Essentials in the theory of framed structures . erpendicular distance from a point is called themoment of the force about thepoint. The perpendicular dis-tance is called the arm of the force,and the point is known as the cen-ter of moments. If the magnitudeof a force P is 8 lb., and its armfrom a point 0 is 3 ft., the mo-ment of the force P about the point0 is 24 In 1687, or about 80 years after Stevin had published hisdemonstration of the triangles of forces, Pierre Varignon pre-sented before the Paris Academy, the Principle of principle is known as Varignons Theorem. He
Essentials in the theory of framed structures . erpendicular distance from a point is called themoment of the force about thepoint. The perpendicular dis-tance is called the arm of the force,and the point is known as the cen-ter of moments. If the magnitudeof a force P is 8 lb., and its armfrom a point 0 is 3 ft., the mo-ment of the force P about the point0 is 24 In 1687, or about 80 years after Stevin had published hisdemonstration of the triangles of forces, Pierre Varignon pre-sented before the Paris Academy, the Principle of principle is known as Varignons Theorem. He statedthat the moment of the diagonal of a parallelogram of forcesequals the sum of the moments of the other two sides. Hisproof is somewhat as follows: The parallelogram CADB (Fig. 3) represents the forces P andQ and their resultant R. The lengths of the perpendicularsfrom the center of moments 0 to the lines representing theforces P, Q and R, are p, q and r respectively: ^The symbol S represents the idea expressed in the words algebraic 12 THEORY OF FRAMED STRUCTURES Chap. I area ^OCD = area WCA + area \OAD + area ^ACDor - rR = - pP + - sQ + -tQ 2 2^2 2 hence, rR = pP -\- qQ Varignon showed that the center of moments O may belocated outside the parallelogram, within it, or on one of itssides. It is obvious that the moment rR of the resultant R, actingcounter-clockwise about the point 0, may be balanced by themoment rE of the equilibrant E acting clockwise. If oppositesigns are given to the moments of the two forces according asthey act clockwise or counter-clockwise, their algebraic sum iszero, or rR - rE = otherefore, pP -\- qQ — rE = o P, Q and E represent three coplanar, concurrent forces inequilibrium; but the principles may be generalized so as toinclude any number of concurrent forces by combining theminto partial resultants after the manner of Fig. 2. Hence, if asystem of coplanar, concurrent forces is in equilibrium, thealgebraic sum of the moments of all the
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