Chapter Objectives. To show how to determine the forces in the members of a truss using: the method of joints and the method of sections.

Transcription

1 Structural Analysis

2 Chapter Objectives To show how to determine the forces in the members of a truss using: the method of joints and the method of sections.

3 Chapter Outline Two-force members Planar (Simple) Trusses The Method of Joints Zero-Force Members The Method of Sections

4 Two-Force Members When a member is subjected to no couple moments and forces are applied at: only two points on a member, then the member is called a two-force member

5 Two-Force Members Example Forces at A and B are summed to obtain their respective resultants F A and F B These two forces will maintain translational and force equilibrium provided F A is of equal magnitude and opposite direction to F B Line of action of both forces is known and passes through A and B

6 When any member is subjected to: No couple moments and TWO EQUAL, OPPOSITE and F A B F COLLINEAR forces are applied at: only two points on a member, F A two-force member (in equilibrium) two-force member (in equilibrium) B F then this member is called a two-force member and this member is in equilibrium.

7 When any member is subjected to: No couple moments and TWO EQUAL, OPPOSITE and COLLINEAR forces are applied at: only two points on a member, then this member is called a two-force member and this member is in equilibrium. F F F F F A B NOT a two-force member (NOT in EQUILIBRIUM) F A B M 0 NOT a two-force member (NOT in EQUILIBRIUM) F A B NOT a two-force member (NOT in EQUILIBRIUM) F A B F NOT a two-force member (NOT in EQUILIBRIUM) M 0 A B M 0 NOT a two-force member (NOT in EQUILIBRIUM) F A B NOT a two-force member (NOT in EQUILIBRIUM)

8 Two-Force Members

9 PLANAR Trusses OR SIMPLE Trusses OR 2-D Trusses

10 Simple Trusses A simple truss is constructed starting with a basic triangular element such as ABC and connecting two members (AD and BD) to form an additional element.

11 Simple Trusses To prevent collapse, the form of a truss must be rigid The four bar shape ABCD will collapse if a diagonal member AC is not added The simplest form that is rigid or stable is a triangle

12 Simple Trusses ALL the truss members, member-end forces and applied forces are in one plane (2D), (in the same plane). That is why it is called PLANAR or 2-D TRUSSES.

13 Simple Trusses JOINT (NODE) MEMBER A truss is a structure composed of slender members joined together at their end points

14 Simple Trusses Joint connections are formed by bolting or welding the ends of the members to a common plate, called a gusset plate, or by simply passing a large bolt or pin through each of the members

15 Simple Trusses Planar trusses lie on a single plane and are used to support roofs and bridges. The truss ABCDE shows a typical roof-supporting truss Roof load is transmitted to the truss at joints by means of a series of purlins, such as DD

16 Simple Trusses

17 Simple Trusses The loads transmitted at the joints are resisted by the members of the truss and forces are developed at the ends of each member. Finally all the loads transmitted to the truss are in turn transmitted to the supports and then to the supporting soil.

18 Simple Trusses K Bridge Truss Warren Roof Truss

19 Simple Trusses The loads transmitted at the joints are resisted by the members of the truss and forces are developed at the ends of each member Member-end force Member-end force forces are developed at the ends of each member is called the Member-end force Each member-end force is either or TENSION COMPRESSION

20 Simple Trusses If the force tends to elongate the member, it is a tensile force, If the force tends to shorten the member, it is a compressive force.

21 Simple Trusses Each member is in equilibrium under the action of only two forces that are: - equal in magnitude, - opposite in direction and - collinear. Therefore truss members are called: TWO-FORCE MEMBERS

22 Simple Trusses Analysis of Planar Trusses means: Find ALL the member-end forces developed in each member, Find the reactions at the supports

23 Simple Trusses Assumptions for Analysis and Design 1. All loadings are applied at the joints Assumption is true for most of the applications of bridge and roof trusses Weight of the members are neglected since the forces supported by the members are large in comparison If member s weight is considered, apply it as a vertical force, by giving half of the magnitude to each end of the member.

24 Simple Trusses Assumptions for Analysis and Design 2. The members are joined together by smooth pins Assumption true when bolted or welded joints are used, provided the center lines of the joining members are concurrent smooth pin

25 Simple Trusses Each truss member acts as a two force member, therefore the forces at the ends must be directed along the axis of the member. If the force tends to elongate the member, it is a tensile force, If the force tends to shorten the member, it is a compressive force.

26 Simple Trusses Important to state the nature of the force in the actual design of a truss: tensile or compressive. Compression members must be made thicker than tensile member to account for the buckling or column effect during compression.

