Mechanics of engineeringComprising statics and dynamics of solids: and the mechanics of the materials of constructions, or strength and elasticity of beams, columns, arches, shafts, etc . the spacing of the rivets, if any, is considered. Secondly, considering the weight of the beam, or anyuniformly distributed loading, weighing w lbs. per unit oflength of beam, in addition to P, Fig. 264, we have thereactions * *>=rM-- »\\ then for a portion Om of length #<^moments about m give FLEXURE. SPECIAL PROBLEMS. 303 X RoX -\-wx.~=0 pi — ^ , _a , on OC, M=E2x — y2 wx2 Evidently for x = 0 (
Mechanics of engineeringComprising statics and dynamics of solids: and the mechanics of the materials of constructions, or strength and elasticity of beams, columns, arches, shafts, etc . the spacing of the rivets, if any, is considered. Secondly, considering the weight of the beam, or anyuniformly distributed loading, weighing w lbs. per unit oflength of beam, in addition to P, Fig. 264, we have thereactions * *>=rM-- »\\ then for a portion Om of length #<^moments about m give FLEXURE. SPECIAL PROBLEMS. 303 X RoX -\-wx.~=0 pi — ^ , _a , on OC, M=E2x — y2 wx2 Evidently for x = 0 ( at 0) M = 0, while for a; at (7) we have, putting w = W -f- I Jfc=RA w?= PW2 , Wk +- JPZ2 (2)Z2 ( (3) « 2 IIt remains to be seen whether a value of M may not existin some section between 0 and (7, (, for a value of x<l2 in eq. (2)), still greater than Mc. Since (2) gives M asa continuous function of x between 0 and C, we putdM-Sr dx = 0, an4 obtain, substituting the value of the con-stants B2 and w, { max. \ pti R2—wx=0 W This must be forR4 w=wj luiiiiii . xQ 1 for M or \ =~l+ % I( min. ) w M max., since d2M -*- dec2 is negative Ri. when this value of x is sub-stituted. If the particularvalue of x given by (4) is l2 Fig. 264. we are not concerned withthe corresponding value ofM, and the greatest Mon OCwould be Mc. For the short portion BC,which has moment and sheardiagrams of its own not con-tinuous with those for OC, itmay easily be shown thatMc is the greatest moment ofany section. Hence the M 304 MECHANICS OF ENGI^EEKING. max., or Mm, of the whole beam is either MG or Mn,according as xu is > or r; PJ P • = i!fn, when —- is < ~? JF ?i . . (6) Seeeqs. (3) and (5)for MP and ilf n If either P
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Keywords: ., bookcentury1800, bookdecade1880, booksubjectenginee, bookyear1888
