shear moment diagram quintic - rice universitypresented to show that this element gives exact...

Quintic Beam Element for Shear and Moment Diagrams

Copyright J,E, Akin. All rights reserved. of 23

1 Introduction

The classic cubic beam element is the most widely used element for analytic and numerical approximations of beam systems. They will yield analytically exact nodal deflections and rotations, but do not typically give accurate results along the interior of the element. That is particularly true for the bending moment and transverse shear force (2‐nd and 3‐rd derivatives) which are needed to design beam members. A review of tables of solutions of common beam loading and support conditions shows that most analytic solutions are third to fifth degree polynomials. That suggests that a better choice for a beam element is a quintic polynomial instead of the common cubic polynomial. Since a 5‐th degree polynomial takes six constants there are two clear choices. A three node element with deflection and slope at each as degrees of freedom (dof), or a two node element with the deflection, slope, and curvature dof at each end. Here the three node element choice is used.

The use of this element as an analytic tool is presented in another document. Here a simple numerical example is presented to show that this element gives exact deflection, slope, moment, and shear diagrams for a proper mesh. A proper mesh should have the end node of an element at any support point, any point force or couple, and at any point of a discontinuity along a distributed line load. A complete Matlab script is listed in the appendix and is available for download.

2 Example

Figure 1 shows a beam for which the program inputs, outputs and graphs will be shown. In this case, not only are the nodal results exact, but the graphs of the interior deflection, slope, moment, and shear are exact. Figure 2 shows the calculated results for the example beam system. The numerical input echos and output are listed below. Note that the two reactions are automatically computed. That is also true for statically indeterminant beams.

Figure 1 Example beam, mesh, and shear and moment diagrams

Rice University, MEMS Department

Copyright J.E. Akin 2008. All rights reserved. of 23

Figure 2 Computed (exact) beam graphs



Sample run listing: >> L3_Quintic_BoEF(1) Read 11 nodes with bc_flag & 1 coordinates. (Echo of file msh_bc_xyz.tmp) 10 0 00 5 00 10 00 15 00 20 00 25 10 30 00 35 00 40 00 45 00 50 Note: expecting 2 displacement BC values. Read 5 elements with (ignored) type & 3 nodes each. (Echo of file msh_typ_nodes.tmp) 1 1 2 3 1 3 4 5 1 5 6 7 1 7 8 9 1 9 10 11 Read 2 EBC data (Echo of file msh_ebc.tmp) Node, DOF, Value. 1 1 0 7 1 0 Read 2 point sources. (Echo of msh_load_pt.tmp) Node, DOF (1=force, 2=couple), Source_value 5 1 -100 9 2 3000 (Echoing file msh_properties.tmp) Properties for element 1 Elastity modulus (N/m^2) = 1e+09 Moment of inertia (m^4) = 0.01 Line Load (N/m) = [ 0 0 ] Foundation stiffness (N/m^2) = 0 Properties for element 2 Elastity modulus (N/m^2) = 1e+09 Moment of inertia (m^4) = 0.01 Line Load (N/m) = [ -10 -10 ] Foundation stiffness (N/m^2) = 0 Properties for element 3 Elastity modulus (N/m^2) = 1e+09 Moment of inertia (m^4) = 0.01 Line Load (N/m) = [ -10 -10 ] Foundation stiffness (N/m^2) = 0



Properties for element 4 Elastity modulus (N/m^2) = 1e+09 Moment of inertia (m^4) = 0.01 Line Load (N/m) = [ 0 0 ] Foundation stiffness (N/m^2) = 0 Properties for element 5 Elastity modulus (N/m^2) = 1e+09 Moment of inertia (m^4) = 0.01 Line Load (N/m) = [ 0 0 ] Foundation stiffness (N/m^2) = 0 Resultant input sources: Node, DOF, Resultant input sources 3 1 -23.3333 3 2 -16.6667 4 1 -53.3333 5 1 -146.667 6 1 -53.3333 7 1 -23.3333 7 2 16.6667 9 2 3000 Totals = -300 3000 Calculated Displacements Node, Y_displacement, Z_rotation at 11 nodes 1 0 -0.00272222 2 -0.0131944 -0.00247222 3 -0.0238889 -0.00172222 4 -0.0296094 -0.000493056 5 -0.0281944 0.00111111 6 -0.0182899 0.00284028 7 0 0.00444444 8 0.0259722 0.00594444 9 0.0594444 0.00744444 10 0.0966667 0.00744444 11 0.133889 0.00744444 Max deflection value is 0.133889 Min deflection value is -0.0300147 Created Beam_deflections.png Max slope value is 0.00744448 Min slope value is -0.00272222 Created Beam_slope.png Max moment value is 1749.94 Min moment value is -0.3 Created Beam_moments.png Max shear value is 50.049 Min shear value is -24.9615 Created Beam_shears.png



