Chapter - 1 Analysis of Truss Dr. Rajesh Sathiyamoorthy Department of Civil Engineering, IIT Kanpur hsrajesh@iitk.ac.in; http://home.iitk.ac.in/~hsrajesh/ Simple Trusses A truss is a structure composed of slender rigid members joined together at their end points. Planar trusses lie in a single plane and if the loading acts in the same plane as the truss, the analysis of the forces developed in the truss members will be two-dimensional. System Idealisation The roof load is transmitted to the truss at the joints 1
Simple Trusses Simple Trusses 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. Assumptions for analysis and design 1) All loadings are applied at the joints. Frequently the weight of the members is neglected because the force supported by each member is usually much larger than its weight. 2) 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, or by simply passing a large bolt or pin through each of the members 2
Formation of Simple Trusses The basic element of a plane truss is the triangle. Three bars joined by pins at their ends, constitute a rigid frame. Attaching two more members and connecting these members to a new joint D forms a larger truss If a truss can be constructed by expanding the basic triangular truss in this way, it is called a simple truss. The term rigid is used to mean non-collapsible and also to mean that t deformation of the members due to induced internal strains is negligible. Method of Joints Method of analysis Method of Sections Analysis of Truss Method of Joint This method is based on the fact that if the entire truss is in equilibrium, then each of its joints is also in equilibrium. The FBD of each joint is drawn. The force equilibrium equations can then be used to obtain the member forces acting on each joint. Method of Section This method is based on the fact that if the entire truss is in equilibrium, then any segment of the truss is also in equilibrium. A section is passed through the truss and the FBD of either of its two parts is drawn. 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 FBD of any segment, then we should try to select a section passes through not more than three members in which the forces are unknown. 3
Internal and External Redundancy If a plane truss has more external supports than are necessary to ensure a stable equilibrium configuration, the truss as a whole is statically indeterminate, and the extra supports constitute external redundancy. If a truss has more internal members than are necessary to prevent collapse when the truss is removed from its supports, then the extra members constitute internal redundancy and the truss is statically indeterminate. For the plane truss composed of m two-force members and having the maximum of three unknown support reactions, there are in all m + 3 unknowns. At each joint, 2 equilibrium conductions exist. Thus, for any plane truss m + 3 (i.e. unknowns) = 2j (i.e. equilibrium condition). Hence (m + 3 = 2j) will be satisfied if the truss is statically determinate internally. If m + 3 > 2j, there are more members than independent equations, and the truss is statically indeterminate internally with redundant members present. If m + 3 < 2j, there is a deficiency of internal members, and the truss is unstable and will collapse under load. For the space truss composed of m two-force members and having the maximum of six unknown support reactions, there are m + 6 unknowns. At each joint, 3 equilibrium conductions exist. Thus, for any space truss, the equation (m + 6 = 3j) will be satisfied if the truss is statically determinate internally. Determine the force in each member of the truss and indicate whether the members are in tension or compression.. Method of Joints 4
Determine the force in the members GE, GC and BC of the truss and indicate whether the members are in tension or compression.. Method of Section Chapter - 1 Friction Dr. Rajesh Sathiyamoorthy Department of Civil Engineering, IIT Kanpur hsrajesh@iitk.ac.in; http://home.iitk.ac.in/~hsrajesh/ 5
Friction Friction is a force that resists the movement of two contacting surfaces that slide relative to one another. This force always acts tangent to the surface at the points of contact and is directed so as to oppose the possible or existing motion between the surfaces. Dry friction (Coulomb friction) occurs between the contacting surfaces of bodies when there is no lubricating fluid Theory of dry friction Friction Impending motion Motion coefficient of static friction angle of static friction coefficient of kinetic friction angle of kinetic friction 6
FBD The uniform 10-kg ladder rests against the smooth wall at B, and the end A rests on the rough horizontal plane for which the coefficient of static friction is 0.3. Determine the angle of inclination of the ladder and the normal reaction at B if the ladder is on the verge of slipping. 7
Find the range of values of W which will hold the block of weight (w 0 ) equal to 500 N in equilibrium on the inclined plane if the coefficient of friction is 0.5. Direction? Blocks A and B have a mass of 3 kg and 9 kg, respectively, and are connected to the weightless links. Determine the largest vertical force P that can be applied at the pin C without causing any movement. The coefficient of static friction between the blocks and the contacting surfaces is 0.3. 8
A light step ladder is resting on floor. Estimate the force in link AB when a man of weight equal to 800N stands on top of the ladder. Without considering friction Considering friction F 1 F 2 Screw Jack In most cases, screws are used as fasteners; however, in many types of machines they are incorporated to transmit power or motion from one part of the machine to another. Square-threaded screw θ 9
Screw Jack Square-threaded screw Upward impending screw motion Screw Jack Square-threaded screw Downward impending screw motion when (ϕs > θ), self locking screws! For general case ± Efficiency of the screw jack 10
The turnbuckle has a square thread with a mean radius of 5 mm and a pitch of 2 mm. If the coefficient of static friction between the screw and the turnbuckle is 0.25, determine the moment M that must be applied to draw the end screws closer together. Since friction at two screws must be overcome Frictional Forces on Flat Belts Whenever belt drives or band brakes are designed, it is necessary to determine the frictional forces developed between the belt and its contacting surface. Consider the flat belt shown in figure which passes over a fixed curved surface. The total angle of belt to surface contact in radians is β, and the coefficient of friction between the two surfaces is μ. We wish to determine the tension T 2 in the belt, which h is needed d to pull the belt counter clockwise over the surface, and thereby overcome both the frictional forces at the surface of contact and the tension T 1 in the other end of the belt. Obviously, T 2 > T 1. 11
Frictional Forces on Flat Belts T 2 > T 1 FBD of Flat belt T 1 opposes the direction of motion (or impending motion) of the belt measured relative to the surface, T 2 acts in the direction of the relative belt motion (or impending motion) T 2 is a function of the angle of belt to surface contact but independent of the radius of the drum. As a result, this equation is valid for flat belts passing over any curved contacting surface. If the pulley at A is free to rotate and the coefficient of static friction at the fixed drums B and C is μs = 0.25, determine the largest weight of the cylinder that can be lifted by the cord. The maximum tension that can be developed in the cord is 500 N. Ans: 24 12
A hawser from a ship is wrapped four times around a rotating capstan. The dockworker pulls with a force of 200 N. What is the maximum force the man can exert on the boat if the coefficient of friction between the capstan and hawser is 0.3? 13