Trusses. Introduction. Plane Trusses Local and global coordinate systems. Plane Trusses Local and global coordinate systems
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1 Introduction russs Lctur Nots Dr Mohd Afndi nivrsiti Maaysia Pris EN Finit Ent Anaysis h discussion of truss wi covr: wo dinsiona trusss (pan trusss) sction. hr dinsiona trusss sction. So considration in truss: A oads and joints ar connctd togthr at thir nds by pin joints Friction wi b ignord Pan russs Loca and goba coordinat systs In pan trusss th is orintations of ach nts Loca and goba coordinat systs wi b introducd to account th diffrnt orintations Pan russs Loca and goba coordinat systs Ent dispacnt q q, q Ent dispacnt q q, q, q, q q q q q q q q q q q = q cos + q sin q = q cos + q sin q q q q = q cos + q sin q = q cos + q sin Loca coordinat Goba coordinat Loca coordinat Goba coordinat Pan russs Loca and goba coordinat systs Pan russs Foruas for cacuating Loca Goba Ent dispacnt Ent dispacnt q q, q q q q q q,,, Dirction cosins (x, y ) q q cos q sin q q cos q sin cos sin q Lq L cos (x, y ) x - x x x y y cos sin y - y x x y y
2 Pan russs Ent Stiffnss Matrix Pan russs Ent Stiffnss Matrix Dirction cosins E A k q kq (x, y ) Fro prvious chaptr (x, y ) x - x q Lq y - y Strain nrgy in goba coordinat q kq k L kl E A k q L klq E A k Strain nrgy in goba coordinat q kq Pan russs Strss cacuation Ex.. Dirction cosins E (x, y ) (x, y ) x - x y - y q Lq Strss in goba coordinat E q q E q E q E Lq q 9 Considr th four-bar truss shown bow. Givn data E = 9. x psi and A = in for a nts a) Dtrin th nt stiffnss atrix for ach nt b) Assb th structura stiffnss atrix K for th ntir truss c) sing th iination approach, sov for th noda dispacnt d) Rcovr th strss in ach nt ) Cacuat th raction forcs Ex.. b Soution b Q Q Q Q E = 9. x psi E = 9. x psi in A =. in in A =. in Q Q b Q Q b in in
3 Soution Soution a) Dtrin th nt stiffnss atrix for ach nt It is rcondd to us a tabuar h noda coordinat data: a) Dtrin th nt stiffnss atrix for ach nt It is rcondd to us a tabuar h nt connctivity: Nod x y Ent Soution Soution a) Dtrin th nt stiffnss atrix for ach nt It is rcondd to us a tabuar h dirction cosins: Ent = cos = sin -.. a) Dtrin th nt stiffnss atrix for ach nt sing Eq.. th nt stiffnss atrics for nt k E A k 9. Goba dof Soution Soution a) Dtrin th nt stiffnss atrix for ach nt sing Eq.. th nt stiffnss atrics for nt k 9. E A k Goba dof a) Dtrin th nt stiffnss atrix for ach nt sing Eq.. th nt stiffnss atrics for nt & Goba dof 9. k Goba dof 9. k
4 Soution Soution Fixd support:,,, and b) Assb th structura stiffnss atrix K for th ntir truss c) sing th iination approach, sov for th noda dispacnt K K Soution Soution c) sing th iination approach, sov for th noda dispacnt 9... Q. Q. Q c) sing th iination approach, sov for th noda dispacnt Dispacnts: Q. Q. in. Q. Dispacnts: Q. Q. in. Q. Dispacnts vctor for th ntir structur: Q,,.,,.,.,, in Soution Soution d) Rcovr th strss in ach nt d) Rcovr th strss in ach nt E q E q For nt : nod and nod For nt : nod and nod Dispacnts: Q, Q, Q, Q 9.. psi Dispacnts: Q, Q, Q, Q psi
5 Soution Soution d) Rcovr th strss in ach nt d) Rcovr th strss in ach nt E q E q For nt : nod and nod For nt : nod and nod Dispacnts: Q, Q, Q, Q psi Dispacnts: Q, Q, Q, Q 9... psi Soution Soution ) Cacuat th raction forcs R = KQ - F d) Cacuat th raction forcs R = KQ - F Raction forcs aong dofs,,,, and R.. R.. 9. R R R Raction forcs aong dofs,,,, and R R R 9 b R R hr-dinsiona russs Loca and goba coordinat systs h -D truss nt can b tratd as a straightforward gnraization of D truss nt. Z q = q cos + q sin q = q cos + q sin q q q (q i + q j + q k) Dford nt hr-dinsiona russs Loca and goba coordinat systs Loca Ent dispacnt q q, q Goba Ent dispacnt q Lq q q, q, q, q, q, q L n n (q i + q j + q k) Loca coordinat Goba coordinat 9
6 hr-dinsiona russs Foruas for cacuating hr-dinsiona russs Ent Stiffnss Matrix Dirction cosins (x, y ) Strain nrgy in goba coordinat q kq cos (x, y ) x - x x x y y cos sin z z n y - y x x y y z z E A k n n n n n n n n n n n n n n n n n n n n hr-dinsiona russs Assby of Goba stiffnss atrix h soution shoud tak th advantag of sytry and sparsity of th goba stiffnss atrix. h bandd approach h nts of ach nt stiffnss atrix k ar dircty pac in a bandd atrix S h skyin approach h nts of k ar pacd in a vctor for with vrtain indntification pointrs Q P. Dispacnt Strss Raction at nod
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