Reactions at Displacement BCs Node, DOF, Reaction Value 1 1 200 7 1 100 Totals = 300.0000 0 Individual Element Load and Reaction Summaries: (F_1, M_1, F_2, M_2) Given Resultant Loading on Member 1, 0 0 0 0 0 0 Net Resultant End Reactions on Member 1, 1.0e+03 * 0.2000 -0.0000 0 0.0000 -0.2000 2.0000 Given Resultant Loading on Member 2, -23.3333 -16.6667 -53.3333 0 -123.3333 16.6667 Net Resultant End Reactions on Member 2, 1.0e+03 * 0.2000 -2.0000 -0.0000 0.0000 -0.0000 3.5000 Given Resultant Loading on Member 3, -23.3333 -16.6667 -53.3333 0 -23.3333 16.6667 Net Resultant End Reactions on Member 3, 1.0e+03 * 0.0000 -3.5000 -0.0000 0 0.1000 3.0000 Given Resultant Loading on Member 4, 0 0 0 0 0 3000 Net Resultant End Reactions on Member 4, 1.0e+03 * -0.0000 -3.0000 0.0000 -0.0000 0.0000 -0.0000 Given Resultant Loading on Member 5, 0 0 0 0 0 0 Net Resultant End Reactions on Member 5, 0 0 0 0 0 0 >> quit



3 Appendix

function L3_Quintic_BoEF (load_pt) % load_pt = 1 flags that point forces and/or couples will be % input via file msh_load_pt.tmp %............................................................. % quintic beam on elastic foundation, % for point loads & couples, line load % Mech 417, Finite Element Analysis, Rice University % Copyright J.E. Akin 2008. All rights reserved. %............................................................. if ( nargin == 0 ) ; % check for optional data flag load_pt = 0 ; % no point source data end % if from argument count % Application and element dependent controls n_g = 2 ; % number of DOF per node (deflection, slope) n_q = 0 ; % number of quadrature points required n_r = 1 ; % number of rows in B_e matrix % Read mesh nodal input data files [n_m, n_s, P, x, y, z] = get_mesh_nodes ; % Extract EBC flags from packed integer flag P [EBC_flag] = get_ebc_flags (n_g, n_m, P) ; % unpack flags EBC_count = sum( sum ( EBC_flag > 0 ) ) ; % # of EBC if ( EBC_count == 0 ) fprintf ('WARNING: No displacement boundary condition given. \n') else fprintf ('Note: expecting %g displacement BC values. \n', EBC_count) end % if % Read mesh element input data files [n_e, n_n, n_t, el_type, nodes] = get_mesh_elements ; n_d = n_g*n_m ; % system degrees of freedom (DOF) n_i = n_g*n_n ; % number of DOF per element S = zeros (n_d, n_d) ; M = zeros (n_d, n_d) ; % stiff & mass C = zeros (n_d, 1) ; % force & moments T = zeros (n_d, 1) ; % displacements % Read EBC values, if any if ( EBC_count > 0 ) ; % need EBC data [EBC_value] = get_ebc_values (n_g, n_m, EBC_flag) ; % read data end % if any EBC data expected % Read point loads & couples data, if any, and insert in C if ( load_pt > 0 ) ; % need point loads data [C] = get_and_add_point_sources (n_g, n_m, C); % add point loads end % if any point source expected C_react = C ; % save before BC for later reaction use % Load the element properties array load msh_properties.tmp ; % one row per element fprintf ('(Echoing file msh_properties.tmp) \n') n_prop = size(msh_properties, 1) ; % == 1 if homogeneous



F_k (1:n_prop) = msh_properties (1:n_prop, 5) ; % extract foundation k's if ( any ( F_k > 0 )) % BEF pressures exist BEF = 1 ; printf ('WARNING: BoEF moment and shear inaccurate for crude meshes \n') else BEF = 0 ; end % if reaction flag if ( n_prop == 1 ) E = msh_properties (1, 1) ; % material modulus I = msh_properties (1, 2) ; % section inertia Line_e (1:2) = msh_properties (1, 3:4) ; % trapezoidal line load k_f = msh_properties (1, 5) ; % foundation stiffness Rho = msh_properties (1, 6) ; % beam mass per unit length % ECHO PROPERTIES fprintf (' \n') fprintf ('Homogeneous Element Properties: \n' ) fprintf ('Elastity modulus (N/m^2) = %g \n', E) fprintf ('Moment of inertia (m^4) = %g \n', I) fprintf ('Line Load (N/m) = [ %g %g ] \n', ... Line_e(1), Line_e(2)) fprintf ('Foundation stiffness (N/m^2) = %g \n', k_f) fprintf ('Mass per unit length (Kg/m) = %g \n', Rho) end % if if ( n_prop > n_e ) error ('ERROR: number of property sets exceeds number of elements') end % if % GENERATE ELEMENT MATRICES AND ASSYMBLE INTO SYSTEM % Assemble n_d by n_d square matrix terms from n_e by n_e for j = 1:n_e ; % loop over elements ====>> ====>> ====>> ====>> S_e = zeros (n_i, n_i) ; % clear array NtN = zeros (n_i, n_i) ; % clear array M_e = zeros (n_i, n_i) ; % clear array C_e = zeros (n_i, 1) ; % clear arrays e_nodes = nodes (j, 1:n_n) ; % connectivity % SET ELEMENT PROPERTIES & GEOMETRY L = x(e_nodes(n_n)) - x(e_nodes(1)) ; % beam length Line_e (1:2) = 0 ; if ( n_prop > 1 ) E = msh_properties (j, 1) ; % material modulus I = msh_properties (j, 2) ; % section inertia Line_e (1:2) = msh_properties (j, 3:4) ; % trapezoidal line load k_f = msh_properties (j, 5) ; % foundation stiffness Rho = msh_properties (j, 6) ; % beam mass per unit length % ECHO PROPERTIES fprintf ('\nProperties for element %g \n', j) fprintf ('Elastity modulus (N/m^2) = %g \n', E) fprintf ('Moment of inertia (m^4) = %g \n', I) fprintf ('Line Load (N/m) = [ %g %g ] \n', ... Line_e(1), Line_e(2)) fprintf ('Foundation stiffness (N/m^2) = %g \n', k_f) fprintf ('Mass per unit length (km/m) = %g \n', Rho) end % if



