1. 6.7 Analysis of trusses By the method of sections
The method of joints is most effective when the forces in all the members of a truss are to be
determined. If, however, the force in only one member of the forces in a very few members are desired,
an other method, the method of sections, is more efficient.

2. Assume for example, that we want to determine the force in member BD of the truss shown in figure 1(a).
To do this, we must determine the forces with which member BD acts on either joint B or joint D. If we were to
use the method of joints, we would choose either joint B or joint D as a free body.
However, we can also choose as a free body a larger portion of the truss, composed of several joints and members,
provided that the desired force is one of the external forces acting on that portion. If, in addition, the portion of the truss
is chosen so that there is a total of only three unknown forces acting upon it, the desired force can be obtained by solving
the equations of equilibrium for this portion of the truss.
In practice, the portion of the truss to be utilized is obtained by passing a section through three members of the truss,
one of which is the desired member, that is, by drawing a line which divides the truss into two completely separate parts but
does not intersect more than three members.
Either of the two portions of the truss obtained after the intersected members have been removed can then be used as a
free body.
In figure 1 (a), the section nn has been passed through members BD, BE, and CE, and the portion ABC of the truss is
chosen as the free body ( figure 1b). The forces acting on the free body are the loads P1 and P2 at points A and B and
the three unknown forces FBD, FBE, and FCE. Since it is not known whether the members removed were in tension or
compression, the three forces have been arbitrarily drawn away from the free body as if the members were in tension.

3. Sample problems 6.2
Determine the force in members EF and GI of the truss shown.

4. Free body : Entire Truss
Free body : Entire Truss. A free body diagram of the entire truss is drawn;
external forces acting on this free body consist of the applied loads and the
reactions at B and J. We write the following equilibrium equations.
= 0

5. = 0

6. Force in member EF : Section nn is passed through the truss so that it intersects member EF
and only two additional members. After the intersected members have been removed, the the left hand
portion of the truss is chosen as a free body. Three unknown are involved; to eliminate the two horizontal
forces, we write
The sense of FEF was chosen assuming member EF to be in tension ; the negative sign obtained
indicates that the number is in compression.
FEF = 22.5 kN

7. Force in member GI . Section mm is passed through the truss so that it intersects
member GI and only two additional members. After the intersected members have been removed,
we choose the right-hand portion of the truss as a free body. Three unknown forces are again involved;
to eliminate the forces passing through point H, we write
= 0
FGI = 46.0 kN

8. Sample problems 6.3
Determine the force in members FH, GH, and GI of the roof truss shown

9. SOLUTION:
Take the entire truss as a free body. Apply the conditions for static equilib-rium to solve for the reactions at A and L.
30 m
tan α = GL
15 m
8 m FG =
0.5333
α = 28.07º

10. Force in member GI
Pass a section through members FH, GH, and GI and take the right-hand section as a free body.
Apply the conditions for static equilibrium to determine the desired member forces.

11. Force in member FH
Force in member GH 15

12. Problems

13. Problems 6.42
A Warren bridge truss is loaded as shown.
Determine the force in members CE, DE and DF

14. FBD Truss Ky Kx Ay
3kN

15. Ky Kx Ay
3kN A B C D E
FDE
FDF 5m
3kN
4.2kN
FCE 6m 5 13 12

16. Ky Kx Ay
3kN A B C D E
FDE
FDF 5m
3kN
4.2kN
FCE 6m

17. 6.10 Analysis of a Frame
Frames and machines are structures with at least one multiforce member. Frames are designed to support loads and are usually stationary. Machines contain moving parts and are designed to transmit and modify forces.
A free body diagram of the complete frame is used to determine the external forces acting on the frame.
Internal forces are determined by dismembering the frame and creating free-body diagrams for each component.
Forces on two force members have known lines of action but unknown magnitude and sense.
Forces on multiforce members have unknown magnitude and line of action. They must be represented with two unknown components.
Forces between connected components are equal, have the same line of action, and opposite sense.

18. 6.11 Frames which cease to be rigid when detached from their support
Some frames may collapse if removed from their supports. Such frames can not be treated as rigid bodies.
A free-body diagram of the complete frame indicates four unknown force components which can not be determined from the three equilibrium conditions.
The frame must be considered as two distinct, but related, rigid bodies.
With equal and opposite reactions at the contact point between members, the two free-body diagrams indicate 6 unknown force components.
Equilibrium requirements for the two rigid bodies yield 6 independent equations.

