Truss: Types and Classification. Theory of Structure - I

Transcription

1 Truss: Types and Classification Theory of Structure - I

2 Lecture Outlines Common Types of Trusses Classification of Trusses Method of Joints Zero Force Members Method of Sections 2

3 TRUSS ANALYSIS A truss is a structure comprising one or more triangular units constructed with straight members whose ends are connected at joints or nodes. If all the bars lie in a plane, the structure is a planar truss. The main parts of a planar truss. In other words, Trusses are designed to form a stable structure.

4 Why are trusses strong? Trusses derive their strength from the triangle. The simplest of plane polygons, a triangle is unique in that it is defined by the length of its sides. That is, one and only one triangle can be drawn if the length of all three sides is given.

5 TRUSS ANALYSIS Why are triangles used in trusses? Rectangles and squares are not very strong because the middle of each side would tend to bend or buckle easily. And these are not used in truss. A truss is a structure made up of triangles. Because triangles are strong because when you define the length of the three sides the relationship between the nodes is fixed. Similarly when you identify any two angles an a side or two sides and a common angle all other properties are fixed. In any other shape there are more degrees of rigidity required to create a fixed structure. Triangles have sides that reinforce each other. They divide up the load.

6 Truss Bridges A metal truss bridge is a bridge whose main structure comes from a triangular framework of structural steel or iron.

7 Iron and Steel Due to their variety of designs, there is a system that is used to classify metal truss bridges by design.

8 Truss Basics If the trusses run beside the deck, with no cross bracing above the deck, it is called a pony truss bridge. Pony Truss Through Truss If cross-bracing is present above the deck of the bridge, then the bridge is referred to as a through truss.

9 Trusses may run under the deck: these are called simply Deck truss bridges. Truss Basics Deck Truss

10 Truss Bridge Forces Compression Tension The chords and members of a truss bridge experience strain in the form of tension (stretching apart) and compression (squeezing together). Engineers often picked different types of materials and designs for the different parts of a bridge based on these forces. An example is shown above.

11 Truss Bridge Connections The pieces of the framework of a truss bridge are held together by connections. Most connections on historic bridges are either riveted or pinned.

12 Pinned Connections Pin Pinned connections can be identified by the bolt-like object called a pin going through the loops of the members. They tend to show up on bridges from the first half of the truss bridge era.

13 Riveted connections are identified by a gusset plate which diagonals and vertical members are riveted to, and no pin is present. These connections tend to show up in the second half of the truss bridge era.

14 Common Types of Trusses gusset plate Roof Trusses roof purlins top cord knee brace bottom cord gusset plate span bay 14

15 Howe truss Pratt truss howe truss Warren truss saw-tooth truss Fink truss three-hinged arch

16 trough Pratt truss Warren truss deck Pratt truss parker truss (pratt truss with curved chord) Howe truss baltimore truss K truss

17 Bridge Trusses top lateral bracing portal bracing stringers portal end post sway bracing panel floor beam top cord bottom cord deck 17

18 Assumptions for Design All members are connected at both ends by smooth frictionless pins. All loads are applied at joints (member weight is negligible). Notes: Centroids of all joint members coincide at the joint. All members are straight. All load conditions satisfy Hooke s law. 18

19 Classification of Coplanar Trusses Simple Trusses P C D P C D P C A B A B A B a new members d (new joint) b c 19

20 Compound Trusses simple truss simple truss simple truss simple truss Type 1 Type 2 secondary simple truss secondary simple truss main simple truss Type 3 secondary simple truss secondary simple truss 20

21 Complex Trusses Determinacy b + r = 2j b + r > 2j statically determinate statically indeterminate In particular, the degree of indeterminacy is specified by the difference in the numbers (b + r) - 2j. 21

22 Stability b + r < 2j unstable b + r > 2j unstable if truss support reactions are concurrent or parallel or if some of the components of the truss form a collapsible mechanism External Unstable Unstable-parallel reactions Unstable-concurrent reactions 22

23 Internal Unstable F C = 11 < 2(6) D A O B E AD, BE, and CF are concurrent at point O 23

24 Example 3-1 Classify each of the trusses in the figure below as stable, unstable, statically determinate, or statically indeterminate. The trusses are subjected to arbitrary external loadings that are assumed to be known and can act anywhere on the trusses. 24

