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Thursday, 5 December 2019

HOW TO SOLVE TRUSS PROBLEMS USING METHOD OF JOINTS 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 and the basic concept of truss in engineering mechanics in our previous posts.  

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

We will take one example and we will find out the forces in the truss members with the help of method of joint 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 will be focused here with the method of joints with the help of this post and further we will see method of sections in our next post. 

Method of joints 

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

Points to remember during the determination of internal forces in the members of truss by using method of joints 

We will consider the equilibrium of each joint separately and we will also satisfy the condition of equilibrium. 

We will pass one imaginary line or imaginary section to isolate a single joint of the given truss. 

Force system acting on the joint will be coplanar and concurrent. We will use the independent equations of equilibrium in order to determine the internal forces in the truss members. 

We will start to determine the internal forces at a joint where only two unknown forces are acting. 

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

Determination of internal forces by using method of joints 

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 = 2 j) is satisfied here with the data given for the truss problem that we are analyzing 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: Equilibrium of joint 

Now, we will select a joint in the given truss problem where only two forces are unknown. We can start here with joint A or joint D. Let us start here with the joint A. 

Equilibrium of joint A 

We will see here now the equilibrium of joint A. We will isolate the joint A by considering the imaginary cut. 

We will see here now the joint A and we will assume the forces in the members AF and AB as FAF and FAB respectively as displayed here in following figure. We have assumed here the direction of forces FAF and FAB 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. for force FAF and force FAB

If we are securing the answers for forces FAF and FAB positive, it indicates that we have assumed the correct direction for forces. 

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

∑ Fx = 0,
Therefore, FAB + FAF Cos 450 = 0 

∑ Fy = 0,
Therefore, 2 + FAF Sin 450 = 0 

We will have the following result for these two unknown forces as mentioned here.


FAB = 2 KN
FAF = - 2.83 KN 

Let us observe the result obtained here for these two unknown forces, we have secured here negative value for the force FAF

Therefore, the direction of this force FAF, that we have assumed earlier, is wrong and we need to reverse the direction for this force. 

Therefore, we will have the following forces at joint A with its magnitude and direction too. 



Considering the Newton’s third law of motion, we can indicate the forces in the truss members AB and AF as displayed here in above figure. 

Force away from the joint will represent the tension in the member of truss. Therefore, member AB will be in tension. 

Force towards the joint will represent the compression in the member of truss. Therefore, member AF will be in compression. 

Equilibrium of joint F  

Similarly, now we will go ahead to find out the equilibrium of joint F as there are only two unknown reaction forces at joint F. 


We have assumed here the direction of forces in the truss members FB and FE as displayed in the above figure. Once we will get the result of these forces, we will correct the direction of these forces, if we will be found wrong in selecting the direction of these forces FFB and FFE

After applying the equations of equilibrium here, we will have the following values for these forces as mentioned below. 

FFB = 2 KN and FFE = -2 KN 

We have secured the value for the force FFE in negative sign and it will indicate that our assumption for the direction for this force FFE is wrong and its direction must be reversed.   

Therefore, we will have the following forces at joint F with its magnitude and direction too.


Similarly, we will find out the internal forces in each member of the given truss by considering the equilibrium of each joint separately. 

Finally, we will get the following values for the internal forces in the truss members and these are displayed in the following truss diagram. 



Therefore, we have seen here the complete procedure of method of joints to determine the internal forces in the truss members with the help of this post.   

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Further we will find out, in our next post, method of sections to determine the internal forces in the truss members.    

Reference:  

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

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