3: Forces in Statically Determinate Trusses

3.4: Method of Sections

This section by G. A. Hartley, damaged by N. M. Holtz.

3.4.1: Introduction

In the Method of Joints, we considered each joint in the structure in turn and the forces acting at each point. We had available 2 equations which could be used in this procedure.

Now we look at a method in which we isolate a portion of the truss by making a cut through it. The purpose of the procedure is to calculate only one, or perhaps a small number of bar forces in the interior of the structure. Whereas the method of joints is a routine procedure for calculating all bar forces in a truss, we may sometimes wish to calculate one or two bar forces. This can be important, for example, during stages of design of the structure when we are trying design configurations. We find that with some trusses, particularly tied trusses, we sometimes can’t get the method of joints process started because each external joint has 3 or more unknowns. The trick is to start the process using the method of sections, and then to proceed by method of joints.

We do the sectioning in a judicious way, and once more we will not have many rules to guide us. The skill is developed through experience and we will try to acquire some of this experience in the remainder of this chapter.

In the method of sections process, we will usually find all three equations of equilibrium (in a plane) useable since the forces systems are normally general (i.e., neither concurrent nor parallel). The strategy will be to try to generate in one step the equation in the one unknown we are seeking. It may not always be possible to accomplish this. It may require the calculation of another bar force first, or we may have to solve simultaneous equations.

Before we look at an example this is the rule of play:

**Procedure:** In applying sections, the cut must go right through the truss. It is preferable not to cut through joints, but rather the cut should pass through members. Care must be taken that each FBD of the isolated section has all approriate forces applied to it.

This is similar to taking sections in beams and frames where we are trying to calculate internal shear and bending moment. We create, in this way, a free body in isolation from the rest of the structure with the exposed internal forces as unknowns in equilibrium equations.

3.4.2 Example S-1

Figure

Figure 3.4-1: Example S-1

The truss in Fig. 3.4-1 is a compound truss similar to the tied arch trusses of Section 3.2. The “tie” in this case can be the tension member A, or its symmetrical counterpart, or the members in the two triangle unit at the top can be considered compression ties. We can solve for the reactions at the simple supports, but will not be able to proceed further until we have calculated a bar force using the method of sections.

Figure

Figure 3.4-2: FBD of portion

The section cut is taken just to the left of the loaded joint at the top and through member “A”. In the FBD of the structure to the right of the cut, we notice that there are four unknown member forces. However, three of them pass through the top loaded joint, so we can take moments about that joint to solve for one of the unknowns. To do that, we resolve the unknown force, $F_A$ into horizontal and vertical components at the support joint:

\[\begin{split} &\sum M = 0~~~~~(\text{+CCW})\\ &150\times2 - \frac{2}{\sqrt{5}}\times F_A \times 8 + \frac{1}{\sqrt{5}}\times F_A \times 2 = 0\\ &F_A = 47.92 \text{kN}~~~~~(\therefore T) \end{split}\]

Numerically we will not proceed further with Example S-1, but it should be evident that the solution can now be completed using the method of joints.

We have just seen method of sections used as a means of “kick starting” the method of joints where there are no available joints with only 2 unknowns at the outset. It can also be used when only a limited number of bar forces are to be calculated, and this could have a number of applications in structural engineering.

3.4.3 Example S-2

Figure

Figure 3.4-3: Cantilever Truss

Consider the example shown in Fig. 3.4-3. Member 1 appears to be a critical member, from the design perspective, as it has a relatively high compression force from the given load case.

Figure

Figure 3.4-4: Modified Design

Consider the design modification in Fig. 3.4-4. Member 2 has been relocated to the other diagonal of the square panel it is within.

Figure

Figure 3.4-5: Cut through member 1

This apparently does lead to an improvement in the force in member 1, as we can verify by applying the method of sections to the first design of Fig. 3.4-3, as shown in Fig. 3.4-5. Here we see that there the forces in member 1 and 2 together provide equilibrium for $\sum M$ about the upper joint. If we drew a comparable section FBD for the original design, we would see that only member 1 would resist the moment, and thus the force in member 1 would be higher in that design.

Figure

Figure 3.4-6: Cut through member 5

What about member 5, however? By considering the section of Fig. 3.4-6, of the original design, we see that the compressive force in 5 is $\sqrt{2}$ times the force in member 1, and it is longer as well. Therefore, member 5 is actually the more critical member, and the modified design does not help that at all.

This example demonstrates that the method of sections is more than a numerical procedure. Judicious use can help you can insight into the behaviour of a structure, and can be used to qualitatively evaluate different design configurations.

3.4.3 Example S-3

Figure

Figure 3.4-7: K Truss

Continuing the discussion of the method of sections, consider the “K” truss shown in Fig. 3.4-7.

Figure

Figure 3.4-8: Section

The forces in the four members i-j, k-j, k-m, and l-m can be calculated by the method of sections as follows. First, the force in member l-m can be found by taking moments about joint i in the sections shown in Figs. 3.4-8 and 3.4-9.

Figure

Figure 3.4-9: FBD

Figure

Figure 3.4-10: Section

The remaining three forces can be calculated by taking the section shown in Figs. 3.4-10 and 3.4-11.

Figure

Figure 3.4-11: FBD

3.4.4 Example S-4

As a final qualitative example, consider the truss of Fig. 3.4-12.

Figure

Figure 3.4-12: Truss

As a general guideline we should try to avoid taking a section as shown through a sloping member as the perpendicular distance from the moment point to that member adds an unnecessary complication.

Figure

Figure 3.4-13: FBD

The usual techniques is to resolve the unknown force in A into horizontal and vertical components, and compute moments using those components.

Figure

Figure 3.4-14: FBD

In this case we can also avoid having to deal with two components of the force by taking the section up close to the joint at the right end of the member. The moment about the lower joint in this section will involve only the horizontal component of FA in the equilibrium equation for solving FA.