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Sunday, 8 December 2019

December 08, 2019

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.

Do you have any suggestions? Please write in comment box and also drop your email id in the given mail box which is given at right hand side of page for further and continuous update from www.hkdivedi.com.   

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  

Also read  

Thursday, 5 December 2019

December 05, 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.   

Do you have any suggestions? Please write in comment box and also drop your email id in the given mail box which is given at right hand side of page for further and continuous update from www.hkdivedi.com.   

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    

Also read   

Sunday, 1 December 2019

December 01, 2019

FORCE SYSTEM IN ENGINEERING MECHANICS

Posts based on basics of force system  
There are the desired pages links which will provide the complete information about the basics of force system in engineering mechanics. 











Now, we will be interested here to understand the basic concept of truss in engineering mechanics, method of joints and method of sections to determine the internal forces in the members of the given truss with the help of this post. 
December 01, 2019

TRUSSES IN ENGINEERING MECHANICS

We have started a new topic in our previous post i.e. engineering mechanics. We have seen there the basic concept of force system with the help of our recent post. 

Now, we will be interested here to understand the basic concept of truss in engineering mechanics with the help of this post. 

Basic concept of truss 

First of all, we must know that what is the origin of the word truss. Truss word comes from the French word trousse and its means collection of things bound together or we can say that collection of two force members bound together. 

Trusses are basically defined as the collection of two force members subjected with axial loads. 

Trusses are basically used in order to support the transverse loads. Members of truss will be subjected with axial loads even the external loading on truss will be in the transverse direction. Members of the truss might be subjected with the tensile load or the compressive load. 

Let us see here a very basic truss as displayed here in following figure. We can see here that external loading on the given truss is in the transverse direction, while members of the given truss are subjected with axial loads. 

There will be some members in the truss which will not transfer any load and those members of the truss will be termed as zero force members. We must note it here that zero force members are very important part of the truss. 

Just because of it is a zero force member, we can not remove it. If we will remove the zero force member then truss will be collapsed. 

You might be thinking that if a member of truss is not transferring any load then why we should keep such member in the truss. Zero force members are very important part of the truss. 

We must know that with the application of the concept of truss, we can save quite enough cost in providing the solution for a given problem. 

Let us see here few examples of structure made with following the principle of truss. You can see that quite enough materials are saved with the application of truss. Therefore, we can say that truss provides the economical solutions for the specific problems. 


If we analyse the various examples of trusses, we will be able to say that most structures are made of several trusses joined together to form a space framework. 

Each truss will be designed to carry those loads which acts in its plane and therefore each truss could be treated as two-dimensional structure and could be considered as a plane truss for the purpose of analysis. 

We have used here one term i.e. plane truss. What is the meaning of the plane truss? When the truss members lie in a single plane, the truss will be termed as plane truss. Plane trusses will be usually supported at its two ends. 

Now we will see here the joint in a truss. 

Joint in a truss 

Let us see here how the members of a truss are joined together to form a space frame work and to support the transverse loads. 

Members of truss could be joined together by welding, bolting, pin joints, riveting etc. 

Bolted and riveted connections or joints permit small amount of rotation while welded connections are more rigid and do not permit the rotation at the joints. 

Welded connections in truss will be used where we need to build less critical structure. It will cost less and its manufacturing process will be relatively less time consuming and easy. 

Riveted connections in truss will be used where we need to build critical structure. Riveted connections in truss will be quite costly and it will take relatively more time as riveted connections need to have drilling and reaming action. In case riveted connections, there must be very small clearance between the drilled holes and rivet pins. 

The cross-section of truss members could be channels, angles, seamless pipes, ERW pipes, I-section or circular rods as displayed here in following figure. 


Let us see the following truss joint as displayed here in following figure. Ends of the members of the truss are joined together with the help of rivets with the foundation plate. The foundation plate over which ends of truss members are fixed by riveting will be called as gusset plate. 

Centroidal axis of the truss members must meet with each other as displayed in the figure. 


In a truss, no member of the truss should be continuous through a joint.  If any member of a truss will be continuous through a joint then it will behave as a beam not as the truss member and there will be high probability that it will be bent. 

When we will solve the truss problems and we will draw the truss, we will have to mark the joints as small circles in order to display that no truss member is continuous through the joint and we will also mention the co-ordinate axis as displayed here in following figure. 


How to construct a truss?

Let us see the following structure ABCD made of four members those are joined with the help of pin joints. Once we will apply the load at point C, this structure will be collapsed and we tried here to display the condition of structure after application of external load at point C by AC’D’B. 

Do you have any idea that why structure will be collapsed after application of external loading at point C.? 

As we know very well that in case of pin joint, translation motion will be arrested but there will be some rotational motion. Once we apply the load at point C, structure will have the rotation as displayed in the figure by AC’D’B. 


So, how we will make this structure stable? 

We will simply add a diagonal member AD and structure will become strong and stable. 


Now we must know that when we need to accept the given truss, there are few check points that we need to check in order to accept the truss for the given specific problems. So what are the check points? 

Check points for accepting the given truss for a specific problem

We need to check the following points in the given truss as mentioned below. 

There must not be any member continuous through the joint. If any member of a truss will be continuous through a joint, then it will behave as a beam not as the truss member. 

Truss members must be properly aligned on the gusset plate so that their centroidal axis meet with each other at a common point. 

External loads must be applied only at the joints in order to avoid the moment reaction at the joints. 

Provision for thermal expansion or thermal contraction and deformation due to the application of eternal loads must be there in the design of truss. For example, if a truss is supported at its one end by pin joints then it should be supported at its other end with the help of roller support. 

Types of trusses 

There are following basic types of trusses as mentioned below 

Simple truss 

Simple truss could be considered as a single triangular truss. Roof trusses are one of the best example of simple truss. 

Planar truss

When the truss members lie in a single plane, the truss will be termed as plane truss or planar truss. Planar trusses will be usually supported at its two ends. 

Space frame truss 

In case of space frame truss, the truss members and nodes will be located in three dimensional space. Electrical and telecommunication towers are one of the best example of space frame truss. 

There are some basic forms of trusses as displayed in following figure. 


Classification of truss

Let us see here now very important point that is the classification of truss on the basis of the relation between the number of members, number of joints and the reaction forces. 

We will have following equations that will provide the relation between the number of members, number of joints and the reaction forces as mentioned below. 

Statically determinate truss 

m + r = 2j 

Statically determinate, we can determine the forces in the members and reaction forces with the help of the statics principle alone. 

We must note it here that above equation is necessary but not sufficient to classify a structure. 

Statically indeterminate truss 

m + r > 2j 

Statically indeterminate, we will not be able to determine the forces in the members and reaction forces with the help of the statics principle alone. 

Mechanism 

m + r < 2j 

The framework will have rigid body motion. 

Therefore, we have seen here the meaning and basics of truss, the manufacturing methods and joints for truss, the importance of supports in the truss, the consideration of thermal expansion and contraction during the construction of truss, some check points for a given truss, types and forms of the truss and finally we have also looked here classification of the truss with the help of this post. 

Do you have any suggestions? Please write in comment box and also drop your email id in the given mail box which is given at right hand side of page for further and continuous update from www.hkdivedi.com.  

Now, we will be interested here to understand the basic concept of truss in engineering mechanicsmethod of joints and method of sections to determine the internal forces in the members of the given truss with the help of this post. 

Reference:  

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

Also read