ANALYSIS OF PLANE TRUSSES BY THE METHOD OF SECTIONS

The method of joints, presented in the preceding section, proves to be very e‰cient when forces in all the members of a truss are to be de- termined. However, if the forces in only certain members of a truss are desired, the method of joints may not prove to be e‰cient, because it may involve calculation of forces in several other members of the truss SECTION 4.6 Analysis of Plane Trusses by the Method of Sections 121

before a joint is reached that can be analyzed for a desired member force. The method of sectionsenables us to determine forces in the spe- cific members of trusses directly, without first calculating many un- necessary member forces, as may be required by the method of joints.

The method of sections involves cutting the truss into two portions by passing an imaginary section through the members whose forces are de- sired. The desired member forces are then determined by considering the equilibrium of one of the two portions of the truss. Each portion of the truss is treated as a rigid body in equilibrium, under the action of any applied loads and reactions and the forces in the members that have been cut by the section. The unknown member forces are determined by applying the three equations of equilibrium to one of the two portions of the truss. There are only three equilibrium equations available, so they cannot be used to determine more than three unknown forces. Thus, in general,sections should be chosen that do not pass through more than three members with unknown forces. In some trusses, the arrangement of members may be such that by using sections that pass through more than three members with unknown forces, we can determine one or, at most, two unknown forces. Such sections are, however, employed in the analysis of only certain types of trusses (see Example 4.9).

Procedure for Analysis

The following step-by-step procedure can be used for determining the mem- ber forces of statically determinate plane trusses by the method of sections.

1. Select a section that passes through as many members as possible whose forces are desired, but not more than three members with unknown forces. The section should cut the truss into two parts.

2. Although either of the two portions of the truss can be used for computing the member forces, we should select the portion that will require the least amount of computational e¤ort in determining the unknown forces. To avoid the necessity for the calculation of reactions, if one of the two portions of the truss does not have any reactions act- ing on it, then select this portion for the analysis of member forces and go to the next step. If both portions of the truss are attached to external supports, then calculate reactions by applying the equations of equili- brium and condition (if any) to the free body of the entire truss. Next, select the portion of the truss for analysis of member forces that has the least number of external loads and reactions applied to it.

3. Draw the free-body diagram of the portion of the truss selected, showing all external loads and reactions applied to it and the forces in the members that have been cut by the section. The unknown member forces are usually assumed to betensileand are, therefore, shown on the free-body diagram by arrowspulling awayfrom the joints.

4. Determine the unknown forces by applying the three equations of equilibrium. To avoid solving simultaneous equations, try to apply the equilibrium equations in such a manner that each equation in- volves only one unknown. This can sometimes be achieved by using the alternative systems of equilibrium equations (P

Fqẳ0, PMAẳ0,P

MBẳ0 or P

MAẳ0, P

MBẳ0, P

MCẳ0) de- scribed in Section 3.1 instead of the usual two-force summations and a moment summation (P

Fxẳ0,P

Fyẳ0,P

Mẳ0) system of equations.

5. Apply an alternative equilibrium equation, which was not used to compute member forces, to check the calculations. This alternative equation should preferably involve all three member forces de- termined by the analysis. If the analysis has been performed cor- rectly, then this alternative equilibrium equation must be satisfied.

Example 4.7

continued

Determine the forces in membersCD;DG, andGHof the truss shown in Fig. 4.22(a) by the method of sections.

3 m

3 m 120 kN 120 kN a

a 4 at 4 m = 16 m

(a)

120 kN 60 kN E

A B C D

F G H I

120 kN

3 4 5

60 kN

4 m

FGH H

D

I FDG

FCD

(b)

x y

FIG.4.22

SECTION 4.6 Analysis of Plane Trusses by the Method of Sections 123

Solution

Section aa As shown in Fig. 4.22(a), a sectionaa is passed through the three members of interest,CD;DG, and GH, cutting the truss into two portions,ACGEandDHI. To avoid the calculation of support reactions, we will use the right-hand portion,DHI, to calculate the member forces.

Member Forces The free-body diagram of the portionDHIof the truss is shown in Fig. 4.22(b). All three unknown forcesFCD;FDG, andFGH, are assumed to be tensile and are indicated by arrows pulling away from the corresponding joints on the diagram. The slope of the inclined force,FDG, is also shown on the free-body diagram. The desired member forces are calculated by applying the equilibrium equations as follows (see Fig. 4.22(b)).

þ’P

MDẳ0 60ð4ị ỵFGHð3ị ẳ0

FGHẳ80 kNðTị Ans.

þ "P

Fyẳ0 12060ỵ3 5FDGẳ0

FDGẳ300 kNðTị Ans.

þ !P

Fxẳ0 804

5ð300ị FCDẳ0 FCDẳ 320 kN

The negative answer forFCDindicates that our initial assumption about this force being tensile was incorrect, andFCDis actually a compressive force.

FCDẳ320 kNðCị Ans.

