STATICS. Equivalent Systems of Forces. Vector Mechanics for Engineers: Statics VECTOR MECHANICS FOR ENGINEERS: Contents 9/3/2015.

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1 3 Rigid CHPTER VECTR ECHNICS R ENGINEERS: STTICS erdinand P. eer E. Russell Johnston, Jr. Lecture Notes: J. Walt ler Teas Tech Universit odies: Equivalent Sstems of orces Contents Introduction Eternal and Internal orces Principle of Transmissibilit: Equivalent orces Vector Products of Two Vectors oment of a orce bout a Point Varigon s Theorem Rectangular Components of the oment of a orce Sample Problem 3.1 Scalar Product of Two Vectors Scalar Product of Two Vectors: pplications ied Triple Product of Three Vectors oment of a orce bout a Given is Sample Problem 3.5 oment of a Couple ddition of Couples Couples Can e Represented Vectors Resolution of a orce Into a orce at and a Couple Sample Problem 3.6 Sstem of orces: Reduction to a orce and a Couple urther Reduction of a Sstem of orces Sample Problem 3.8 Sample Problem

Introduction Treatment of a bod as a single particle is not alwas possible. In general, the sie of the bod and the specific points of application of the forces must be considered.

Current chapter describes the effect of forces eerted on a rigid bod and how to replace a given sstem of forces with a simpler equivalent sstem.

2 Introduction Treatment of a bod as a single particle is not alwas possible. In general, the sie of the bod and the specific points of application of the forces must be considered. ost bodies in elementar mechanics are assumed to be rigid, i.e., the actual deformations are small and do not affect the conditions of equilibrium or motion of the bod. Current chapter describes the effect of forces eerted on a rigid bod and how to replace a given sstem of forces with a simpler equivalent sstem. moment of a force about a point moment of a force about an ais moment due to a couple n sstem of forces acting on a rigid bod can be replaced b an equivalent sstem consisting of one force acting at a given point and one couple. 3-3 Eternal and Internal orces orces acting on rigid bodies are divided into two groups: - Eternal forces - Internal forces Eternal forces are shown in a free-bod diagram. If unopposed, each eternal force can impart a motion of translation or rotation, or both. 3-4

Principle of Transmissibilit: Equivalent orces Principle of Transmissibilit - Conditions of equilibrium or motion are not affected b transmitting a force along its

oving the point of application of the force to the rear bumper does not affect the motion or the other forces acting on the truck.

3-5 Vector Product of Two Vectors Concept of the moment of a force about a point is more easil understood through applications of the vector product or cross

3 Principle of Transmissibilit: Equivalent orces Principle of Transmissibilit - Conditions of equilibrium or motion are not affected b transmitting a force along its line of action. NTE: and are equivalent forces. oving the point of application of the force to the rear bumper does not affect the motion or the other forces acting on the truck. Principle of transmissibilit ma not alwas appl in determining internal forces and deformations. 3-5 Vector Product of Two Vectors Concept of the moment of a force about a point is more easil understood through applications of the vector product or cross product. Vector product of two vectors P and Q is defined as the vector V which satisfies the following conditions: 1. Line of action of V is perpendicular to plane containing P and Q.. agnitude of V is V PQsin 3. Direction of V is obtained from the right-hand rule. Vector products: - are not commutative, - are distributive, - are not associative, Q P P Q P Q1 Q P Q1 P Q P Q S P Q S 3-6 3

Vector Products: Rectangular Components Vector products of Cartesian unit vectors, i i 0 j i k k i j i j k j j 0 k j i i k j j k i k k 0 Vector products in terms of rectangular coordinates V P i P j

4 Vector Products: Rectangular Components Vector products of Cartesian unit vectors, i i 0 j i k k i j i j k j j 0 k j i i k j j k i k k 0 Vector products in terms of rectangular coordinates V P i P j P k Qi Q j Qk P Q P Q i P Q P Q j P Q PQ k i j k P Q P Q P Q 3-7 oment of a orce bout a Point force vector is defined b its magnitude and direction. Its effect on the rigid bod also depends on it point of application. The moment of about is defined as r The moment vector is perpendicular to the plane containing and the force. agnitude of measures the tendenc of the force to cause rotation of the bod about an ais along. r sin d The sense of the moment ma be determined b the right-hand rule. n force that has the same magnitude and direction as, is equivalent if it also has the same line of action and therefore, produces the same moment

oment of a orce bout a Point Two-dimensional structures have length and breadth but negligible depth and are subjected to forces contained in the plane of the structure.

If the force tends to rotate the structure clockwise, the sense of the moment vector is out of the plane of the structure and the magnitude of the moment is positive.

