Engineering Mechanics Chapter Three (Equilibrium) PDF

ADAMA SCIENCE AND TECHNOLOGY UNIVERSITY School of Civil Engineering and Architecture Engineering Mechanics CHAPTER THREE (EQUILIBRIUM) Instructor name : Israel W. January, 2024...

ADAMA SCIENCE AND TECHNOLOGY UNIVERSITY School of Civil Engineering and Architecture Engineering Mechanics CHAPTER THREE (EQUILIBRIUM) Instructor name : Israel W. January, 2024 Adama, Ethiopia Ceng 2201/2024 1  Chapter three: 3.1 introduction In the previous chapter, we have seen systems of forces. In this chapter stability of force systems, named as equilibrium of a body. Thus a body is said to be in equilibrium when the resultant of all the forces acting on it is zero. If all forces and moments applied to it are in balance. The vector expression of equilibrium equation for two dimensional problems may be written in scalar forms as , , and The three equations must be satisfied for complete equilibrium in two dimensions. Ceng 2201/2024 2  3.2 Equilibrium in Two-Dimensions 3.2.1 Mechanical system isolation and free body diagram (FBD) Before considering equilibrium conditions, it is very much essential and absolutely necessary to define unambiguously the particular body or mechanical system to be analyzed and represent clearly and completely all forces which act on the body Modeling the action of forces in Two – Dimensional Analysis The most important step in drawing the free body diagram will be to show the external forces exerted on the rigid body. On of the external forces will be forces exerted by contacts with supports and reactions. The different support and contact forces are shown in the Figure-1. A diagram showing a body/group of bodies considered in the analysis with all forces and relevant dimensions is called free body diagram (FBD). Ceng 2201/2024 3  Cont’d It is after such diagram is clearly drawn that the equilibrium equations be used to determine some of the unknown forces. Therefore free body diagram is the most important single step in the solution of problems in mechanics. Steps for the construction of free body diagram Decide which body or combination of bodies to be considered. The body or combination chosen is isolated by a diagram that represents its complete external boundary. All forces that act on the isolated body by the removed contacting and attracting bodies and known forces represented in their proper positions on the diagram of the isolated body. Each unknown force should be represented by a vector arrow with the unknown magnitude and direction. Ceng 2201/2024 4  Cont’d The force exerted on the body to be isolated by the body to be removed is indicated and its sense shall be opposite to the movement of the body which would occur if the contacting or supporting member were removed The choice of coordinate axes should be indicated directly on the diagram and relevant dimensions should be represented. 3.2.2 Equilibrium Conditions It was stated that a body is in equilibrium if the resultant force vector and the resultant couple vector are both zero. These requirements can be stated in the form of vector equation of equilibrium, which in two dimensions can be written as Ceng 2201/2024 5 Figure-1 Ceng 2201/2024 6  Cont’d Ceng 2201/2024 7  Cont’d Ceng 2201/2024 8  Cont’d Ceng 2201/2024 9 Supports for Rigid Bodies Subjected to Two-Dimensional Force Systems(table5.1) Ceng 2201/2024 10  Cont’d Ceng 2201/2024 11  Cont’d Ceng 2201/2024 12. Ceng 2201/2024 13 Important points Ceng 2201/2024 14  Procedure for analysis of equilibrium problems Coplanar force equilibrium problems for a rigid body can be solved using the following procedure. Free-Body Diagram. Establish the x, y coordinate axes in any suitable orientation. Draw an outlined shape of the body. Show all the forces and couple moments acting on the body. Label all the loadings and specify their directions relative to the x or y axis. The sense of a force or couple moment having an unknown magnitude but known line of action can be assumed. Indicate the dimensions of the body necessary for computing the moments of forces. Ceng 2201/2024 15  Cont’d Equations of Equilibrium. Apply the moment equation of equilibrium, = 0, about a point (O) that lies at the intersection of the lines of action of two unknown forces. In this way, the moments of these unknowns are zero about O, and a direct solution for the third unknown can be determined. When applying the force equilibrium equations, = 0 and = 0, orient the x and y axes along lines that