Statically indeterminate stresses in stiff framed structures . ange in slope from B to A is expressed by ©B - 9^, A EI MB Ma £sib r + f- t jr Combining equations (c) and (d) to eliminate MB, gives 2©AL + ©BL 3D s EI /-MAL - Pa2b 2L , or 2EI 2©A + 9B- 3L/lj Pa2b - %T ? • • 18. Similarly, combining equations (c) and (d) to eliminate M .gives 20BL + ©AL - 3D EI —+-T~\ 2 _ L IT 2n whence Eeb2 L2 Substituting Kr i/l, and R» D/L, the equations for and M-g MA = -2EK ( 2©A + 9B T-3R) - Pa2b/L2 ^M-g s 2EK ( 2©fi + ©A -*r 3R) - Pab2/L2 J become (2). Case 3. Member in flexure carrying a series of loa
Statically indeterminate stresses in stiff framed structures . ange in slope from B to A is expressed by ©B - 9^, A EI MB Ma £sib r + f- t jr Combining equations (c) and (d) to eliminate MB, gives 2©AL + ©BL 3D s EI /-MAL - Pa2b 2L , or 2EI 2©A + 9B- 3L/lj Pa2b - %T ? • • 18. Similarly, combining equations (c) and (d) to eliminate M .gives 20BL + ©AL - 3D EI —+-T~\ 2 _ L IT 2n whence Eeb2 L2 Substituting Kr i/l, and R» D/L, the equations for and M-g MA = -2EK ( 2©A + 9B T-3R) - Pa2b/L2 ^M-g s 2EK ( 2©fi + ©A -*r 3R) - Pab2/L2 J become (2). Case 3. Member in flexure carrying a series of loadssymmetrical sbout the middle of the member. Fig. 17 shows a mem-ber carrying a series of loadswhich is symmetrical about themiddle of the member. The M/EIdiagram for this member issimilar to that of ,withthe M/EI diagram of a simplebeam carrying the same loads,superimposed upon it. Let thearea of this superimposed dia-gram be represented by deflections at the point B will now be considered. The tangential deviation is given by. D - s Z~ A EI -nil F B + 1 + 6 3 2 » • • • • • (e). 19. The change in slope from B to A is given by Q 1 f MJ& M,L EI B%;^r f F (f). Combining equations (e) and (f) to eliminate MA, gives E9 L t 64L - 3D s 1B A EI 2 T2 whence and similarly MB s 2EK ( 2©B + 9A - 3R) - F/LMA s -2EK( 20A + ©B - 3R) - F/L, ..(3). It is seen that these equations can be applied to a membercarrying a uniform load, concentrated load at middle, equal con-centrated loads at the third point, or any loading which is sym-metrical about the middle of the beam. The preceding equations give an expression for the momentsat the ends of a member. It may be desirable to know the momentat the middle of a member, or under a concentrated load. From thegeometrical construction of the moment diagram, it is seen from Fig. 19 that in Case 1, the moment at any point c is equal to MAX + u2{L-*] T in Case L 2, referring to Fig. 20, the moment under th
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