% ELEMENT, FOUNDATION STIFFNESSES AND SOURCE RESULTANT MATRICES L2 = L^2 ; L3 = L^3 ; % Constant cubic element & foundation stiffnesses & mass matrices S_e = [5092, 1138*L, -3584, 1920*L, -1508, 242*L ; 1138*L, 332*L2, -896*L, 320*L2, -242*L, 38*L2 ; -3584, -896*L, 7168, 0, -3584, 896*L ; 1920*L, 320*L2, 0, 1280*L2, -1920*L, 320*L2 ; -1508, -242*L, -3584, -1920*L, 5092, -1138*L ; 242*L, 38*L2, 896*L, 320*L2, -1138*L, 332*L2 ] ... *E*I/(35 * L^3) ; % stiffness NtN = L*[2092, 114*L, 880, -160*L, 262, -29*L ; 114*L, 8*L2, 88*L, -12*L2, 29*L, -3*L2 ; 880, 88*L, 5632, 0, 880, -88*L ; -160*L, -12*L2, 0, 128*L2, 160*L, -12*L2 ; 262, 29*L, 880, 160*L, 2092, -114*L ; -29*L, -3*L2 -88*L, -12*L2, -114*L, 8*L2 ]/13860; S_e = S_e + k_f * NtN ; % add foundation stiffness before assembly M_e = Rho * NtN ; % mass matrix %b disp(S_e) %b disp(NtN) % Map line load to node forces & moments; C_e = p_To_F * Line_e C_e = zeros (n_i, 1) ; % clear arrays if ( any (Line_e) ) ; % then form forcing vector p_To_F = L * [ 79, 19 ; 5*L, 2*L ; 112, 112 ; -8*L, 8*L ; 19, 79 ; -2*L, -5*L ] / 420 ; % cubic H, linear Line % 6 x 2 * 2 x 1 = 6 x 1 result C_e = p_To_F (1:n_i, 1:2) * Line_e' ; % force moment @ nodes %b C_e = C_e + C_p ; % scatter end % if or set up resultant node loads for line load % SCATTER TO (ASSEMBLE INTO) SYSTEM ARRAYS % Insert completed element matrices into system matrices [rows] = get_element_index (n_g, n_n, e_nodes); % eq numbers S (rows, rows) = S (rows, rows) + S_e ; % add to system stiffness M (rows, rows) = M (rows, rows) + M_e ; % add to system mass C (rows) = C (rows) + C_e ; % add to sys force/couples end % for each j element in mesh <<==== <<==== <<==== <<==== <<==== % ALLOCATE STORAGE FOR OPTIONAL REACTION RECOVERY if ( EBC_count > 0 ) ; % reactions occur [EBC_row, EBC_col] = save_reaction_matrices (EBC_flag, S, C); save_resultant_load_vectors (n_g, C) end % if essential BC exist (almost always true) % ENFORCE ESSENTIAL BOUNDARY CONDITIONS if ( EBC_count > 0 ) ; % reactions occur [S, C] = enforce_essential_BC (EBC_flag, EBC_value, S, C); end % if essential BC exist (almost always true)



