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Recall that the external forces acting on a rigid body can be grouped into applied forces and support reactions. Also the entire system of forces acting on a rigid body can be reduced to a single force and a couple (the force-couple system) at some arbitrary point. A rigid body in which the external forces acting reduce to zero force and zero couple at an arbitrary point 0 is said to be in equilibrium. Hence a body is in equilibrium if and only if: Fi = FR = 0 and Moi = MOR = 0 Fx=0, Fy=0 and Fz=0 Mx=0, My=0 and Mz=0
Fi = FR = 0 and Moi = MOR = 0 Fx=0, Fy=0 and Fz=0 Mx=0, My=0 and Mz=0 Sum of the rectangular components of the forces and moments along the three orthogonal axis must be equal to zero. The equations above imply that a body in equilibrium must be at rest since there is no translation or rotation of the body.
In considering the equilibrium of rigid bodies, it is important to draw a free-body diagram showing the entire rigid-body or part of it that is being analyzed and all the external forces acting on the free-body. The external forces acting on a free-body diagram will consist of all the forces applied to it, the weight of the free-body (which is the force the earth applies to the free-body), reactions exerted on the free-body by the supports and the action exerted on the free-body by any portion of the rigid body that has been detached.
Classification of Supports Supports can be classified into three groups based on the type of reaction associated with the support. • 1 - Supports whose reactions consist of a single force with known line of action. • These kinds of supports are generally called roller supports and they prevent translation along the known direction of the reaction while allowing translation along other directions as well as rotation. Supports that fall into this group are: • - rollers - rockers - frictionless surfaces - cable support - link support
2 -Supports whose reactions consist of a single force with unknown line of action. The general name for these supports is pinned or hinged supports and they prevent translation in all directions but permit rotation. For convenience, the single force with unknown line of action is normally resolved into it's rectangular components resulting in two unknown reactions.
3 - Supports whose reactions consist of a force with unknown line of action and a couple. These kinds of supports are generally called fixed supports and they prevent any kind of motion (translation and rotation) at the support. Fixed supports have three unknown reactions in 2-D, the x- and y- components of the force with unknown line of action, as well as the moment of the couple. Always keep in mind that the direction of a moment vector (couple vector) is always perpendicular to a plane containing the moment/couple and it's sense is given by the right hand rule.
Equilibrium of a Rigid Body in 2-Dimensions When considering the equilibrium of a rigid body in 2-D, say the x-y plane, the six equations of equilibrium reduce to three which are Fx=0; Fy=0 and MA=0 where A is any point in the plane of the body. Note that the three equations can be solved for a maximum of three unknowns. Although different forms of the equilibrium equations can be written, e.g. Fx=0; MA=0 and MB=0
For a plane for a plane structure , only three independent equilibrium equations exists and hence only three unknown reactions can be determined. If the number of unknown reactions is equal to the number of independent equilibrium equations available and the rigid body is properly constrained, then the reactions can be determined from the equilibrium equations and the structure is said to be statically determinate. Hence a statically determinate structure is one in which all the unknown reactions can be determined from the equilibrium equations.
Conversely, if the number of unknown reactions in a structure is greater than the number of independent equilibrium equations available or the rigid body is not properly constrained, then all the unknown reactions cannot be determined by using the equilibrium equations alone and the structure is said to be statically indeterminate. Statically indeterminate structures contain more reactions (called redundant reactions) than are necessary to completely restrain the structure. Analysis of statically indeterminate structures can be carried out by using additional equations that are derived from compatibility (deformation) of the structure.
If there are fewer unknown reactions than the number of equilibrium equations available, then all the equilibrium equations will not be satisfied and the structure will not be at rest. Such a structure is said to be unstable or partially restrained. Statically Determinate Structures - Sufficient number of reactions provided to ensure full restraint. Structure is also properly constrained. Statically Indeterminate Structures - These have at least one redundant reaction in each case.
Unstable or Partially Restrained Structures - Fewer number of reactions than are necessary to ensure full restraint. Improperly Constrained Structures - (Both Unstable & Statically Indeterminate)
Note that a rigid-body is improperly constrained whenever its supports are arranged in such a way that the reactions are either concurrent or parallel. Note: Given a structure you have to be able to identify if it is statically determinate, statically indeterminate or unstable.
