Chapter -4- Force System Resultant
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1 Ishik University / Sulaimani Civil Engineering Department Chapter -4- Force System Resultant 1 2 1
2 CHAPTER OBJECTIVES To discuss the concept of the moment of a force and show how to calculate it in two and three dimensions. To provide a method for finding the moment of a force about a specified axis. To define the moment of a couple. To present methods for determining the resultants of nonconcurrent force systems. To indicate how to reduce a simple distributed loading to a resultant force having a specified location. 3 CHAPTER OUTLINE Moment of a Force Scalar Formation Principle of Moments Moment of a Force about a Specified Axis Moment of a Couple Equivalent System Resultants of a Force and Couple System Reduction of a Simple Distributed Loading 4 2
3 4.1 MOMENT OF A FORCE SCALAR FORMATION Moment of a force about a point or axis a measure of the tendency of the force to cause a body to rotate about the point or axis. Case 1 Consider horizontal force F x, which acts perpendicular to the handle of the wrench and is located d y from the point O. 5 F x tends to turn the pipe about the z axis. The larger the force or the distance d y, the greater the turning effect. Torque tendency of rotation caused by F x or simple moment (M o ) z Moment axis (z) is perpendicular to shaded plane (x-y) F x and d y lies on the shaded plane (x-y) Moment axis (z) intersects the plane at point O 6 3
4 Case 2 Apply force F z to the wrench. Pipe does not rotate about z axis. Tendency to rotate about x axis. The pipe may not actually rotate F z creates tendency for rotation so moment (M o ) x is produced. Moment axis (x) is perpendicular to shaded plane (y-z). Fz and dy lies on the shaded plane (y-z). 7 Case 3 Apply force F y to the wrench. No moment is produced about point O. Lack of tendency to rotate as line of action passes through O. 8 4
5 In General Consider the force F and the point O which lies in the shaded plane. The moment M O about point O, or about an axis passing through O and perpendicular to the plane, is a vector quantity. Moment M O has its specified magnitude and direction. 9 Magnitude For magnitude of M O, M O = Fd where d = moment arm or perpendicular distance from the axis at point O to its line of action of the force. Units for moment is N.m. Direction Direction of MO is specified by using right hand rule. - fingers of the right hand are curled to follow the sense of rotation when force rotates about point O. - Thumb points along the moment axis to give the direction and sense of the moment vector. - Moment vector is upwards and perpendicular to the shaded plane. 10 5
6 Direction M O is shown by a vector arrow with a curl to distinguish it from force vector. Example (Fig b) M O is represented by the counterclockwise curl, which indicates the action of F. 11 Direction Arrowhead shows the sense of rotation caused by F Using the right hand rule, the direction and sense of the moment vector points out of the page. In 2D problems, moment of the force is found about a point O. Moment acts about an axis perpendicular to the plane containing F and d. Moment axis intersects the plane at point O. 12 6
7 Resultant Moment of a System of Coplanar Forces Resultant moment, M Ro = addition of the moments of all the forces algebraically since all moment forces are collinear. M Ro = Fd taking clockwise to be negative. o A clockwise curl is written along the equation to indicate that a positive moment if directed along the + z axis and negative along the z axis
8 Moment of a force does not always cause rotation. Force F tends to rotate the beam clockwise about A with moment. M A = Fd A Force F tends to rotate the beam counterclockwise about B with moment. M B = Fd B Hence support at A prevents the rotation
9 17 Example 4.1 For each case, determine the moment of the force about point O. 18 9
10 Solution; Line of action is extended as a dashed line to establish moment arm d. Tendency to rotate is indicated and the orbit is shown as a colored curl. ( a) M ( b) M o o (100N)(2m) 200N. m( CW ) (50N)(0.75m) 37.5N. m( CW ) 19 Solution; ( c) M ( d) M ( e) M o o o (40N)(4m 2cos 30 m) 229N. m( CW ) (60N)(1sin 45 m) 42.4N. m( CCW ) (7kN)(4m 1m) 21.0kN. m( CCW ) 20 10
11 Example 4.2 Determine the moments of the 800N force acting on the frame about points A, B, C and D. 21 Solution; Scalar Analysis M (800N)(2.5m) 2000N.m(CW) M M A B C (800N)(1.5m) 1200N.m(CW) (800N)(0m) 0kN.m Line of action of F passes through C M (800N)(0.5m) 400N.m(CCW) D 22 11
12 Example 4.3 Determine the resultant moment of the four forces acting on the rod shown in figure about point O
13 4.2 PRINCIPLES OF MOMENTS The guy cable exerts a force F on the pole and creates a moment about the base at A, M A = Fd If the force is replaced by F x and F y at point B where the cable acts on the pole, the sum of moment about point A yields the same resultant moment. 25 F y create zero moment about A M A = F x h Apply principle of transmissibility and slide the force where line of action intersects the ground at C, F x create zero moment about A M A = F y b 26 13
14 Example 4.3 The force F acts at the end of the angle bracket. Determine the moment of the force about point O. 27 Solution Method 1: M O = 400sin30 N(0.2m)-400cos30 N(0.4m) = -98.6N.m = 98.6N.m (CCW) As a Cartesian vector, M O = {-98.6k}N.m 28 14
15 Solution Method 2: Express as Cartesian vector r = {0.4i 0.2j}N F = {400sin30 i 400cos30 j}n = {200.0i 346.4j}N For moment, M O rxf 98.6k N. m i j k Example 4.4 The member is subjected to a force of F=6 kn. If θ= 45º, determine the moment produced by F about point A. Ans; 30 15
16 Example 4.5 The two boys push on the gate with forces of FA=30 Ib and FB=50 Ib as shown. Determine the moment of each force about C. Which way will the gate rotate, clockwise or counterclockwise? Neglect the thickness of the gate. Ans; The gate will rotate C.C.W
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