Engineering Mechanics: Statics in SI Units, 12e 5 Equilibrium of a Rigid Body Chapter Objectives Develop the equations of equilibrium for a rigid body Concept of the free-body diagram for a rigid body Solve rigid-body equilibrium problems using the equations of equilibrium 1
Chapter Outline 1. Conditions for Rigid Equilibrium 2. Free-Body Diagrams 3. Equations of Equilibrium 4. Two and Three-Force Members 5. Free Body Diagrams 6. Equations of Equilibrium 7. Constraints and Statical Determinacy 5.1 Conditions for Rigid-Body Equilibrium The equilibrium of a body is expressed as FR = F = ( MR) = O 0 M O = 0 Consider summing moments about some other point, such as point A, we require ( M ) MA = r FR + R O = 0 2
5.2 Free Body Diagrams Support Reactions If a support prevents the translation of a body in a given direction, then a force is developed on the body in that direction. If rotation is prevented, a couple moment is exerted on the body. 5.2 Free Body Diagrams 3
5.2 Free Body Diagrams 5.2 Free Body Diagrams Internal Forces External and internal forces can act on a rigid body For FBD, internal forces act between particles which are contained within the boundary of the FBD, are not represented Particles outside this boundary exert external forces on the system 4
5.2 Free Body Diagrams Weight and Center of Gravity Each particle has a specified weight System can be represented by a single resultant force, known as weight W of the body Location of the force application is known as the center of gravity 5.2 Free Body Diagrams Procedure for Drawing a FBD 1. Draw Outlined Shape Imagine body to be isolated or cut free from its constraints Draw outline shape 2. Show All Forces and Couple Moments Identify all external forces and couple moments that act on the body 5
5.2 Free Body Diagrams 3. Identify Each Loading and Give Dimensions Indicate dimensions for calculation of forces Known forces and couple moments should be properly labeled with their magnitudes and directions Example 5.1 Draw the free-body diagram of the uniform beam. The beam has a mass of 100kg. 6
Solution Free-Body Diagram Solution Free-Body Diagram Support at A is a fixed wall Three forces acting on the beam at A denoted as A x, A y, MA, drawn in an arbitrary direction Unknown magnitudes of these vectors Assume sense of these vectors For uniform beam, Weight, W = 100(9.81) = 981N acting through beam s center of gravity, 3m from A 7
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Examples and Problems Resolve the following problems for Homework: 5.1, 5.4 and 5.10. 9
5.3 Equations of Equilibrium Please refer to the Companion CD for the animation: Equilibrium of a Free Body For equilibrium of a rigid body in 2D, F x = 0; F y = 0; M O = 0 F x and F y represent sums of x and y components of all the forces M O represents the sum of the couple moments and moments of the force components 5.3 Equations of Equilibrium Please refer to the Companion CD for the animation: Equilibrium of a Free Body Alternative Sets of Equilibrium Equations For coplanar equilibrium problems, F x = 0; F y = 0; M O = 0 2 alternative sets of 3 independent equilibrium equations, F a = 0; M A = 0; M B = 0 10
5.3 Equations of Equilibrium Please refer to the Companion CD for the animation: Equilibrium of a Free Body Procedure for Analysis Free-Body Diagram 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 5.3 Equations of Equilibrium Please refer to the Companion CD for the animation: Equilibrium of a Free Body Procedure for Analysis Equations of Equilibrium Apply M O = 0 about a point O Unknowns moments of are zero about O and a direct solution the third unknown can be obtained Orient the x and y axes along the lines that will provide the simplest resolution of the forces into their x and y components Negative result scalar is opposite to that was assumed on the FBD 11
Example 5.5 Determine the horizontal and vertical components of reaction for the beam loaded. Neglect the weight of the beam in the calculations. Solution Free Body Diagrams 600N represented by x and y components 200N force acts on the beam at B 12
Solution Equations of Equilibrium o + Fx = 0 ; 600cos45 N Bx = 0 Bx = 424N M 100N(2m) + (600sin 45 A + F 319N 600sin 45 B y y B = 0; = 319N y = 0; = 405N o o N)(5m) (600cos45 N 100N 200N+ B y o = 0 N)(0.2m) A (7m) = 0 y 13
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Examples and Problems Resolve the following problems for Homework: 5.21, 5.41, 5.42 and 5.91. 16
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5.4 Two- and Three-Force Members Two-Force Members When forces are applied at only two points on a member, the member is called a two-force member Only force magnitude must be determined 18
5.4 Two- and Three-Force Members Three-Force Members When subjected to three forces, the forces are concurrent or parallel Example 5.13 The lever ABC is pin-supported at A and connected to a short link BD. If the weight of the members are negligible, determine the force of the pin on the lever at A. 19
Solution Free Body Diagrams BD is a two-force member Lever ABC is a three-force member Equations of Equilibrium 0.7 tan 1 θ = = 60.3 0.4 + F + F y x = 0; = 0; o o o F cos 60.3 Fcos45 + 400N = 0 o o F sin 60.3 Fsin45 = 0 A A Solving, F A = 1.07kN F = 1. 32kN 5.5 Free-Body Diagrams Support Reactions As in the two-dimensional case: A force is developed by a support A couple moment is developed when rotation of the attached member is prevented The force s orientation is defined by the coordinate angles α, β and γ 20
5.5 Free-Body Diagrams 5.5 Free-Body Diagrams 21
Example 5.14 Several examples of objects along with their associated free-body diagrams are shown. In all cases, the x, y and z axes are established and the unknown reaction components are indicated in the positive sense. The weight of the objects is neglected. Solution 22
