Analysis of statically determinate trusses.
Trusses can be found in buildings,
bridges,
towers,
and other types of structural systems.
We classify a structural system as a truss
based on how it responds to the applied loads.
For instance, a beam when subjected to a load bends like this:
resulting in the development of two internal forces:
a shear force
and a bending moment.
A truss however responds differently to such a load.
Its members displace and their lengths change,
but they don't bend the way a beam bends.
Therefore in theory no shear force or bending moment develops in a truss member.
The member carries an axial force only.
In this lecture, we're going to examine the analysis of statically determinate trusses that are defined in a two dimensional space.
We shall employ the method of joints for this analysis.
a truss can be viewed as a network of interconnected slender members.
A point at which two or more members are connected together is called a joint.
A truss can carry loads that are placed at its joints only.
If a load is placed directly on a member like this, then it needs to be treated as a beam,
since the load causes internal shear and moment forces to develop.
Such a structure can no longer be viewed as a truss.
Trusses need to be properly supported using pins and rollers.
In theory, we consider a truss joint as a mechanism that allows the attached members to rotate relative to each other like this:
However the amount of joint rotation is restricted by the geometry and topology of the structure.
For example, consider this simple truss.
Under the applied load, the top joint is going to displace downward,
forcing the two inclined members to rotate inward.
The important point to keep in mind is
that although the members change length to accommodate the new truss configuration
they remain straight bars, they do not bend.
So this kind of deformation does not take place in trusses.
The purpose of a truss analysis is to calculate the axial force in each member.
For statically determinate trusses, we can use the method of joints to analyze the structure.
This method involves first isolating the joints of the truss,
then applying the equilibrium equations to each joint
in order to determine the unknown member forces.
Let's examine the method of joints in the context of this truss.
To simplify the process of calculating the member forces,
we can always start by determining the support reactions.
This is done by applying the equilibrium equations to the entire truss.
In this case, there are three unknown reaction forces.
To determine them, we need to write the three static equilibrium equations, they are:
Solving these equations for the unknowns, we get:
As I mentioned before, the method of joints involves isolating each joint of the truss
before applying the equilibrium equations to the joints.
How do we isolate the truss joints?
Consider member AD.
Imagine cutting and separating the member from the structure like this:
We know that the member carries an axial force.
Let's assume it is a tensile force
which we can show this way.
As this pair of arrow suggests, the member is being pulled at either end.
It is being elongated under the assumed tensile force.
Since the member has been cut from the truss near joints A and D,
the tension force needs to appear at the other end of the cut as well.
Therefore we need to place the force here, and here, like this:
It is important to keep in mind that these two forces must cancel each other out.
And these forces must do the same.
That is why they are shown in opposite directions.
Further, since these arrows refer to the same force they all have the same magnitude.
Let's refer to the force in member AD as Fad.
Now consider member AB.
Assuming it carries a tensile force,
we can show the force acting at joints A and B pointing away from the joint.
using the same scheme, we can replace each remaining truss member with two arrows, acting as its end joints.
Now that we have isolated and shown the member forces acting at each truss joint,
we are in a position to start calculating the unknown forces.
Since the truss is assumed to be in the state of static equilibrium,
the equilibrium equations must be satisfied for each truss joint.
What are the equilibrium equations?
There are three of them:
The sum of the forces in the X direction must be zero.
The sum of the forces in the Y direction must be zero.
thee moments about the joint must be zero.
Note that the third equation is automatically satisfied.
That is, since every force that acts at a joint passes through the joint, its moment about the joint is zero.
Therefore we are left with these two equations for calculating the unknown member forces.
Consider joint A.
Four forces are acting on it but only two of them are unknown.
Applying the equilibrium equations to the joint, therefore enables us to determine the unknown forces.
The equations are:
Solving them for Fad and Fab, we get:
Let's write these values on the force diagram.
There are 7 additional unknown member forces to be determined.
We can determine 2 of them
by applying the equilibrium equations to another joint that has at most 2 unknown forces.
At this point, 2 joints qualify: D and F.
We can use either one in the next step.
Let's use joint D.
The unknowns at the joint are Fbd and Fde.
The equilibrium equations applied to the joint are:
These equations yield:
What does the negative sign for Fbd mean?
It means the direction of the force is opposite to what was assumed initially.
We assumed all the truss members were in tension.
The negative sign tells us that member BD is in compression.
Therefore we can either keep the direction of the force arrow as is,
and write -7.5 kN next to it.
Or we can reverse the direction of the arrow, and write +7.5 kN.
Either way, we are stating that the member is carrying a compressive force of 7.5 kN.
For the sake of simplicity, I'm going to keep the original direction of the force arrow,
and write -7.5 as the force magnitude
to indicate that member BD is in compression.
This eliminates the need for redrawing the diagram
thereby simplifying the steps involved in solving the problem.
Here is the revised force diagram.
Next, we can consider either joint B or F, since each is subjected to two unknown forces only.
Let's go with joint F.
We can write the equilibrium equations for the joint this way:
Solving them for the unknown forces we get:
Now each remaining joint shows only two unknown forces.
We can use either joint B, C, or E to determine two more forces
I am going to use joint C to calculate Fce and Fbc.
The equilibrium equations for the joint are:
Solving them for the unknowns, we get:
Given the negative sign here,
we know that member CE is in compression,
carrying a force of 7.5 kN.
We keep the direction of the force as is,
and write -7.5 kN as its magnitude indicating the compressive nature of the force.
At this point only one unknown force remains Fbe.
We can either use joint B or E to determine the force.
Let's use join E.
Here are the two equilibrium equations.
Although we only need one of them to determine the unknown force.
We can use either equation for this purpose.
Since this is a simpler equation, I am going to use it to determine Fbe.
This concludes our analysis of the truss as we have determined all the member forces.
Let's summarize the results by writing the force magnitude on the members,
and indicate whether a member is in tension or compression.
So the truss has 2 compression members, 6 tension members, and 1 zero-force member.
Now it is your turn.
Analyze the following statically determinate trusses using the method of joints