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Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,*,*,Equilibrium of,a Rigid Body,Engineering Mechanics,1,51 Introduction,52 Equations of Equilibrium,53 Equilibrium of a Rigid Body,54 Statical Determinacy and Constraint,Equilibrium of a Rigid Body,Equilibrium of a Rigid Body,2,5-1 Introduction,1. Support conditions for bodies in a plane,The supports develop reactions in response to the weight of the body and to loads (external forces or moments) that are applied to the body. They prevent the body from moving. That is, the body is in equilibrium under the action of the loads and reactions. There are several types of supports for bodies loaded by forces acting in a plane. See page 142 in textbook.,Equilibrium of a Rigid Body,3,2. Construction of free-body diagrams,Successful application of the equation of equilibrium requires a complete specification of all the known and unknown external forces that act on the body. It is necessary to show all the forces and couple moments that the surroundings exert on the body so that these effects can be accounted for when the equations of equilibrium are applied. For this reason, a thorough understanding of how to draw a free-body diagram is of primary importance for solving problems in mechanics.,Equilibrium of a Rigid Body,4,Procedure for drawing a free-body diagram,To construct a free-body diagram for a rigid body or group of bodies considered as a single system, the following steps should be performed:,Draw outlined shape.,Imagine the body to be isolated or cut free from its constraints and connections and draw its outlined shape.,Show all forces and couple moments.,Identify all the external forces and couple moments that act on the body. Those generally encountered are due to,Equilibrium of a Rigid Body,5,(1) Applied loadings, (2) reactions occurring at the supports or at points of contact with other bodies (see Table 4-1, 4-2), and (3) the weight of the body.,Identify each loading and give dimensions.,The forces and couple moments that are known should be labelled with their proper magnitudes and directions. Letters are used to represent the magnitudes and direction angles of unknown forces and couple moments. Establish an,x,y,coordinate system so that these unknowns can be identified.,Equilibrium of a Rigid Body,6,IMPORTANT POINTS,No equilibrium problem should be solved without first drawing the free-body diagram, so as to account for all the forces and couple moments that act on the,body.,If a support prevents translation of a body in a particular direction, then the support exerts a force on the body in that direction.,If rotation is prevented, then the support exerts a couple moment on the body.,Equilibrium of a Rigid Body,7,Study Table 4-1,Internal forces are never shown on the free-body diagram, since they occur in equal but opposite collinear pairs and therefore cancel,out.,The weight of a body is an external force, and its effect is shown as a single resultant force acting through the bodys centre of gravity,G,.,Couple moment can be placed anywhere on the free-body diagram, since they are free vectors. Forces can act at any point along their lines of action, since they are sliding vectors.,Equilibrium of a Rigid Body,8,5-2 Equations of Equilibrium,Equations for equilibrium of a rigid body,When a body is subjected to a system of forces, which all lie in the,x,-,y,plane, then the forces can be resolved into their,x,and,y,components. Consequently, the conditions for equilibrium in two dimensions are:,Equilibrium of a Rigid Body,(1),9,Alternative sets of equilibrium equations,Although equations (1) are most often used for solving coplanar equilibrium problems, two alternative sets of three independent equilibrium equations may also be used. One such set is:,Equilibrium of a Rigid Body,When using these equations, it is required that a line passing through points,A,and,B,is,not perpendicular,to the,a,axis.,(2),10,A second alternative set of equations is:,Equilibrium of a Rigid Body,Here it is necessary that points,A,B,and,C,do not lie on the same line.,Any one of the three sets of equations may be used to solve an equilibrium problem. The problem can be simplified if we select equations that result in only one unknown in each equation. Only three of the equations can be independent.,(3),11,5-3 Equilibrium of a Rigid Body,Equilibrium of a two-force body,When a body is