5 oment of a orce bout a Point Two-dimensional structures have length and breadth but negligible depth and are subjected to forces contained in the plane of the structure. The plane of the structure contains the point and the force., the moment of the force about is perpendicular to the plane. If the force tends to rotate the structure clockwise, the sense of the moment vector is out of the plane of the structure and the magnitude of the moment is positive. If the force tends to rotate the structure counterclockwise, the sense of the moment vector is into the plane of the structure and the magnitude of the moment is negative. 3-9 Varignon s Theorem The moment about a give point of the resultant of several concurrent forces is equal to the sum of the moments of the various moments about the same point. r 1 r 1 r Varigon s Theorem makes it possible to replace the direct determination of the moment of a force b the moments of two or more component forces of

6 Rectangular Components of the oment of a orce k j i k j i k j i The moment of about, k j i k j i r r, 3-1 Rectangular Components of the oment of a orce The moment of about, r / k j i k j i r r r / k j i

Rectangular Components of the oment of a orce or two-dimensional structures, k Z Z k 3-13 Sample Problem 3.1 100-lb vertical force is applied to the end of a lever which is attached to a shaft at.

7 Rectangular Components of the oment of a orce or two-dimensional structures, k Z Z k 3-13 Sample Problem lb vertical force is applied to the end of a lever which is attached to a shaft at. Determine: a) moment about, b) horiontal force at which creates the same moment, c) smallest force at which produces the same moment, d) location for a 40-lb vertical force to produce the same moment, e) whether an of the forces from b, c, and d is equivalent to the original force

8 Sample Problem 3.1 a) oment about is equal to the product of the force and the perpendicular distance between the line of action of the force and. Since the force tends to rotate the lever clockwise, the moment vector is into the plane of the paper. d d 4in. cos lb1 in. 1 in. 100 lbin 3-15 Sample Problem 3.1 c) Horiontal force at that produces the same moment, d 100 lbin. 0.8 in. d 4 in. sin in. 100 lbin. 0.8 in lb

9 Sample Problem 3.1 c) The smallest force to produce the same moment occurs when the perpendicular distance is a maimum or when is perpendicular to. d 100 lbin. 4 in. 100 lbin. 4 in. 50 lb 3-17 Sample Problem 3.1 d) To determine the point of application of a 40 lb force to produce the same moment, d 100 lbin. 40 lb d 100 lbin. d 5 in. 40 lb cos60 5 in. 10 in

Sample Problem 3.1 e) lthough each of the forces in parts b), c), and d) produces the same moment as the 100 lb force, none are of the same magnitude and sense, or on the same line of action.

10 Sample Problem 3.1 e) lthough each of the forces in parts b), c), and d) produces the same moment as the 100 lb force, none are of the same magnitude and sense, or on the same line of action. None of the forces is equivalent to the 100 lb force Sample Problem 3.4 SLUTIN: The moment of the force eerted b the wire is obtained b evaluating the vector product, r C The rectangular plate is supported b the brackets at and and b a wire CD. Knowing that the tension in the wire is 00 N, determine the moment about of the force eerted b the wire at C

Sample Problem 3.4 SLUTIN: r r C r r C i j 0.3 0 0.08 10 96 18 C 00 N 0.3 mi 0.08 m j r 00 N r C D C D 0.3 mi 0.4 m j 0.

8 Nmk k 3-1 Scalar Product of Two Vectors The scalar product or dot product between two vectors P and Q is defined as P Q PQ cos scalar result

11 Sample Problem 3.4 SLUTIN: r r C r r C i j C 00 N 0.3 mi 0.08 m j r 00 N r C D C D 0.3 mi 0.4 m j 0.3 m 96 N k 10 Ni j 18 Nk 0.5 m 7.68 Nmi 8.8 Nm j 8.8 Nmk k 3-1 Scalar Product of Two Vectors The scalar product or dot product between two vectors P and Q is defined as P Q PQ cos scalar result Scalar products: - are commutative, - are distributive, - are not associative, P Q Q P P Q Q P Q P Q S 1 1 P Q undefined Scalar products with Cartesian unit components, P Q P i P j P k Q i Q j Q k i i 1 j j 1 k k 1 i j 0 j k 0 k i 0 P Q P Q P Q P Q P P P P P P 3-11

Scalar Product of Two Vectors: pplications ngle between two vectors: P Q PQ cos P Q P Q P Q P Q P Q P Q cos PQ Projection of a vector on a given ais: PL P cos projection of P along L P Q PQ cos P Q P