27 The Method of Joints For design and analysis of a truss, we need to obtain the force in each of the members. Dis-assemble the truss and draw the FBD of each pin and each member. Pin at joint C Considering the equilibrium of a pin at a joint of the truss, a member-end force becomes an external force on the pin at a joint and equations of equilibrium can be applied. This forms the basis for the method of joints Action of member-end forces on the pins at the joints: EQUAL and OPPOSITE force is applied on the pin

28 The Method of Joints y 1 2 x 3 Pin at joint C A X A Y C Y Action of member-end forces on the pins at the joints: EQUAL and OPPOSITE force is applied on the pin

29 The Method of Joints The force system acting at each pin at a joint is coplanar and concurrent F x = 0 and F y = 0 must be satisfied for equilibrium F x = 0 and F y = 0 F x = 0 and F y = 0 F x = 0 and F y = 0

30 SOLUTION BY METHOD OF JOINT Method of Analyses 1) If x and y coordinates are not given show it on FBD. This will help to obtain the x and y components of all the forces acting on that joint. 2) Assume all the unknown forces acting on the members to be in TENSION. 3) Draw FBD for each joint and attempt to solve it by using Equilibrium equations. To start with select that joint that has: - at least (minimum) one known force and - at most (maximum) two unknown forces. 4) If the joint has a support reaction you can find it earlier and then apply method of Joint

31 SOLUTION METHOD OF JOINT Method of Analyses 5) Apply equilibrium equations F x = 0 and F y = 0 6) Solve the unknown forces based on that joints FBD. 7) APPLY THE SAME METHODOLOGY FOR THE REST OF THE JOINTS.

32 Example: The Method of Joints Determine the force in each member of the truss and state if the members are in tension or in compression. Use method of joints. y x Ay

33 5.125 m Example: The Method of Joints 1. Determine the forces in all the members of the truss and state whether they are in tension T or compression C. use the method of joints. give your results in tabular form 2. Find the reactions at the supports A and E. y P 1 B 6 7 C D x P P 1 = kn P 2 = kn A 1 E = m 3.5 m 2 m

34 The reactions and member forces of the truss: Ax= -5 kn Ay= kn Ey= kn member F1 F2 F3 F4 F5 F6 F7 force 2.17 kn kn kn kn kn kn kn type T T C C C T T

35 1. Find the reactions at the pin supports A and B. 2. Determine the forces in all the members of the truss and state whether they are in tension (T) or compression (C). Use the method of joints. Give your results in tabular form. Note that members 2 and 6 are collinear! Example: The Method of Joints y 1 2 x

36 The reactions and member forces of the truss: Ax= 0 Ay= kn Bx= kn By=2100 kn member F1 F2 F3 F4 F5 F6 force 1200 kn kn 0 kn 2100 kn kn kn type T C ---- T C C

37 2.85 m Example T: The Method of Joints y x 1. Find the reactions at the pin supports A and B. 2. Determine the forces in all the members of the truss and state whether they are in tension (T) or compression (C). Use the method of joints. Give your results in tabular form. Note that member 1 and 5 are NOT COLLINEAR!

38 Example: The Method of Joints 1. Find the reactions at the pin supports A and G. 2. Determine the forces in all the members of the truss and state whether they are in tension (T) or compression (C). Use the method of joints. Give your results in tabular form. 12 kn 7 kn 2.1 m L M N y 6 kn x 3.5 m A B C E 2.5 m 3.2 m 3.2 m 2.5 m G

39 Example: The Method of Joints Finding the effective forces acting on the joints L and M F y = -7-11= -19 kn M L = kn m F at M = /3.2 = kn So F at L = -19 ( ) = kn 1.1 m 12 kn 7 kn 2.1 m kn kn L 3.2 m M L 3.2 m M

40 Example: The Method of Joints y kn kn x L M N 6 kn 3.5 m A B C E 2.5 m 3.2 m 3.2 m 2.5 m G

41 Example T: The Method of Joints 1. Find the reactions at the pin supports A and F. 2. Determine the forces in all the members of the truss and state whether they are in tension (T) or compression (C). Use the method of joints. Give your results in tabular form m 1.60 m 0.75 m C 1.60 m 6 D kn kn B 3 E 8 F kn A 1.85 m

42 ZERO-FORCE Members Method of joints is simplified when the members which support no loading are determined. Within a truss, zero-force member exists due to: Shape of the member connections Support reaction types.

43 ZERO-FORCE Members Shape of the truss Zero-force members (support no loading ) are used to increase the stability of the truss during construction and to provide support if the applied loading is changed.