% COMPUTE SOLUTION & SAVE T = S \ C ; % Compute displacements & rotations list_save_beam_displacements (n_g, n_m, T) ; % save and print % GRAPHS if ( BEF == 1 ) % plot foundation pressure C1_L3_BoEF_pressure (F_k) % graph pause (5) % display for 5 seconds end % if C1_L3_beam_plots(1) % deflection pause (5) % display for 5 seconds C1_L3_beam_plots(2) % slope %p pause (5) % display for 5 seconds C1_L3_beam_plots(3) % moment diagram %p pause (5) % display for 5 seconds C1_L3_beam_plots(4) % transverse shear diagram %p pause (5) % display for 5 seconds % OPTIONAL REACTION RECOVERY & SAVE if ( EBC_count > 0 ) ; % reactions exist ? [EBC_react] = recover_reactions_print_save (n_g, n_d, ... EBC_flag, EBC_row, EBC_col, T); % reaction to EBC end % if EBC exist % POST-PROCESS ELEMENT REACTIONS (MEMBER FORCES) if ( BEF == 0 ) fprintf ('\nIndividual Element Load and Reaction Summaries: \n') fprintf ('(F_1, M_1, F_2, M_2) \n') for j = 1:n_e ; % loop over elements ====>> ====>> ====>> ====>> ====>> e_nodes = nodes (j, 1:n_n) ; % connectivity [rows] = get_element_index (n_g, n_n, e_nodes) ; % eq numbers T_e (1:n_i) = T(rows) ; % gather element displacements % SET ELEMENT PROPERTIES & GEOMETRY L = x(e_nodes(n_n)) - x(e_nodes(1)) ; % beam length Line_e (1:2) = 0 ; if ( n_prop > 1 ) E = msh_properties (j, 1) ; % material modulus I = msh_properties (j, 2) ; % section inertia Line_e (1:2) = msh_properties (j, 3:4) ; % trapezoidal line load k_f = msh_properties (j, 5) ; % foundation stiffness end % if % ELEMENT, FOUNDATION STIFFNESSES AND SOURCE RESULTANT MATRICES L2 = L^2 ; L3 = L^3 ; % Constant cubic element & foundation stiffnesses & mass matrices S_e = [5092, 1138*L, -3584, 1920*L, -1508, 242*L ; 1138*L, 332*L2, -896*L, 320*L2, -242*L, 38*L2 ; -3584, -896*L, 7168, 0, -3584, 896*L ; 1920*L, 320*L2, 0, 1280*L2, -1920*L, 320*L2 ; -1508, -242*L, -3584, -1920*L, 5092, -1138*L ; 242*L, 38*L2, 896*L, 320*L2, -1138*L, 332*L2 ] ... *E*I/(35 * L^3) ; % stiffness



% Map line load to node forces & moments; C_e = p_To_F * Line_e C_e = zeros (n_i, 1) ; % clear arrays if ( any (Line_e) ) ; % then form forcing vector p_To_F = L * [ 79, 19 ; 5*L, 2*L ; 112, 112 ; -8*L, 8*L ; 19, 79 ; -2*L, -5*L ] / 420 ; % cubic H, linear Line % 6 x 2 * 2 x 1 = 6 x 1 result C_e = p_To_F (1:n_i, 1:2) * Line_e' ; % force moment @ nodes end % if or set up resultant node loads for line load % Assign any point force/couple to the first element with that node if ( j == 1 ) % first element only gets sources from 3 nodes C_e = C_e + C_react (rows) ; else % last 2 nodes C_e (3:6) = C_e (3:6) + C_react (rows(3:6)) ; % quintic C1 beam ONLY !! end % if fprintf ('\n Given Resultant Loading on Member %g, \n', j) disp (C_e') % Finally, get the reactions C_m = S_e * T_e' - C_e ; fprintf ('Net Resultant End Reactions on Member %g, \n', j) disp (C_m') end % for each j element in mesh <<==== <<==== <<==== <<==== <<==== end % if no BEF % End finite element calculations. % See /home/mech517/public_html/Matlab_Plots for graphic options % http://www.owlnet.rice.edu/~mech517/help_plot.html for help % end of L3_Quintic_BoEF % +++++++++++++ functions in alphabetical order +++++++++++++++++ function C1_L3_BoEF_pressure (F_k) % Copyright 2008 J.E. Akin. All rights reserved. % ------------------------------------------------------ % Matlab graph of foundation pressurelue at mesh nodes % for a mesh of 3 node quintic Hermite line elements % ------------------------------------------------------ % c_x = x coordinates of nod_per_el line element % msh_typ_nodes = connectivity list , nt x nod_per_el % loop = corners for nod_per_el line element % nod_per_el = Nodes per element % np = Number of Points % nt = Number of elements % pre_e = Element items before connectivity list pre_e = 0 ; % pre_p = Nodal items before coordinates pre_p = 1; % msh_bc_xyz = Nodal coordinates (with preceeding data) % t_x = x coordinates of nod_per_el corners



