www.kutub.info 2500

 ‫א‬‫א‬‫א‬ ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<í‰‰ö¹] ‫א‬‫א‬‫א‬‫א‬   ١٠٣  ‫א‬  ١٠٣        W،،‫א‬‫א‬،‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬   ‫א‬ ‫א‬  ،‫א‬   ‫א‬ ‫א‬ ‫א‬ ‫א‬ ‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K ‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬   ‫א‬        ،‫א‬ ‫א‬‫א‬   ‫א‬  ‫א‬ ‫א‬ ‫א‬ ‫א‬ ‫א‬   ‫א‬    ،   ‫א‬،‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬      ‫א‬  ‫א‬   ،‫א‬     K‫א‬‫א‬،‫א‬‫א‬  ‫א‬  ?   ?  ?  ? ‫א‬ ‫א‬    K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ،‫א‬،‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬     W    ‫א‬ ‫א‬  ‫א‬    ‫א‬  K‫א‬   ‫א‬‫א‬‫א‬‫א‬          ١٠٣       ‫א‬‫א‬‫א‬‫א‬Statics ‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬EStaticsF‫א‬ ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬ ،‫א‬‫א‬‫א‬ ‫א‬،‫א‬‫א‬‫א‬،E‫א‬‫א‬F‫א‬  K‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬  W K‫א‬‫א‬ J E‫א‬F‫א‬‫א‬ J  KFree Body Diagram‫א‬‫א‬ J K J ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ J K‫א‬ KE‫א‬F‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ J  Truss System‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ J  ‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫• א‬ ‫א‬‫א‬‫• א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬ • ‫א‬‫א‬ • ‫א‬‫א‬‫א‬‫• א‬ ‫א‬‫א‬‫• א‬  ‫א‬‫• א‬  ‫א‬‫א‬‫א‬ ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<í‰‰ö¹] ‫א‬‫ א‬‫א‬‫א‬   ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬ ١  ‫א‬‫א‬ ‫א‬‫א‬‫א‬  ١٠٣        W‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬   W‫א‬‫א‬  W‫א‬‫א‬  ‫א‬‫א‬‫א‬ • ‫א‬‫א‬ • ‫א‬‫א‬‫א‬ •  K٪100‫א‬‫א‬‫א‬W‫א‬‫א‬‫א‬    ٧W‫א‬‫א‬   W‫א‬‫א‬  •  •  •   W‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  KVectors‫א‬ -١-  ‫א‬‫א‬ ‫א‬‫א‬‫א‬  ١٠٣        Vector‫א‬ K ‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬‫א‬ ‫א‬  ،‫א‬‫א‬ ‫א‬‫א‬  K‫א‬،‫א‬،‫א‬،‫א‬W‫א‬‫א‬ K١ ‫א‬  ‫א‬‫א‬٢{١  Coplanar Vectors‫א‬‫א‬E١  ‫א‬‫א‬‫א‬ F3 F1 F2 ١ ‫ﺷﻜﻞ‬  Colinear Vectors ‫א‬E٢  K٢‫א‬‫א‬ F2 F1   ٢  Concurrent Vectors ‫א‬E3  K٣‫א‬‫א‬‫א‬ F3 F2 F1  ٣ -٢-  ‫א‬‫א‬ ‫א‬‫א‬‫א‬  ١٠٣        ٣{١  K٤‫א‬‫א‬‫א‬‫א‬   F2 F1 F1 + F2 F1 F2   ٤  W‫א‬‫א‬ A+B=B+A  K  ‫א‬٤{١ ‫א‬ (٥ ‫ )أﻧﻈﺮ ﺷﻜﻞ رﻗﻢ‬α‫א‬BA  A.B = A.B.Cos(α)  W   Kvector scalar‫א‬  A   α O  B ٥ -٣-   ‫א‬‫א‬ ‫א‬‫א‬‫א‬  ١٠٣        K‫א‬٥{١  ‫א‬‫א‬ J ١ ‫א‬‫א‬ E٦FC, B,A  W A = Acos(α1) + ASin(α1) B = BCos(α2) + BSin(α2) C = CCos(α3) + CSin(α3)   K‫א‬‫א‬C, B, A ‫א‬‫א‬‫א‬α3،α2،α1 y Cy C D Dy By Ay O B A Ax Bx Cx x Dx  ٦ ‫א‬  ‫א‬y‫א‬x‫א‬D‫א‬ ٦‫א‬  W‫א‬‫א‬ Dx = A.Cos(α1) + B.Cos(α 2) + C.Cos(α3) Dy = A.Sin(α1) + B.Sin(α 2) + C.Sin(α3) -٤-  ‫א‬‫א‬ ‫א‬‫א‬‫א‬  ١٠٣        WD‫א‬‫א‬ D = D x2 + D y2 Tan (α ) = Dy α = Tan ( −1 Dx Dy Dx)   K‫א‬‫א‬D‫א‬‫א‬‫א‬‫א‬α   ‫א‬٦{١ K  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬ vector‫א‬‫א‬‫א‬ ‫א‬‫א‬KN‫א‬kN ‫א‬،  K‫א‬‫א‬، ‫א‬‫א‬‫א‬،‫א‬  KE‫א‬FEF‫א‬W‫א‬‫א‬   W ‫א‬‫א‬F‫א‬ FxWW ‫א‬،٧‫א‬  K‫א‬‫א‬F‫א‬‫א‬‫ هﻲ א‬α،٧‫א‬Fy -٥-  ‫א‬‫א‬ ‫א‬‫א‬‫א‬  ١٠٣        F Fy α Fx  ٧ Fx = FCos (α ) W‫א‬‫א‬‫א‬ Fy = FSin(α )  WF ‫א‬‫א‬ F = 2 F x + F y2 α = Tan −1 ( Fy Fx )  W١  W٨‫א‬‫א‬‫א‬‫א‬  F1 = 25 kN, F2 = 50 kN, F3 = 100 kN  W‫א‬ ‫א‬x‫א‬‫א‬ ٨‫א‬  KEFKy -٦-  ‫א‬‫א‬  ١٠٣       ‫א‬‫א‬‫א‬  y  50 kN F2y   30 F3 100 kN  F2x F2  25 kN  F1y F1 60 45 F1x x F3y   F3x ٨   F1x = F1cos 45 = 25cos 45 = 17.68 kN F1y = F1sin 45 = 25sin 45 = 25 kN F2x = J 50cos (60 ) = -25 kN F2y = 50sin (60 ) = 43.30 kN F3x = 100cos( 30 ) = -86.60 F3y = 100 sin( 30 ) = - 50 kN  ‫א‬  ‫א‬ ‫א‬F2x, F3x, F3y ‫א‬  K Rx = F1x + F2x + F3x = 17.68 – 25 -86.60 = - 93.92 kN Ry = F1y + F2y + F3y = 17.68 + 43.30 – 50 = 10.98 kN  WR‫א‬     -٧-  ‫א‬‫א‬ ‫א‬‫א‬‫א‬ R =   ١٠٣       2 R x + R y2 α = Tan −1 ( Ry Rx )  W‫א‬   R = (−93.92)2 + (10.98)2 = 94.56 kN α = Tan−1(10.98/ 93.92) = 6.670  Graphical Method ‫א‬‫א‬‫א‬٧{١ ٦‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬   W F2 = 30 kN ‫א‬‫א‬F1= 20 kN‫א‬‫א‬  K‫א‬‫א‬‫א‬٩ F1 = 20 kN F2 = 30 kN  ٩  10 kN -٨-  W‫א‬     ١  ‫א‬‫א‬ ‫א‬‫א‬‫א‬  ١٠٣       ‫א‬‫א‬ 20 kNO‫א‬ R‫א‬K١٠‫א‬30 kN‫א‬‫א‬ K١٠‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬ ‫א‬R‫א‬‫א‬K  ‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬   R F1 = 20 kN O F1 = 20 kN O OF1 = 20 kN ١٠ α F1 = 20 kN F1 = 20 kN        W ،‫א‬‫א‬‫א‬ ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬   K‫א‬‫א‬ Particle‫א‬‫א‬٨{١ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬K‫א‬ -٩-  ‫א‬‫א‬ ‫א‬‫א‬‫א‬  ١٠٣        ‫א‬‫א‬‫א‬‫א‬٩{١ ‫א‬‫א‬،E‫א‬‫א‬F‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬KE‫א‬F‫א‬‫א‬‫א‬‫א‬   ‫א‬z, y, x‫א‬(Fx, Fy, Fz)  WE١١‫א‬F  F = Fx + Fy + Fz Fx = F Cos(αx ) Fy = F Cos(αy) Fz = F Cos(αz)  W x‫א‬‫א‬‫א‬‫א‬αx y‫א‬‫א‬‫א‬‫א‬αy z‫א‬‫א‬‫א‬‫א‬ αz  W‫א‬  F = Fx2 + Fy2 + Fz2 y Fy  A   F  αy αx αz Fz Fx O   B z x ١١ -١٠-   ‫א‬‫א‬  ١٠٣       ‫א‬‫א‬‫א‬   K‫א‬‫א‬١٢‫א‬‫א‬‫א‬ K١    ١٢٠ kN ١٢٠ kN 110 ٤٥ E F  ١٥ kN ٨0 kN ١٠٠ kN  20 kN  EF E F   ١٢   ‫א‬‫א‬‫א‬K ‫א‬‫א‬١٣‫א‬‫א‬‫א‬ K٢ K‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬ ٢٠٠ kN ١٥٠ kN ٣٥٠ ٩٠ ٠ ١٣ ١٨٠ kN     ‫א‬‫א‬K  ‫א‬‫א‬١٤‫א‬‫א‬‫א‬ K٣ K‫א‬‫א‬‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬ ٢٠٠ kN ٩٠ kN ٦٥ ٠ ٤٥ ١٨٠ kN ١٤ -١١-    ‫א‬‫א‬ ‫א‬‫א‬‫א‬  ١٠٣        K١٥‫א‬A‫א‬  K‫א‬‫א‬‫א‬α‫א‬‫א‬EF  K‫א‬‫א‬‫א‬EF  40 kN ٣0 kN  α A α ٦0 kN  ١٥ -١٢-   ‫א‬‫א‬‫א‬ ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<í‰‰ö¹] ‫א‬‫א‬‫א‬‫א‬   ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬ ٢  ‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      W‫א‬‫א‬ ‫א‬  K   W‫א‬‫א‬  W‫א‬‫א‬   • ‫א‬ • ‫א‬‫א‬ • ‫א‬‫א‬ •  K٪100‫א‬‫א‬‫א‬W‫א‬‫א‬‫א‬    ٥W‫א‬‫א‬   W‫א‬‫א‬  •  •  •   W‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ - ١٣ -  ‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      ‫א‬‫א‬١{٢ KO‫א‬‫א‬‫א‬‫א‬ E١‫א‬F O  W‫א‬‫א‬   E‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬FxE‫א‬FZ‫א‬ Moment = Force x Perpendicular Distance M=Fxd    W kN.mN.mO‫א‬ F ‫א‬M  KmO‫א‬F ‫א‬‫א‬‫א‬d  ‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬  K‫א‬‫א‬‫א‬‫א‬   O - + d   F ١ - ١٤ -    ‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      W١  K٢‫א‬O‫א‬F= 400 N‫א‬‫א‬‫א‬ :‫א‬ F = 400 N 2m O    ٢    Mo = + 400 x 2 = + 800 N.m  WO‫א‬‫א‬   K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  W٢  K٣‫א‬‫א‬F= 250 kNO‫א‬‫א‬  10 m 250 kN O ٣    WO‫א‬‫א‬ Mo = - 250 x 10 = - 2500 N.m  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ - ١٥ -  ‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      W٣  K٤‫א‬O‫א‬‫א‬‫א‬  500 kN  O  ٤  W‫א‬ ،d‫א‬ O‫א‬‫א‬٤‫א‬  WKO‫א‬‫א‬  Mo = 500 x 0 = 0 kN.m   ٢{٢ ‫א‬ ،‫א‬‫א‬‫א‬  K‫א‬‫א‬  W٤   K٥‫א‬‫א‬‫א‬‫א‬A ‫א‬‫א‬  50 kN 100 kN 100 kN 50 kN 1m A 1m 3m 3m ٥  - ١٦ - 50 kN 2m 2m    ‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      W‫א‬ ،A‫א‬‫א‬  ‫א‬‫א‬‫א‬A‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ KA‫א‬  MA = J 50x0 H100x2 J 100x3 H50x6 J 50x1 = = 0 H200 J 300 H300 J 50 = H150 kN.m  ‫א‬‫א‬‫א‬‫א‬50 kN‫א‬‫א‬ ‫א‬‫א‬A‫א‬‫א‬‫א‬‫א‬A‫א‬  KA ‫א‬  Varignon’s Theorem ??٣{٢ OFE ‫א‬F?? ‫א‬‫א‬‫א‬،O ‫א‬‫א‬  K‫א‬‫א‬   W٥  K٦‫א‬O‫א‬F = 100 kN‫א‬  100 kN ٣٠٠ 10 m O 40 m ٦  - ١٧ -    ‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬       30٠ 10 m O ٨٧ kN 40 m ٧    W‫א‬ ٣٠‫א‬ ‫א‬F = 100 kN‫א‬ ٦‫א‬ ‫א‬‫א‬ O‫א‬‫א‬،‫א‬‫א‬  KO‫א‬  WE٧FF‫א‬‫א‬‫א‬   Fx = Fcos(30) = 100xcos(30) = 87 kN W‫א‬‫א‬ Fy = Fsin(30) = 100xsin(30) = 50 kNW‫א‬‫א‬   ZO‫א‬‫א‬‫א‬ H87 x 10 = H870 kN.m  ‫א‬‫א‬‫א‬‫א‬  WO‫א‬‫א‬‫א‬  - 50x40 = -2000 kN.m K‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬O‫א‬F ‫א‬‫א‬‫א‬ Mo = -2000 +870 = -1130 kN.m - ١٨ -   ‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      W٦  KEF٨‫א‬A‫א‬F = 1000 N‫א‬ 3m 3m Fx = 1000Cos(50) 50  500 ٠ ١٠٠٠ N ١٠٠٠ N Fy = 1000Sin(50) 10 m 10 m     A A  EF  E F  ٨      W‫א‬ EF٨‫א‬ Fy  FxF‫א‬  KA‫א‬   Fx = Fcos(50) =1000xCos(50) = 642.78 N Fy = Fsin(50) = 1000xSin(50) = 766 N Z‫א‬‫א‬ Z‫א‬‫א‬  WA‫א‬‫א‬‫א‬   Mx = +642.78x10 = +6427.8 N.m  WA‫א‬‫א‬‫א‬ My = +766x3 = +2298 N.m  W‫א‬‫א‬‫א‬‫א‬A‫א‬F‫א‬‫א‬‫א‬ MA = +6427.8 +2298= +8725.8 N.m  K‫א‬‫א‬‫א‬‫א‬‫א‬ - ١٩ -  ‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      Couple‫א‬‫א‬٤{٢ ‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬،‫א‬‫א‬ Mo‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K٩‫א‬ A F   h F    B    ٩  ‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬  WK٩‫א‬ h/2  ‫א‬ M1 = Fxh/2 M2 = Fxh/2  Z‫א‬‫א‬ M1 + M2ZF(h/2 + h/2)ZFxh/2 + Fxh/2 ZFxh  ‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬K  ‫א‬‫א‬‫א‬‫א‬  K١٠‫א‬‫א‬‫א‬‫א‬ - ٢٠ -  ‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬       F  + d    M = Fd F     ١٠  ‫א‬،‫א‬‫א‬ ‫א‬‫א‬‫א‬d‫א‬F‫א‬‫א‬ ‫א‬  K١١‫א‬  F O h  d   F  ١١    Kh‫א‬E١١F‫א‬‫א‬‫א‬O  WO‫א‬‫א‬‫א‬‫א‬‫א‬ Mo = Fx(d + h) – Fxh = Fxd + Fxh –Fxh = Fxd  ‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬ - ٢١ -  ‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      W١  K١٢‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  50 kN 50 kN 10 m       ١٢   W‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬ ،١٢‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬‫א‬  W C = J 50 x 10 = -500 kN.m   ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬ ‫א‬K ‫א‬‫א‬‫א‬‫א‬‫א‬ K١٣‫א‬‫א‬‫א‬500 kN‫א‬‫א‬‫א‬ 500 kN.m  ١٣ - ٢٢ -   ‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      ‫א‬‫א‬٥K٢ FBAF‫א‬ ‫א‬‫א‬ B‫א‬–F  F  ‫א‬EEF١٤ KEF١٤‫א‬A‫א‬‫א‬F‫א‬‫א‬‫א‬ ‫א‬B‫א‬‫א‬–F‫א‬A‫א‬‫א‬F‫א‬ ‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬  WEF١٤‫א‬‫א‬‫א‬  C = -Fxd  ‫א‬‫א‬  KEF١٤‫א‬‫א‬‫א‬  F -F B B d A  EF F d A EF  ١٤ F B d F A C =F.d EF         W٢ A‫א‬‫א‬‫א‬B‫א‬‫א‬F = 20 kN‫א‬‫א‬‫א‬  KEF١٥‫א‬    1`  - ٢٣ -  ‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬     B 