This lecture covers the analysis of statically determinate trusses using the method of sections.
Please review the previous lectures before continuing with this presentation.
More specifically, you need to know how to draw free-body diagrams,
write and solve the static equilibrium equations,
and be familiar with truss analysis using the method of joints.
Consider this statically determinate truss.
Suppose we want to calculate the force in member CD only, without analyzing the entire structure.
We can do this by cutting through a few truss members in order to split the structure into two parts.
When we cut a truss member, we expose its internal force.
In this case, since we have cut through three members, we've exposed three member forces.
We know that when a truss is in the state of equilibrium,
the equilibrium equations are satisfied for the entire structure, and for any isolated part of it.
So in this case, we can argue that since the entire truss is in the state of equilibrium
these two parts must also be in equilibrium.
That is, the three equilibrium equations must be satisfied for each part.
To compute the force in member CD, we only need to consider one of the diagrams.
We don't need both of them to do the calculations.
Let's consider this diagram
for which we can write three equations in terms of F-cd, F-ch, and F-gh.
The equations are:
the sum of the forces in the X direction must be zero.
The sum of the forces in the Y direction must be zero.
The sum of the moments, say, about point H must be zero.
We can solve these equations for the unknown member forces.
If we want to calculate F-cd only, we can use the third equation.
It gives: F-cd = - 36 k.
If necessary, we can use the other equations to determine the other unknown forces.
The second equation gives F-ch = -50.9 k
Then, we can use the first equation to get F-gh.
Note that for this particular problem we did not need to know the support reactions in order to determine the force in member CD.
But in some other cases, we may have to calculate the support reactions before we can apply the method of section to solve the problem.
For example, suppose we want to calculate the tension force in member BC.
We cut through members FG, FC, and BC dividing the truss into two parts.
This leads to a total of five unknown forces for the left diagram.
There are three unknown member forces and two unknown support reactions.
And the right diagram is having a total of four unknowns.
But we can only write three equilibrium equations for each part.
So we end up with more unknowns than equations
which means we cannot solve the equations for the unknowns.
Therefore, it is a good practice to calculate the support reactions
before splitting the truss into two parts.
We can calculate the support reactions by writing the static equilibrium equations for the entire truss.
The sum of the forces in the X direction must be zero.
The sum of the forces in the Y direction must be zero.
The sum of the moments about point A must be zero.
Solving these equations for the unknowns , we get:
Knowing the support reactions, we are now ready to apply the method of sections for determining the force in member BC
We cut the truss like this:
Now we can use the equilibrium equations to calculate the unknown member forces.
Since we are interested in determining F-bc only,
we can calculate that force by writing the moment equilibrium equation about point F.
Solving this equation for F-bc, we get: F-bc = -36 K
Let's summarize what we've learned so far.
The method of sections is suitable for calculating a few member forces without needing to analyze the entire truss
Four main steps are involved in using this method.
1) We calculate the support reactions for the truss.
2) We cut the truss through the target member, and a few other members, in order to divide the structure into two separate parts.
This exposes the internal forces in the cut members.
Hence, one force per cut member is exposed.
3) We draw the free-body diagram for one of the parts
showing the support reactions and the exposed member forces
and relevant distances and angles on the diagram.
4) We write and solve the equilibrium equations for the free-body diagram
in order to determine the unknown forces.
Let's apply this procedure to solve another problem.
Consider this statically determinate truss.
Suppose we wish to calculate the force in member EF.
We start by calculating the support reactions for the entire truss.
Here is the free-body diagram of the system for which we can write three equilibrium equations
Solving these equations for the unknowns, we get the values for By, Ay, and Ax.
Let's cut the truss horizontally like this:
The cut divides the trust into two separate parts
exposing three member forces
one of which is our target force.
We can use either of these two diagrams to calculate the target force.
Let's take the lower free-body diagram.
Note the angle that F-ef makes with the horizontal axis.
We have labeled the angle alpha.
We need to know the angle in order to calculate the unknowns.
We can determine alpha using simple trigonometry.
tangent of alpha equals 2 over 5.
Hence, alpha = 21.8 degrees.
To determine F-ef, we can set the sum of the forces in the x-direction to zero
Solving this equation for the unknown force, we get:
In summary, using the method of sections
we can calculate the force in a member without needing to analyze the entire truss.
For most statically determinate trusses
we can easily apply the method of sections to calculate member forces.
However, there are a few cases that you need to be aware of when dealing with this method.
Case one: How to cut a K-truss for calculating a member force?
Case 2: What if the number of unknowns is more than the number of equations?
Case 3: when using the method of sections, can we cut away a single joint?
These cases are discussed in the supplementary material for this lecture.
Here are a few exercise problems.