Example 1 A gardener is using a 50N wheelbarrow to transport a 220N bag of fertilizer. What force must she exert on each handle?
Definitions A structure is statically determinate if all the support reactions can be determined using only equilibrium equations. For such a structure, the number of unknown reactions must be equal to the number of independent equations of equilibrium available. Conversely, a structure is statically indeterminate if all the support reactions cannot be determined by using only the equilibrium equations because the number of unknown reactions is more than the number of available independent equations of equilibrium or the structure is improperly constrained.
Definitions continued If a structure has less number of reactions than the available equations of equilibrium, the structure is said to be unstable or partially constrained. A structure is that has sufficient number of unknown reactions to ensure that the structure is completely restrained but these reactions are improperly arranged such that they are either parallel or concurrent, such a structure is geometrically unstable and is said to be improperly constrained
Example 2 A 2800-kg forklift truck is used to lift a 1500-kg crate. Determine the reaction of each of the two (a) front wheels A. (b) rear wheels B.
Example 3 The gardener of example 1wishes to transport a second 220 N bag of fertilizer at the same time as the first one. Determine the maximum allowable horizontal distance from the axle A of the wheelbarrow to the center of gravity of the second bag if she can only hold 65 N with each arm.
Example 4 A load of lumber of weight W = 25 kN is being raised by a mobile crane. The weight of the boom ABC and the combined weight of the truck and driver are as shown (See text book). Determine the reaction at each of the two (a) front wheels H (b) rear wheels K.
Example 5 A load of lumber of weight W = 25 kN is being raised by a mobile crane. Knowing the tension is 25 kN in all portions of the cable AEF and that the weight of the boom ABC is 3 kN, determine (a) the tension in rod CD (b) the reaction at pin B.
Example 6 A truck - mounted crane is used to lift a 3-kN compressor. The weights of the boom and AB and of the truck are as shown (see text book), and the angle the boom forms with the horizontal is a = 40 degrees. Determine the reaction at each of the two (a) rear wheels C. (b) front wheels D.
Example 7 For the truck - mounted crane in the previous question, determine the smallest allowable value of a if the truck is not to tip over when a 13 kN load is lifted.
Example 8 Three loads are applied as shown to a light beam supported by cables attached to at B and D. Neglecting the weight of the beam, determine the range of values of Q for which neither cable becomes slack when P=0.
Example 9 The three loads are applied as shown to a light beam supported by cables attached at B and D. Knowing that the maximum allowable tension in each cable is kN and neglecting the weight of the beam, determine the range of values of Q for which loading is safe when P=0.
Example 10 For the beam in the previous problem, determine the range of values of Q for which the loading is safe when P=5 kN.
Example 11 Determine the maximum load which may be raised by the mobile crane of problem (see above) without tipping over, knowing that the largest force which can be exerted by the hydraulic cylinder D is 100 kN, and that the maximum allowable tension in cable AEF is 35 kN.
Example 12 Determine the tension in cable ABD and the reaction at the support C.
Example 13 Two links AB and DE are connected by a bell crank as shown (see text book). Knowing that the tension in link AB is 900 N, determine (a) the tension in link DE. (b) the reaction at C.
Example 14 Two links AB and DE are connected by a bell crank as shown (see text book). Determine the maximum force which may be safely exerted by link AB on the bell crank if the maximum allowable value for the reaction at C is 2 kN.
Example 15 The lever AB is hinged at C and attached to a control cable at A. If the lever is subjected at B to a 500 N horizontal force, determine (a) the tension in the cable (b) the reaction at C.
Example 16 A light bar AD is suspended from a cable BE and supports a 20 kg block at C. The extremities A and D of the bar are in contact with frictionless, vertical walls. Determine the tension in cable BE and the reactions at A and D.
Example 17 A movable bracket is held at rest by a cable attached at C and by frictionless rollers at A and B. For the loading shown, determine (a) the tension in the cable (b) the reactions at A and B.
