The elasticity and resistance of the materials of engineering . ch a length of fibre at units distancefrom that surface will be itP. If the beam were originally straight and horizontal, n wouldbe equal to dx. P being supposed small, the efiect of the strain fiP at any Art. 22.] GENERAL FLEXURE FORMULAE. 145 section, B, is to cause the end K of the tangent BK, to movevertically through the distance nPx, If BK and BR (taken equal) are the positions of the tan-gents before and after flexure, nPx will be the vertical dis-tance between K and R. By precisely the same kinematical principle, the expre
The elasticity and resistance of the materials of engineering . ch a length of fibre at units distancefrom that surface will be itP. If the beam were originally straight and horizontal, n wouldbe equal to dx. P being supposed small, the efiect of the strain fiP at any Art. 22.] GENERAL FLEXURE FORMULAE. 145 section, B, is to cause the end K of the tangent BK, to movevertically through the distance nPx, If BK and BR (taken equal) are the positions of the tan-gents before and after flexure, nPx will be the vertical dis-tance between K and R. By precisely the same kinematical principle, the expres-sion nPywWi be the horizontal movement oi A in referenceto B, Let ^itPx and ^nPy represent summations extendingfrom A to C, then will those expressions be the vertical andhorizontal deflections, respectively, of A in reference to C, Itis evident that these operations are perfectly general, and thatX and y may be taken in any direction whatever. The following general, but strictly approximate equations,relating to the subject of flexure, may now be written :. D^ represents horizontal deflection. 10 146 THEORY OF FLEXURE. [Art. 23. The summation ^Pz must extend from ^4 to a point of nobending; or from ^ to a point at which the bending momentis M^, In the latter case : M, = 2Pz -{-M/ (7) J// may be positive or negative. Art. 23.—The Theorem of Three Moments. The object of this theorem is the determination of the re-lation existing between the bending moments which are foundin any continuous beam at any three adjacent points of sup-port. In the most general case to which the theorem applies,the section of the beam is supposed to be variable, the pointsof support are not supposed to be in the same level, and atany point, or all points, of support there may be constraintapplied to the beam external to the load which it is to carry ;or, what is equivalent to the last condition, the beam may notbe straight at any point of support before flexure takes place. Before establishing the
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