Analysis of structures to find its internal force is the first step in design of them. Method of sections is a powerful tool to determine forces in statically determinate plane structures. In this video lecture we will see how and when to use this tool in structural analysis.
Why Method of Sections?
Fig.1 Using Method of joints, to find out force in the blue member you have to proceed as shown
In short you have to analyze 5 joints, to determine force in the member. So you have to solve for 10 equations. But using method of sections just by solving 3 equations we can solve for load in any truss member. That is the power of method of sections.
Steps in Method of Sections
Steps involved in method of section are as follows.
Determination of reaction forces
We can use 3 equilibrium equations of truss to find this. But for cantilever truss you can skip this step.
Cutting the Truss into 2 Sections
This is the most important step. Cutting the entire truss in to 2 parts by a section line. We will learn how to draw a section line in coming sections.
Fig.2 Separating out the truss into 2 parts, using a section line
Concept of Equilibrium to Any of Cut Part
Third step is to apply concept of equilibrium to any of the cut part, so that we can determine the member forces.
How to Draw a Section Line ?
The big question. How to draw a section line?.Following are the rules to draw it.
Section line should pass through three members, whose internal force has to be determined.
Section line should not cut more than 3 truss members. This is because we have got only 3 equilibrium equations, so we can solve only 3 unknown forces.
Drawing Section Line – Few Examples
For the problem given below if you want to determine force in member 1 and 2,you can draw the section line as shown.
Here both the lines are passing only 3 members, so this line is good.
Method of Section Example – Cantilever Case
For this cantilever example, for determining force in member 1, you can draw the section line as shown.
To find out internal unknown forces you can separate out sections.
Fig.4 Section line drawn for a cantilever case
Section at right hand side does not have a support. So if I apply concept of equilibrium to that section, we need not solve for reaction forces. Assume all 3 internal forces are tensile in nature.
Fig.5 This section is under equilibrium
Since this section is under equilibrium, we can say sum of horizontal forces is zero, sum of vertical forces is zero, and moment produced by truss forces at any point is zero.
Using these equations we can solve for 3 unknown forces. It is interesting to see that, in method of section we can completely omit effect of forces in uncut members. While solving if you get sign of any force as negative, that means member is under compression.
Example of Simply Supported Beam
Now let’s consider the previous example. The first step determination of reaction forces from free body diagram.
This section is good; it has got only 3 unknown forces.
Fig.7 This section is under equilibrium
Remaining procedure is same as the previous problem.
Advantages of Method of Section
If you want to determine forces in few members instead of all the members, you can definitely go for method of sections. Also if members are away from support, you can go for this method. So it is obvious that method of sections is quick, while method of joint is tedious.