44 ZERO-FORCE Members Shape of the truss Consider the truss shown From the FBD of the pin at point A, members AB and AF become zero force members

45 ZERO-FORCE Members Shape of the truss Zero-force members (support no loading ) are used to increase the stability of the truss during construction and to provide support if the applied loading is changed. 0 0

46 ZERO-FORCE Members Shape of the truss Consider FBD of joint D DC and DE are zero-force members.

47 ZERO-FORCE Members Shape of the truss Consider FBD of joint D DC and DE are zero-force members. 0 0

48 ZERO-FORCE Members Shape of the truss General rule 1: IF only two members form a truss joint and no external load or support reaction is applied to the joint, THEN BOTH members must be zero-force members

49 ZERO-FORCE Members Shape of the truss Consider the truss shown: From the FBD of the pin of the joint D, DA is a zero-force member From the FBD of the pin of the joint C, CA is a zero-force member

50 ZERO-FORCE Members Shape of the truss Consider the truss shown: From the FBD of the pin of the joint D, DA is a zero-force member From the FBD of the pin of the joint C, CA is a zero-force member 0 0

51 General Rule 2: ZERO-FORCE Members Shape of the truss IF three members form a truss joint for which two of the members are collinear, also having the same magnitude and sign; THEN the third member is a zero-force member provided no external force or support reaction is applied to the joint

52 ZERO-FORCE Members Shape of the truss FOR ANY JOINT IF 1- NO SUPPORT 2- NO EXTERNAL LOADING ONLY TWO MEMBERS BOTH ARE ZERO MEMBERS ONLY THREE MEMBERS BUT TWO OF THEM ARE COLLINEAR THE REMAINING MEMBER IS ZERO MEMBER

53 ZERO-FORCE Members Shape of the truss

54 ZERO-FORCE Members Shape of the truss 0

55 ZERO-FORCE Members Shape of the truss Example: Determine all the zero-force members of the Fink roof truss. Assume all joints are pin connected.

56 ZERO-FORCE Members Shape of the truss 0

57 ZERO-FORCE Members Shape of the truss 0 0

58 ZERO-FORCE Members Shape of the truss 0 0 0

59 ZERO-FORCE Members Shape of the truss

60 0 ZERO-FORCE Members Shape of the truss

61 ZERO-FORCE Members Shape of the truss 0 0

62 ZERO-FORCE Members Shape of the truss 0 0 0

63 ZERO-FORCE Members Shape of the truss

64 ZERO-FORCE Members Shape of the truss

65 ZERO-FORCE Members Shape of the truss

66 ZERO-FORCE Members Shape of the truss

67 ZERO-FORCE Members Shape of the truss

68 ZERO-FORCE Members Shape of the truss 0

69 ZERO-FORCE Members Shape of the truss 0 0 0

70 ZERO-FORCE Members Support types 10 kn B 2 kn 5 kn C 4 m A D 2 kn 5 kn 2 kn 5 kn 10 kn B C 10 kn B C 4 m A D 4 m A 0 D -F CD = D y Dy Dy

71 ZERO-FORCE Members Support types 2 kn 5 kn 10 kn B C 4 m A D 2 kn 5 kn 2 kn 5 kn 10 kn B C 10 kn B C 4 m A D Dx 4 m A -F AD = D x 0 D Dx

72 ZERO-FORCE Members Support types 2 kn 5 kn 10 kn B C 4 m A D 10 kn B 2 kn 5 kn C 4 m A -F AD = D x D -F CD = D y Dx No zero member Dy

73 ZERO-FORCE Members Support types 2 kn 5 kn 10 kn B C 4 m A D 4 m 10 kn B A 2 kn 5 kn C D No zero member Ay

74 ZERO-FORCE Members Support types 5 kn 10 kn B C 4 m A D 4 m 10 kn -F AB = A y A B kn C -F CD = D y D Dx = 0 Ay Dy

75 ZERO-FORCE Members Shape of the truss 4 m 4 m 5 kn 10 kn 10 kn C B D 5 kn E F 4 m A G 4 m

76 ZERO-FORCE Members Shape of the truss 4 m 4 m 5 kn 10 kn 10 kn C B D kn E F 4 m A G 4 m

77 ZERO-FORCE Members Shape of the truss 4 m 4 m 5 kn 10 kn 10 kn C B D kn E F 4 m A G 4 m

78 ZERO-FORCE Members Shape of the truss D 4 m 5 kn 10 kn C kn E 4 m 10 kn B 0 F 4 m A 0 G 4 m

79 Summary Truss Analysis A simple truss consists of triangular elements connected by pin joints. The force within determined by assuming all the members to be two force member, connected concurrently at each joint.

80 Summary Method of Joints For a coplanar truss, the concurrent force at each joint must satisfy force equilibrium For numerical solution of the forces in the members, select a joint that has FBD with at most 2 unknown and 1 known forces

81 Summary Method of Joints Once a member force is determined, use its value and apply it to an adjacent joint. Forces that PULL the joint are in Tension. Forces that PUSH the joint are in Compression.

82 Summary Method of Joints To avoid simultaneous solution of two equations, sum the force in a direction that is perpendicular to one of the unknown. To simplify problem-solving, first identify all the zero-force members.