Design of members with combined actions

When a member is loaded with combined actions, the member should also be checked for its combined utilisation. The common combined actions are Compression-Bending, Shear-Torsion and Shear- Tension. Checking for such utilisation is called Unity Check or Interaction Check. This article will answer the question like, What is the difference between bending and compression? This article will show how a member with bending and compression should be dealt with. I will explain the process of unity check 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.

In the previous articles I have explained, in brief, how different capacities of a member against compression, bending and tension are computed. In order to check the combined capacity against combined actions, we calculate the combined load capacity ratios (unity checks) as shown below. By carrying out these unity checks we make sure that the adopted member size is adequate for carrying the combined stresses caused by the combined actions. 

Unity check for Axial force and Moment

(N*/фN)+(Mx*/фMx)+(My*/фMy) =< 1

Unity check for Shear and Torsion

(V*/фV)+(T*/фT) =< 1

Unity check for Shear and Tension

(V*/фV)+(Nt*/фNt) =< 1

Where,

N*: Factored Compression force

фN: Compression capacity

Nt*: Factored Tension force

фNt: Tension capacity

Mx*: Factored Bending moment about axis x

фMx: Bending capacity about axis x

My*: Factored Bending moment about axis y

фMy: Bending capacity about axis y

фT: Torsional capacity of member

T*: Factored Torsion 

V*: Factored Shear force

фV: Shear capacity

Following example will describe how members with combined actions are checked. This structure has different actions, i.e., Bending, Torsion, Compression and Shear. In this example we will check only the members with combined actions.We won't design the connections for this structure now.

Let's calculate the actions first.

Structure with combined actions

Let's assume Square Hollow Section for the members with following parameters:

89x89x6 SHS- C350

According to ASI Design capacity tables Vol-2, Weight= 14.7 kg/m

Let's apply a load factor of 1.2 for self weight and 1.5 for imposed load

Bending Moments

Member DC, Mcd*=1.2*(14.7*10/1000*900/1000*(900/1000)/2)+1.5*(3000/1000*900/1000)= 4.12 kNm

Similarly, Mbc*=1.2*(14.7/100*(0.9*1.2+1.2*1.2/2))+1.5*3*1.2=5.72 kNm

Mabx*=1.2*14.7/100*(0.9*1.2+1.2*1.2/2)+1.5*3*1.2=5.72 kNm

Mabz*=1.2*14.7/100*0.9*0.9/2+1.5*3*0.9= 4.12 kNm

Torsion at member CB, Tbc*=1.2*14.7/100*0.9*0.9/2+1.5*3*0.9=4.12 kNm

Shear force

Vcd*=1.2*14.7/100*0.9+1.5*3=4.66 kN

Vbc*=1.2*14.7/100*(0.9+1.2)+1.5*3=4.87 kN

Compression in member AB, Nab*=1.2*14.7/100*(3+1.2+0.9)+1.5*3=5.40 kN

Effective length, Le=2.2*3000/1000=6.6 m

Different member capacities of the steel sections can be calculated in the traditional way, however in order to save time we use the available capacity tables for that purpose. Here we use the Australian Steel Institute design capacity tables Vol-2.

Let's use the above mentioned capacity table to read the following capacities:

Approximate compression capacity, фN=80 kN

Assuming the member does not fail in buckling prior to bending,

Bending capacity about axis x,фMx=17.9 kNm

Bending capacity about axis z,фMz=17.9 kNm

Shear capacity,фV=172 kN

Now let's see the unity check for member AB,

(N*/фN)+(Mx*/фMx)+(Mz*/фMz) =

(5.4/80+5.72/17.9+4.12/17.9)=0.62 < 1 Good. 

Member BC has bending, torsion and shear actions in it. Since the factored bending and shear forces are less than half the respective capacities, we won't carry out the Moment-Shear unity check for it.

Since Torsion and Shear both cause shear stress in the member, a unity check for Torsion and Shear may practically be carried out as below:

(V*/фV+T*/фT) =< 1

 According to the Square Hollow Section capacity table, Torsional capacity of the section, фT= 13.6 kNm

Therefore, Unity check for Shear-Torsion,

(4.87/172+4.12/13.6)=0.33 < 1 Good.

There is a possibility for using a thinner or smaller section.

Since the member CD does not have combined actions, we won't deal with this member here.

Different national standards recommend improved version of above mentioned unity checks. Since this article only introduce the fundamental concept of unity check, we won't deal with the improved version of the unity check. Related national standards are recommended to be referenced for the design purposes. 

In the next article I will explain, in brief, how a table of Section Properties is used in the design of steel structures.

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