12 Scalar Product of Two Vectors: pplications ngle between two vectors: P Q PQ cos P Q P Q P Q P Q P Q P Q cos PQ Projection of a vector on a given ais: PL P cos projection of P along L P Q PQ cos P Q P cos PL Q or an ais defined b a unit vector: P P L P cos P cos P cos 3-3 ied Triple Product of Three Vectors ied triple product of three vectors, S PQ scalar result The si mied triple products formed from S, P, and Q have equal magnitudes but not the same sign, S P Q P Q S Q S P S Q P P S Q Q P S Evaluating the mied triple product, S P Q S P Q P Q S P Q P Q S P Q P Q S S S P P P Q Q Q 3-4 1

13 oment of a orce bout a Given is oment of a force applied at the point about a point, r Scalar moment L about an ais L is the projection of the moment vector onto the ais, r L oments of about the coordinate aes, 3-5 oment of a orce bout a Given is oment of a force about an arbitrar ais, L r r r r The result is independent of the point along the given ais

14 Sample Problem 3.5 cube is acted on b a force P as shown. Determine the moment of P a) about b) about the edge and c) about the diagonal G of the cube. d) Determine the perpendicular distance between G and C. 3-7 Sample Problem 3.5 oment of P about, r P r ai a j ai j P P i j P i j ai j P i j ap i j k oment of P about, i i ap i j k ap

15 Sample Problem 3.5 oment of P about the diagonal G, G r G ai aj ak 1 i j k r G a 3 3 ap i j k 1 ap G i j k i j k 3 ap G ap Sample Problem 3.5 Perpendicular distance between G and C, P P 0 1 P j k i j k 0 11 Therefore, P is perpendicular to G. G ap Pd 6 3 d a

16 oment of a Couple Two forces and - having the same magnitude, parallel lines of action, and opposite sense are said to form a couple. oment of the couple, r r r r r r sin d The moment vector of the couple is independent of the choice of the origin of the coordinate aes, i.e., it is a free vector that can be applied at an point with the same effect oment of a Couple Two couples will have equal moments if 1 d1 d the two couples lie in parallel planes, and the two couples have the same sense or the tendenc to cause rotation in the same direction

ddition of Couples Consider two intersecting planes P 1 and P with each containing a couple 1 r 1 in plane P1 r in plane P Resultants of the vectors also form a couple r R r 1 Varigon s theorem r 1 r

17 ddition of Couples Consider two intersecting planes P 1 and P with each containing a couple 1 r 1 in plane P1 r in plane P Resultants of the vectors also form a couple r R r 1 Varigon s theorem r 1 r 1 Sum of two couples is also a couple that is equal to the vector sum of the two couples 3-33 Couples Can e Represented b Vectors couple can be represented b a vector with magnitude and direction equal to the moment of the couple. Couple vectors obe the law of addition of vectors. Couple vectors are free vectors, i.e., the point of application is not significant. Couple vectors ma be resolved into component vectors

The three forces ma be replaced b an equivalent force vector and couple vector, i.e, a force-couple sstem.

18 Resolution of a orce Into a orce at and a Couple orce vector can not be simpl moved to without modifing its action on the bod. ttaching equal and opposite force vectors at produces no net effect on the bod. The three forces ma be replaced b an equivalent force vector and couple vector, i.e, a force-couple sstem Resolution of a orce Into a orce at and a Couple oving from to a different point requires the addition of a different couple vector r ' The moments of about and are related, ' r' r s r s s oving the force-couple sstem from to requires the addition of the moment of the force at about

19 Sample Problem 3.6 SLUTIN: ttach equal and opposite 0 lb forces in the + direction at, thereb producing 3 couples for which the moment components are easil computed. Determine the components of the single couple equivalent to the couples shown. lternativel, compute the sum of the moments of the four forces about an arbitrar single point. The point D is a good choice as onl two of the forces will produce non-ero moment contributions Sample Problem 3.6 ttach equal and opposite 0 lb forces in the + direction at The three couples ma be represented b three couple vectors, 30 lb18 in. 540 lb 0 lb1 in. 40lb 0 lb9 in. 180 lbin. in. in. 540 lbin. i 40lbin. 180 lbin. k j

Sample Problem 3.6 lternativel, compute the sum of the moments of the four forces about D. nl the forces at C and E contribute to the moment about D. D 18 in.

k j 3-39 Sstem of orces: Reduction to a orce and Couple sstem of forces ma be replaced b a collection of force-couple sstems acting a given point The force and couple

