Design of compression members

Compression members buckles when loaded with more than its capacity. Capacity of a compression member mostly depends on its length and then moment of inertia. Whereas capacity of a tensile member depends mostly on its cross section area. In this article answers will be available to questions like, what are the factors affecting design of compression members?, etc. In this article I will describe the concept of buckling phenomena and capacity of a rod against buckling. Explanation will be provided in simple language so that the people with limited knowledge can also get benefit from this article. Engineering students and those who want to learn structural engineering will get benefit from this article.

Behaviour of Tension and Compression members

Upon application of tension in a rod the length of the rod increases to some extent. The increase in length is called an elongation. Capacity of a tension member is calculated by multiplying its cross section area by permissible stress. Tensile capacity of a rod does not depend on its length in general.

On the other hand when we apply compression in a rod the rod starts buckling when applied compression force is greater than its compression capacity. Buckling is a phenomena when a compressed slender member deflects sidewise. A compressed slender member fails first by buckling before it fails by crushing. Therefore, capacity of a slender member is also called member capacity of a compression member. Longer the member lesser is the compression capacity. Therefore, member capacity of a compression member depends very much on its length.

Failure by crushing is very rare and can happen only in short, non-slender members. Capacity of a slender compressed member must not be calculated by multiplying its cross section area by permissible stress. This capacity would be a crushing capacity only which is also called a section capacity of a compression member. Following figures show the basic behaviour of rods in tension and in compression. 

Difference between behaviour of rods in tension and compression

Capacity of a compression member 

Different national standards define different methods for calculating compression capacity of a member depending on material and cross section shape of the compression member. All the methods are based on a general relation ship provided by Euler which is as follows:

This relationship shows a Critical Stress beyond which the compression member starts buckling.

Where, 

E: Young's modulus

L:Unsupported length

r: Radius of gyration =(I/A)^0.5

I: Moment of inertia

A:Cross section area

fy: Yield stress

K:Coefficient of effective length. Effective length, Le=K.L. Some values of K is shown below:

Coefficient of effective length

As the relationship shows, the value provided by this relationship may show higher value but shall not be taken as greater than the yield stress.

Design using Permissible Stress

This stress may be converted to a permissible stress if we are working according to permissible stress method.

For example, Compression capacity of a member =0.6*σcr*A >= N

Where N is applied load.

0.6: Coefficient for permissible stress

Deign using Limit State method

For Limit State method the following form of the relationship shall be used for the Critical load:

This value shall be multiplied by an appropriate capacity reduction factor to use in the Limit State method.

For example, Compression capacity of a member фN= 0.8*Pcr >= N*
Where N* is factored load.
0.8: Capacity reduction factor

In order to increase member capacity, the unsupported length of the member must be reduced by adding lateral supports on both directions of the section. The lateral support is called Lateral Restraints. With application of lateral restraints the value of K.L, in Euler's formula, can be reduced, which basically results in increasing compression capacity of a member significantly. Following figure shows how lateral restraints are applied to increase the compression capacity of a member.

Lateral restraint applied to increase compression capacity

Let's look at an example of designing a truss member.We will design the member using Limit State method.

In the case of a roof truss wind load is a significant load as compared to the self (dead) load. Wind loads have usually a load factor of 1. Whereas, same for the dead load is about 1.3. Therefore, it is a good practice to apply the load factors in load combination prior to analysis. For example, load at joints of truss = 1.0*Wind Load+1.3*Dead load. 

Let's assume the factored axial compression force for the vertical member of the truss, Fc=N*=6.8 kN

Let's use a tubular steel section with fy=250 MPa., Outer Diameter, Do=21.3 mm, Wall thickness, t=2.6 mm

A=π/4*(21.3^2-(21.3-2*2.6)^2)=3.14/4*(21.3^2-16.1^2)=152.67 mm2

I = π/64*(Do^4-Di^4)= 3.14/4*(21.3^4-16.1^4)=68002 mm4

L= 1.2 m= 1.2*1000=1200 mm

Truss members are assumed to be pin jointed, therefore, use K=1.00

Effective length, Le=K.L=1.00*1200=1200 mm

Elastic modulus of steel, E= 200000 MPa

Pcr = 200000*68002* π^2 /(1.00*1200)^2=9314.94 N=9.3kN < 250*152.67=38167 N =38.167 kN Okay

Capacity/Load= 0.8*9.3/6.8=1.09 >1 Good.

Use 21.3 mm OD tube with 2.6 mm wall thickness.

Purpose of this calculation is only to demonstrate how a compression member is designed. Most of the structural codes (National standards) do not calculate the compression capacity directly like this. Some codes involve yield stress in calculating compression capacity of a member. However, the code suggested capacity is more or less similar to the capacity calculated above.

With this, I have explained basic concept of designing a compression member. Codes suggest different methods for different shapes of cross section. Related national standards are recommended to be referenced for precise calculation of the compression members.

In the next article I will explain briefly how beams are designed.

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