for the vertical components and 1, 2", 3", 4", 5" that for the horizontal components. Take a pole P at unit distance from yy, and draw the link-polygon, close this by the line CD (since the vertical Fig. 121 components must be in equilibrium), this is then the bending moment diagram. Through P draw PO' parallel to CD, then 0'1 is the vertical reaction at A and 5'0 that at B. But the horizontal force 15" is also acting on the beam at A, hence the resultant force at A is given by 5"1. The shearing force diagram offers no difficulty. 185. As a final example of the use of the theorems in this chapter, consider the case of a loaded crane as in figure. B Fig. 122. M Suppose the crane kept in position by a smooth collar at C and a supporting piece at D. Then there are three external forces acting on the crane, W at P, the horizontal reaction at C and the reaction at D. Since there is equilibrium these must pass through the point M. The vectortriangle for the point M will therefore determine the forces at C and D. The forces along AP and BP are determined by finding the components of W along those lines. We thus know the forces at A, B, C and D, and can therefore find the shearing force and bending moment at any point by our constructions of § 182. If the load had been applied in AP produced, then the theorem of § 174 (3) could have been applied to determine the forces in AP and BP. EXERCISES XI. (1) Decompose a rotor of magnitude 10 into two parallel ones (a) distant 3 and 5 ft. on either side, (2) Find the components of a force of 12 lbs. in the line AB along the three bars indicated. 12 A -10" (3) A horizontal beam freely supported at the ends carries loads of 5, 10 and 5 tons at points distant 5, 10 and 5 ft. from one end. Find graphically the reactions at the supports and draw the bending moment and shearing force diagram. (4) A horizontal beam 30 ft. long freely supported at ends is loaded as in table, determine the reactions at the ends and the bending moment and shearing force at points distant 6, 11, 15, 23 ft. from end, graphically and by calculation. Distance from one end in ft. 5 12 14 21 25 4.1 3.5 2.8 1.6 3.8 (5) A horizontal beam 40 ft. long is propped in the middle and freely supported at ends; if the reaction at the middle is equivalent to an upward force of 10 tons, find the reaction at the supports and the bending moment and shearing force at points distant 10, 15, 25, 30 ft. from end both graphically and by calculation. (6) A horizontal beam pinned at one end and freely supported at the other is loaded as indicated, find reactions at pin and support, and draw the bending moment and shearing force diagram. (7) A ladder 30 ft. long and weighing 200 lbs. is placed against a smooth vertical wall as indicated. The friction at the ground prevents the ladder from slipping. A man weighing 12 stone is 30 ft of the ladder. (a) 10 ft. from the top, (b) 10 ft. from bottom Find in each case the reaction of 8f the wall and ground. Find also the bending moment and shearing force along the ladder at points 5, 12, 18 and 25 ft. from the ground. (8) A beam AB is pivoted at A and fixed to C by a cord CB and loaded as indicated. the string. Find the reaction at A and the tension in (9) AB is a crowbar 6 ft. long, C is the fulcrum, AC=1.6 ft. The force P of 150 lbs. is applied vertically downwards, the load W of 320 lbs. is applied perpendicular to the bar. Find the reaction at C and the bending moment and shearing force diagram for crowbar. B 30° (c) No collar at D but a tie-rod SR where RB=7 ft. (13) Decompose a given rotor into two, satisfying the following conditions: (i) Components having given direction and one passing through a fixed point. (ii) One component lying in a given line, the other passing through a given point. (iii) One component completely determined. (iv) Components through given points and parallel to given rotor. (v) One component having a given direction, the other passing through a given point and being as small as possible. (vi) The components passing through given points and one being as small as possible, see § 174 (2). (14) A set of rotors is given, find two rotors one passing through a given point and one in a given direction such that all the rotors may be in equilibrium. (Draw the first line of the link-polygon through the given point.) (15) A set of rotors is given, find rotors in three given lines which will be in equilibrium with the given ones. (Take one of the given lines as a line of the link-polygon and the pole of the vector-polygon on the corresponding line.) (16) Four rotors are in equilibrium, given the magnitude of one and the axes of all, find the rotors. |