20 Sample Problem 3.6 lternativel, compute the sum of the moments of the four forces about D. nl the forces at C and E contribute to the moment about D. D 18 in. j 30 lbk 9 in. j 1 in. k 0 lb i 540 lbin. i 40lbin. 180 lbin. k j 3-39 Sstem of orces: Reduction to a orce and Couple sstem of forces ma be replaced b a collection of force-couple sstems acting a given point The force and couple vectors ma be combined into a resultant force vector and a resultant couple vector, R R r The force-couple sstem at ma be moved to with the addition of the moment of R about, R R s R ' Two sstems of forces are equivalent if the can be reduced to the same force-couple sstem

urther Reduction of a Sstem of orces If the resultant force and couple at are mutuall perpendicular, the can be replaced b a single force acting along a new

The resultant force-couple sstem for a sstem of forces will be mutuall perpendicular if: 1) the forces are concurrent, ) the forces are coplanar, or 3) the

21 urther Reduction of a Sstem of orces If the resultant force and couple at are mutuall perpendicular, the can be replaced b a single force acting along a new line of action. The resultant force-couple sstem for a sstem of forces will be mutuall perpendicular if: 1) the forces are concurrent, ) the forces are coplanar, or 3) the forces are parallel urther Reduction of a Sstem of orces Sstem of coplanar forces is reduced to a R force-couple sstem R and that is mutuall perpendicular. Sstem can be reduced to a single force b moving the line of action of until its moment about becomes R R In terms of rectangular coordinates, R R R 3-4 1

Note: Since the support reactions are not included, the given sstem will not maintain the beam in equilibrium. b) ind an equivalent force-couple sstem at based on the forcecouple sstem at.

22 Sample Problem 3.8 SLUTIN: a) Compute the resultant force for the forces shown and the resultant couple for the moments of the forces about. or the beam, reduce the sstem of forces shown to (a) an equivalent force-couple sstem at, (b) an equivalent force couple sstem at, and (c) a single force or resultant. Note: Since the support reactions are not included, the given sstem will not maintain the beam in equilibrium. b) ind an equivalent force-couple sstem at based on the forcecouple sstem at. c) Determine the point of application for the resultant force such that its moment about is equal to the resultant couple at Sample Problem 3.8 SLUTIN: a) Compute the resultant force and the resultant couple at. R 150 N j 600 N j 100 N j 50 N j R 600 N j R r 1.6 i 600 j.8 i 100 j 4.8 i 50 j R 1880 Nmk 3-44

Sample Problem 3.8 b) ind an equivalent force-couple sstem at based on the force-couple sstem at. The force is unchanged b the movement of the force-couple sstem from to.

23 Sample Problem 3.8 b) ind an equivalent force-couple sstem at based on the force-couple sstem at. The force is unchanged b the movement of the force-couple sstem from to. R 600 N j The couple at is equal to the moment about of the force-couple sstem found at. R R r R 1880 N mk 4.8 mi 600 N j 1880 N mk 880 N mk 1000 Nmk R 3-45 Sample Problem 3.10 SLUTIN: Determine the relative position vectors for the points of application of the cable forces with respect to. Resolve the forces into rectangular components. Three cables are attached to the bracket as shown. Replace the forces with an equivalent forcecouple sstem at. Compute the equivalent force, R Compute the equivalent couple, r R

Sample Problem 3.10 SLUTIN: Determine the relative position vectors with respect to. r 0.075i 0.050k m rc 0.075i 0.050k m r 0.100i 0.100 j m D Resolve the forces into rectangular components.

24 Sample Problem 3.10 SLUTIN: Determine the relative position vectors with respect to. r 0.075i 0.050k m rc 0.075i 0.050k m r 0.100i j m D Resolve the forces into rectangular components. 700 N re 75i 150 j 50k re i j 0.89k 300i 600 j 00k N C D 1000 Ncos45i cos45 j 707i 707 j N 100 Ncos60i cos30 j 600i 1039 j N 3-47 Sample Problem 3.10 Compute the equivalent force, R i j k R 1607i 439 j 507k N Compute the equivalent couple, R r i j k r i 45k r C r D R 300 i c D 707 i j k j j k k i j 118.9k

STATICS. Equivalent Systems of Forces. Vector Mechanics for Engineers: Statics VECTOR MECHANICS FOR ENGINEERS: Contents & Objectives.

STATICS. Equivalent Systems of Forces. Vector Mechanics for Engineers: Statics VECTOR MECHANICS FOR ENGINEERS: Contents & Objectives. 3 Rigid CHATER VECTOR ECHANICS FOR ENGINEERS: STATICS Ferdinand. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Teas Tech Universit Bodies: Equivalent Sstems of Forces Contents & Objectives