20 kN B 20 kN B d A  EF A 450 20 kN  ٢0 m ٢0 m ٢0 m A 450 20 kN EF  ١٥ C = 282.8 kN m 450   20 kN    EF      W‫א‬ A ‫א‬ ‫א‬F‫א‬‫א‬ ‫א‬B‫א‬‫א‬KEF١٥‫א‬20 kN‫א‬ ‫א‬‫א‬ ‫א‬‫א‬A‫א‬  WC C = +20xd kN.m  W‫א‬‫א‬‫א‬d   0 d = 20xSin(45 ) = 20x0.707 = 14.14 m  W‫א‬‫א‬ C = +20x14.14 = +282.8 kN.m K ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬   ‫א‬‫א‬ A‫א‬B‫א‬F ‫א‬ ‫א‬،EF١٥‫א‬+282.8 kN.m‫א‬A ‫א‬  KEF١٥‫א‬EF١٥‫א‬ - ٢٤ -  ‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      ٦{٢   F = 50 kN‫א‬‫א‬O‫א‬‫א‬  K١٦‫א‬ E١   ٥٠ kN 30٠ O 5m  ١٦   K١٧‫א‬A‫א‬F‫א‬E٢ ١٠ ٤٥٠  F = 100 kN  ٢٥    A   ١٧  O‫א‬‫א‬ F‫א‬  K١٨‫א‬ E٣    - ٢٥ -  ‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬       50 kN 20 kN 5m 5m  O  15 m F   ١٨  K١٩‫א‬A‫א‬‫א‬‫א‬‫א‬‫א‬ E٤    20 kN A  ٥m  ٥m ٥m ٥m  B 10 kN 25 kN    ١٩  K٢٠‫א‬‫א‬‫א‬‫א‬B‫א‬‫א‬ E٥   100 200 ٥٠ kN A ٣٠٠  ٥٠ kN B 20 m  ٢٠  - ٢٦ -  ١٠m    ‫א‬‫א‬‫א‬ ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<í‰‰ö¹] ‫א‬‫א‬‫א‬‫א‬   ‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬ ٣  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬  ١٠٣        W‫א‬‫א‬ free ‫א‬‫א‬،‫א‬‫א‬‫א‬E‫א‬F‫א‬‫א‬  K‫א‬‫א‬body digram   W‫א‬‫א‬  W‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ •  ‫א‬‫א‬‫א‬‫א‬ • ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ • K٪100‫א‬‫א‬‫א‬W‫א‬‫א‬‫א‬    ١١W‫א‬‫א‬   W‫א‬‫א‬  •  •  •   W‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬ - ٢٧ -  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬  ١٠٣        ١{٣ ‫א‬‫א‬K  ‫א‬‫א‬‫א‬E ‫א‬F‫א‬‫א‬ ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬ K  ‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ Roller Support ‫א‬‫א‬٢{٣ ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬  ‫א‬‫א‬K١ ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬ ‫א‬K‫א‬‫א‬‫א‬ (Ax = 0F‫א‬‫א‬‫א‬ EA‫א‬‫א‬‫א‬FA ‫א‬‫א‬  K١‫א‬  y  x A Ay  A A Ay ١ Ay     Hinge Support ‫א‬‫א‬٣{٣ ‫א‬‫א‬ ‫א‬‫א‬ ‫א‬‫א‬‫א‬ ‫א‬KEF٢‫א‬A‫א‬  KEF٢‫א‬‫א‬‫א‬ - ٢٨ -  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬  ١٠٣        Ax  A A Ay EF E F ٢    Fixed or Cantiliver Support ‫א‬‫א‬‫א‬٤{٣ A ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  W  ‫א‬KEF3‫א‬ KEF٣‫א‬A‫א‬‫א‬  MA  Ax  A Ay (‫)أ‬ ٣ (‫)ب‬   Equilibrium Conditions‫א‬‫א‬٥{٣  ‫א‬‫א‬‫א‬  ‫א‬‫א‬K‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬K ‫א‬‫א‬‫א‬ ‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬  ‫א‬K‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬ - ٢٩ -  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬  ١٠٣        ‫א‬‫א‬‫א‬٦{٣  ،4‫א‬‫א‬ F2y F1   F2  y  F2x F4 F٣ R1 x    R2  ‫א‬ –٤ y‫א‬x‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬x‫א‬‫א‬‫א‬E١  ∑F x =0 F4 – F2x – F3 =0  W‫א‬‫א‬‫א‬y‫א‬‫א‬‫א‬E2 ∑ Fy = 0  R1 + R2 – F1 –F2y = 0 ‫א‬E  ‫א‬‫א‬‫א‬F‫א‬E3 ∑M = 0  W‫א‬‫א‬   K‫א‬??‫א‬‫א‬‫א‬‫א‬‫א‬  - ٣٠ -  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬  ١٠٣       W‫א‬‫א‬‫א‬٧{٣ Colinear Forces ‫א‬‫א‬‫א‬E K‫א‬٥‫א‬‫א‬‫א‬‫א‬   ‫א‬ ‫א‬‫א‬ ‫א‬  ‫א‬‫א‬ F2‫א‬F1‫א‬K‫א‬‫א‬‫א‬‫א‬  KF1‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬  ∑F x  =0 F2 -F1= 0 F1 = F2    F2 F1  ‫א‬ W٥    Concurrent Forces ‫א‬‫א‬‫א‬E K  ‫א‬‫א‬ ‫א‬‫א‬‫א‬  ‫א‬‫א‬  ‫א‬‫א‬K٦‫א‬ ‫א‬O‫א‬‫א‬‫א‬‫א‬W  ‫א‬‫א‬  Kyx‫א‬ - ٣١ -  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬  ١٠٣       F2 F3 O F1  W٦   Parallel Forces ‫א‬‫א‬‫א‬E  ‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬،K٧‫א‬‫א‬   ∑F x =0 ∑M = 0 K ∑Fy =0‫א‬‫א‬ ‫א‬ : ٧  - ٣٢ -   ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬  ١٠٣       Opposite Couples ‫א‬‫א‬E   ‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K٨‫א‬‫א‬‫א‬‫א‬   +M -M    ‫א‬‫ א‬W ٨    Free-Body Diagram‫א‬‫א‬٨{٣ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬Free-Body Diagram‫א‬‫א‬  KE‫א‬F‫א‬‫א‬   W‫א‬‫א‬١  KEF٩‫א‬‫א‬AB‫א‬‫א‬‫א‬  W‫א‬ ‫א‬‫א‬‫א‬‫א‬AB ‫א‬‫א‬ hinge A‫א‬‫א‬K‫א‬‫א‬‫א‬‫א‬ B‫א‬،AyAxsupport ‫א‬FByroller support KEF٩‫א‬AB‫א‬‫א‬،E  - ٣٣ -  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬  ١٠٣       B‫א‬A ‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬  ∑F x  W‫א‬‫א‬ = 0 W‫א‬‫א‬E  Ax = 0   WA‫א‬‫א‬E  ∑MA = 0 +100x2 +100x4 –Byx6 = 0 +200 +400 – 6By = 0 +6By = 600 By = + 600/6 = + 100 kN ‫א‬‫א‬‫א‬‫א‬+100 kN B‫א‬By‫א‬‫א‬‫א‬‫א‬‫א‬  Kyx‫א‬‫א‬ ‫א‬‫א‬A‫א‬E٩FA ‫א‬‫א‬  KAy ‫א‬ W،‫א‬‫א‬E ∑ Fy =0  Ay + By -100 -100 =0 Ay = 200 -By   WAy‫א‬‫א‬EF‫א‬By = +100 kN  Ay = 200 – 100 = +100 kN  100 kN 100 kN  D C A 2m 2m – EF٩  - ٣٤ - B 2m     ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬  ١٠٣        100 kN 100 kN y Ax A Ay x B D C   By ‫א‬‫א‬‫א‬    ‫א‬‫ א‬WEF٩   W٢ A‫א‬‫א‬EF١٠‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬B‫א‬  100  100 kN   450 B A 2m 2m 2m  WEF ١٠        W‫א‬ ‫א‬‫א‬‫א‬‫א‬AB ‫א‬‫א‬ A‫א‬K  ‫א‬‫א‬‫א‬‫א‬‫א‬ B‫א‬،AyAxhinge support Byroller support AB‫א‬‫א‬،E‫א‬F - ٣٥ -  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬  ١٠٣        ٤٥‫א‬‫א‬‫א‬‫א‬KEF١٠‫א‬  W  +70.7 kN = +100Cos(45) W‫א‬‫א‬ +70.7 kN = +100Sin(45) W‫א‬‫א‬   ∑F x  Ax +70.7 = 0  W‫א‬‫א‬‫א‬ = 0 Wx‫א‬‫א‬‫א‬EF Ax = - 70.7 kN  ‫א‬  ‫א‬ ‫א‬،A‫א‬‫א‬  Kx‫א‬‫א‬ ‫א‬‫א‬ ∑ M A = 0 WA‫א‬‫א‬E  K‫א‬‫א‬ +100x2 + 70.7x4 – Byx6 = 0 +200 + 282.8 = 6By By = +482.8/6 = + 80.47 kN    KA‫א‬A‫א‬‫א‬   ∑F =0W،‫א‬‫א‬E y  Ay + By -100 -70.7 = 0 Ay + By = +170.7 Ay = +170.7 – By = +170.7 – 80.47 = +90.23 kN Ay = +90.23 kNWA‫א‬Ay‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬ - ٣٦ -  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬  ١٠٣         100 100 kN 50cos(45) Ax 450 B  A Ay 2m 2m 2m  y 50sin(45) By x    ‫א‬‫ א‬WEF ١٠    W٣ 50 ‫א‬‫א‬‫א‬،EF١١‫א‬‫א‬‫א‬‫א‬‫א‬  KC‫א‬kN  10 m 2m A   C 1000 5m B ‫א‬ WEF١١      W‫א‬  EF١١‫א‬EF١١‫א‬‫א‬‫א‬‫א‬  ∑F  W‫א‬‫א‬E١ x = 0 Wx‫א‬‫א‬‫א‬ ‫א‬  Bx – Ax =0 - ٣٧ -  W‫א‬  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬  ١٠٣       ‫א‬‫א‬  ‫א‬‫א‬‫א‬ ‫א‬ Wy  ∑F =0  y Ay – 50 -1000 = 0 Ay = + 1050 kN  E‫א‬‫א‬‫א‬F ∑ M A = 0 WA‫א‬‫א‬‫א‬‫א‬ +1000x10 + 50x2 – Bxx5 = 0 +10000 +100 -5Bx = 0 +10100 = 5Bx  Bx = +10100/5 = +2020 kN  E٢   WAx‫א‬‫א‬Bx  Bx – Ax = 0 Ax = Bx = +2020 kN  ‫א‬‫א‬‫א‬‫א‬Ax‫א‬‫א‬Ax‫א‬‫א‬  K‫א‬x‫א‬Ax ‫א‬،‫א‬‫א‬  10 m 2m Ax A  ١  C Ay  5m Bx 50 kN 1000   B  ‫א‬‫ א‬: EF١١ - ٣٨ -   ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬  ١٠٣        ٩{٣ EF١٢EF١٢‫א‬‫א‬‫א‬‫א‬‫א‬ E١  K‫א‬‫א‬  500 kN A C 5m  A 5m 500 kN  ٤٥٠  B 3m  C B 3m 3m EF    3m  EF  ١٢ E،،،F١٣‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬ E٢   100 N  1٥0 N  A B 6m 6m  EF  6m      - ٣٩ -  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬  ١٠٣        1٢0 N 240 N   100 N A  B ١٠ m ١٠ m ١٠ m   EF  240 N M = +100 kN.m A  ١٠ m ١٠ m  EF  300 kN 250 kN 600 A B ٥m 5m ٥m EF ١٣   B ١٠ m          - ٤٠ -  ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬  ١٠٣       ‫א‬‫א‬K١٤١‫א‬  ‫א‬‫א‬‫א‬‫א‬E٣  K‫א‬‫א‬   800 kN 3m     3m 500 kN  A B 2m ٢m  1m   ١٤ - ٤١ -  ‫א‬‫א‬‫א‬ ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<í‰‰ö¹] ‫א‬‫א‬‫א‬‫א‬   ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬ ٤  ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      W‫א‬‫א‬ ‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬  K‫א‬‫א‬‫א‬   W‫א‬‫א‬  W‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬ •  ‫א‬‫א‬‫א‬‫א‬ • ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ • ‫א‬‫א‬‫א‬‫א‬ • K٪100‫א‬‫א‬‫א‬W‫א‬‫א‬‫א‬    ١٦W‫א‬‫א‬   W‫א‬‫א‬  •  •  •   W‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬ - ٤٣ -  ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      ‫א‬١{٤ EF ‫א‬‫א‬  K‫א‬‫א‬‫א‬E‫א‬‫א‬F‫א‬‫א‬   ‫א‬‫א‬‫א‬٢{٤ ‫א‬K‫א‬E‫א‬F‫א‬‫א‬‫א‬  W١‫א‬‫א‬‫א‬‫א‬ EF١‫א‬Simple Beam • ‫א‬KE٣F K‫א‬‫א‬ EF١‫א‬Continuous Beam • ‫א‬‫א‬‫א‬ K‫א‬،‫א‬‫א‬ EF١‫א‬Cantiliver Beam  • ‫א‬KE٣F K‫א‬‫א‬ EF١‫א‬Overhanging Beam • ‫א‬KE٣F K‫א‬‫א‬ EF١‫א‬Built-in Beam‫א‬ • ‫א‬٦‫א‬ K‫א‬،‫א‬‫א‬‫א‬ End-supported Cantiliver Beam •  EF١‫א‬ ‫א‬٤‫א‬  K‫א‬،‫א‬‫א‬‫א‬ - ٤٤ -  ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬     Compound Beam  •  EF١EF١‫א‬ hinge  ‫א‬‫א‬‫א‬‫א‬‫א‬،roller ‫א‬ ‫א‬‫א‬‫א‬  K‫א‬‫א‬   Simple BeamWEF١      Continuous Beam WEF١       Cantiliver BeamWEF١         - ٤٥ -   ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬          ‫א‬WEF١  Built-in Beam      WEF١ End-supported cantiliver Beam        ‫א‬ WEF١ Compound Beam composed of cantiliver and simple beam connected by internal hinge       WEF١ Compound Beam composed of cantiliver and simple beam connected by internal roller - ٤٦ -     ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      ‫א‬‫א‬‫א‬‫א‬٣{٤ W‫א‬‫ א‬E١ ،EF٢‫א‬  K  W‫א‬‫א‬‫ א‬E٢ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬KEF٢‫א‬‫א‬  K‫א‬‫א‬‫א‬ W‫א‬ E٣ ‫א‬‫א‬  KEF٢‫א‬ W‫א‬‫ א‬E٤ ‫א‬  KEF٢‫א‬ W E٥  KEF٢‫א‬  KE‫א‬FW‫א‬‫א‬‫א‬ K‫א‬،‫א‬،‫א‬‫א‬،‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬K‫א‬،‫א‬،‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ - ٤٧ -  ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      F kN y   x  WEF    w kN/m   ‫א‬WEF w2 kN/m     w1 kN/m   ‫א‬WEF w (kN/m)        WEF - ٤٨ -   ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      M (kN.m)     WEF   ‫א‬‫א‬‫א‬‫א‬W٢   ‫א‬‫א‬‫א‬٤{٤ ‫א‬Kxy‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬Kyx‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬  ‫א‬‫א‬‫א‬‫א‬٥{٤ ‫א‬‫ א‬E١ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬ ∑F =0  ∑Fy =0  ∑M = 0    K‫א‬‫א‬ x  K‫א‬‫א‬  BA‫א‬‫א‬  - ٤٩ -  ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬     W‫א‬‫ א‬E٢  ‫א‬‫א‬‫א‬‫א‬‫א‬E‫א‬‫א‬F‫א‬‫א‬‫א‬ ‫א‬O‫א‬F‫א‬‫א‬ Kx‫א‬‫א‬ ‫א‬‫א‬EEF٣‫א‬A ‫א‬x   KEF٣‫א‬OB ‫א‬AO‫א‬F‫א‬  WVx‫א‬ A‫א‬x‫א‬‫א‬‫א‬‫א‬ Vx ‫א‬O‫א‬Vx‫א‬ K٣‫א‬ KEF٣‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  KEF٣‫א‬‫א‬‫א‬  WMx ‫א‬ C‫א‬‫א‬ C‫א‬Mx‫א‬  KO‫א‬‫א‬OB ‫א‬‫א‬AO‫א‬ W‫א‬‫ א‬E٣  K‫א‬‫א‬‫א‬ - ٥٠ -  ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      y P x A  w C  B O  C x   WEF٣ P Vx Mx   w Mx B A RA O Vx EF٣ +V RB         +M    ‫א‬ ‫א‬WEF٣ - ٥١ -    ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      W١  KEF٤‫א‬‫א‬C‫א‬‫א‬‫א‬  10 kN 5m A  5m  B C  3m   WEF4   10 kN 5m  5m A B C    RB = 5 kN RA = 5 kN   10 kN MC A  MC 5m 3m B C VC  VC RB = 5 kN RA = 5 kN ‫א‬‫א‬WEF4 - ٥٢ -      ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      W‫א‬  ‫א‬‫א‬‫א‬‫א‬C‫א‬‫א‬ E١  WAC ∑F =0  y RA – Vc =0 Vc = RA = + 5 kN   WAC‫א‬C‫א‬‫א‬C‫א‬‫א‬E٢  ∑M = 0  RA.3 – Mc = 0 Mc = 5x3 = + 15 kN.m  ‫א‬‫א‬‫א‬‫א‬٦{٤  ‫א‬‫א‬‫א‬E١ ‫א‬‫א‬‫א‬‫א‬‫א‬ ، ‫א‬ ‫א‬‫א‬‫א‬E ‫א‬‫א‬F‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬   W٢  KE١FEF٤‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K٥‫א‬RA = RB = +5kN W‫א‬١‫א‬  10 kN 5m A x  a C a b 5m B b x RB = 5 kN RA = 5 kN W٥ - ٥٣ -      ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬     ‫א‬A‫א‬xCA‫א‬a-a ‫א‬‫א‬ K٦‫א‬‫א‬‫א‬‫א‬‫א‬٥‫א‬ ‫א‬‫א‬ ‫א‬ ‫א‬K ‫א‬‫א‬‫א‬  W‫א‬‫א‬ ∑F =0 y 5–V=0 V = 5 kN  K+5kN٥x‫א‬‫א‬ ∑M a = 0 Wa-a‫א‬‫א‬ 5.x – M = 0 M = 5x ‫א‬‫א‬‫א‬ Kx ‫א‬‫א‬  WE٥m٠xFx   ٥  ٤  ٣  ٢  ١  ٠ EmFx  25  20  15  10  ٥  ٠ M‫א‬ kN.m   x M V 5 kN a-a‫א‬‫א‬W٦  - ٥٤ -  ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬     ‫א‬A‫א‬xBC‫א‬b-b ‫א‬‫א‬ K7‫א‬‫א‬‫א‬‫א‬‫א‬ ٥  ‫א‬ b-‫א‬‫א‬‫א‬ ‫א‬‫א‬ ‫א‬K ‫א‬‫א‬‫א‬  KE١٠٥x‫א‬FWb  ∑F =0 y 5 – 10 -V = 0 V = -5 kN   K-5 kN١٠٥x‫א‬‫א‬  ∑M =0 b Wb-b‫א‬‫א‬ 5.x – ١٠.(x – 5) -M = 0 M = 5x -10.x +50 = 50 – 5.x = 5(10 – x)  10 kN  5m M x 5 kN   V b-b‫א‬‫א‬W7   ‫א‬‫א‬‫א‬ Kx ‫א‬‫א‬  WE10 m٥xFx  - ٥٥ -  ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      ١٠  ٩  ٨  ٧  ٦  ٥  EmFx  ٠  ٥  ١٠  ١٥  ٢٠  ٢٥ M‫א‬ kN.m  x ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ 10 kN A 5m  K٨‫א‬ C 5 kN 5m  B   5 kN   +5 kN ‫א‬ + - Shear Force V    - 5 kN   ‫א‬  Bending Moment M +    +25 kN.m    ‫א‬‫א‬W٨ - ٥٦ -  ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      W٢  K٩‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬  ‫א‬ E١  W‫א‬‫א‬‫א‬‫א‬‫א‬ P = 5x10 = 50 kN ‫א‬،‫א‬‫א‬‫א‬  W،E‫א‬‫א‬F ‫א‬‫א‬ RA = RB = 50/2 = +25 kN  a-a‫א‬‫א‬‫א‬ E٢ ‫א‬9‫א‬‫א‬A‫א‬xa-a  K‫א‬‫א‬‫א‬K10‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ x‫א‬‫א‬ a-a‫א‬x ‫א‬‫א‬‫א‬  P = 5.x WE١٠F  KA‫א‬x/2‫א‬ ∑F =0 y 25 – 5.x - V = 0 V = 25 – 5x  xFx‫א‬‫א‬ ∑M - ٥٧ -  KE 10 m ٠ a = 0 Wa-a‫א‬‫א‬  ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬     25 . x − 5 x . x − M = 0 2 x2 − M = 0 2 5 M = 25 x − x 2 2 25 x − 5 ‫א‬‫א‬ Kx ‫א‬‫א‬‫א‬ E١٠٠xFx‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬  10  9 -25  0  8  7  EmFx 6  ٥ ٤  ٣ ٢  ١ ٠ -5 5 15 25 ‫א‬ 0 M‫א‬ -20 -15 -10 0 10 20 22.5 40 52.5 60 62.5 60 52.5 40 22.5 kN kN.m  x ‫א‬‫א‬‫א‬‫א‬ ‫א‬ ‫א‬‫א‬١١‫א‬‫א‬ K١١‫א‬ ‫א‬K ‫א‬x=5 m‫א‬ ‫א‬K ‫א‬‫א‬‫א‬ ‫א‬‫א‬F‫א‬‫א‬‫א‬  KE‫א‬‫א‬‫א‬‫א‬‫א‬       - ٥٨ -  ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      a 5 kN/m A B 10 m a x 25 kN 25 kN W9 5 kN/m M x V 25 kN 5.x kN x/2 x M V 25 kN a-a‫א‬‫א‬W10 - ٥٩ -  ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      5 kN/m   A 5m C B 5m 25 kN 25 kN     +25 kN  ‫א‬ +   - 25 kN  ‫א‬ 0  0 ++      Mmax =+62.5 kN.m   ‫א‬‫א‬W١١ - ٦٠ -   ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ٧{٤   ‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬K ‫א‬‫א‬‫א‬ ‫א‬K   ‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬   ‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬K  ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ K ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬Dr. BeamPro  K١٢‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬               ‫א‬‫א‬‫א‬‫א‬ Dr. BeamPro ‫א‬ :١٢ - ٦١ -   ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬      W٨{٤ ،١٦،١٥،١٤،١٣‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K٢٣،٢٢،،٢١،٢٠،١٩،١٨،١٧ 10 kN/m A 3m ١٣ 20 kN 3m 5m 3m A  15 kN B D C ١٤ 10 kN 6m   5 kN 6m A  12 m ١٥ 10 kN/m A 3m C D 5m B 3m ١٦ 10 kN/m A 5m 5m ١٧ - ٦٢ -  ‫א‬‫א‬‫א‬  ١٠٣    ‫א‬‫א‬‫א‬     ١٠ kN ٢0 kN/m A 5m 5m ١٨ 20 kN 20 kN 10 kN/m 2m A 2m 2m C 2m B D 5m ١٩ 20 kN 10 kN/m A C B 5m 3 m ٢٠  5 kN/m  A  B 10 m  ٢١ 10 kN 5 kN C B A 2m  2m D 2m ٢٢ - ٦٣ - 2m E      ‫א‬‫א‬‫א‬ ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<í‰‰ö¹] ‫א‬‫א‬‫א‬‫א‬   ‫א‬ ‫א‬ ٥  ‫א‬‫א‬  ١٠٣    ‫א‬      W‫א‬‫א‬ ‫א‬‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬   W‫א‬‫א‬  W‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬ • ‫א‬‫א‬‫א‬‫א‬ •  E‫א‬‫א‬F‫א‬‫א‬‫א‬ • ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ • K٪100‫א‬‫א‬‫א‬W‫א‬‫א‬‫א‬    ١١W‫א‬‫א‬   W‫א‬‫א‬  •  •  •  •   W‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬ - ٦٥ -  ‫א‬‫א‬  ١٠٣    ‫א‬      ‫א‬١{٥ ‫א‬‫א‬E F‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬joint‫א‬‫א‬‫א‬‫א‬K  ‫א‬‫א‬  K١‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬K  ‫א‬‫א‬‫א‬ ‫א‬‫א‬K٢‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬  K   ‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬K٥،٤، ٣‫א‬  K‫א‬‫א‬statically determinate trusses               ‫א‬W١    - ٦٦ -  ‫א‬‫א‬  ١٠٣    ‫א‬       ‫א‬‫א‬‫א‬٢        ٣        PRATT        ENGLISH HOWE ‫א‬‫א‬‫א‬‫א‬W٤ - ٦٧ -     ‫א‬‫א‬  ١٠٣    ‫א‬        PRATT        WARREN ‫א‬‫א‬‫א‬‫א‬WEF٤         PRATT       BAILEY ‫א‬‫א‬‫א‬‫א‬W٥ - ٦٨ -     ‫א‬‫א‬  ١٠٣    ‫א‬           WARREN ‫א‬‫א‬‫א‬‫א‬WEF٥   Determinacy‫א‬‫א‬‫א‬‫א‬٣{٥ ‫א‬‫א‬EF‫א‬ K١‫א‬‫א‬joints‫א‬ ‫א‬،‫א‬‫א‬  W‫א‬  ‫א‬‫א‬HE‫א‬‫א‬‫א‬F‫א‬Z‫א‬   ‫א‬EFbW‫א‬‫א‬ ‫א‬‫א‬r  ‫א‬jointsE‫א‬F‫א‬J  b + r Z‫א‬‫א‬‫א‬  ∑F =0 ∑F =0W‫א‬‫א‬ y x  Kb + rZ‫א‬2JZ‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ - ٦٩ -  ‫א‬‫א‬  ١٠٣    ‫א‬       ‫א‬  ‫א‬‫א‬ unstableEF‫א‬ b + r ⟨ 2J  b + r = 2J  statically determinate ‫א‬  EF F statically indeterminate‫א‬  ‫א‬‫א‬‫א‬ b + r⟩ 2J  E‫א‬‫א‬   KE‫א‬‫א‬F‫א‬‫א‬‫א‬ b + r ≤ 2J W‫א‬ ‫א‬‫א‬‫א‬ EF٦‫א‬‫א‬‫א‬Krigid frameEF٦‫א‬ ‫א‬ nonrigid frameEF‫א‬ ‫א‬EF٦‫א‬B‫א‬ F‫א‬EF‫א‬‫א‬،   KD‫א‬B‫א‬C‫א‬AE‫א‬   B B  C   A C D A EF E F ‫א‬‫א‬W٦ - ٧٠ -     ‫א‬‫א‬  ١٠٣    ‫א‬      W١  K٧‫א‬‫א‬‫א‬‫א‬       ٧    W‫א‬ b  ‫א‬٦J‫א‬‫א‬٧‫א‬  W3 r‫א‬٩ b + r = 9 + 3 = 12 2J = 2x6 = 12 b + r = 2JW‫א‬‫א‬  KstableEF‫א‬   W٢  K٨‫א‬‫א‬‫א‬‫א‬  W‫א‬ J = 5 E‫א‬F‫א‬ b = 6‫א‬ r = 3‫א‬ b+r=6+3=9 2J = 2x5 = 10  - ٧١ - W b + r ⟨ 2J W‫א‬‫א‬  ‫א‬‫א‬  ١٠٣    ‫א‬      KunstableEF‫א‬ ٣   ٤ ٢    ٥ ١   ٨   ‫א‬‫א‬J ٤{٥  Section Method‫א‬Joints Method‫א‬W‫א‬‫א‬ ‫א‬ ٩‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬EF‫א‬‫א‬‫א‬‫א‬  K‫א‬   Joints Method E‫א‬F‫א‬‫א‬E١ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬F‫א‬ ‫א‬ ‫א‬‫א‬‫א‬،E ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬ ‫א‬‫א‬F‫א‬‫א‬K ‫א‬‫א‬ ‫א‬ ‫א‬KE ‫א‬‫א‬  KE٢‫א‬F ٢‫א‬  W‫א‬‫א‬‫א‬‫א‬‫א‬  - ٧٢ -  ‫א‬‫א‬  ١٠٣    ‫א‬      W١ ‫א‬‫א‬E  ‫א‬‫א‬F‫א‬‫א‬‫א‬  Kjoints method‫א‬‫א‬٩‫א‬    3 kN E  F 15 m C B A 20 m RA 20 m 9 kN D 20 m 12 kN     RB ٩   W‫א‬  K‫א‬‫א‬‫א‬‫א‬ ‫א‬ ∑ M D = 0 WD‫א‬‫א‬E١  ‫א‬ RA.60 – 9x40 – 3x40 – 12x20 = 0 Rax60 = 720 RA = 720/60 = 12 kN ∑F =0W‫א‬‫א‬ y RA + RB – 9 – 3 -12 = 0 RA + RB = +24 RB = 24 - RA = 24 – 12 = 12 kN - ٧٣ -  ‫א‬‫א‬  ١٠٣    ‫א‬      W‫א‬‫א‬‫א‬‫א‬E٢  EFA ‫א‬E ‫א‬AE‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  K   FAE  ٥ A FAB  ٣  ٤   RA = 12 kN   FAEyFAExFAE ‫א‬ FAEy = 3/5 FAE  4/5 FAE ZFAEx  WA‫א‬‫א‬‫א‬  RA + FAEy = 0 12 + (3/5)xFAE = 0 FAE = - (5/3)x12 = - 20 kN ∑F =0 y ‫א‬AE‫א‬‫א‬‫א‬‫א‬‫א‬  K   WA‫א‬‫א‬‫א‬ FAB + FAEx = 0 FAB + (4/5)xFAE =0  FAB = - (4/5)xFAE = -(4/5)x (-20) = +16 kN - ٧٤ -  ∑Fx=0  ‫א‬‫א‬  ١٠٣    ‫א‬     ‫א‬AB‫א‬‫א‬‫א‬‫א‬‫א‬  K   B‫א‬E   FBE   B  FBC FBA = 16 kN   ٩ kN FBC – 16 = 0 FBC = + 16 kN    ∑ Fx = 0 Tension  FBE J 9 = 0 FBE = + 9 kN Tension ∑F =0 y  E‫א‬E   3 kN ٥ 3  FEF E   20 kN ٤ 9 kN FEC   ∑F =0 y - ٧٥ -  ‫א‬‫א‬  ١٠٣    ‫א‬     -9 -3 – (3/5)xFEC + (3/5)x20 = 0 (3/5)xFEC = 12 -12 = 0 kN FEC = 0   ∑Fx=0 FEF + (4/5) FEC + (4/5)x20 = 0 = -(4/5)x20 – (4/5)xFEC = - 16 kN Compression FEF   C‫א‬E   FCF  FCE = 0 ٥ C FBC FCD FCD   ٤ 12 kN   3 FCD - FBC = 0 = FBC = + 16 kN Tension FCF - 12 = 0 FCF = +12 Tension ∑Fx=0    ∑F =0 y   F‫א‬E F   FEF 12 kN - ٧٦ - FFD    ‫א‬‫א‬  ١٠٣    ‫א‬      FFD ∑Fx=0 (4/5) xFFD + FEF = 0 (4/5)xFFD = -FEF = - 16 kN = - (5/4)x16 = - 20 kN Compression  ‫א‬‫א‬‫א‬‫א‬١٠‫א‬  K–‫א‬H‫א‬  3 kN E 12 kN 0 kN B + 16 kN + 16 kN 9 kN  H F + 12 kN A - 16 kN + 9 kN - 20 kN  W‫א‬  J - 20 kN C ١٠     D + 16 kN 12 kN  12 kN       Section Method ‫א‬‫א‬E٢ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬K‫א‬ ‫א‬ ‫א‬ ‫א‬K   K‫א‬ ‫א‬‫א‬‫א‬‫א‬ ‫א‬ ‫א‬ ‫א‬‫א‬‫א‬‫א‬  K٣‫א‬‫א‬‫א‬  W‫א‬‫א‬‫א‬‫א‬‫א‬ - ٧٧ -  ‫א‬‫א‬  ١٠٣    ‫א‬      W٢ ‫א‬‫א‬CEEF BC‫א‬E ‫א‬‫א‬F‫א‬‫א‬‫א‬  Ksection method‫א‬‫א‬٩‫א‬   W‫א‬ ‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬ ‫א‬ CEEFBC‫א‬a-a١١‫א‬K   K١١‫א‬  3 kN   a E  F  A RA C B 9 kN a ١١ 12 kN D RB       ‫א‬‫א‬a-a ‫א‬ KCEEFBC‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،١٢‫א‬ ‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬  KE‫א‬‫א‬‫א‬F‫א‬ - ٧٨ -  ‫א‬‫א‬  ١٠٣    ‫א‬      3 kN FEF E FEF F  15 m   FEC FEC A FBC B C 20 m  D FBC 12 kN 9 kN 12 kN   12 kN  12  C‫א‬‫א‬a-a‫א‬‫א‬‫א‬‫א‬  WFEF‫א‬‫א‬‫א‬ WC‫א‬‫א‬ E١  ∑MC =0 -FEF x 15 -12x20 =0 FEF = -240/15 = - 16 kN Compression ‫ ﻹﻳﺠﺎد‬E‫א‬‫א‬a-a‫א‬‫א‬‫א‬‫ א‬E٢ ∑ME =0  W FBC ‫ﻗﻴﻤﺔ اﻟﻘﻮة‬ +12x20 – FBCx15 = 0 FBC = +240/15 = + 16 kN Tension ‫א‬‫א‬a-a‫א‬‫א‬‫א‬‫ א‬E٣  ∑Fy=0 W FEC ‫اﻟﻘﻮة‬ 12 - 9 -3 - (3/5) FEC = 0  12 - 12 - (3/5) FEC= 0  FEC= 0 ‫א‬EC‫א‬‫א‬  K - ٧٩ -  ‫א‬‫א‬  ١٠٣    ‫א‬      ٥{٥ ‫א‬‫א‬‫א‬،١٣‫א‬‫א‬‫א‬E١  Kjoints‫א‬  A  B  C 5m D E 5m F 5m G 5m    10 kN  13  K١٤‫א‬‫א‬‫א‬‫א‬E٢ A    B 14 ‫א‬‫א‬،١٥‫א‬‫א‬‫א‬ E٣ K‫א‬‫א‬joints‫א‬‫א‬     - ٨٠ -  ‫א‬‫א‬  ١٠٣    ‫א‬     C 20 kN 5m A 10 m B 10 m 20 kN ١٥ ‫א‬‫א‬،١٦‫א‬‫א‬‫א‬ E٤   K‫א‬‫א‬joints‫א‬‫א‬ E  F    10 m A 10 m 10 m B D C 10 m 10 Kn 20 Kn ١٦ ‫א‬‫א‬،١٧‫א‬‫א‬‫א‬ E٥  