Tài liệu Bai toan phang

§Ò bµi sè 36 Cho kÕt cÊu chÞu lùc nh trªn h×nh 01. Trong ®ã: a = 3m, a/2 t = 10cm, E = 1.2x104 N/cm2, a/2 ν= 0.18, P = 20 kN, q = 5 kN/m. Yªu cÇu: - TÝnh chuyÓn vÞ c¸c nót, - X¸c ®Þnh vÐc t¬ øng suÊt trong c¸c phÇn tö. a P t q 2a H×nh 01 - S¬ ®è kÕt cÊu phÇn tÝnh to¸n bµI 36 1. C¸c sè liÖu ban ®Çu . ChiÒu dÇy tÊm t = 0.10m . KÝch thíc cña kÕt cÊu a = 3.00m . M« ®un ®µn håi E = 1.2E+07kN/m2 . HÖ sè Po¸t - x«ng ν = 0.18 2. Chia kÕt cÊu thµnh c¸c phÇn tö vµ c¸c th«ng tin cho tÝnh to¸n KÕt cÊu ®îc chia thµnh 12 phÇn tö tam gi¸c nh h×nh 02. Liªn kÕt biªn díi cña tÊm ®îc m« h×nh hãa bëi c¸c liªn kÕt gèi cè ®Þnh t¹i c¸c nót däc theo biªn nµy. y 6 9 8 4 3 1 2 2 4 11 8 11 10 6 3 1 12 7 5 12 5 9 7 H×nh 02 - S¬ ®å rêi r¹c hãa kÕt 10 x C¸c th«ng tin vÒ nót: 1 0 0 Sè hiÖu nót To¹ ®é X To¹ ®é Y 2 3 1.5 1.5 0 1.5 4 3 0 5 3 1.5 6 3 3 7 8 9 10 11 12 4.5 4.5 4.5 6 6 6 0 1.5 3 0 1.5 3 TÝnh to¸n dêi t¶i träng ph©n bå vÒ c¸c nót qa/ 24 = a = qa/ 12 qa/ 8 qa/ 24 qa/ 8 q q / qa/ 24 P qa/ 4 qa/ 12 5qa/ 24 q / H×nh 03 - S¬ ®å chia t¶i träng vµ dêi t¶i träng vÒ nót H×nh 04 - S¬ ®å t¶i träng nót C¸c th«ng tin vÒ t¶i träng: Sè hiÖu nót 1 2 3 4 5 6 7 8 9 10 Px ? ? 0 ? 0 0 ? 0 0 ? PY ? ? -20 ? 0 0 ? 0 0 ? v9 u11 v11 u12 Sau khi xö lý ®iÒu kiÖn biªn ta cã vÐc t¬ chuyÓn vÞ cÇn t×m: ∆ = { u3 v 3 u5 v5 u6 v6 u8 v8 u9 v12 } 11 12 -3.75 -0.63 0 0 3. LËp ma trËn ®é cøng 3.1. LËp ma trËn ®é cøng phÇn tö Ma trËn ®é cøng cña phÇn tö ®îc x¸c ®Þnh bëi c«ng thøc: [k ij ] [k im ]  [ ] [k jj ] [k jm ] [k mj ] [k mm ]  [k ii ] [k ] =  k ji [k mi ]  Trong ®ã: 1 −ν 1 −ν    br bs + 2 c r c s νbr c s + 2 c r b s  [k rs ] =   1 −ν 4 1 − ν 2 ∆ νc b + 1 − ν b c cr cs + br b s  r s r s 2 2   (r = i, j, m; s = i, j, m ) ( Et ) Hay ma trËn [k] cã thÓ viÕt l¹i nh sau: 1 −ν 1 −ν 1 −ν 1 −ν 1 −ν 1 −ν   c i bi bi b j + ci c j ci b j bi b m + ci c m c i bm  νbi c i + νbi c j + νbi c m +  bi bi + 2 c i c i 2 2 2 2 2   1 −ν 1 −ν 1 −ν 1 −ν 1 −ν 1 −ν  νc i bi + bi c i c i ci + bi bi bi c j ci c j + bi b j bi c m ci ci + bi bm  νc i b j + νc i b m + 2 2 2 2 2 2   1 −ν 1 −ν 1 −ν 1 −ν 1 −ν 1 −ν   b b c c b c c b b b c c b c c b b b c c b c c b + + + + + + ν ν ν j i j i j i j i j j j j j j j j j m j m j m j m  2 2 2 2 2 2 [k ] = Et 2   1 −ν 1 −ν 1 −ν 1 −ν 1 −ν 4 1 − ν ∆  νc b + 1 − ν b c c c b b c b b c c c b b c b b c c c b b + + + + + ν ν j i j i j i j i j j j j j j j j j m j m j m j m  2 2 2 2 2 2   1 −ν 1 −ν 1 −ν  b b + 1 − ν c c