( π^2 E*I) /Le^2

                                                    le = ke * L (values of ke is calculated from AS 4100).

ke for some of the cases is given below.

Effect of Initial Curvature and Residual stress

Initial Curvature - Real structural members are not perfectly straight but have initial crookedness in them. The initial curvature causes it to bend from commencement of application of the axial load.

Residual stress - Residual stress causes significant reduction in buckling strength as there some stress present in the member. This stress is established because of different rates of cooling of different member sections.

DESIGN According to Australian Standards

Section Capacity - is the yield/squash capacity of the net section. It is concerned with yielding member.

                                                                       or
The load load at which very stocky members will fail in yielding

                                                         Ns = Ny = An*fy

where,  Ns is section capacity
             Ny is squash load
             fy is yield stress


Section capacity Ns of short compression members comprising of slender elements is reduced below its squash load:

                                                      Ns = Ny = kf*An*fy

where kf = Ae/Ag ( Area effective/Area gross).


Member capacity - Section capacity reduced by member slenderness ratio factor αc. Member capacity is concerned with resistance to column buckling.

Nc = αc *Ns = αc *Ns * kf*An*fy <= Ns

αc is calculated as follows:

Design Inequalities

For applied load N,

• N < ϕ Ns            (check for yielding failure) 
• N < ϕ Nc            (check for buckling failure) ​​

References

• Trahair, N. S and M. A Bradford. The Behavior And Design Of Steel Structures To AS 4100. London: E & FN Spon, 1998. Print.
• Salmon, Charles G and John Edwin Johnson. Steel Structures. New York: Harper & Row, 1980. Print.
• Gorenc, B, R Tinyou, and A Syam. Steel Designers' Handbook. Sydney, NSW: UNSW Press, 2005. Print.
• "DESIGN OF TENSION MEMBERS". http://www.steel-insdag.org/. Web. 25 July 2016.
• ​"NPTEL :: Civil Engineering - Design Of Steel Structures I". Nptel.ac.in. N.p., 2016. Web. 25 July 2016.