Essentials in the theory of framed structures . independent moment equation when ZF = oLet us suppose that the magnitudes and locations of the fivevertical forces, acting upon the body illustrated in Fig. 13, aresuch that IMp = oand IMg = o when F and G are any two points not in a line parallel to thesystem. The two equations signify that the moments of thefive forces are balanced about F and G, or {k + a)A + {k + h)B + {k + c)C = ()fe + d)D -\-{k + e)£(i)and aA -\-hB-\-cC = dD ^- eE (2) Subtract whence or kA+kB + kC = kD-\- kEA +5 -\-C = D + EZF = o (3) Sec. V EQUILIBRIUM OF COPLANAR FORCES 2
Essentials in the theory of framed structures . independent moment equation when ZF = oLet us suppose that the magnitudes and locations of the fivevertical forces, acting upon the body illustrated in Fig. 13, aresuch that IMp = oand IMg = o when F and G are any two points not in a line parallel to thesystem. The two equations signify that the moments of thefive forces are balanced about F and G, or {k + a)A + {k + h)B + {k + c)C = ()fe + d)D -\-{k + e)£(i)and aA -\-hB-\-cC = dD ^- eE (2) Subtract whence or kA+kB + kC = kD-\- kEA +5 -\-C = D + EZF = o (3) Sec. V EQUILIBRIUM OF COPLANAR FORCES 27 Hence, if the moments of the forces in a parallel system balanceabout any two points not in a line parallel to the system, themagnitudes will also balance and equilibrium is Only two of the Eqs. (i), (2) and (3) are independent, sinceeach may be derived from the other two. 21. Illustrative Problems.—(i) Seven vertical forces holdthe body (Fig. 14a) in equilibrium. The magnitudes and 28 THEORY OF FRAMED STRUCTURES Chap. I locations of six forces are known: the magnitude and locationof the seventh force P is desired. t i IV i 20 IS 16 25 24 12 60 52 52 8 ZM about the line AB 0 P 15X0 = 0 20 X 5 = 100 25X8 = 200 16 X 14 = 224 12X31 = 372 24 X 19 = 456 572 780 572 8)208 26 The sum of the known magnitudes acting upward is 60 lb., andthe sum of the known magnitudes acting downward is 52 lb. Inorder to balance the magnitudes, P must act downward havinga magnitude of 8 lb. Since any point may be chosen as thecenter of moments, it is expedient to select a point in the lineof one of the forces, thereby eliminating that force from themoment calculations. The sum of the counter-clockwisemoments about the line AB is 208 greater than the sumof the clockwise m
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