Tài liệu Bai tap he thanh

§Ò bµi sè 06 q 2P VÏ biÓu ®å néi lùc kÕt cÊu cho trªn h×nh vÏ. Trong ®ã: q = 5 kN/m, P = 20 kN, a = 3m, E = 1.2x106 N/cm2, ν = 0.18. MÆt c¾t ngang cña c¸c thanh lµ h×nh vu«ng cã c¹nh b = 15 cm. 2q a P a a H×nh 01 - S¬ ®å kÕt cÊu phÇn tÝnh to¸n bµi 06 1. C¸c sè liÖu ban ®Çu . MÆt c¾t ngang thanh lµ h×nh vu«ng víi chiÒu dµi c¹nh b = 0.15m . 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 . DiÖn tÝch mÆt c¾t ngang thanh A = 0.0225m2 .M« men qu¸n tÝnh cña mÆt c¾t J = 0.000042m4 y y 3 4 3 2 4 2 5 7 1 5 x 6 1 6 2. Chia kÕt cÊu thµnh c¸c PTHH S¬ ®è rêi r¹c ho¸ kÕt cÊu ®îc thÓ hiÖn trªn h×nh 02. C¸c th«ng tin vÒ phÇn tö: H×nh 02 - S¬ ®å c¸c nót vµ phÇn tö Sè hiÖu phÇn tö Nót ®Çu Nót cuèi 1 1 2 2 2 3 3 3 4 4 5 4 5 6 5 6 1 6 7 2 5 1 0 0 2 0 3 3 0 6 4 3 6 5 3 3 6 3 0 C¸c th«ng tin vÒ nót: Sè hiÖu nót To¹ ®é X To¹ ®é Y C¸c th«ng tin vÒ t¶i träng: TÝnh to¸n dêi t¶i träng ph©n bè vÒ nót q q = 3 a 3 a 2q "Tr¹ng th¸i thùc" qa2/12 qa/2 7 a qa/2 3 a qa2/6 qa qa qa2/6 + "Tr¹ng th¸i cè ®Þnh" qa/2 qa2/12 + 2q = 7 a qa2/12 7 a "Tr¹ng th¸i tù do" qa/2 qa2/12 2P qa2/6 qa qa qa2/6 P Sè hiÖu nót 1 2 3 Px PY ? ? ? MZ 0.0 ? -15.0 -7.5 (0) (0) (1) (0) (2) (3) (0) (4) (5) (6) (7) (8) u'5 v'5 (9) (10) (11) (0) (0) (12) θ'5 u'6 v'6 θ'6 5 6 40.0 20.0 ? -7.5 -15.0 ? -7.5 -3.75 3.75 H×nh 03 - S¬ ®å t¶i träng nót C¸c th«ng tin vÒ chuyÓn vÞ nót: . Sè chuyÓn vÞ nót: 18 . Sè chuyÓn vÞ nót b»ng 0: 6 . Sè Èn chuyÓn vÞ nót: 12 . §¸nh sè chuyÓn vÞ nót: {∆}= u'1 v'1 θ'1 u'2 v'2 θ'2 u'3 v'3 θ'3 u'4 v'4 θ'4 4 7.5 0.0 3. LËp ma trËn ®é cøng: Ma trËn ®é cøng cña c¸c phÇn tö trong hÖ to¹ ®é ®Þa ph¬ng: [k]1 = [k]2 = [k]3 = [k]4 = [k]5 = [k]6 = [k]7 =[k]pt [k]pt = [k]pt = EA/a 0 0 -EA/a 0 0 0 12EJ/a3 6EJ/a2 0 -12EJ/a3 6EJ/a2 0 6EJ/a2 4EJ/a 0 -6EJ/a2 2EJ/a 90000 0 0 -90000 0 0 0 225 338 0 -225 338 0 338 675 0 -338 338 -EA/a 0 0 3 0 -12EJ/a 6EJ/a2 0 -6EJ/a2 2EJ/a EA/a 0 0 3 0 12EJ/a -6EJ/a2 0 -6EJ/a2 4EJ/a -90000 0 0 90000 0 0 0 -225 -338 0 225 -338 0 338 338 0 -338 675 0 0 0 cosϕ -sinϕ 0 0 0 0 sinϕ cosϕ 0 0 0 0 0 0 1 Ma trËn biÕn ®æi to¹ ®é cña c¸c phÇn tö: Ma trËn biÕn ®æi to¹ ®é cña phÇn tö cã d¹ng: [T] = cosϕ -sinϕ 0 0 0 0 sinϕ cosϕ 0 0 0 0 0 0 1 0 0 0 Ma trËn biÕn ®æi to¹ ®é cña phÇn tö 1 90o Gãc xoay gi÷a hÖ to¹ ®é ®Þa ph¬ng vµ hÖ to¹ ®é kÕt cÊu,ϕ = [T]1 = 0 -1 0 0 0 0 1 0 0 0 0 0 Ma trËn biÕn ®æi to¹ ®é cña phÇn tö 2 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 1 Gãc xoay gi÷a hÖ to¹ ®é ®Þa ph¬ng vµ hÖ to¹ ®é kÕt cÊu,ϕ [T]2 = 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 