subjected to no couple moments and forces are applied at only two points on a body, the body is called a two-force body. For equilibrium of a two-force body the force acting at one point must be,equal in magnitude, opposite in direction, and have the same line of action,as the force acting at the other point.,Equilibrium of a Rigid Body,12,Equilibrium of a three-force body,A body acted on by three forces is called a three-force body. If a body is subjected to only three forces, then it is necessary that the forces be either concurrent or parallel for the body to be in equilibrium.,Many mechanical elements act as two- or three-force body, and the ability to recognize them in a problem will considerably simplify an equilibrium analysis.,Equilibrium of a Rigid Body,13,静力学,Example 1, A crane as shown in figure,P,=700kN,W,=200kN (The maximum lifting weight). Determine (a) The range of the weight of the balance member,Q,. (b) when,Q,=180kN, the reactive forces at wheel,A,and,B,.,14,静力学,Solution,: (a) When the crane is lifting the maximum weight, i.e.,W,=200,kN, to avoid the crane turn to right side, the following equation must be satisfied:,and also,15,静力学,Therefore:,When the crane is free of work, to avoid the crane turn to left side, the following equation must be satisfied:,and also,16,静力学,(b),when,Q,=180kN, find out the reactive forces at wheel,A,and,B,. According to equations of equilibrium, we have:,Solve these two equations, we have:,17,Equilibrium of a Rigid Body,Example 2,P,=20kN,m,=16,kN,m,q,=20kN/m,a,= 0.8 m. Calculate the reactions at,A,and,B,.,Solution,: The free-body diagram of the beam is shown in figure.,18,Equilibrium of a Rigid Body,19,静力学,Example 3,OA=R, AB= l, when,OA,is in horizontal position, the compressive force is,P,. Determine: ,M,=,？, the reactive force at point,O,. the force in member,AB,. the force from the rail to member,B,.,Solution,: Member B is chosen first,20,静力学,The negative sign means the direction of the force is opposite to the direction shown in figure.,Then the wheel is chosen,21,Equilibrium of a Rigid Body,Solution,：,1. Free-body diagram,Example 4,The links shown in Figure are all pin-connected,B,is a fixed end,P,=1000N,，,AE,=,BE,=,CE,=,DE,=1m. If the weight of the members is negligible, determine the reactions at,B,and the tension or compression in member,AC,.,22,Equilibrium of a Rigid Body,2. Equations of equilibrium,Summing moments about point,B,23,Equilibrium of a Rigid Body,Study member,CD, the free-body diagram of member,CD,is shown in figure. Summing moments about point,E, equations of equilibrium:,24,Equilibrium of a Rigid Body,Example 5,P,=100N.,AC,=1.6m,BC,=0.9m,CD=EC,=1.2m,AD,=2m. Determine the force in member,BD,and the reactions at,A,.,Solution,：,The free-body diagram is shown in figure. Equations of equilibrium are:,25,Equilibrium of a Rigid Body,26,Equilibrium of a Rigid Body,Free-body diagram of member,AB,27,Equilibrium of a Rigid Body,Example 6,P,= 10,kN,Q,= 50,kN,. Determine the reactive forces at,A, B,and,D,.,28,Equilibrium of a Rigid Body,Solution,：,The free-body diagram of the crane is shown in figure. The equation of equilibrium is:,29,Equilibrium of a Rigid Body,Secondly, the free-body diagram of beam,CD,. The equations of equilibrium is:,30,Equilibrium of a Rigid Body,For the system, equations of equilibrium are:,31,5-4 Statical Determinacy and Constraint,In general, structural or solid-body mechanics involves determination of unknown forces within the structure or body.,If the number of equations available from statements of equilibrium is the same as the number of unknown forces (including reactions) then the problem is,statically determinate,.,Equilibrium of a Rigid Body,32,If the number of unknown reactions or internal forces in the structure or component is greater than the number of equilibrium equations available, then the problem is said to be,statically indeterminate,.,In order to solve a statically indeterminate problem it is necessary to consider additional equations relating to the displacement or deformation of the body.,Equilibrium of a Rigid Body,33,If support of the body exceeds that required for complete constraint and one of the support is redundant, with three equilibrium equations and four unknown reactions, the body is statically indeterminate to the first degree.,Equilibrium of a Rigid Body,34,The End,Equilibrium of a Rigid Body,35,
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