25 SOLUTION b = 19, r = 3, j = 11, then b + r = 2j or 22 = 22. Therefore, the truss is statically determinate. By inspection the truss is internally stable. b = 15, r = 4, j = 9, then b + r > 2j or 19 > 18. The truss is statically indeterminate to the first degree. By inspection the truss is internally stable. 25

26 Since b = 9, r = 3, j = 6, then b + r = 2j or 12 = 12. The truss is statically determinate. By inspection the truss is internally stable. b = 12, r = 3, j = 8, then b + r < 2j or 15 < 16. The truss is internally unstable. 26

27 Method of Joints If a truss is in equilibrium, then each of its joints must also be in equilibrium The method of joints consists of satisfying the equilibrium conditions for the forces exerted on the pin at each joint of the truss 27

28 Truss members are all straight two-force members lying in the same plane The force system acting at each pin is coplanar and concurrent (intersecting) Rotational or moment equilibrium is automatically satisfied at the joint, only need to satisfy Fx = 0, Fy = 0 28

29 Procedure for Analysis Draw the free-body diagram of a joint having at least one known force and at most two unknown forces (may need to first determine external reactions at the truss supports) 29

30 Establish the sense of the unknown forces Always assume the unknown member forces acting on the joint s free-body diagram to be in tension (pulling on the pin ) Assume what is believed to be the correct sense of an unknown member force In both cases a negative value indicates that the sense chosen must be reversed 30

31 Orient the x and y axes such that the forces can be easily resolved into their x and y components Apply Fx = 0 and Fy = 0 and solve for the unknown member forces and verify their correct sense Continue to analyze each of the other joints, choosing ones having at most two unknowns and at least one known force 31

32 Members in compression push on the joint and members in tension pull on the joint Mechanics of Materials and building codes are used to size the members once the forces are known 32

33 The Method of Joints B 500 N 2 m A x = 500 N A 45 o C A y = 500 N 2 m C y = 500 N Joint B y B 500 N x + SF x = 0: F BC sin45 o = 0 F BC = N (C) F BA 45 o F BC + SF y = 0: - F BA + F BC cos45 o = 0 F BA = 500 N (T) 33

34 B 500 N 2 m A x = 500 N A 45 o C A y = 500 N 2 m C y = 500 N Joint A 500 N + SF x = 0: 500 N F AC F AC = 0 F AC = 500 N (T) 500 N 34

35 Zero Force Members Truss analysis using the method of joints is greatly simplified if one is able to determine those members which support no loading (zero-force members) These zero-force members are used to increase stability of the truss during construction and to provide support if the applied loading is changed 35

36 If only two members form a truss joint and no external load or support reaction is applied to the joint, the members must be zero-force members. If three members form a truss for which two of the members are collinear, the third member is a zero-force member provided no external force or support reaction is applied. 36

37 Zero-force members Frequently the analysis can be simplified by identifying members that carry no load two typical cases are found When only two members form a non-collinear joint and there is no external force or reaction at that joint, then both members must be zero-force A E P B D C T CB T CD C If either T CB or T CD 0, then C cannot be in equilibrium, since there is no restoring force towards the right. Hence both BC and CD are zero-load members here. 37

38 When three members form a truss joint for which two members are collinear and the third is at an angle to these, then this third member must be zero-force in the absence of an external force or reaction from a support A B P C Here, joint B has only one force in the vertical direction. D T AB B T BC Hence, this force must be zero or B would move (provided there are no external loads/reactions) E T BD Also T AB = T BC 38

39 While zero-force members can be removed in this configuration, care should be taken any change in the loading can lead to the member carrying a load the stability of the truss can be degraded by removing the zeroforce member A B P C You may think that we can remove AD and BD to make a triangle This satisfies the statics requirements E D However, this leaves a long CE member to carry a compressive load. This long member is highly susceptible to failure by buckling. 39

40 P B C A E D D x E y D y F C + CB SF x = 0: F CB = 0 + SF y = 0: F CD = 0 F CD F AB + SF y = 0: F AB sinq = 0, F AB = 0 A q F AE + SF x = 0: F AE + 0 = 0, F AE = 0 40

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