νb c + 1 − ν c b b b + 1 − ν c c  b c c b b b c c b c c b + + + ν ν m i m i m i m i m j m j m j m j m m m m m m m m   2 2 2 2 2 2   1 −ν 1 −ν 1 −ν 1 −ν 1 −ν 1 −ν bm c i c m c i + bm bi νc m b j + bm c j c m c j + b m b j νc m b m + bm c m c m c m + bm b m  νc m bi + 2 2 2 2 2 2   ( ) Ta ph©n c¸c phÇn tö tam gi¸c cña kÕt cÊu thµnh hai lo¹i: Lo¹i 1 gåm c¸c phÇn tö: 3, 6, 8, 10, 12 (h×nh 05) Lo¹i 2 gåm c¸c phÇn tö: 1, 2, 4, 5, 7, 9, 11 (h×nh 06) LËp ma trËn ®é cøng cña phÇn tö lo¹i 1 y m i j x H×nh 05 - PhÇn tö lo¹i 1 Tªn nót To¹ ®é X To¹ ®é Y i 0 0 j 1.5 1.5 TÝnh c¸c hÖ sè: ai= xjym-xmyj bi = yj-ym ci = xm-xj aj = xmyi-xiym bj = ym-yi = = == = 2.3 0.0 1.5 0.0 1.5 cj = xi-xm = 0.0 m 0 1.5 am = xiyj-xjyi bm = yi-yj cm = xj-xi = 0.0 = - 1.5 = 1.5 DiÖn tÝch tam gi¸c ijm: ∆ = 0.5* 1 1 1 xi xj xm yi yj ym ∆ = 0.5* 1 1 1 0 1.5 0 0 1.5 1.5 = 1.1 LËp ma trËn ®é cøng cña phÇn tö lo¹i 2 y m j Tªn nót To¹ ®é X To¹ ®é Y x i i 0 0 j 1.5 0 TÝnh c¸c hÖ sè: H×nh 06 - PhÇn tö lo¹i 2 ai = xjym-xmyj bi = yj-ym ci = xm-xj aj = xmyi-xiym bj = ym-yi cj = xi-xm am = xiyj-xjyi bm = yi-yj cm = xj-xi = == = = == = = 2.3 1.5 0.0 0.0 1.5 1.5 0.0 0.0 1.5 DiÖn tÝch tam gi¸c ijm: ∆ = 0.5* 1 1 1 0 1.5 1.5 0 0 1.5 = 1.1 m 1.5 1.5 Ma trËn ®é cøng phÇn tö tam gi¸c lo¹i 1 (12) (10) (8) (6) (3) [k]1 = u8 0 u5 0 0 v8 0 v5 0 0 u12 u11 u9 u8 u5 v12 v11 v9 v8 v5 u9 u8 u6 u5 u3 v9 v8 v6 v5 v3 254237 0 0 -254237 -254237 254237 0 620091 -111616 0 111616 -620091 0 -111616 620091 0 -620091 111616 -254237 0 0 254237 254237 -254237 -254237 111616 -620091 254237 874328 -365854 254237 -620091 111616 -254237 -365854 874328 0 0 u5 v5 u3 v3 (3) 0 0 u8 v8 u5 v5 (6) u5 v5 u9 v9 u6 v6 (8) 0 0 u11 v11 u8 v8 (10) u8 v8 u12 v12 u9 v9 (12) Ma trËn ®é cøng phÇn tö tam gi¸c lo¹i 2 (11) (9) (7) (5) (4) (2) (1) [k]2 = u3 u8 0 u5 0 u3 0 0 v8 0 v5 0 v3 0 0 u11 0 u8 0 u5 0 0 v11 0 v8 0 v5 0 0 u12 u11 u9 u8 u6 u5 u3 v12 v11 v9 v8 v6 v5 v3 620091 0 -620091 111616 0 -111616 0 254237 254237 -254237 -254237 0 -620091 254237 874328 -365854 -254237 111616 111616 -254237 -365854 874328 254237 -620091 0 -254237 -254237 254237 254237 0 -111616 0 111616 -620091 0 620091 v3 u5 v5 u6 v6 u8 0 0 0 0 u3 v3 (1) 0 0 0 0 u5 v5 (2) u3 v3 u5 v5 u6 v6 (4) 0 0 0 0 u8 v8 (5) u5 v5 u8 v8 u9 v9 (7) 0 0 0 0 u11 v11 (9) u8 v8 u11 v11 u12 v12 (11) v8 u9 v9 u11 v11 u12 v12 1748656 -365853.7 -1240182 365853.7 -365853.7 1748656 365853.7 -1240182 365853.7 365853.7 0 -111616.4 0 0 0 0 0 0 0 0 -508474.6 -254237.3 0 0 0 0 0 0 0 0 0 3497313 -731707.3 -508474.6 365853.7 -1240182 365853.7 0 -365853.7 0 0 0 0 -508474.6 -731707.3 3497313 365853.7 -1240182 365853.7 -508474.6 -365853.7 0 0 0 0 0 0 -254237.3 -508474.6 365853.7 1128566 -365853.7 0 0 -620090.9 254237.3 0 0 0 0 -111616.4 0 365853.7 -1240182 -365853.7 1494419 0 0 111616.4 -254237.3 0 0 0 0 0 0 -1240182 365853.7 0 0 3497313 -731707.3 -508474.6 365853.7 -1240182 365853.7 0 -365853.7 0 0 365853.7 -508474.6 0 0 -731707.3 3497313 365853.7 -1240182 365853.7 -508474.6 -365853.7 0 0 0 111616.4 -508474.6 365853.7 1748656 -365853.7 0 0 -620090.9 254237.3 0 0 -365853.7 0 254237.3 -254237.3 365853.7 -1240182 -365853.7 1748656 0 0 111616.4 -254237.3 0 0 0 0 0 0 -1240182 365853.7 0 0 1748656 -365853.7 -254237.3 111616.4 0 0 0 0 0 0 365853.7 -508474.6 0 0 -365853.7 1748656 254237.3 -620090.9 0 0 0 0 0 0 0 -365853.7 -620090.9 111616.4 -254237.3 254237.3 874328.2 0 0 0 0 0 0 0 -365853.7 0 254237.3 -254237.3 111616.4 -620090.9 0 874328.2 -365853.7 -620090.9 0 u3 v3 u5 v5 u6 v6 u8 v8 u9 v9 u11 v11 u12 v12 1748656 -365854 -1240182 365854 0 -111616 0 0 0 0 0 0 0 0 -365854 1748656 365854 -508475 -254237 0 0 0 0 0 0 0 0 0 -1240182 365854 3497313 -731707 -508475 365854 -1240182 365854 0 -365854 0 0 0 0 365854 -508475 -731707 3497313 365854 -1240182 365854 -508475 -365854 0 0 0 0 0 0 -254237 -508475 365854 1128566 -365854 0 0 -620091 254237 0 0 0 0 -111616 0 365854 -1240182 -365854 1494419 0 0 111616 -254237 0 0 0 0 0 0 -1240182 365854 0 0 3497313 -731707 -508475 365854 -1240182 365854 0 0 0 365854 -508475 0 0 -731707 3497313 365854 -1240182 365854 -508475 -365854 0 0 0 -365854 -620091 111616 -508475 365854 1748656 -365854 0 0 -620091 254237 0 0 -365854 0 254237 -254237 365854 -1240182 -365854 1748656 0 0 111616 -254237 0 0 0 0 0 0 -1240182 365854 0 0 1748656 -365854 -254237 111616 0 0 0 0 0 0 365854 -508475 0 0 -365854 1748656 254237 -620091 0 0 0 0 0 0 0 -365854 -620091 111616 -254237 254237 874328 0 0 0 0 0 0 0 -365854 0 254237 -254237 111616 -620091 0 874328 Tªn nót 1 2 3 4 5 6 7 8 9 10 11 12 u 0.00E+00 0.00E+00 -5.18E-06 0.00E+00 -3.69E-06 -1.03E-05 0.00E+00 -4.59E-06 -9.28E-06 0.00E+00 -6.60E-06 -9.88E-06 v 0.00E+00 0.00E+00 -1.41E-05 0.00E+00 -3.03E-06 -3.75E-06 0.00E+00 5.03E-08 3.34E-07 0.00E+00 2.19E-06 3.27E-06 -365854 x 0 0 u3 -20 v3 0 u5 0 v5 0 u6 0 v6 u8 v8 u9 v9 u11 v11 u12 v12 = 0 0 0 0 -3.75 0 -0.63 0 u3 v3 u5 v5 u6 v6 u8 v8 u9 v9 u11 