K‫א‬‫א‬joints‫א‬‫א‬ - ٨١ -     ‫א‬‫א‬  ١٠٣    ‫א‬     15 kN 25 kN A 45 45 45 5m 5m 45 B 10 kN ١٧  Kjoints‫א‬‫א‬١٨‫א‬‫א‬‫א‬‫א‬E٦ 150 Kn  E  2m D  4m  150 Kn C B A  120 Kn 4m  8m  ١٨  ‫א‬‫א‬cd, Cd, CD, BC, cC‫א‬‫א‬ E٧ ‫א‬‫א‬ section method‫א‬‫א‬١٩‫א‬  K - ٨٢ -   ‫א‬‫א‬  ١٠٣    ‫א‬     d c b  g e 40 m A C B E D H G 30 Kn 40 Kn 40 Kn   ٢٠‫א‬‫א‬‫א‬ CH, CI, BJ‫א‬‫א‬ E٨  K‫א‬‫א‬section method‫א‬‫א‬ 10 kN 10 kN 10 kN I 10 m 10 kN H 10 kN G A B C D 5@5 = 25 m 20 - ٨٣ -   ١٩ J   6x30 = 180 m K  E F   ‫א‬‫א‬  ١٠٣    ‫א‬     K٢١‫א‬‫א‬‫א‬ AD DB‫א‬‫א‬E٩ A 6m B 50 kN 5m D 50 kN C E 5m F G 2 3 3 3 ٢١ ‫א‬‫א‬‫א‬٢٢‫א‬‫א‬‫א‬‫א‬E١٠ K 5 kN 20 kN 5 kN C B A 1.6 m D 3m 3m E 12 kN ٢٢ - ٨٤ - F  ‫א‬‫א‬‫א‬ ã¹]<gè…‚jÖ]æ<ËÖ]<Üé×Ãj×Ö<íÚ^ÃÖ]<í‰‰ö¹] ‫א‬‫א‬‫א‬‫א‬   ‫א‬‫א‬ ‫א‬ ‫א‬ ٦  C,B,A   ١٠٣        W‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬K‫א‬  K‫א‬‫א‬‫א‬‫א‬   W‫א‬‫א‬  W‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ • ‫א‬‫א‬‫א‬‫א‬‫א‬ • ‫א‬‫א‬‫א‬‫א‬ •  ‫א‬‫א‬ •  K٪100‫א‬‫א‬‫א‬W‫א‬‫א‬‫א‬    ٨W‫א‬‫א‬   W‫א‬‫א‬  •  • ‫א‬‫א‬‫א‬‫א‬ •  W‫א‬‫א‬ ‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬ - ٨٥ -  C,B,A   ١٠٣        ‫א‬‫א‬‫א‬‫א‬‫א‬١{٦ ‫א‬‫א‬‫א‬‫א‬‫א‬ ‫א‬ ‫א‬  ‫א‬‫א‬‫א‬‫א‬ ‫א‬‫א‬‫א‬K‫א‬  F‫א‬‫א‬‫א‬K‫א‬‫א‬ ‫א‬‫א‬  F‫א‬K١‫א‬  F  ‫א‬‫א‬ ‫א‬ ‫א‬F‫א‬‫א‬  N/mm2 N/m2σ‫א‬ ‫א‬  W‫א‬  F σ = A  W kN/mm2, N/m2, N/mm2 ‫ ﺑـ‬‫א‬‫א‬‫א‬ :σ kg, kN , N  E‫א‬F‫א‬‫א‬‫א‬‫א‬WF m2 , cm2, mm2‫א‬WA a F F a F  σ Section a-aW١ - ٨٦ -   C,B,A   ١٠٣        W١ ‫א‬‫א‬‫א‬،P = 30 kN‫א‬  K3cmx2cm‫א‬‫א‬      W‫א‬  WA‫א‬ A = 3x 2 = 6 cm2= 6x10 m2  -4  W30 kN ‫א‬‫א‬‫א‬‫א‬ σ= 3 P 30 x10 = = 5 x10 7 N = 50 N − 4 m2 mm2 A 6 x10     ‫א‬‫א‬‫א‬‫א‬‫א‬٢{٦ ‫א‬FL0‫א‬ F‫א‬،٢‫א‬∆L E‫א‬F‫א‬  W‫א‬ε‫א‬‫א‬‫א‬E‫א‬  ε= ∆L L0   W m , cm, mm‫א‬‫א‬∆L mm, cm, m ‫א‬‫א‬L0 KEF‫א‬ε - ٨٧ -  C,B,A   ١٠٣        F F ∆L L0  ‫א‬W٢       W٢ 30 L0   ‫א‬‫א‬،K٣‫א‬‫א‬‫א‬ ‫א‬‫א‬70 kN‫א‬ F10 cm cm  K‫א‬‫א‬‫א‬‫א‬K0.02 cm   W‫א‬  70 kN 10 cm 70 kN 30 cm   ٣ A=  π D2 = π A‫א‬E (0.1) 2 = 7.85 x10−3 m 2 4 4 70 x103 F σ= = = 8.92 x106 N 2 = 8.92 MN 2 −3 m m A 7.85 x10   ‫א‬σ‫א‬‫א‬E  - ٨٨ -  C,B,A   ١٠٣       70 x103 F  σ= = = 8.92 x106 N 2 = 8.92 MN 2 −3 m m A 7.85 x10  ‫א‬ε‫א‬E ∆L 0 . 02 x 10 − 2 = = 0 . 67 x 10  ε = L0 30 x 10 − 2  −3   ‫א‬‫א‬‫א‬٣{٦ K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ high-strength ‫א‬‫א‬‫א‬‫א‬  ‫א‬ ٤‫א‬ ‫א‬‫א‬‫א‬‫א‬ ‫א‬Ksteel ‫א‬ ‫א‬‫א‬200 N/mm2  ‫א‬٠ ‫א‬‫א‬ ‫א‬‫א‬‫א‬ ‫א‬Hooke ‫א‬‫א‬EElastic modulus (Young’s modulus)‫א‬  K‫א‬‫א‬‫א‬E‫א‬١‫א‬‫א‬ (Ν/mm2) 300 σ 200 ‫ﻣﻨﻄﻘﺔ اﻟﻠﺪوﻧﺔ‬ ‫ﻧﻘﻄﺔ اﻻﻧﻜﺴﺎر‬ 250 150 100 50 0 ‫ﻣﻨﻄﻘﺔ اﻟﻤﺮوﻧﺔ‬ Stress ‫اﻹﺟﻬﺎد‬ 350 0 .0 5 S tra in ε ‫اﻻﻧﻔﻌﺎل‬ 0 .1  ‫א‬‫א‬‫א‬‫א‬W4 - ٨٩ -  C,B,A   ١٠٣       E=   WE‫א‬ σ ε σ = E.ε kN/m2 , N/mm2 W‫א‬‫א‬   E F‫א‬ ‫א‬‫א‬‫א‬ ٤‫א‬ K‫א‬‫א‬‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬‫א‬‫א‬E‫א‬   W‫א‬  ‫א‬‫א‬‫א‬‫א‬WF  ‫א‬WA   ‫א‬‫א‬WL0 : F ‫א‬‫א‬‫א‬‫ א‬: ∆L  E=  σ ε   Wεσ   F σ A = F .L 0 E = = ∆L ε ∆ L.A L0 - ٩٠ -  C,B,A   ١٠٣       L0‫א‬‫א‬A‫א‬E‫א‬F ‫א‬‫א‬  W    ∆L = F .L0 E. A   ‫א‬‫א‬‫א‬‫א‬‫א‬٤‫א‬‫א‬   ‫א‬‫א‬‫א‬  ‫א‬     K٤‫א‬‫א‬ - ٩١ -  C,B,A   ١٠٣        ‫א‬‫א‬‫א‬‫א‬W١  EGpaFE‫א‬  ‫א‬  ٢١٠–١٩٠  ‫א‬  ٧٩J ٧٠  ‫א‬  ١١٠J ٩٦  ‫א‬  ١٤–٠{٧  ‫א‬  ٣١–١٧  E‫א‬F‫א‬  ١٣J ١١  ‫א‬   W٣ Fy yield load‫א‬‫א‬2 cm‫א‬‫א‬ ‫א‬ Fmax = 150 kN‫א‬= 80 kN  K٥‫א‬Fb = 70 kN‫א‬ Fmax Load (κΝ) 150 80 Fy 70 Fb 0  Elongation  ‫א‬‫א‬W٥ - ٩٢ -    C,B,A   ١٠٣        W‫א‬‫א‬  ‫א‬‫א‬ E١ ‫א‬‫א‬‫ א‬E٢ ‫א‬‫א‬‫א‬‫א‬‫א‬ E٣ K1 cm   W‫א‬  ‫א‬‫א‬E١  A0 = A0‫א‬‫א‬ π 4 (0.02) 2 = 0.314 x10 −3 m 2   W‫א‬‫א‬  σy = Fy A0 = 80 x103 = 254 x10 6 N m2 0.314 x10 −3  W‫א‬‫א‬E٢   σ max = Fmax 150 x103 = = 477 x10 6 N m2 0.314 x10 −3 A0   ‫א‬‫א‬E٣  A= π 4  ‫א‬‫א‬ (0.01) 2 = 0.0785 x10 −3 m 2  W‫א‬‫א‬   σb = Fb 70 x103 = = 892 x10 6 N m2 A 0.0785 x10 −3   - ٩٣ -  C,B,A   ١٠٣        ‫א‬‫א‬‫א‬٤{٦   ‫א‬ E  F‫א‬‫א‬‫א‬‫א‬ ،٧٦‫א‬shear stress‫א‬  W‫א‬‫א‬   τ= V A  W N/m2, kg/cm2, N/mm2 ‫א‬‫א‬‫א‬‫א‬τ kG, kN, N ‫א‬‫א‬‫א‬V m2, cm2, mm2‫א‬A   WE٧Fγ‫א‬‫א‬  ∆L  γ = L  W  m, cm, mm‫א‬‫א‬‫א‬‫א‬W∆L m, cm, mm ‫א‬‫א‬L  K‫א‬‫א‬γ K ‫א‬‫א‬‫א‬‫א‬‫א‬  W‫א‬‫א‬  τ = Gγ - ٩٤ -   C,B,A   ١٠٣        W ،‫א‬kN/m2, N/mm2 ‫א‬G ‫א‬K ‫א‬‫א‬‫א‬G‫א‬ 2‫א‬  W E E‫א‬F‫א‬‫א‬G  E G =  2(1 + υ )   W Poisson’s ratio ?‫א‬?ν   ‫א‬‫א‬‫א‬‫א‬W2  EGpaFG‫א‬  ‫א‬  80–75  ‫א‬  30J 26  ‫א‬  41J 36  ‫א‬   10 kN 10 kN    10 kN  ‫א‬W٦ - ٩٥ - 10 kN    C,B,A   ١٠٣          ‫א‬ ‫א‬‫א‬W٧   ‫א‬٥{٦  8‫א‬  W‫א‬،‫א‬‫א‬‫א‬‫א‬‫א‬ σ=  M y I   W kG.m , N.m , kN.m ‫א‬‫א‬‫א‬‫א‬‫א‬WM  mm4 , cm4, m4  moment of inertia ‫א‬WI  K‫א‬‫א‬‫א‬‫א‬neutal axis‫א‬‫א‬‫א‬Wy  - ٩٦ -  C,B,A   ١٠٣                 KE‫א‬F‫א‬‫א‬‫א‬‫א‬‫א‬A ‫א‬٣‫א‬  ‫א‬‫א‬‫א‬‫א‬‫א‬W٣  ‫א‬  ‫א‬  bh3 Ix = 12  hb3 Iy = 12 y  h x b y Ix = Iy = πR 4 4 =  4 πD  64 x R   - ٩٧ - D  C,B,A   ١٠٣        W  K٩‫א‬‫א‬‫א‬‫א‬‫א‬   10 kN   A B ١٠ cm ١٠ m ٥ cm ‫א‬      ٩   W‫א‬  RA = RB = +٥kNZ‫א‬‫א‬ M = 5 x5 = 25kN .m  W   b.h3 5 x103 I= = = 0.42 x103 cm 4 12 12     W‫א‬   W‫א‬‫א‬‫א‬‫א‬ 10 y = h = = 5cm 2 2 - ٩٨ -    W‫א‬  C,B,A   ١٠٣        25 x10 3 x10 N N M σ = x 5 2976 29 . 