19. Sample problem 6.4
In the frame shown, members ACE and BCD are connected by a pin
at C and by the link DE. For the loading shown,
Determine the force in link DE and the components of the force
exerted at C on member BCD

20. SOLUTION:
Create a free-body diagram for the complete frame and solve for the support reactions.

21. Members. We now dismember the frame. Since only two members are
connected at C, the components of the unknown forces acting on ACE
and BCD are, respectively, equal and opposite and are assumed directed
as shown. We assume that link DE is in tension and exerts equal and
opposite forces at D and E, directed as shown.
Free Body : member BCD Using the free body BCD, we write
FDE = 561 N C
From the signs obtained for Cx and Cy we conclude that the force components Cx and Cy exerted on member
BCD are directed, respectively, to the left and up. We have
Cx = 795 N
Cy = 216 N

22. Check Free body : member ACE ( check )
The computations are checked by considering the free body ACE.
For example,

23. Sample problem 6.5
Determine the components of the forces acting on each member of the frame shown.

24. FBD
Free body : Entire Frame. Since the external reactions involve only three unknowns,
we compute the reactions by considering the free body diagram of the entire frame.
F = 1800 N
Ey = 600 N
Ex = 0

25. Cy = + 3600 N By = + 1200 N Members. The frame is now dismembered;
since only two members are connected at each joint,
equal and opposite components are shown on each
member at each joint.
Free Body : Member BCD
Cy = N
By = N
We note that neither Bx nor Cx can be obtained by considering
only member BCD. The positive values obtained for By and Cy
indicate that the force components By and Cy are directed as assumed

26. Bx = 0 Ax = 0 Ay = + 1800 N Cx = 0 Free Body : Member ABE
Free Body : Member BCD
Returning to member BCD, we write
Cx = 0

27. Free Body : Member ACF ( Check )
All unknown components have now been found;
to check the results, we verify that member ACF is in equilibrium.
Check

28. Sample problem 6.6
2.7 kN
0.75 m
0.75 m
0.75 m
0.75 m
1.8 m
A 2.7 kN horizontal force is applied to pin A of the frame shown.
Determine the force acting on the two vertical members of the frame.

29. 2.7 kN Free body: Entire Frame
The entire frame is chosen as a free body although the reaction involve
four unknowns, Ey and Fy may be determined by writing
2.7 kN
1.8 m
3.0 m

31. 2.7 kN
0.75 m
1.5 m
4.5 kN
2.25 m

32. 2.7 kN
0.75 m
1.5 m
4.5 kN
2.25 m

33. PROBLEMS

34. Determine the force in member AC and the reaction at B when = 30˚
Problems 6.77
90 N
100mm
250mm
50mm
200mm
Determine the force in member AC and the reaction at B when
= 30˚
= 60˚

35. Solution
90 N
100mm
250mm
50mm
200mm θ
90 N
100mm
250mm Bx By
FAC C D

36. Solution
90 N
100mm
250mm
50mm
200mm θ
90 N
100mm
250mm Bx By
FAC C D

37. Solution
90 N
100mm
250mm
50mm
200mm θ
90 N
100mm
250mm Bx By
FAC C D

39. 6.12 Machines
Photo 1 : The lamp shown can be replaced in many positions.
By considering various free bodies, the force in the springs and
the internal forces at the joints can be determined.

40. Given the magnitude of P, determine the magnitude of Q.
6.12 Machines
Machines are structures designed to transmit and modify forces. Their main purpose is to transform input forces into output forces.
Given the magnitude of P, determine the magnitude of Q.
Create a free-body diagram of the complete machine, including the reaction that the wire exerts.
The machine is a nonrigid structure. Use one of the components as a free-body.
Taking moments about A,

41. Sample Problem 6.7
A hydraulic-lift table is used to raise a 1000 kg crate. It consists of a platform and two identical
linkages on which hydraulic cylinders exert equal forces. (Only one linkage and one cylinder are
shown. Members EDB and CG are each of length 2a, and member AD is pinned to the midpoint
of EDB. If the crate is placed on the table, so that half of its weight is supported by the system
shown,
Determine the force exerted by each cylinder in raising the create for θ= 60˚ , a = 0.70 m, and
L = 3.20 m.
Show that the result obtained is independent of the distance d.

42. Solution
The machine considered consists of the platform and the linkage. Its free body diagram includes a force FDH exerted
by the cylinder, the weight ½ W, and reactions at E ad G that we assume to be directed as shown. Since more than three
unknowns are involved, this diagram will not be used. The mechanism is dismembered and a free body diagram is drawn
for each of its component parts. We note that AD, BC and CG are two force members. we already assumed member CG
to be in compression. We now assume that AD and BC arein tension; the forces exerted on them are then directed as shown.
Equal and opposite vectors will be used to represent the forces exerted by the two force members on the platform, on
member BDE, and on roller C.

47. PROBLEMS

48. Problem 6.139
Determine the magnitude of the gripping forces exerted along line aa on the nut when two 240-N forces
are applied to the handles as shown.
Assume that pins A and D slide freely in slots cut in the jaws.

49. Solution
0.01m
0.03m A By Bx G

50. 240N
0.09m
0.015m Cx Cy
Ex = 0
Ey = 4G 3
A = G A C E

51. Problem 6.140
100mm
25mm
500mm
12.5mm
340N
In using the bolt cutter shown, a worker applies two 340 N forces to the handles.
Determine the magnitude of the forces exerted by the cutter on the bolt.

52. II

53. THE END