if ( nargin == 0 ) i_p = 1 ; end % if no arguments n_fk = size (F_k, 1) ; % number of different foundationS format short % Read coordinate file and connectivity file % integer bc code, real xy pairs for np points (pre_p = 1) load msh_bc_xyz.tmp ; % Set control data: number of points np = size (msh_bc_xyz,1) ; % number of nodal points %b fprintf ('Read %g mesh coordinates \n', np) ns = size (msh_bc_xyz,2) - pre_p ; % space dimension if ( ns ~= 1) error ('This is not a 1D mesh') end % if not 1D data % Set control data: number elements load msh_typ_nodes.tmp ; % nod_per_el nodes per element nt = size (msh_typ_nodes,1) ; % number of elements in mesh nod_per_el = size (msh_typ_nodes,2) - pre_e -1 ;% nodes per elem %b fprintf ('Read %g elements connections \n', nt) if ( nod_per_el ~= 3 ) error ('This is not a mesh of 3 node line elements') end % if load node_results.tmp nr = size (node_results, 1); if ( nr == 0 ) error ('Error missing file node_results.tmp') end % if error max_p = size (node_results, 2) ; % number of columns %b fprintf ('Read %g nodal solution values \n', nr) %b fprintf (' with %g components each \n', max_p) H (3) = 0. ; HC1 (6) = 0. ; DHC1 (6) = 0. ; D2HC1 (6) = 0. ; D3HC1 (6) = 0. ; D4HC1 (6) = 0. ; x (np) = 0. ; % pre-allocate array x y (np) = 0. ; % pre-allocate array y dy (np) = 0. ; % pre-allocate array y t_nodes (nod_per_el) = 0 ; % Optional pre-allocation t_x (nod_per_el) = 0 ; % Optional pre-allocation t_y (nod_per_el) = 0 ; % Optional pre-allocation t_dy (nod_per_el) = 0 ; % Optional pre-allocation c_x (nod_per_el) = 0 ; % Optional pre-allocation c_y (nod_per_el) = 0 ; % Optional pre-allocation % set constants loop = [1:nod_per_el] ; % default to sequential order % msh_bc_xyz has: pre_p items then: x, y x = msh_bc_xyz (1:np, (pre_p+1)) ; % extract x column xmax = max (x) ; xmin = min (x) ; y = node_results(:, 1) ; dy = node_results(:, 2) ;



clf % clear graphics hold on % hold image for plots xlabel (['X, Node at 45 deg (', int2str(nod_per_el), ... ' per element), Element at 90 deg']) % Are properties needed ? load msh_properties.tmp n_prop = size (msh_properties, 2) ; % Loop over all elements max_all = -1e9 ; min_all = 1e9 ; for it = 1:nt ; % Element foundation stiffness, if any if ( n_fk == 1 ) % then homogeneous k_f = F_k (1) ; else k_f = F_k (it) ; end % if % Extract element connectivity t_nodes = msh_typ_nodes (it, (pre_e+2):(nod_per_el+pre_e+1)); % Skip point elements, if any if ( all (t_nodes) ) % then valid line % Extract element coordinates & values t_x = x (t_nodes) ; % x at those nodes, only %b if ( t_x(2) ~= (t_x(1)+t_x(3))/2 ) %b fprintf ('Bad Jaconian element %g \n', it) %b end % if non-constant Jacobian t_y = y (t_nodes) ; % y at those nodes, only t_dy = dy (t_nodes) ; % dy at those nodes, only D (1:2:6) = t_y ; D (2:2:6) = t_dy ; % Loop over local points on the quadratic polynomial element n_poly = ceil ( 95 / nt) ; for k = 1: (n_poly + 1) % points in parametric space % get element parametric interpolation functions R = (k - 1)/n_poly ; % on 0 to 1 X = 2*R - 1 ; % on -1 to 1 % H = ELEMENT SHAPE FUNCTIONS % X = LOCAL COORDINATE OF POINT, -1 TO +1 % LOCAL NODE COORD. ARE -1,0,+1 1-----2-----3 H (1) = 0.5*(X*X - X) ; H (2) = 1. - X*X ; H (3) = 0.5*(X*X + X) ; x_el (k) = H * t_x ; % true x value A = t_x(3) - t_x(1) ; P = X ; P_2 = P * P ; P_3 = P * P_2 ; P_4 = P * P_3 ; P_5 = P * P_4 ; % deflection HC1(1) = (4*P_2 - 5*P_3 - 2*P_4 + 3*P_5) * 0.25 ; HC1(2) = (P_2 - P_3 - P_4 + P_5) * 0.125 * A ; HC1(3) = 1 - 2*P_2 + P_4 ;