v11 u12 v12 -5.18E-06 -1.41E-05 -3.69E-06 -3.03E-06 -1.03E-05 -3.75E-06 = -4.59E-06 5.03E-08 -9.28E-06 3.34E-07 -6.60E-06 2.19E-06 -9.88E-06 3.27E-06 6. X¸c ®Þnh c¸c vÐc t¬ øng suÊt trong c¸c phÇn tö VÐc t¬ øng suÊt trong c¸c phÇn tö ®îc x¸c ®Þnh th«ng qua vÐc t¬ chuyÓn vÞ nót cña phÇn tö theo c«ng thøc sau: {σ} e σ x    = σ y  = [D][B]{δ }e τ   xy   1 0 E  [D][B] = 2 ν 0 1−ν  1−ν 0 2   ui  v   i u  {δ }e =  j  v j  um     vm   [D][B] = 0 ν 0 1−ν 2 1 12401819 2232327 0 −1 −ν 1−ν 0 − 2 0 0 5084746  −ν  −1  1−ν  −  2  0 0 5084746 0 1 0 -1.00 -0.18 -0.41 -0.18 -1 -0.41 TÝnh cho phÇn tö thø nhÊt: 0.00E+00 0.00E+00 0.00E+00 σx {σ} e 1 = σy = τxy 1 [D][B] x 0.00E+00 -5.18E-06 -1.41E-05 7.72E-06 = 1.51E-05 7.91E-06 (kN/m2) TÝnh cho phÇn tö thø hai: 0.00E+00 0.00E+00 0.00E+00 σx {σ} e 2 = σy = [D][B] x τxy 0.00E+00 4.23E-06 = -3.69E-06 -3.03E-06 2 3.69E-06 (kN/m2) 2.75E-06 TÝnh cho phÇn tö thø ba: 0.00E+00 0.00E+00 -3.69E-06 σx {σ}e 3 = σy = [D][B] x τxy -3.03E-06 7.17E-06 = -5.18E-06 -1.41E-05 3 1.20E-05 (kN/m2) -1.88E+01 TÝnh cho phÇn tö thø t: -5.18E-06 -1.41E-05 -3.69E-06 σx {σ}e 4 = σy = [D][B] x τxy -3.03E-06 -6.42E+01 = -1.03E-05 -3.75E-06 4 -1.16E+01 (kN/m2) -9.06E+01 TÝnh cho phÇn tö thø n¨m: 0.00E+00 0.00E+00 0.00E+00 σx {σ}e 5 = σy = τxy 5 TÝnh cho phÇn tö thø s¸u: [D][B] x 0.00E+00 -4.59E-06 5.03E-08 4.58E-06 = 7.76E-07 1.86E-06 (kN/m2) 0.00E+00 0.00E+00 -4.59E-06 σx {σ} e 6 = σy = [D][B] x τxy 5.03E-08 4.24E-06 = -3.69E-06 -3.03E-06 6 3.74E-06 (kN/m2) -2.34E+01 TÝnh cho phÇn tö thø b¶y: -3.69E-06 -3.03E-06 -4.59E-06 σx {σ} e 7 = σy = [D][B] x τxy 5.03E-08 -4.58E+01 = -9.28E-06 3.34E-07 7 -8.24E+00 (kN/m2) -3.87E+01 TÝnh cho phÇn tö thø t¸m: -3.69E-06 -3.03E-06 -9.28E-06 σx {σ}e 8 = σy = [D][B] x τxy 3.34E-07 -4.58E+01 = -1.03E-05 -3.75E-06 8 -8.24E+00 (kN/m2) -6.26E+01 TÝnh cho phÇn tö thø chÝn: 0.00E+00 0.00E+00 0.00E+00 σx {σ}e 9 = σy = τxy 9 [D][B] x 0.00E+00 -6.60E-06 2.19E-06 TÝnh cho phÇn tö thø mêi: 0.00E+00 0.00E+00 6.20E-06 = -1.00E-06 1.81E-06 (kN/m2) σx {σ}e 10 = σy -6.60E-06 = [D][B] x τxy 2.19E-06 4.98E-06 = -4.59E-06 5.03E-08 10 2.97E-06 (kN/m2) -3.36E+01 TÝnh cho phÇn tö thø mêi mét: -4.59E-06 5.03E-08 -6.60E-06 σx {σ}e 11 = σy = [D][B] x τxy 2.19E-06 -5.70E+01 = -9.88E-06 3.27E-06 11 -1.03E+01 (kN/m2) -3.33E+01 TÝnh cho phÇn tö thø mêi hai: -4.59E-06 5.03E-08 -9.88E-06 σx {σ} e 12 = σy = τxy 12 [D][B] x 3.27E-06 -9.28E-06 3.34E-07 -5.70E+01 = -1.03E+01 -5.00E+01 (kN/m2)