76 y= = = I 0 .42 x10 3 cm 2 mm 2     ٦{٦ ،F = 20 kN‫א‬ 2.5 cm‫א‬‫א‬ K‫א‬‫א‬E = 70 GN/m2‫א‬ 6 cm 40 cm‫א‬‫א‬ E١  E٢ ‫א‬ ‫א‬‫א‬،1 MN/m2‫א‬‫א‬ ‫א‬K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  E = 200 GN/m2.  6 cm P 40 cm  6 mm19 mm ‫א‬‫א‬‫א‬‫א‬،‫א‬  K116 kN  K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬ - ٩٩ - E٣   C,B,A   ١٠٣       P = 116 kN 19 m m 6 mm   ABC BC‫א‬50 mmAB‫א‬،P‫א‬ E٤  ‫א‬40 MpaAB‫א‬‫א‬‫א‬‫א‬K38 mm .BC ‫ ﻓﻲ اﻟﻤﻨﻄﻘﺔ‬σ‫א‬ A C B P 13.3 ‫א‬12 mm1m‫א‬ E٥ ‫א‬‫א‬‫א‬‫א‬‫א‬،‫א‬kN K‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬،0.5 mm‫א‬ 12 mm 13.3 kN  1m - ١٠٠ -  13.3 kN  C,B,A   ١٠٣       q = 6.4 kN/m 3.5 m E٦ ‫א‬ ‫א‬  ‫א‬‫א‬‫א‬K ‫א‬ (b = 140 mm, h = 240 mmF‫א‬ K‫א‬ q kN/m 240 mm 3.5 m  140 mm    ‫א‬q = 60 kN/m 2 m K‫א‬(b = 50 mm, h = 150 mmF‫א‬ E٧ K‫א‬‫א‬‫א‬‫ א‬E١  c = 500 mm‫א‬‫א‬B‫א‬‫א‬E٢  K‫א‬‫א‬d = 25 mm ‫א‬‫א‬‫א‬  q kN/m B 150 mm d 50 mm c    L=2m - ١٠١ -  C,B,A   ١٠٣       P ‫א‬ ‫א‬ E٨ ‫א‬‫א‬‫א‬K‫א‬= 16 kN  K‫א‬(b = 100 mm, h = 250 mmF  W‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬‫א‬‫א‬‫א‬ E١ K‫א‬‫א‬‫א‬ E٢  ‫א‬‫א‬‫א‬ E٣ 5.5 kN/m3‫א‬‫א‬‫א‬  K‫א‬‫א‬‫א‬ P P B A C D 250 mm 100 mm 2.5 m 0.5 m 0.5 m - ١٠٢ -    C,B,A   ١٠٣        ‫א‬‫א‬‫א‬J ‫א‬ Area Moments of Inertia      - ١٠٣ -  C,B,A   ١٠٣        ‫א‬‫א‬‫א‬‫א‬J B  Areas and Volume                     ‫א‬‫א‬‫א‬ - ١٠٤ -  C,B,A   ١٠٣        1. Right Angles B c A b a C a = cos B c b cos A = = sin B c a tan A = = cot B b c sec A = = cosec B b c cosec A = = sec B a b cot A = = tan B a sin A =  2. Derived Relationships a = c sin A = c cos B = b tan A = b cot B = c 2 - b 2 b = c cos A = c sin B = a cot A = a tan B = c 2 - a 2 c= a a b b = = = = sin A cos B sin B cos A - ١٠٥ - a 2 + b2  C,B,A  ١٠٣        3. Oblique Triangles B c a A C b a- Sine Law a b c = = sin A sin B sin C b- Cosine Law a 2 = b 2 + c 2 - 2 bc cos A b 2 = a 2 + c 2 - 2 ac cos B c 2 = a 2 + b 2 - 2 ab cos C 4. General Trigonometric Formulas sin A = 2 sin 1 2 1 A cos A = 2 1 - cos 2 A = tan A cos A 1 1 2 2 cos A = 2 cos 2 A - 1 = 1 - sin 2 A 1 1 2 2 cos A = cos 2 A - sin 2 A = 1 - sin 2 A tan A = sin 2A sin A = = cos A 1 + cos 2A a- Addition and Subtraction Identities - ١٠٦ - sec 2 A - 1  C,B,A   ١٠٣       sin (A ± B) = sin A . cos B ± sin B . cos A cos (A ± B) = cos A . cos B ± sin A . sin B tan (A ± B) = tan A ± tan B 1 ± tan A . tan B 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 sin A + sin B = 2 sin ( A + B) . sin (A - B) sin A − sin B = 2 cos ( A + B) . sin (A - B) cos A + cos B = 2 cos ( A + B) . cos (A - B) cos A − cos B = 2 sin ( A + B) . sin (A - B) b- double-Angle Identities sin 2A = 2 sin A . cos A cos 2A = cos 2 A - sin 2 A = 1 - 2 sin 2 A = 2 cos 2 A - 1 tan 2A = 2 tan A 1 − tan 2 A c- Half-Angle Identities sin A = 2 1 - cos A 2 cos A = 2 1 + cos A 2 tan A sin A = 2 1 + cos A - ١٠٧ -  ‫א‬‫א‬   ١٠٣        K١٩٩٠،‫א‬‫א‬‫א‬‫א‬،‫א‬W١‫א‬‫א‬،‫א‬‫א‬KK١  ‫א‬–‫א‬‫א‬‫א‬K٢  ‫א‬،،‫א‬‫א‬  KE٠٠٤٠J ٤J ١٢F١٩٩٩،‫א‬  ‫א‬،،‫א‬‫א‬K٣  KE٠٠٤٠J ٤J ١٢F١٩٩٩،‫א‬  . Engineering Mechanics – Statics, 2nd Edition.٤ By Anthony Bedford and Wallace Fowler, Addison-Wesley, 1999.   Chapman & Hall, New York, 1986.. Jafar Vossoughi, Statics for Architects,٥   for Engineers,. Ferdinand P. Beer and E. Russell Johnston, Vector Mechanics٦ McGraw-Hill Book Company, Singapore, 1990. - ١٠٨ -  ‫א‬   ١٠٣       ‫א‬  ١ ٢ ٢ ٢ ٢ ٤ ٥ ٨ ٩ ١٠ ١١ ١٣ ١٤ ١٦ ١٧ ٢٠ ٢٣ ٢٥ ٢٧ ٢٨ ٢٨ KKKKKKK‫א‬‫א‬‫א‬W‫א‬‫א‬  Vector‫א‬ ١{١  ‫א‬‫א‬٢{١  ٣{١  ‫א‬٤{١  ‫א‬٥{١  ‫א‬٦{١ Graphical ‫א‬‫א‬‫א‬٧{١  Method Particle‫א‬‫א‬٨{١  ‫א‬‫א‬‫א‬‫א‬٩{١  ١٠{١  ………………….………………….. ‫א‬‫א‬‫א‬W‫א‬‫א‬  ‫א‬‫ א‬١{٢  ٢{٢ Varignon’s Theorem ??٣{٢  Couple‫א‬‫א‬٤{٢  ‫א‬‫א‬٥K٢  ٦{٢  .………………….. KKK……K‫א‬‫א‬‫א‬‫א‬W‫א‬‫א‬  ١{٣ Hinge Support ‫א‬‫א‬٣{٣  ‫א‬  ٢٩ ٢٩ ٣٠ ٣١ ٣٣ ٣٩ ٤٣ ٤٤ ٤٤ ٤٧ ٤٩ ٤٩ ٥٣ ٦١ ٦٢ ٦٥ ٦٦ ٦٦ ٦٩ ٧٢ ٧٢ ٧٧ ٨٠ ٨٥  ١٠٣       Fixed or Cantiliver Support ‫א‬‫א‬‫א‬٤{٣ Equilibrium Conditions‫א‬‫א‬٥{٣  ‫א‬‫א‬‫א‬٦{٣  ‫א‬‫א‬‫א‬٧{٣ Free-Body Diagram‫א‬‫א‬٨{٣  ٩{٣  KKKKKKKKKKKK‫א‬‫א‬‫א‬W‫א‬‫א‬‫א‬  ‫א‬١{٤  ‫א‬‫א‬‫א‬٢{٤  ‫א‬‫א‬‫א‬‫א‬٣{٤  ‫א‬‫א‬‫א‬٤{٤  ‫א‬‫א‬‫א‬‫א‬٥{٤  ‫א‬‫א‬‫א‬‫א‬٦{٤ ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬٧{٤  ٨{٤  KK‫א‬W‫א‬‫א‬  ‫א‬١{٥  ‫א‬‫א‬ ٢{٥  Determinacy‫א‬‫א‬‫א‬‫א‬J ٣  ‫א‬‫א‬J ٤{٥  Joints Method E‫א‬F‫א‬E١  Section Method ‫א‬E٢  ٥{٥  K‫א‬‫א‬W‫א‬‫א‬  ‫א‬  ٨٦ ٨٧ ٨٩ ٩٤ ٩٦ ٩٩ ٩٩ ١٠٣  ١٠٣        ‫א‬‫א‬‫א‬‫א‬‫א‬١{٦  ‫א‬‫א‬‫א‬‫א‬‫א‬٢{٦  ‫א‬‫א‬‫א‬٣{٦  ‫א‬‫א‬‫א‬٤{٦  ‫א‬٥{٦  ٦{٦  ‫א‬‫א‬‫א‬WA ١٠٤ KKKKKK‫א‬‫א‬‫א‬‫א‬WB ١٠٨  KKKKKKKKKKKKKK‫א‬‫א‬              ‫א‬‫א‬‫א‬‫א‬‫א‬‫א‬  ‫א‬E‫א‬F‫א‬‫א‬ GOTEVOT appreciates the financial support provided by BAE SYSTEMS

www.kutub.info 2500

Related documents

Products

Support

www.kutub.info 2500

Related documents

Add this document to collection(s)

Add this document to saved

Suggest us how to improve StudyLib