Checking Computations (See Fig. 4.22(b).) þ’P

MIẳ120ð4ị ð320ị34

5ð300ịð3ị 3

5ð300ịð4ị ẳ0 Checks

Example 4.8

Determine the forces in membersCJ andIJof the truss shown in Fig. 4.23(a) by the method of sections.

Solution

Section aa As shown in Fig. 4.23(a), a sectionaais passed through membersIJ;CJ, andCD, cutting the truss into two portions,ACI andDGJ. The left-hand portion,ACI, will be used to analyze the member forces.

Reactions Before proceeding with the calculation of member forces, we need to determine reactions at supportA.

By considering the equilibrium of the entire truss (Fig. 4.23(b)), we determine the reactions to be Axẳ0, Ayẳ100 kN", andGyẳ100 kN".

continued

Member Forces The free-body diagram of the portionACI of the truss is shown in Fig. 4.23(c). The slopes of the inclined forces,FIJ andFCJ, are obtained from the dimensions of the truss given in Fig. 4.23(a) and are shown on the free-body diagram. The unknown member forces are determined by applying the equations of equilibrium, as follows.

BecauseFCJ andFCD pass through pointC, by summing moments about C, we obtain an equation containing onlyFIJ:

þ’P

MCẳ0 100ð8ị ỵ40ð4ị 4 ffiffiffiffiffi

p17FIJð5ị ẳ0 FIJẳ 132 kN

FIG.4.23

continued

SECTION 4.6 Analysis of Plane Trusses by the Method of Sections 125

The negative answer forFIJ indicates that our initial assumption about this force being tensile was incorrect. ForceFIJ is actually a compressive force.

FIJ ẳ132 kNðCị Ans.

Next, we calculateFCJ by summing moments about pointO, which is the point of intersection of the lines of action ofFIJ and FCD. Because the slope of memberIJ is 1:4, the distance OCẳ4ðICị ẳ4ð5ị ẳ20 m (see Fig. 4.23(c)).

Equilibrium of moments aboutOyields þ’P

MOẳ0 100ð12ị 40ð16ị 40ð20ị ỵ 3 ffiffiffiffiffi

p13FCJð20ị ẳ0

FCJ ẳ14:42 kNðTị Ans.

Checking Computations To check our computations, we apply an alternative equation of equilibrium, which in- volves the two member forces just determined.

þ "P

Fyẳ1004040 1 ffiffiffiffiffi

p ð132ị ỵ17 3 ffiffiffiffiffi

p ð1413 :42ị ẳ0 Checks

Example 4.9

FIG.4.24

Determine the forces in membersFJ;HJ, andHKof the K truss shown in Fig. 4.24(a) by the method of sections.

continued

Solution

From Fig. 4.24(a), we can observe that the horizontal sectionaapassing through the three members of interest,FJ;HJ, and HK, also cuts an additional member FI, thereby releasing four unknowns, which cannot be determined by three equations of equilibrium. Trusses such as the one being considered here with the members arranged in the form of the letter K can be analyzed by a section curved around the middle joint, like sectionbbshown in Fig. 4.24(a). To avoid the calculation of support reactions, we will use the upper portionIKNLof the truss above sectionbbfor analysis. The free- body diagram of this portion is shown in Fig. 4.24(b). It can be seen that although sectionbbhas cut four members, FI;IJ;JK, andHK, forces in membersFI andHKcan be determined by summing moments about pointsKandI, re- spectively, because the lines of action of three of the four unknowns pass through these points. We will, therefore, first computeFHK by considering sectionbband then use sectionaato determineFFJandFHJ.

Section bb Using Fig. 4.24(b), we write þ’P

MIẳ0 25ð8ị FHKð12ị ẳ0 FHK ẳ 16:67 kN

FHK ẳ16:67 kNðCị Ans.

Section aa The free-body diagram of the portionIKNLof the truss above sectionaais shown in Fig. 4.24(c). To determineFHJ, we sum moments aboutF, which is the point of intersection of the lines of action ofFFI andFFJ. Thus,

þ’P

MFẳ0 25ð16ị 50ð8ị ỵ16:67ð12ị 3

5FHJð8ị 4

5FHJð6ị ẳ0 FHJ ẳ 62:5 kN

FHJẳ62:5 kNðCị Ans.

By summing forces in the horizontal direction, we obtain þ !P

Fxẳ0 25ỵ503 5FFJ3

5ð62:5ị ẳ0

FFJ ẳ62:5 kNðTị Ans.

FIG.4.24 (contd.)

continued

SECTION 4.6 Analysis of Plane Trusses by the Method of Sections 127

Checking Computations Finally, to check our calculations, we apply an alternative equilibrium equation, which involves the three member forces determined by the analysis. Using Fig. 4.24(c), we write

þ’P

MIẳ 25ð8ị 4

5ð62:5ịð6ị ỵ4

5ð62:5ịð6ị ỵ16:67ð12ị ẳ0 Checks