Essentials in the theory of framed structures . res. There are several methods by which deflections caused bybending may be determined. In the oldest and most widelyknown method, the second differential equation of the elasticcurve is derived. This equation must be integrated twicebefore the deflection at any point may be found. The methodis long and greatly involved, except for the simplest conditionsof loading. The simpler and less known method of area-moments establishes a relation between a tangent to theelastic curve and the bending moment diagram. 137. Method of Area-moments.—^Let A and
Essentials in the theory of framed structures . res. There are several methods by which deflections caused bybending may be determined. In the oldest and most widelyknown method, the second differential equation of the elasticcurve is derived. This equation must be integrated twicebefore the deflection at any point may be found. The methodis long and greatly involved, except for the simplest conditionsof loading. The simpler and less known method of area-moments establishes a relation between a tangent to theelastic curve and the bending moment diagram. 137. Method of Area-moments.—^Let A and B (Fig. 127)represent any two points on the neutral axis of a beam, whichis bent by any arrangement of loading. Through A and Bdraw the tangents AD and BC intersecting at C, and the 208 Sec. I DEFLECTION OF BEAMS 209 normals AI and BI intersecting at /. Then AAIB = LBCD= <^. Let QPRS represent the bending moment diagram forthe portion of the beam between A and B. Let EFHG repre-sent an element of the beam between two right sections EG and. Pig. 127. Fig. 128. FH (drawn to a larger scale in Fig. 128) which were parallel and adistance ds apart before the element was bent by the bendingmoment M. Let r (Fig. 128) represent the radius of curvatureof the neutral axis for this element. The fiber at the neutralaxis remains unchanged in length, while the fiber KL at a 2IO THEORY OF FRAMED STRUCTURES Chap. V distance y below the neutral plane has been increased in length from ds — rd(t> to (r + y)dcl). Hence the total strain (change in ydlength) in the length ds is yd and the unit strain is -5— Let / represent the unit stress on the fiber KL; and let E represent themodulus of elasticity. Then ^ = ^°^^=ir dsLet I = the moment of inertia of the cross-section about theneutral plane; then My? I . Eyd(l> My whence -^1— = —:r~ ds I , Mdsd = ^ and 0 = I J0 = I Ja Ja ^Mds,A EI If the beam in its natural state is straight (not arched) andis properly designed, the curvature wil
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Keywords: ., bookcentury1900, bookdecade1920, booksubjectstructu, bookyear1922