5.6 Equations of Equilibrium Vector Equations of Equilibrium For two conditions for equilibrium of a rigid body in vector form, F = 0 M O = 0 Scalar Equations of Equilibrium If all external forces and couple moments are expressed in Cartesian vector form F = F x i + F y j + F z k = 0 M O = M x i + M y j + M z k = 0 5.7 Constraints for a Rigid Body Redundant Constraints More support than needed for equilibrium Statically indeterminate: more unknown loadings than equations of equilibrium 23
5.7 Constraints for a Rigid Body Improper Constraints Instability caused by the improper constraining by the supports When all reactive forces are concurrent at this point, the body is improperly constrained 5.7 Constraints for a Rigid Body Procedure for Analysis Free Body Diagram Draw an outlined shape of the body Show all the forces and couple moments acting on the body Show all the unknown components having a positive sense Indicate the dimensions of the body necessary for computing the moments of forces 24
5.7 Constraints for a Rigid Body Procedure for Analysis Equations of Equilibrium Apply the six scalar equations of equilibrium or vector equations Any set of non-orthogonal axes may be chosen for this purpose Equations of Equilibrium Choose the direction of an axis for moment summation such that it insects the lines of action of as many unknown forces as possible Example 5.15 The homogenous plate has a mass of 100kg and is subjected to a force and couple moment along its edges. If it is supported in the horizontal plane by means of a roller at A, a ball and socket joint at B, and a cord at C, determine the components of reactions at the supports. 25
Solution Free Body Diagrams Five unknown reactions acting on the plate Each reaction assumed to act in a positive coordinate direction Equations of Equilibrium F x F y F z = 0; B x = 0; B y = 0 = 0 = 0; A + B z z + T C 300N 981N = 0 Solution Equations of Equilibrium M M = 0; T = 0; 300N(1.5m) + 981N(1.5m) B (3m) A (3m) 200Nm. = 0 Components of force at B can be eliminated if x, y and z axes are used M M x y x' y' = 0;981N(1m) + 300N(2m) A(2m) = 0 = 0; C (2m) 981N(1m) + B (2m) = 0 300N(1.5m) 981N(1.5m)200Nm. + T z Z z z C (3m) = 0 26
Solution Solving, A z = 790NB z = -217N T C = 707N The negative sign indicates B z acts downward The plate is partially constrained as the supports cannot prevent it from turning about the z axis if a force is applied in the x-y plane QUIZ 1. If a support prevents translation of a body, then the support exerts a on the body. A) Couple moment B) Force C) Both A and B. D) None of the above 2. Internal forces are shown on the free body diagram of a whole body. A) Always B) Often C) Rarely D) Never 27
QUIZ 3. The beam and the cable (with a frictionless pulley at D) support an 80 kg load at C. In a FBD of only the beam, there are how many unknowns? A) 2 forces and 1 couple moment B) 3 forces and 1 couple moment C) 3 forces D) 4 forces QUIZ 4. Internal forces are not shown on a free-body diagram because the internal forces are. A) Equal to zero B) Equal and opposite and they do not affect the calculations C) Negligibly small D) Not important 28
QUIZ 5. The three scalar equations F X = F Y = M O = 0, are equations of equilibrium in two dimensions. A) Incorrect B) The only correct C) The most commonly used D) Not sufficient 6. A rigid body is subjected to forces. This body can be considered as a member. A) Single-force B) Two-force C) Three-force D) Six-force QUIZ 7. For this beam, how many support reactions are there and is the problem statically determinate? A) (2, Yes) B) (2, No) C) (3, Yes) D) (3, No) F F F F 8. The beam AB is loaded as shown: a) how many support reactions are there on the beam, b) is this problem statically determinate, and c) is the structure stable? A) (4, Yes, No) B) (4, No, Yes) C) (5, Yes, No) D) (5, No, Yes) A Fixed support F B 29
QUIZ 9. Which equation of equilibrium allows you to determine FB right away? 100 lb A) FX = 0 B) FY = 0 A X A B C) MA = 0 D) Any one of the above. 10. A beam is supported by a pin joint and a roller. How many support reactions are there and is the structure stable for all types of loadings? A) (3, Yes) B) (3, No) C) (4, Yes) D) (4, No) AY F B QUIZ 11. If a support prevents rotation of a body about an axis, then the support exerts a on the body about that axis. A) Couple moment B) Force C) Both A and B. D) None of the above. 12. When doing a 3-D problem analysis, you have scalar equations of equilibrium. A) 3 B) 4 C) 5 D) 6 30
QUIZ 13. The rod AB is supported using two cables at B and a ball-and-socket joint at A. How many unknown support reactions exist in this problem? A) 5 force and 1 moment reaction B) 5 force reactions C) 3 force and 3 moment reactions D) 4 force and 2 moment reactions QUIZ 14. If an additional couple moment in the vertical direction is applied to rod AB at point C, then what will happen to the rod? A) The rod remains in equilibrium as the cables provide the necessary support reactions. B) The rod remains in equilibrium as the ball-and-socket joint will provide the necessary resistive reactions. C) The rod becomes unstable as the cables cannot support compressive forces. D) The rod becomes unstable since a moment about AB cannot be restricted. 31
QUIZ 15. A plate is supported by a ball-and-socket joint at A, a roller joint at B, and a cable at C. How many unknown support reactions are there in this problem? A) 4 forces and 2 moments B) 6 forces C) 5 forces D) 4 forces and 1 moment QUIZ 16. What will be the easiest way to determine the force reaction BZ? A) Scalar equation F Z = 0 B) Vector equation M A = 0 C) Scalar equation M Z = 0 D) Scalar equation M Y = 0 32