Chapter 6 Structural Analysis pdf slides notes
- 1. CHAPTER-6 Structural Analysis Book: ▪ Engineering Mechanics Statics, R. C. Hibbeler, 14th Ed
- 2. STRUCTURAL ANALYSIS • Simple Trusses • The Method of Joints • The Method of Sections
- 3. SIMPLE TRUSSES • A truss is a structure composed of slender members joined together at their end points • The members commonly used in construction consist of wooden struts/tie or metal bars • In particular, planar trusses lie in a single plane and are often used to support roofs and bridges • The truss shown is an example of a typical roof-supporting truss • In this figure, the roof load is transmitted to the truss at the joints by means of a series of purlins Since this loading acts in the same plane as the truss, the analysis of the forces developed in the truss members will be two-dimensional
- 4. SIMPLE TRUSSES: POWER TRANSMISSION TOWER
- 5. SIMPLE TRUSSES • In the case of a bridge, such as shown, the load on the deck is first transmitted to stringers, then to floor beams, and finally to the joints of the two supporting side trusses • Like the roof truss, the bridge truss loading is also coplanar • When bridge or roof trusses extend over large distances, a rocker or roller is commonly used for supporting one end, for example, joint A in figure • This type of support allows freedom for expansion or contraction of the members due to a change in temperature or application of loads
- 6. SIMPLE TRUSSES Assumptions for Design: • To design both the members and the connections of a truss, it is necessary first to determine the force developed in each member when the truss is subjected to a given loading • To do this we will make two important assumptions: i) All loadings are applied at the joints ii) The members are joined together by smooth pins
- 7. SIMPLE TRUSSES – Assumptions for Design i) All loadings are applied at the joints: – In most situations, such as for bridge and roof trusses, this assumption is true – Frequently the weight of the members is neglected because the force supported by each member is usually much larger than its weight – However, if the weight is to be included in the analysis, it is generally satisfactory to apply it as a vertical force, with half of its magnitude applied at each end of the member
- 8. SIMPLE TRUSSES – Assumptions for Design ii) The members are joined together by smooth pins: – The joint connections are usually formed by bolting or welding the ends of the members to a common plate, called a gusset plate, as shown in figure (a), or by simply passing a large bolt or pin through each oft the members, figure (b) – We can assume these connections act as pins provided the center lines of the joining members are concurrent
- 9. SIMPLE TRUSSES – Assumptions for Design • Because of these two assumptions, each truss member will act as a two force member, and therefore the force acting at each end of the member will be directed along the axis of the member • If the force tends to elongate the member, it is a tensile force (T), whereas if it tends to shorten the member, it is a compressive force (C) • In the actual design of a truss it is important to state whether the nature of the force is tensile or compressive • Often, compression members must be made thicker than tension members because of the buckling or column effect that occurs when a member is in compression
- 10. SIMPLE TRUSSES – Assumptions for Design • If three members are pin connected at their ends they form a triangular truss that will be rigid • Attaching two more members and connecting these members to a new joint D forms a larger truss • This procedure can be repeated as many times as desired to form an even larger truss • If a truss can be constructed by expanding the basic triangular truss in this way, it is called a simple truss.
