Text-book of mechanics . EIf2 ™? + Cl,dx3 21 ,#2 TO0 E17i = -1CT + Cix + C*>dx2 0/ (1) (2) 88 MECHANICS OF MATERIALS „Tdy wxA CiX2 . n n (s EIrf = -A + Cix + C3, ... (3) ax 241 2 £,y__i«L + c£ + ^ + c* + c4. (4) I20t 6 2 d3vNow as £7 —~ represents the shearing force Qx, at thedx? ft V 7£J£ point a;, Ci can be found by putting EI -7^ = —- when x = o, for the shearing force at the left end equals thereaction at that point; hence Again, £7 —^ is the bending moment at any section xunits from the left origin, and as the bending moment EI -r% = o when x = o,oar C2 = o. As the slope is not known a
Text-book of mechanics . EIf2 ™? + Cl,dx3 21 ,#2 TO0 E17i = -1CT + Cix + C*>dx2 0/ (1) (2) 88 MECHANICS OF MATERIALS „Tdy wxA CiX2 . n n (s EIrf = -A + Cix + C3, ... (3) ax 241 2 £,y__i«L + c£ + ^ + c* + c4. (4) I20t 6 2 d3vNow as £7 —~ represents the shearing force Qx, at thedx? ft V 7£J£ point a;, Ci can be found by putting EI -7^ = —- when x = o, for the shearing force at the left end equals thereaction at that point; hence Again, £7 —^ is the bending moment at any section xunits from the left origin, and as the bending moment EI -r% = o when x = o,oar C2 = o. As the slope is not known at any point of the beam,pass to equation (4). Here y = o when x = o, .. C4 = y = o when x = I, A, „ . w/4 wZ4 7 w/4 so that L3I = H — o = — -—-! 120 36 300 or C3 = - i-r- • 360 Substituting the values of the constants of integra-tion just found in equations (1), (2), (3), and (4), weobtain w° = EId* = -T (5) DEFLECTION OF BEAMS o-«£- )3X>-12 slope -% — i6oEIl deflection = y = -. Fig. 47 From equations (8) and (9) the slopes and deflectionsincluding the greatest and maximum values can beobtained as before. 90 MECHANICS OF MATERIALS In Fig. 47 the equations (5) to (9) are plotted; notetheir relations as derivative curves, and compute allprincipal values and their location and indicate same onFig- 47- Exercise 103. Deduce the above results from the equa- <Pvtion Mx= EI -A- CHAPTER IV STATICALLY INDETERMINATE BEAMS Section IX PROPPED AND BUILT-IN BEAMS Whenever a free body is in equilibrium under theaction of coplanar forces, three independent equations,and only three, can be written expressive of the condi-tions of equilibrium. Thus only three unknown quan-tities can in general exist in such a problem. Any problem in planar equilibrium involving morethan three unknown quantities cannot be solved bythe principles of statics alone, but requires in additionthe use of the principles of mechanics of materials; suchproblems are said to be sta
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