HC1(4) = (P - 2*P_3 + P_5) * A * 0.5 ; HC1(5) = (4*P_2 + 5*P_3 - 2*P_4 - 3*P_5) * 0.25 ; HC1(6) = (-P_2 - P_3 + P_4 + P_5) * 0.125 * A ; r_el (k) = HC1 * D' ; % true y value end % for k plot points r_el = -r_el* msh_properties(it,5) ; % elem max, min values [V_X, L_X] = max (r_el) ; [V_N, L_N] = min (r_el); if ( V_X > max_all ) max_all = V_X ; end if ( V_N < min_all ) min_all = V_N ; end plot (x_el, r_el) end % if zero in connectivity % Plot the element number x_bar = sum (t_x' )/nod_per_el ; y_bar = mean (r_el) ; t_text = sprintf (' (%g)', it); % offset # from pt text (x_bar, y_bar, t_text) % incline end % for over all elements fprintf ('Max foundation pressure value is %g \n', max_all) fprintf ('Min foundation pressure value is %g \n', min_all) null (1:np) = 0.5*(V_N + V_X) ; if ( V_N <= 0 & V_X >= 0 ) null (1:np) = 0. ; end % if % finalize axes ymax = max_all ; ymin = min_all ; diff = abs(ymax-ymin) ; ymax = ymax + abs (diff)/10. ; ymin = ymin - abs (diff)/10. ; axis ([xmin, xmax, ymin, ymax]) % set axes title(['Beam Foundation Pressure from: ', ... int2str(nt),' Elements, ', int2str(np),' Nodes']) ylabel (['Pressure: max=', num2str(max_all), ... ', min=', num2str(min_all)]) % plot node points on axis for i = 1:np t_text = sprintf (' %g', i); % offset # from pt text (x(i), null(i), t_text, 'Rotation', 45) % incline end % for all plot (x, null, 'k*') grid % -depsc -tiff % for an eps version %P print ('-dpng', ['BoEF_pressure']) v_text = ['Created BoEF_pressure.png'] ;



fprintf (1,'%s', v_text) ; fprintf (1, ' \n' ) hold off % end of C1_L3_BoEF_pressure function C1_L3_beam_plots (i_p) % Copyright 2008 J.E. Akin. All rights reserved. % ------------------------------------------------------ % Matlab graph of i_p-th component value at mesh nodes % for a mesh of 3 node quintic Hermite line elements % i_p = 1 is deflection, = 2 is slope % = 3 is curvature (moment), = 4 is curvature rate (shear) % ------------------------------------------------------ % c_x = x coordinates of nod_per_el line element % msh_typ_nodes = connectivity list , nt x nod_per_el % loop = corners for nod_per_el line element % nod_per_el = Nodes per element % np = Number of Points % nt = Number of elements % pre_e = Element items before connectivity list pre_e = 0 ; % pre_p = Nodal items before coordinates pre_p = 1; % msh_bc_xyz = Nodal coordinates (with preceeding data) % t_x = x coordinates of nod_per_el corners if ( nargin == 0 ) i_p = 1 ; end % if no arguments format short % Read coordinate file and connectivity file % integer bc code, real xy pairs for np points (pre_p = 1) load msh_bc_xyz.tmp ; % Set control data: number of points np = size (msh_bc_xyz,1) ; % number of nodal points %b fprintf ('Read %g mesh coordinates \n', np) ns = size (msh_bc_xyz,2) - pre_p ; % space dimension if ( ns ~= 1) error ('This is not a 1D mesh') end % if not 1D data % Set control data: number elements load msh_typ_nodes.tmp ; % nod_per_el nodes per element nt = size (msh_typ_nodes,1) ; % number of elements in mesh nod_per_el = size (msh_typ_nodes,2) - pre_e -1 ;% nodes per elem %b fprintf ('Read %g elements connections \n', nt) if ( nod_per_el ~= 3 ) error ('This is not a mesh of 3 node line elements') end % if load node_results.tmp nr = size (node_results, 1); if ( nr == 0 ) error ('Error missing file node_results.tmp') end % if error max_p = size (node_results, 2) ; % number of columns



if ( i_p > 4 ) fprintf ('Data requested for component i_p = %g \n', i_p) error ('i_p > available data') end % if error H (3) = 0. ; HC1 (6) = 0. ; DHC1 (6) = 0. ; D2HC1 (6) = 0. ; D3HC1 (6) = 0. ; D4HC1 (6) = 0. ; x (np) = 0. ; % pre-allocate array x y (np) = 0. ; % pre-allocate array y dy (np) = 0. ; % pre-allocate array y t_nodes (nod_per_el) = 0 ; % Optional pre-allocation t_x (nod_per_el) = 0 ; % Optional pre-allocation t_y (nod_per_el) = 0 ; % Optional pre-allocation t_dy (nod_per_el) = 0 ; % Optional pre-allocation c_x (nod_per_el) = 0 ; % Optional pre-allocation c_y (nod_per_el) = 0 ; % Optional pre-allocation % set constants loop = [1:nod_per_el] ; % default to sequential order % msh_bc_xyz has: pre_p items then: x, y x = msh_bc_xyz (1:np, (pre_p+1)) ; % extract x column xmax = max (x) ; xmin = min (x) ; y = node_results(:, 1) ; dy = node_results(:, 2) ; clf % clear graphics hold on % hold image for plots xlabel (['X, Node at 45 deg (', int2str(nod_per_el), ... ' per element), Element at 90 deg']) % Are properties needed ? if ( i_p > 2 ) load msh_properties.tmp n_prop = size (msh_properties, 2) ; else n_prop = 0; end % if % Loop over all elements max_all = -1e9 ; min_all = 1e9 ; for it = 1:nt ; % Extract element connectivity t_nodes = msh_typ_nodes (it, (pre_e+2):(nod_per_el+pre_e+1)); % Skip point elements, if any if ( all (t_nodes) ) % then valid line % Extract element coordinates & values t_x = x (t_nodes) ; % x at those nodes, only %b if ( t_x(2) ~= (t_x(1)+t_x(3))/2 ) %b fprintf ('Bad Jaconian element %g \n', it) %b end % if non-constant Jacobian t_y = y (t_nodes) ; % y at those nodes, only t_dy = dy (t_nodes) ; % dy at those nodes, only



