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Example Structure. Selecting Node Positions. Stiffness Matrix Method. EMTH , Assignment 1 Cable-Stayed Bridge Analysis

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Exaple Structure EMTH7-8, Assignent Cable-Stayed Bridge Analysis Y/ C dia Steel Cable Dr B. Dea eicester Steen EQC ecturer in Earthquake Engineering Office E8 Pin Connection B UB. D kn/ UB 8. E A X/ Stiffness
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Exaple Structure EMTH7-8, Assignent Cable-Stayed Bridge Analysis Y/ C dia Steel Cable Dr B. Dea eicester Steen EQC ecturer in Earthquake Engineering Office E8 Pin Connection B UB. D kn/ UB 8. E A X/ Stiffness Matrix Method Selecting Node Positions. Define nodes for the coponent connections.. Create coponent stiffness atrices.. Connect coponent stiffness atrices at nodes. Constrain deflections and rotations at the support nodes.. Apply loads to the nodes.. Calculate the displaceents and rotations of free nodes. 7. Calculate coponent forces fro end nodes oeents. Nodes are used:. at the structure s supports or foundations;. at connections between coponents;. at discontinuities in coponent properties;. at discontinuities in coponent direction;. at loads or distributed load discontinuities;. where deforations or forces are needed. Exaple Structure. Node Definitions Y/ C %. Define nodes for the coponent equation % or connection nubers Pin Connection B UB. dia Steel Cable D kn/ UB 8. E eax ; eay ; eat ; ebx ; eby ; ebt ; ebt 7; ecx 8; ecy 9; ect ; edx ; edy ; edt ; edx ; edy ; edt ; eex 7; eey 8; eet 9; neqn eet; A X/ Exaple Structure. Stiffness atrices Y/ C %. Create the coponent stiffness atrices Pin Connection dia Steel Cable kn/ [K_AB, kt_ab] bead( UB,, ); [K_BC, kt_bc] bead( UB,, ); [K_BD, kt_bd] bead( UB8,, ); [K_DE, kt_de] bead( UB8,, ); [K_CD, kt_cd] tied(pi/*.^, 8E9,,-); B D UB 8. E UB. A X/ function tied function bead function [K, kt] tied(a, E, lx, ly) % Create a global and a transfored local % atrix for a -diensional tie coponent. % Direction cosines sqrt(lx*lx + ly*ly); C lx/; S ly/; % Transforation atrix (X, Y and Theta) T [C S ; -S C ; ; C S ; -S C ; ]; % For the local stiffness atrix k E*A/*[ - ; ; ; - ; ; ]; % Finally, for global stiffness atrix kt k * T'; K T * kt; function [K kt] bead(section, lx, ly) % Create a global and a transfored local % atrix for a -diensional bea coponent. % Direction cosines sqrt(lx*lx + ly*ly); C lx/; S ly/; % Transforation atrix (X, Y and Theta) T [C S ; -S C ; ; C S ; -S C ; ]; % Material properties (all in SI units) if strcp(section, 'UB') E E9; A.E-; I 8.E-; elseif strcp(section, 'UB8') E E9; A.E-; I.8E-; else error('unknown section %s', section) end function bead (cont). Asseble K % For local stiffness atrix fro the axial % and flexural parts (added together) k E*A/*[ - ; ; ; - ; ; ]; k k + E*I/^*[ ; * - *; * *^ -* *^; ; - -* -*; * *^ -* *^]; % Finally, for global stiffness atrix kt k * T'; K T * kt; %. Asseble global stiffness atrix % by adding tower eleent stiffness atrices Kzeros(nEqn); KaddK(K, K_AB, [eax eay eat ebx eby ebt]); KaddK(K, K_BC, [ebx eby ebt ecx ecy ect ]); %... and the bea coponents KaddK(K, K_BD, [ebx eby ebt edx edy edt ]); KaddK(K, K_DE, [edx edy edt edx edy edt]); KaddK(K, K_DE, [edx edy edt eex eey eet ]); %... and the tie stiffness atrix KaddK(K, K_CD, [ecx ecy ect edx edy edt]); function addk. Restraints function K addk(k, newk, eqnnus) % Add a stiffness atrix to the global % stiffness atrix K(eqnNus, eqnnus) K(eqnNus, eqnnus) + newk; A([ ], [ ]) [ ; ] A %. Restrain node A X, Y and T K restraink(k, [eax eay eat]) function K restraink(k, eqnnus) % Restrain the stiffness atrix K by zeroing % the equations in ectore ewnnus zeroeczeros(size(k,),); for i eqnnus K(:,i) zeroec; % Zero row K(i,:) zeroec'; % Zero colun K(i,i).; end. oad Vectors Helper functions % Create the load ectors for the coponents G_BD UDD(-9.8*8., ); G_DE UDD(-9.8*8., ); Q_DE UDD(-, ); function F UDD(w, ) % Calculate fixed end forces for a D bea F w * *[/ / / -/]' %. Create the global load ector and add loads Fzeros(nEqn, ); FaddF(F, G_BD, [eby ebt edy edt]); FaddF(F, G_DE + Q_DE, [edy edt edy edt]); FaddF(F, G_DE + Q_DE, [edy edt eey eet]); function F addf(f, newf, eqnnus) % Add a load ector to the global load ector F(eqnNus) F(eqnNus) + newf; . Sole Equations. Calculate global displaceents, D D K \ F; 7. Coponent forces % 7. Print the forces in the tie % and end cantileer span (as exaples) (kt_ab * D([eAX eay eat ebx eby ebt]))' (kt_cd * D([eCX ecy ect edx edy edt]))' EF(kT_DE * D([eDX edy edt eex eey eet])); % Subtract fixed end forces! (EF([ ]) - G_DE - Q_DE)' oading Exaple + w y y w Cantileer y w w Rearrange y w w Sole y w w y w y w w y 8 Coponent forces w w w w w w w
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