90o = 0 0 0 1 0 0 0 0 0 0 0 1 Ma trËn biÕn ®æi to¹ ®é cña phÇn tö 3 Gãc xoay gi÷a hÖ to¹ ®é ®Þa ph¬ng vµ hÖ to¹ ®é kÕt cÊu,ϕ [T]3= 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0o = 0 0 0 0 1 0 0 0 0 0 0 1 Ma trËn biÕn ®æi to¹ ®é cña phÇn tö 4 Gãc xoay gi÷a hÖ to¹ ®é ®Þa ph¬ng vµ hÖ to¹ ®é kÕt cÊu,ϕ [T]4 = 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 90o = 0 0 0 1 0 0 0 0 0 0 0 1 Ma trËn biÕn ®æi to¹ ®é cña phÇn tö 5 Gãc xoay gi÷a hÖ to¹ ®é ®Þa ph¬ng vµ hÖ to¹ ®é kÕt cÊu,ϕ [T]5 = 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 90o = 0 0 0 1 0 0 0 0 0 0 0 1 Ma trËn biÕn ®æi to¹ ®é cña phÇn tö 6 Gãc xoay gi÷a hÖ to¹ ®é ®Þa ph¬ng vµ hÖ to¹ ®é kÕt cÊu,ϕ = 0o [T]6 = 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 Ma trËn biÕn ®æi to¹ ®é cña phÇn tö 7 Gãc xoay gi÷a hÖ to¹ ®é ®Þa ph¬ng vµ hÖ to¹ ®é kÕt cÊu,ϕ [T]7 = 1 0 0 0 0 0 0o = 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 ChuyÓn trÝ c¸c ma trËn T: [T]T1 = [T]T2= T3 [T] = [T]T4= [T]T5= [T]T6= [T]T7= 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 (2) (3) Ma trËn ®é cøng cña c¸c phÇn tö trong hÖ to¹ ®é kÕt cÊu: [k]' = [T]T*[k]*[T] (0) (0) (1) (0) [k]'1= [k]'2= [k]'3= [k]'4= [k]'5= 225 0 -338 -225 0 -338 0 90000 0 0 -90000 0 -338 0 675 338 0 338 -225 0 338 225 0 338 0 -90000 0 0 90000 0 -338 0 338 338 0 675 (0) (2) (3) (0) (4) (5) 225 0 -338 -225 0 -338 0 90000 0 0 -90000 0 -338 0 675 338 0 338 -225 0 338 225 0 338 0 -90000 0 0 90000 0 -338 0 338 338 0 675 (0) (4) (5) (6) (7) (8) 90000 0 0 -90000 0 0 0 225 338 0 -225 338 0 338 675 0 -338 338 -90000 0 0 90000 0 0 0 -225 -338 0 225 -338 0 338 338 0 -338 675 (9) (10) (11) (6) (7) (8) 225 0 -338 -225 0 -338 0 90000 0 0 -90000 0 -338 0 675 338 0 338 -225 0 338 225 0 338 0 -90000 0 0 90000 0 -338 0 338 338 0 675 (0) (0) (12) (9) (10) (11) 225 0 -338 -225 0 -338 0 90000 0 0 -90000 0 -338 0 675 338 0 338 -225 0 338 225 0 338 0 -90000 0 0 90000 0 -338 0 338 338 0 675 (0) (0) (1) (0) (2) (3) (0) (2) (3) (0) (4) (5) (0) (4) (5) (6) (7) (8) (9) (10) (11) (6) (7) (8) (0) (0) (12) (9) (10) (11) [k]'6= [k]'7= (0) (0) (1) (0) (0) (12) 90000 0 0 -90000 0 0 0 225 338 0 -225 338 0 338 675 0 -338 338 -90000 0 0 90000 0 0 0 -225 -338 0 225 -338 0 338 338 0 -338 675 (0) (2) (3) (9) (10) (11) 90000 0 0 -90000 0 0 0 225 338 0 -225 338 0 338 675 0 -338 338 -90000 0 0 90000 0 0 0 -225 -338 0 225 -338 0 338 338 0 -338 675 (0) (0) (1) (0) (0) (12) (0) (2) (3) (9) (10) (11) Ma trËn ®é cøng K cña kÕt cÊu: [K] = (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) 