Method of Joints | Zero-Force Members
Method of Sections: If only a few of the member forces are of interest, and those members happen to be somewhere in the middle of the truss, it would be very inefficient to use the method of joints to solve for them. In such cases an alternate procedure based on the so-called method of sectionsis used.
Method of sections has its roots in the principle stated earlier, viz., if a body is in the state of equilibrium, then every part of that body is also in the state of equilibrium. With respect to truss structures, this principle makes it possible to cut a truss into two or more sections. Then by considering any section we can solve for the corresponding unknown forces using the equilibrium equations.
The crucial point to remember is that for a coplanar non-concurrent force system, such as those encountered in planar truss problems, only three scalar equilibrium equations may be used. Therefore, as a section is cut from the rest of the truss, care should be taken to avoid cutting more than three members. Otherwise, there will be more unknown forces than there are equilibrium equations to solve for them.
Let us assume that for the truss shown below we are only interested in knowing the forces in members , and .
The method of sections allows us to cut the truss through members , and as shown above ( move the mouse over the image above to see how we cut it). This results in two separate sections. Either the right or the left section could be used to solve for the three member forces of interest. The only criterion to consider is picking the section that would require the least amount of calculations to get to the final answer. Looking at the problem carefully, we find that by using the section to the right of the cut line we only need to know one reaction force (i.e., R_{4-y}) prior to solving for member forces. However, if we were to use the left section, we would need to first solve for two reactions at joint 1. Therefore, in this case it makes sense to use the right section for the internal loads analysis.
We begin the analysis by first drawing the free-body diagram of the truss as shown below.
We then sum moments about joint 1 to solve for reaction force R_{4-y}.
ΣM_{1} = 0 | => | R_{4-y}(x_{4}) - F_{3-y}(x_{3}) - F_{5-x}(y_{5}) = 0 |
=> | R_{4-y} = [F_{3-y}(x_{3}) + F_{5-x}(y_{5})] / (x_{4}) |
Next, we draw the free-body diagram of the right section showing the forces in members 4, 5, and 6. Notice that member forces F, F and F are assumed to be pulling on the corresponding members.
We can solve for the three member forces by using one moment equilibrium and two force equilibrium equations. To find F, we sum moments about joint 6 so to eliminate the contributions of F and F. Then, we use the equilibrium of forces in y direction to solve for F, and the equilibrium of forces in x direction to solve for F, as shown below.
ΣM_{6} = 0 | => | R_{4-y}(x_{4} - x_{6}) - F_{3-y}(x_{3} - x_{6}) - F(y_{6}) = 0 |
ΣF_{y} = 0 | => | R_{4-y} + F [(y_{6}) / l ] - F_{3-y} = 0 |
ΣF_{x} = 0 | => | - F - F [(x_{3} - x_{6}) / l ] - F + F_{5-x} = 0 |
Note that the moment center need not be located in the same section that is being examined. For example, joint 6 is not located in the right section of the truss. What is important is identifying the easiest point in space about which to sum moments.
Alternative Set of Equilibrium Equations:
If only the moment equilibrium equations are used to solve for the member forces, the procedure is referred to as the Method of Moments. Using this method, we begin by summing moments about joint 6 with the counter clockwise direction as positive. This results in one equation and one unknown (F).
ΣM_{6} = 0 | => | R_{4-y}(x_{4} - x_{6}) - F_{3-y}(x_{3} - x_{6}) - F (y_{6}) = 0 |
We then sum moments about joint 3. This also results in one equation and one unknown (F).
ΣM_{3} = 0 | => | R_{4-y}(x_{4} - x_{3}) - F_{5-x}(y_{5}) + F (y_{5}) = 0 |
Finally, by summing moments about joint 5, we solve for F.
ΣM_{5} = 0 | => | R_{4-y}(x_{4} - x_{5}) - F (y_{5}) - F [(x_{3} - x_{6}) / l ] (y_{5}) = 0 |
Caution: When using the method of moments, keep in mind that the three moment centers must not be located along a straight line. Otherwise, the three moment equations will not be independent.
If we were only interested in finding the force in the diagonal member, then by just summing forces in the y direction, we could directly solve for F. This procedure is referred to as the Method of Shears. This name reflects the fact that the diagonal members provide the shear rigidity.
Comparison with the method of joints:
If we were to find the unknown forces in the above example by using the method of joints, then we would need to start from joint 4 and first solve for forces F andF. Then by applying the equilibrium equations to joint 5 we would solve for F and F. Finally, by applying the equilibrium equations to joint 3 we would solve for F andF. It is clear that the method of joints would require more analysis, and should not be used in cases when only a few member forces are to be determined.
Factors to remember about the method of sections:
1. The cut line cannot pass through a member more than once
2. Do not cut through more than three members at a time. Although it is possible to cut through more than three members, there would not be enough equations to solve for forces in all the cut members.