- 11. STRUCTURAL ANALYSIS • Simple Trusses • The Method of Joints • The Method of Sections
- 12. THE METHOD OF JOINTS • In order to analyze or design a truss, it is necessary to determine the force in each of its members • One way to do this is to use the method of joints • This method is based on the fact that if the entire truss is in equilibrium, then each of its joints is also in equilibrium • Therefore, if the free-body diagram of each joint is drawn, the force equilibrium equations can then be used to obtain the member forces acting on each joint • Since the members of a plane truss are straight two-force members lying in a single plane, each joint is subjected to a force system that is coplanar and concurrent • As a result, only ƩFx = 0 and ƩFy = 0 need to be satisfied for equilibrium
- 13. THE METHOD OF JOINTS • For example, consider the pin at joint B of the truss shown • Three forces act on the pin, namely, the 500-N force and the forces exerted by members BA and BC • The free-body diagram of the pin is also shown • Here, FBA is “pulling” on the pin, which means that member BA is in tension; whereas FBC is “pushing” on the pin, and consequently member BC is in compression • These effects are clearly demonstrated by isolating the joint with small segments of the member connected to the pin as shown • The pushing or pulling on these small segments indicates the effect of the member being either in compression or tension
- 14. THE METHOD OF JOINTS • When using the method of joints, always start at a joint having at least one known force and at most two unknown forces • In this way, application of ƩFx = 0 and ƩFy = 0 yields two algebraic equations which can be solved for the two unknowns
- 15. THE METHOD OF JOINTS • When applying these equations, the correct sense of an unknown member force can be determined using one of two possible methods: Method 1: – In simpler cases, the correct sense of direction of an unknown member force can be determined “by inspection” – In more complicated cases, the sense of an unknown member force can be assumed; then, after applying the equilibrium equations, the assumed sense can be verified from the numerical results – A positive answer indicates that the sense is correct, whereas a negative answer indicates that the sense shown on the free-body diagram must be reversed Method 2: – Always assume the unknown member forces acting on the joint’s free-body diagram to be in tension; i.e., the forces “pull” on the pin – If this is done, then numerical solution of the equilibrium equations will yield positive scalars for members in tension and negative scalars for members in compression – Once an unknown member force is found, use its correct magnitude and sense (T or C) on subsequent joint free-body diagrams
- 16. THE METHOD OF JOINTS Procedure for Analysis • Draw the free-body diagram of a joint having at least one known force and at most two unknown forces. (If this joint is at one of the supports, then it may be necessary first to calculate the external reactions at the support) • Use one of the two methods described above for establishing the sense of an unknown force • Orient the x and y axes such that the forces on the free-body diagram can be easily resolved into their x and y components and then apply the two force equilibrium equations. Solve for the two unknown member forces and verify their correct sense • Using the calculated results, continue to analyze each of the other joints. Remember that a member in compression “pushes” on the joint and a member in tension “pulls” on the joint. Also, be sure to choose a joint having at most two unknowns and at least one known force
- 19. EXAMPLE 6-1 Determine the force in each member of the truss shown and indicate whether the members are in tension or compression.
- 22. EXAMPLE 6-2 Determine the force in each member of the truss shown below.
- 24. EXAMPLE 6-3 Determine the force in each member of the truss shown. Indicate whether the members are in tension or compression.
- 29. THE METHOD OF JOINTS Examples: 6.1, 6.2, 6.3, 6.4 Fundamental Problems: F6-2, F6-5, F6-6 Practice Problems: 6-2, 6-3, 6-7, 6-12, 6-15, 6-20, 6-26
- 30. PROBLEM 6-7 Determine the force in each member of the truss and state if the members are in tension or compression.
- 34. PROBLEM 6-20 Determine the force in each member of the truss and state if the members are in tension or compression. Set P1 = 9 kN and P2 = 15 kN.
- 35. PROBLEM 6-26 The maximum allowable tensile force in the members of the truss is (FT)max = 5 kN, and the maximum allowable compressive force is (FC)max = 3 kN. Determine the maximum load P of the two loads that can be applied to the truss.