will provide the simplest resolution of the forces into their x and y components. If the solution of the equilibrium equations yields a negative scalar for a force or couple moment magnitude, this indicates that the sense is opposite to that which was assumed on the free-body diagram. Ceng 2201/2024 16 Categories of Equilibrium in Two Dimensions The following categories of equilibrium conditions can be identified due to the nature of forces considered. Ceng 2201/2024 17  Cont’d Ceng 2201/2024 18  Alternative equilibrium equations In two-dimensional body, the maximum number of unknown variables is three. And the three equilibrium equations are sufficient to solve the unknown variables. Thus, whatever the combination, three total equations are maximally needed. What we have seen is two forces and one moment equations. But we could have the following combinations. Ceng 2201/2024 19 Constraint and statically determinacy Constraint refers to the restriction to movement, which the equation that are adequate to determine all the unknowns depends on the characteristics of constraint against possible movement of the body provided by its support Consider the roller support shown below (Figure 1); it can only provide a constraint normal to the surface but it doesn’t provide a tangential constraint unlike pin supports (Figure 2) Ceng 2201/2024 20 Statically determinate and indeterminate When all the forces in a structure can be determined strictly form these equations, the structure is referred to as statically determinate. On statically determinate structure the number of equations and the number of unknowns is equal. Equations Unknowns Ceng 2201/2024 21  Cont’d A structure is said to be indeterminate when the number of unknowns exceed the number of equilibrium equations As a general rule, a structure can be identified as being either statically determinate or statically indeterminate: by drawing free-body diagrams of all its members, or selective parts of its members, and then comparing the total number of available equilibrium equations. For a coplanar structure there are at most three equilibrium equations for each part, so that if there is a total of n parts and r force and moment reactions components, we have. Ceng 2201/2024 22 Example Ceng 2201/2024 23  examples 1-Determine the magnitudes of the forces C and T, which, along with the other three forces shown, act on the bridge-truss joint Ceng 2201/2024 24  answer Ceng 2201/2024 25 2. Ceng 2201/2024 26  question The 400 kg uniform I beam supports the load shown determine the reactions at the supports Ceng 2201/2024 27 3 Calculate the tension T in the cable which supports the 1000-lb load with the pulley arrangement shown. Each pulley is free to rotate about its bearing, and the weights of all parts are small compared with the load. Find the magnitude of the total force on the bearing of pulley C Ceng 2201/2024 28  answer Ceng 2201/2024 29 4 The uniform 100-kg I-beam is supported initially by its end rollers on the horizontal surface at A and B. By means of the cable at C it is desired to elevate end B to a position 3 m above end A. Determine the required tension P, the reaction at A, and the angle ϴ made by the beam with the horizontal in the elevated position. Ceng 2201/2024 30  answer Ceng 2201/2024 31 5. Determine the horizontal and vertical components of reaction on the beam caused by the pin at B and the rocker at A as shown in below. Neglect the weight of the beam. Ceng 2201/2024 32  answer Ceng 2201/2024 33 6. Determine the support reactions on the member in Figure below. The collar at A is fixed to the member and can slide vertically along the vertical shaft. Ceng 2201/2024 34  Answer Ceng 2201/2024 35  questions 1. Determine the components of reaction at the fixed support A. Neglect the thickness of the beam. Ceng 2201/2024 36  Cont’d 2. Determine the horizontal and vertical components of reaction at the supports. Neglect the thickness of the beam. Ceng 2201/2024 37 3. Ceng 2201/2024 38  3.3Equilibrium condition in three dimensions The two vector equations of equilibrium in three dimensions and their scalar components may be written as Notes; -In applying the vector form of the above equations, we first express each of the forces in terms of the coordinate unit vectors i, j and k. For the first equation, = 0, the vector sum will be zero only if the coefficients of i, j and k in the expression are, respectively, zero. These three sums when each is set equal to zero yield precisely the