1350 0 338 0 0 0 0 0 0 0 0 338 (1) 0 180225 338 -90000 0 0 0 0 0 -225 338 0 (2) 338 338 2025 0 338 0 0 0 0 -338 338 0 (3) 0 -90000 0 90225 338 0 -225 338 0 0 0 0 (4) 0 0 338 338 1350 0 -338 338 0 0 0 0 (5) 0 0 0 0 0 90225 0 338 -225 0 338 0 (6) 0 0 0 -225 -338 0 90225 -338 0 -90000 0 0 (7) 0 0 0 338 338 338 -338 1350 -338 0 338 0 (8) 0 0 0 0 0 -225 0 -338 90450 0 0 338 (9) 0 -225 -338 0 0 0 -90000 0 0 180225 -338 0 (10) 0 338 338 0 338 0 0 0 0 0 338 0 0 0 338 0 0 338 -338 0 2025 338 338 1350 (11) (12) 4. HÖ ph¬ng tr×nh c©n b»ng cña PPPTHH 1350 0 0 180225 338 338 0 0 0 0 0 0 0 0 338 θ1 0.00 θ1 0.0014346 338 -90000 0 0 0 0 0 -225 338 0 v2 -15.00 v2 -0.0002497 338 2025 0 338 0 0 0 0 -338 338 0 θ2 -7.50 θ2 -0.0042667 0 -90000 0 90225 338 0 -225 338 0 0 0 0 v3 -7.50 v3 −0.000333 θ3 -3.75 θ3 -0.0022728 40.00 u4 0 0 338 338 1350 0 -338 338 0 0 0 0 0 0 0 0 0 90225 0 338 -225 0 338 0 0 0 0 -225 -338 0 90225 -338 0 -90000 0 0 v4 -7.50 v4 -0.0003337 0 0 0 338 338 338 -338 1350 -338 0 338 0 θ4 3.75 θ4 0.0022459 0 0 0 0 0 -225 0 -338 90450 0 0 338 u5 20.00 u5 0.000236 0 -225 -338 0 0 0 -90000 0 0 180225 -338 0 v5 -15.00 v5 -0.0002503 0 338 338 0 0 338 0 338 0 -338 2025 338 θ5 7.50 θ5 0.0042157 338 0 0 0 0 0 0 0 338 0 338 1350 θ6 0.00 θ6 −0.001472 x u4 = = 0.0004198 5. TÝnh néi lùc nót cña c¸c phÇn tö: TÝnh cho phÇn tö thø nhÊt: {F}e 1 = U1 V1 0.000000 0.000000 M1 U2 V2 M2 = [k]1 x [T]1 x 0.001435 0.000000 -0.000250 -0.004267 22.4740 -0.9558 = -0.4717 -22.4740 0.9558 -2.3959 TÝnh cho phÇn tö thø hai: {F}e 2 = U2 V2 0.00000 -0.00025 M2 U3 V3 M3 = [k]2 x [T]2 x -0.00427 0.00000 -0.00033 -0.00227 7.4911 -2.2071 = -3.6471 -7.4911 2.2071 -2.9741 TÝnh cho phÇn tö thø ba: {F}e 3 = U3 V3 0.00000 -0.00033 M3 U4 V4 M4 = [k]3 x [T]3 x -0.00227 0.00042 -0.00033 0.00225 -37.7779 -0.0089 = -0.7759 37.7779 0.0089 0.7492 TÝnh cho phÇn tö thø t: U5 V5 {F}e 4 = M5 U4 V4 M4 0.00024 -0.00025 = [k]4 x [T]4 x TÝnh cho phÇn tö thø n¨m: 0.00422 0.00042 -0.00033 0.00225 7.5089 2.2221 = 3.6656 -7.5089 -2.2221 3.0008 {F}e 5 = U6 V6 0.00000 0.00000 M6 U5 V5 M5 = [k]5 x [T]5 x -0.00147 0.00024 -0.00025 0.00422 22.5260 0.9793 = 0.5091 -22.5260 -0.9793 2.4286 TÝnh cho phÇn tö thø s¸u: U1 V1 {F}e 6 = M1 U6 0.00000 0.00000 = [k]6 x [T]6 x V6 M6 0.00143 0.00000 0.0000 -0.0125 = 0.4717 0.0000 0.00000 -0.00147 0.0125 -0.5091 U2 V2 0.00000 -0.00025 -21.2429 -0.0171 M2 U5 V5 M5 = [k]7 x [T]7 x -0.00427 0.00024 -0.00025 0.00422 TÝnh cho phÇn tö thø b¶y: {F}e 7 = = -1.4570 21.2429 0.0171 1.4058