We have started a new topic in our previous post i.e.
engineering mechanics. We have seen thereÂ the basic concept of force system and the basic concept of truss in engineering mechanics in our previous posts.Â Â

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.Â 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.Â

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.Â

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

There are two methods as mentioned below in order to
determine the forces in the various members of the given truss.Â

- Method of joints
- Method of sectionsÂ

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.Â

###
**Method of jointsÂ **

We can determine the forces in all the members of the
truss by using the method of joints.Â 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.Â

###
**Points to remember during the determination
of internal forces in the members of truss by using method of jointsÂ **

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

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

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.Â

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.Â

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

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

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

###
**Determination of internal forces by using
method of jointsÂ **

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.Â

Roller support is provided here in order to compensate the variation due to change in temperature.Â

Roller support is provided here in order to compensate the variation due to change in temperature.Â

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

###
**Step 1: Drawing of free body diagramÂ **

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.Â

###
**Step 2: Checking for determinacy or
indeterminacyÂ **

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.Â

#### m + r = 2jÂ

Where,

m = Number of members in the given truss

r = Number of reactions in the given truss

j = Number of jointsÂ

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.Â

For the given truss problem, we have following data as
mentioned below.

m = 9, j = 6 and r = 3Â

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

###
**Step 3: Determination of reaction forcesÂ **

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

âˆ‘ F

_{x}= 0,
Therefore, R

_{Ax}= 0Â
âˆ‘ F

_{y}= 0,
R

_{Ay}+ R_{Dy}= 4 KNÂ
âˆ‘ M

_{A}= 0,
R

_{Dy}x 3a â€“ 4a â€“ 2a = 0
R

_{Dy}x 3a = 6 aÂ
R

_{Dy}= 2 KNÂ
Therefore, R

_{Ay}= 2 KNÂ

###
**Step 4: Equilibrium of jointÂ **

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.Â

###
**Equilibrium of joint AÂ **

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

We will see here now the joint A and we will assume
the forces in the members AF and AB as F

_{AF}and F_{AB}respectively as displayed here in following figure. We have assumed here the direction of forces F_{AF}and F_{AB}as per my own assumptions.Â Â
Once we will get the result for these forces, we will
have the correct direction for these forces i.e. for force F

_{AF}and force F_{AB}.Â
If we are securing the answers for forces F

_{AF}and F_{AB}positive, it indicates that we have assumed the correct direction for forces.Â
If we are securing the answers for forces F

_{AF}and F_{AB}negative, it indicates that we have assumed the wrong direction for forces and we will have to reverse the direction of forces.Â
âˆ‘ F

_{x}= 0,
Therefore, F

_{AB}+ F_{AF}Cos 45^{0}= 0Â
âˆ‘ F

_{y}= 0,
Therefore, 2 + F

_{AF}Sin 45^{0}= 0Â
We will have the following result for these two
unknown forces as mentioned here.

F

_{AB}= 2 KN
F

_{AF}= - 2.83 KNÂ
Let us observe the result obtained here for these two
unknown forces, we have secured here negative value for the force F

_{AF}.Â
Therefore, the direction of this force F

_{AF}, that we have assumed earlier, is wrong and we need to reverse the direction for this force.Â
Therefore, we will have the following forces at joint
A with its magnitude and direction too.Â

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

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

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

###
**Equilibrium of joint FÂ Â **

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.Â

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 F

_{FB}and F_{FE}.Â
After applying the equations of equilibrium here, we
will have the following values for these forces as mentioned below.Â

F

_{FB}= 2 KN and F_{FE}= -2 KNÂ
We have secured the value for the force F

_{FE}in negative sign and it will indicate that our assumption for the direction for this force F_{FE}is wrong and its direction must be reversed. Â Â
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.Â

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

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

DoÂ 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 fromÂ www.hkdivedi.com.Â Â

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

#### Reference:Â Â

Engineering Mechanics, By Prof K. RameshÂ

Image courtesy: GoogleÂ Â Â

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