- 38. STRUCTURAL ANALYSIS • Simple Trusses • The Method of Joints • The Method of Sections
- 39. THE METHOD OF SECTIONS • When we need to find the force in only a few members of a truss, we can analyze the truss using the method of sections • It is based on the principle that if the truss is in equilibrium then any segment of the truss is also in equilibrium • For example, consider the two truss members shown. If the forces within the members are to be determined, then an imaginary section, indicated by the blue line, can be used to cut each member into two parts and thereby “expose” each internal force as “external” to the free-body diagrams shown on the right • It can be seen that equilibrium requires that the member in tension (T) be subjected to a “pull,” whereas the member in compression (C) is subjected to a “push”
- 40. THE METHOD OF SECTIONS • The method of sections can also be used to “cut” or section the members of an entire truss • If the section passes through the truss and the free-body diagram of either of its two parts is drawn, we can then apply the equations of equilibrium to that part to determine the member forces at the “cut section” • Since only three independent equilibrium equations can be applied to the free-body diagram of any segment, then we should try to select a section that, in general, passes through not more than three members in which the forces are unknown • For example, consider the truss shown:
- 41. THE METHOD OF SECTIONS • If the forces in members BC, GC, and GF are to be determined, then section aa would be appropriate • The free-body diagrams of the two segments are shown • The line of action of each member force is specified from the geometry of the truss, since the force in a member is along its axis • Also, the member forces acting on one part of the truss are equal but opposite to those acting on the other part—Newton’s third law • Members BC and GC are assumed to be in tension since they are subjected to a “pull,” whereas GF in compression since it is subjected to a “push” (a) (b)
- 42. THE METHOD OF SECTIONS • The three unknown member forces FBC, FGC and FGF can be obtained by applying the three equilibrium equations to the free-body diagram in figure (a) • If, however, the free-body diagram in figure (b) is considered, the three support reactions Dx, Dy and Ex will have to be known, because only three equations of equilibrium are available • This can be done in the usual manner by considering a free-body diagram of the entire truss (a) (b)
- 43. THE METHOD OF SECTIONS • When applying the three equations of equilibrium, the correct sense of an unknown member force can be determined using one of two possible methods: Method 1: – In simpler cases, the correct sense of direction of an unknown member force can be determined “by inspection” – In more complicated cases, the sense of an unknown member force can be assumed; then, after applying the equilibrium equations, the assumed sense can be verified from the numerical results – A positive answer indicates that the sense is correct, whereas a negative answer indicates that the sense shown on the free-body diagram must be reversed Method 2: – Always assume the unknown member forces acting on the joint’s free-body diagram to be in tension; i.e., the forces “pull” on the pin – If this is done, then numerical solution of the equilibrium equations will yield positive scalars for members in tension and negative scalars for members in compression – Once an unknown member force is found, use its correct magnitude and sense (T or C) on subsequent joint free-body diagrams
- 44. THE METHOD OF SECTIONS Procedure for Analysis • Make a decision on how to “cut” or section the truss through the members where forces are to be determined • Before isolating the appropriate section, it may first be necessary to determine the truss’s support reactions. If this is done then the three equilibrium equations will be available to solve for member forces at the section • Draw the free-body diagram of that segment of the sectioned truss which has the least number of forces acting on it • Use one of the two methods described above for establishing the sense of the unknown member forces • Moments should be summed about a point that lies at the intersection of the lines of action of two unknown forces, so that the third unknown force can be determined directly from the moment equation • If two of the unknown forces are parallel, forces may be summed perpendicular to the direction of these unknowns to determine directly the third unknown force
- 45. THE METHOD OF SECTIONS Examples: 6.5, 6.6, 6.7 Fundamental Problems: F6-7, F6-9, F6-11 Practice Problems: 6-28, 6-29, 6-32, 6-34, 6-38, 6-42
- 46. EXAMPLE 6-5 Determine the force in members GE, GC, and BC of the truss shown. Indicate whether the members are in tension or compression.
- 49. EXAMPLE 6-6 Determine the force in member CF of the truss shown. Indicate whether the member is in tension or compression. Assume each member is pin connected.
- 50. EXAMPLE 6-7 Determine the force in member EB of the roof truss shown. Indicate whether the member is in tension or compression.
- 51. PROBLEM 6-29 Determine the force in members HG, HE and DE of the truss, and state if the members are in tension or compression.
- 52. PROBLEM 6-34 The Howe truss is subjected to the loading shown. Determine the force in members GH, BC and BG of the truss and state if the members are in tension or compression.
- 53. PROBLEM 6-42 Determine the force in members BC, HC and HG of the truss. State if the members are in tension or compression.