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.

Design of Tension Members According to Australian Standards

Design of Tension Members

Tension members are linear members in which axial forces act so as to elongate (stretch) the member. A rope, for example, is a tension member. Tension members carry loads most efficiently, since the entire cross section is subjected to uniform stress. Unlike compression members, they do not fail by buckling.

Behavior of Tension Members

Load – Elongation of Tension Members

since axially loaded tension members are subjected to uniform tensile stress, their load deformation behaviour is similar to the corresponding basic material stress strain behaviour. Mild steel members exhibit an elastic range (a-b) ending at yielding (b). This is followed by yield plateau (b-c). In the Yield Plateau the load remains constant as the elongation increases to nearly ten times the yield strain. Under further stretching the material shows a smaller increase in tension with elongation (c-d), compared to the elastic range. This range is referred to as the strain hardening range. After reaching the ultimate load (d), the loading decreases as the elongation increases (de) until rupture (e). High strength steel tension members do not exhibit a well-defined yield point and a yield plateau. The 0.2% offset load, T, as shown in figure usually taken as the yield point in such cases. 

Concentrically Loaded Tension members

• Members without holes - Although steel tension members can sustain loads up to the ultimate load without failure, the elongation of the members at this load would be nearly 10-15% of the original length and the structure supported by the member would become un-serviceable. Hence, in the design of tension members, the yield load is usually taken as the limiting load. 

          Force corresponding to yield stress,

                                                                             

                                                           Ny = An* fy 

           where,    An is net area

                          fy is yield strength of material used 

• Members with holes - the presence of small local holes in a tension member causes early yielding around the holes.  This means that area around the holes reaches yielding, while the rest of cross section is below yield stress.  The average stress around the hole is about 3 times the average stress in the net area.  As small length of member (in red) adjacent to the holes reaches ultimate load before the rest of the part. Since only this length ( in red) would stretch a lot at the ultimate stress, and the overall member elongation need not be large, as long as the stresses in the gross section is below the yield stress. Hence Ultimate load is taken as limiting load. 
  

In statically loaded tension members with a hole, the point adjacent to the hole reaches yield stress, fy , first. On further loading, the stress at that point remains constant at the yield stress and the section plastifies progressively away from the hole (b), until the entire net section at the hole reaches the yield stress, fy , (c). Finally, the rupture (tension failure) of the member occurs when the entire net cross section reaches the ultimate stress, fu.

The fracture load for members with significant holes is ,

                                                                             

                                                        Nu = An *fu

where An is the net cross sectional area is perpendicular to the line of

action of the load, and is given by

                                                       An = Ag - Σ d*t

​

where d is the diameter of a hole, t the thickness of the member at the hole, and the summation is carried out for all holes in the cross-section under consideration. Nu is determined by the weakest cross-section, and therefore by the minimum net area An.

A member which fails by fracture before the gross yield load can be reached is not ductile, and there is little warning of failure. 

In many practical tension members with more than one row of holes, the reduction in the

cross-sectional area may be reduced by staggering the rows of holes (see figure below). In this case, the possibility must be considered of failure along a zig-zag path such as ABCDE in the figure, instead of across the section perpendicular to the load.

 Picture

The minimum amount of stagger Spm for which a hole no longer reduces the area of the member depends on the diameter 'd' of the hole and the inclination Sg/sp of the failure path, where Sg is the gauge distance between the rows of holes. An approximate expression for this minimum stagger is: 

                                                       

                                                    Spm = (4*Sg*d)^0.5 

When the actual stagger sp is less than spm, some reduced part of the hole area Ah must be deducted from the member area A , and this can be approximated by , 

                                                      Ah = d*t( 1- Sp^2/4*sg*d)

                                                      An = Ag - Σ d*t + Ah 

Net area has increased because cross sectional length of BC is greater than what it would have been if BC was perpendicular to direction of application of force.  

Design According to Australian Standards 

According to AS4100, the tensile force N* must satisfy the inequality,

                                                         N* <= ϕ Nt

where, ϕ is capacity reduction factor ϕ = 0.9

              Nt is nominal section capacity. 

Nt is lesser of:

                           Nt = Ag*fy                      [limit state of yielding ]

                           Nt = 0.85*Kt *An*fu      [ limit state of fracture of net cross sectional area]

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.
• "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.