Example 18 The spanner shown is used to rotate a shaft. A pin fits in a hole at A, while a flat, frictionless surface rests against the shaft at B. If a 300 N force P is exerted on the spanner at D. Find the reactions at A and B.
Example 19 The spanner shown is used to rotate a shaft. A pin fits in the hole at A, while a flat, frictionless surface rests against the shaft at B. If the moment about C of the force exerted on the shaft at A is to be 90 N . m, find (a) the force P which should be exerted on the spanner at D (b) the corresponding value of the force exerted on the spanner at B.
Example 20 A 70 kg overhead garage door consists of a uniform rectangular panel AC, 2100 mm high, supported by the cable AE attached at the middle of the upper edge of the door and by two sets of frictionless rollers at A and B. Each set consists of two rollers located on either side of the door. The rollers at A are free to move in horizontal channels, while the rollers B are guided by vertical channels. If the door is held in the position for which BD = 1050 mm, determine (a) the tension in cable AE, (b) the reaction at each of the four rollers.
Example 21 In the previous problem, determine the distance BD for which the tension in cable AE is equal to 3 kN.
Example 22 Determine the reactions at B and C when a = 30 mm.
Example 23 A T shaped bracket supports a 300 N load as shown. Determine the reactions at A and C when (a) a = 90 degrees (b) a = 45 degrees
Example 24 One end of the rod AB rests in the corner A and the other is attached to cord BD. If the rod supports a 200 N load at it's midpoint C, find the reaction at A and the tension in the cord.
Example 25 A 300 mm wooden beam weighing 54 kg is supported by a pin and bracket at A and by cable BC. Find the reaction at A and the tension in the table.
Equilibrium of a Two-Force or Three-Force Body A two-force is a rigid body that is subjected to forces acting at only two points. The system of forces acting at each point may be reduced to their respective resultants, FR1 and FR2 . For equilibrium of a two force body, the resultant forces FR1 and FR2 acting at two points on a rigid body must have the same magnitude, the same line of action and opposite sense. (This is a necessary and sufficient condition for equilibrium).
Equilibrium of a Two-Force or Three-Force Body cont’d A three-force body is a rigid body that is subjected to forces acting at only three points. The system of forces acting on each of the three points may be reduced to their respective resultants, FR1, FR2, and FR3. For a three-force body to be in equilibrium, the lines of action of the three forces must be either concurrent or parallel. (Necessary but insufficient condition for equilibrium.)
Two Force Bodies Three Force Bodies
Assignment a) Show that for equilibrium of a two force body, the resultant of the forces acting at the two points on the rigid body must have the same magnitude, same line of action and opposite sense. b) Show that if a three force body is in equilibrium, the resultant forces acting at three points on the body need to be concurrent.
Example 26 A slender rod of length L is lodged between peg C and the vertical wall. It supports a load P at end A. Neglecting friction and the weight of the rod, determine the angle q corresponding to the equilibrium.
Example 27 A 1.2 m rod, of uniform cross section, is held in equilibrium s shown, with one end against a frictionless vertical wall and the other end attached to a cord. Determine the length of the cord. If the rod supports a 200 N load at it's midpoint C, find the reaction at A and the tension in the cord.
Example 28 A worker pushes a 50 kg wheelbarrow at a constant speed up an 18 degree incline. Determine the magnitude and direction of (a) the force he exerts on each handle (b) the reaction at C. (Hint: The wheel is a two-force body.)
Example 29 A slender rod of length 2r and weight W is attached to a collar at B and rests on a circular cylinder of radius r. Knowing that the collar may slide freely along a vertical guide and neglecting friction, determine the value of q corresponding to equilibrium.
Example 30 A uniform rod AB of length 3R and weight W rests inside a hemispherical bowl of radius R as shown. Neglecting friction, determine the angle q corresponding to equilibrium.
Equilibrium of a Body in Three Dimensions The reactions at the supports of a 3 - Dimensional structure may consist of a single force of known direction (single reaction), a single force of unknown direction (three reactions) or a force-couple system at fixed supports (six reactions). The reactions present at a particular support will correspond to the directions in which translations and/or rotations are prevented at the support.
