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HOW TO SOLVE TRUSS PROBLEMS USING METHOD OF SECTIONS STEP BY STEP

We have started a new topic in our previous post i.e. engineering mechanics. We have seen there the basic concept of force system, basic concept of truss in engineering mechanics and we have also discussed there the process to solve the truss problems using method of joints with step by step with the help of our previous post. 

Now, we will be interested here to understand how to solve truss problems using method of sections step by step with the help of this post. We will see here, in this post, the analysis of the forces in the given truss member of the truss by using the method of sections.  

We will take one example and we will find out the force in the given truss member with the help of method of sections with step by step. 

Forces in the truss members are required to calculate for the selection of appropriate size, structural shapes and material to withstand the forces. 

There are two methods as mentioned below in order to determine the forces in the various members of the given truss. 
  • Method of joints
  • Method of sections 

We have already seen the method of joints, so we will be focused here with the method of sections with the help of this post. 

Method of sections  

We can determine the forces in the given members of the truss by using the method of sections. Before going to see the method of sections, we need to see here few very important points in respect of method of sections. 

Points to remember during the determination of internal forces in the specified member of truss by using method of sections 

We will pass one imaginary line or imaginary section to separate the truss in two parts as displayed in the following figure. Sections XX, as shown in the following figure, is cutting and completely separating the given truss in two parts.  

After cutting the truss by an imaginary section XX, we will have to show the forces in the members and these forces will be determined by using method of sections. We will determine here the force in the truss member FE to understand the basics of method of sections. 

Imaginary line cutting the truss completely might be vertical or inclined or of any shape. We need to cut the given truss completely in two parts. 

In case of method of joints, force system acting on the joint was coplanar and concurrent. But if we see the method of sections, force system will not be concurrent. Therefore, we will have to use all the three independent equations of equilibrium in order to determine the internal forces in the truss members. 

Now its time to determine the internal forces in the given truss members by using method of sections with step by step. 

Determination of internal forces by using method of sections   

Let us consider the following figure indicating a truss. There are two supports for the given truss. One support out of these two supports is supported with pin joint or hinged joint and second joint is supported with roller support as displayed in following figure. Roller support is provided here in order to compensate the variation due to change in temperature.  


There are two transverse forces of 2 KN are acting on the members of the given truss as displayed here in above figure. 

Step 1: Drawing of free body diagram 

The first step is to draw the free body diagram for the given truss. We will first show the known forces at its given point or joint. We will show the reaction forces as per suitable force interaction at each support of the given truss.  

Step 2: Checking for determinacy or indeterminacy

After drawing the free body diagram, we will check the given truss for determinacy. We will have to check the given truss with the equation as mentioned below to secure the information whether the given truss problem could be solved by using the principle of equilibrium or equations of static equilibrium or not.

m + r = 2j

Where,
m = Number of members in the given truss
r = Number of reactions in the given truss
j = Number of joints 

So, let us see here for the given truss, whether above equation is satisfied or not. If above equation is satisfied then only, we can say that given truss problem could be solved or determined by using the equations of equilibrium. 

For the given truss problem, we have following data as mentioned below.
m = 9, j = 6 and r = 3 

We can say that equation (m + r = 2j) is satisfied here with the data given for the truss problem that we are analysing here to understand the complete process for determining the internal forces in the truss members. 

Step 3: Determination of reaction forces 

Now we will determine the value of the reaction forces. We will use the equations of equilibrium in order to determine the reaction forces. 

∑ Fx = 0,
Therefore, RAx = 0

∑ Fy = 0,
RAy + RDy = 4 KN

∑ MA = 0,
RDy x 3a – 4a – 2a = 0
RDy x 3a = 6 a

RDy = 2 KN
Therefore, RAy = 2 KN 

Step 4: Determination of force in truss member 

As we have cut the truss by an imaginary section XX, we can show the forces FFE, FBE and FBC as displayed here in following figure. We have assumed here the direction of these forces as per my own assumptions. 

Once we will get the result for these forces, we will have the correct direction for these forces i.e. forces FFE, FBE and FBC.   

If we are securing the answers for forces FFE, FBE and FBC positive, it indicates that we have assumed the correct direction for forces. 

If we are securing the answers for forces FFE, FBE and FBC negative, it indicates that we have assumed the wrong direction for forces and we will have to reverse the direction of forces. 

We can consider either the left portion of the truss or we can take also the right portion of the truss.  


As we have discussed above that force system acting over the truss members will not be concurrent. Therefore, we will have to use all three equation of static equilibrium. 

Hence, we must have to consider such moment centre, during writing the equation ∑M =0, that maximum unknown forces could be nullified. 

Therefore, we will take moment about B.
∑MB =0
Ray x a + FFE x a = 0   

FFE = - 2 KN

As we are securing here the value of force FFE in negative sign and therefore it is understood that the direction of this force is wrong and it need to be reversed.


Therefore, we have seen here a very important concept that is how to solve truss problems using method of sections step by step with the help of this post.

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Further we will find out, in our next post, moment of a couple in engineering mechanics. 

Reference:  

Engineering Mechanics, By Prof K. Ramesh 
Image courtesy: Google  

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1 comment:

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