three scalar equations of equilibrium, 0 Ceng 2201/2024 39  Cont’d For the second equation, =0 where the moment sum may be taken about any convenient point O, we express the moment of each force as the cross product r X F, where r is the position vector from O to any point on the line of action of the force F. Thus, = (r X F) = 0. The coefficients of i, j and k in the resulting moment equation when set equal to zero, respectively, produce the three scalar moment equations Free Body Diagram (FBD) shall always be drawn before analysis of the force system. Usually either pictorial view or orthogonal projects of the FBD are used. Ceng 2201/2024 40. Ceng 2201/2024 41. Ceng 2201/2024 42. Ceng 2201/2024 43  Three-dimensional equilibrium problems for a rigid body can be solved using the following procedure Free-Body Diagram. Draw an outlined shape of the body. Show all the forces and couple moments acting on the body. Establish the origin of the x, y, z axes at a convenient point and orient the axes so that they are parallel to as many of the external forces and moments as possible. Label all the loadings and specify their directions. In general, show all the unknown components having a positive sense along the x, y, z axes. Indicate the dimensions of the body necessary for computing the moments of forces. Ceng 2201/2024 44  Cont’d Equations of Equilibrium. If the x, y, z force and moment components seem easy to determine, then apply the six scalar equations of equilibrium; otherwise use the vector equations. It is not necessary that the set of axes chosen for force summation coincide with the set of axes chosen for moment summation. Actually, an axis in any arbitrary direction may be chosen for summing forces and moments. Choose the direction of an axis for moment summation such that it intersects the lines of action of as many unknown forces as possible. Realize that the moments of forces passing through points on this axis and the moments of forces which are parallel to the axis will then be zero. If the solution of the equilibrium equations yields a negative scalar for a force or couple moment magnitude, it indicates that the sense is opposite to that assumed on the free- body diagram. Ceng 2201/2024 45  Example on 3-D Equilibrium 1. The uniform 7-m steel shaft has a mass of 200 kg and is supported by a ball and- socket joint at A in the horizontal floor. The ball end B rests against the smooth vertical walls as shown. Compute the forces exerted by the walls and the floor on the ends of the shaft. Ceng 2201/2024 46  answer Ceng 2201/2024 47  Cont’d Ceng 2201/2024 48 2. A 200-N force is applied to the handle of the hoist in the direction shown. The bearing A supports the thrust (force in the direction of the shaft axis), while bearing B supports only radial load (load normal to the shaft axis). Determine the mass m which can be supported and the total radial force exerted on the shaft by each bearing. Assume neither bearing to be capable of supporting a moment about a line normal to the shaft axis Ceng 2201/2024 49  Cont’d Ceng 2201/2024 50  answer Ceng 2201/2024 51. 3. The welded tubular frame is secured to the horizontal x-y plane by a ball and-socket joint at A and receives support from the loose-fitting ring at B. Under the action of the 2- kN load, rotation about a line from A to B is prevented by the cable CD, and the frame is stable in the position shown. Neglect the weight of the frame compared with the applied load and determine the tension T in the cable, the reaction at the ring, and the reaction components at A. Ceng 2201/2024 52  Answer Ceng 2201/2024 53  Cont’d Ceng 2201/2024 54  questions 4. Determine the components of reaction that the ball-and-socket joint at A, the smooth journal bearing at B, and the roller support at C exert on the rod assembly in Fig. below. Ceng 2201/2024 55  Cont’d 5. The bent rod in Fig. below is supported at A by a journal bearing, at D by a ball-and- socket joint, and at B by means of cable BC. Using only one equilibrium equation, obtain a direct solution for the tension in cable BC. The bearing at A is capable of exerting force components only in the z and y directions since it is properly aligned on the shaft. In other words, no couple moments are required at this support. Ceng 2201/2024 56  Cont’d 6. If the weight of the boom is negligible compared with the applied 30-kN load, determine the cable tensions and the force acting at the ball joint at A. Ceng 2201/2024 57. THANK YOU Ceng 2201/2024 58