D (1:2:6) = t_y ; D (2:2:6) = t_dy ; if ( i_p == 1 ) plot (t_x, t_y, 'ko') % plot nodal value symbols end % if % Loop over local points on the quadratic polynomial element n_poly = ceil ( 95 / nt) ; for k = 1: (n_poly + 1) % points in parametric space % get element parametric interpolation functions R = (k - 1)/n_poly ; % on 0 to 1 X = 2*R - 1 ; % on -1 to 1 % H = ELEMENT SHAPE FUNCTIONS % X = LOCAL COORDINATE OF POINT, -1 TO +1 % LOCAL NODE COORD. ARE -1,0,+1 1-----2-----3 H (1) = 0.5*(X*X - X) ; H (2) = 1. - X*X ; H (3) = 0.5*(X*X + X) ; x_el (k) = H * t_x ; % true x value A = t_x(3) - t_x(1) ; P = X ; P_2 = P * P ; P_3 = P * P_2 ; P_4 = P * P_3 ; P_5 = P * P_4 ; % select result of interest, r_el if ( i_p == 1 ) % deflection HC1(1) = (4*P_2 - 5*P_3 - 2*P_4 + 3*P_5) * 0.25 ; HC1(2) = (P_2 - P_3 - P_4 + P_5) * 0.125 * A ; HC1(3) = 1 - 2*P_2 + P_4 ; HC1(4) = (P - 2*P_3 + P_5) * A * 0.5 ; HC1(5) = (4*P_2 + 5*P_3 - 2*P_4 - 3*P_5) * 0.25 ; HC1(6) = (-P_2 - P_3 + P_4 + P_5) * 0.125 * A ; r_el (k) = HC1 * D' ; % true y value elseif ( i_p == 2 ) % slope DHC1 (1) = (8*P - 15*P_2 - 8*P_3 + 15*P_4) * 0.5 / A ; DHC1 (2) = (2*P - 3*P_2 - 4*P_3 + 5*P_4) * 0.25 ; DHC1 (3) = (-4*P + 4*P_3) * 2 / A ; DHC1 (4) = (1 - 6*P_2 + 5*P_4) ; DHC1 (5) = (8*P + 15*P_2 - 8*P_3 - 15*P_4) * 0.5 / A ; DHC1 (6) = (-2*P - 3*P_2 + 4*P_3 + 5*P_4) * 0.25 ; r_el (k) = DHC1 * D' ; % true y' value elseif ( i_p == 3 ) % curvature or moment D2HC1 (1) = (8 - 30*P - 24*P_2 + 60*P_3)/A/ A ; D2HC1 (2) = (2 - 6*P - 12*P_2 + 20*P_3) * 0.5/A ; D2HC1 (3) = (-4 + 12*P_2) * 4 / A / A ; D2HC1 (4) = (-12*P + 20*P_3)*2 / A ; D2HC1 (5) = (8 + 30*P - 24*P_2 - 60*P_3)/A/A; D2HC1 (6) = (-2 - 6*P + 12*P_2 + 20*P_3) * 0.5/A ; r_el (k) = D2HC1 * D' ; % true y'' value elseif ( i_p == 4 ) % curvature rate or shear force D3HC1 (1) = (-30 - 48*P + 180*P_2)*2/A/A/A ; D3HC1 (2) = (-6 - 24*P + 60*P_2)/A/A; D3HC1 (3) = (24*P) * 8/A/A/A ; D3HC1 (4) = (-12 + 60*P_2)*4/A/A ; D3HC1 (5) = (30 - 48*P - 180*P_2)*2/A/A/A;



D3HC1 (6) = (-6 + 24*P + 60*P_2)*2/A/A ; r_el (k) = D3HC1 * D' ; % true y''' value end % if plot type end % for k plot points if ( n_prop > 1 ) EI = msh_properties(it,1)*msh_properties(it,2) ; r_el = r_el* EI ; elseif ( n_prop == 1 ) EI = msh_properties(1,1)*msh_properties(1,2) ; r_el = r_el* EI ; end % if % elem max, min values [V_X, L_X] = max (r_el) ; [V_N, L_N] = min (r_el); if ( V_X > max_all ) max_all = V_X ; end if ( V_N < min_all ) min_all = V_N ; end plot (x_el, r_el) end % if zero in connectivity % Plot the element number x_bar = sum (t_x' )/nod_per_el ; y_bar = mean (r_el) ; t_text = sprintf (' (%g)', it); % offset # from pt text (x_bar, y_bar, t_text) % incline end % for over all elements null (1:np) = 0.5*(V_N + V_X) ; if ( V_N <= 0 & V_X >= 0 ) null (1:np) = 0. ; end % if % finalize axes ymax = max_all ; ymin = min_all ; diff = abs(ymax-ymin) ; ymax = ymax + abs (diff)/10. ; ymin = ymin - abs (diff)/10. ; axis ([xmin, xmax, ymin, ymax]) % set axes if ( i_p == 1 ) title(['Qunitic Beam Deflection from: ', ... int2str(nt),' Elements, ', int2str(np),' Nodes']) ylabel (['Deflection: max=', num2str(max_all), ... ', min=', num2str(min_all)]) fprintf ('Max deflection value is %g \n', max_all) fprintf ('Min deflection value is %g \n', min_all) elseif ( i_p == 2 ) title(['Qunitic Beam Slope from: ', ... int2str(nt),' Elements, ', int2str(np),' Nodes']) ylabel (['Slope: max=', num2str(max_all), ... ', min=', num2str(min_all)]) fprintf ('Max slope value is %g \n', max_all) fprintf ('Min slope value is %g \n', min_all)



elseif ( i_p == 3 ) title(['Qunitic Beam Moment from: ', ... int2str(nt),' Elements, ', int2str(np),' Nodes']) ylabel (['Moment: max=', num2str(max_all), ... ', min=', num2str(min_all)]) fprintf ('Max moment value is %g \n', max_all) fprintf ('Min moment value is %g \n', min_all) elseif ( i_p == 4 ) title(['Qunitic Beam Shear from: ', ... int2str(nt),' Elements, ', int2str(np),' Nodes']) ylabel (['Transverse Shear: max=', num2str(max_all), ... ', min=', num2str(min_all)]) fprintf ('Max shear value is %g \n', max_all) fprintf ('Min shear value is %g \n', min_all) end % if % plot node points on axis for i = 1:np t_text = sprintf (' %g', i); % offset # from pt text (x(i), null(i), t_text, 'Rotation', 45) % incline end % for all plot (x, null, 'k*') grid % -depsc -tiff % for an eps version if ( i_p == 1 ) %P print ('-dpng', ['Beam_deflections']) v_text = ['Created Beam_deflections.png'] ; elseif ( i_p == 2 ) %P print ('-dpng', ['Beam_slope']) v_text = ['Created Beam_slope.png'] ; elseif ( i_p == 3 ) %P print ('-dpng', ['Beam_moments']) v_text = ['Created Beam_moments.png'] ; elseif ( i_p == 4 ) %P print ('-dpng', ['Beam_shears']) v_text = ['Created Beam_shears.png'] ; end % if fprintf (1,'%s', v_text) ; fprintf (1, ' \n' ) hold off % end of C1_L3_beam_plots function [S, C] = enforce_essential_BC (EBC_flag, EBC_value, S, C) % modify system linear eqs for essential boundary conditions % (by trick to avoid matrix partitions, loses reaction data) n_d = size (C, 1) ; % number of DOF eqs if ( size (EBC_flag, 2) > 1 ) ; % change to vector copy flag_EBC = reshape ( EBC_flag', 1, n_d) ; value_EBC = reshape ( EBC_value', 1, n_d) ; else flag_EBC = EBC_flag ; value_EBC = EBC_value ; end % if for j = 1:n_d % check all DOF for essential BC if ( flag_EBC (j) ) % then EBC here % Carry known columns*EBC to RHS. Zero that column and row. % Insert EBC identity, 1*EBC_dof = EBC_value. EBC = value_EBC (j) ; % recover EBC value



C (:) = C (:) - EBC * S (:, j) ; % carry known column to RHS S (:, j) = 0 ; S (j, :) = 0 ; % clear, restore symmetry S (j, j) = 1 ; C (j) = EBC ; % insert identity into row end % if EBC for this DOF end % for over all j-th DOF % end enforce_essential_BC (EBC_flag, EBC_value, S, C) function [C] = get_and_add_point_sources (n_g, n_m, C) load msh_load_pt.tmp ; % node, DOF, value (eq. number) n_u = size(msh_load_pt, 1) ; % number of point sources if ( n_u < 1 ) ; % missing data error ('No load_pt data in msh_load_pt.tmp') end % if user error fprintf ('\nRead %g point sources. (Echo of msh_load_pt.tmp) \n', n_u) fprintf ('Node, DOF (1=force, 2=couple), Source_value \n') for j = 1:n_u ; % non-zero Neumann pts node = msh_load_pt (j, 1) ; % global node number DOF = msh_load_pt (j, 2) ; % local DOF number value = msh_load_pt (j, 3) ; % point source value fprintf ('%g %g %g \n', node, DOF, value) Eq = n_g * (node - 1) + DOF ; % row in system matrix C (Eq) = C (Eq) + value ; % add to system column matrix end % for each EBC % end get_and_add_point_sources (n_g, n_m, C) function [EBC_flag] = get_ebc_flags (n_g, n_m, P) EBC_flag = zeros(n_m, n_g) ; % initialize for k = 1:n_m ; % loop over all nodes if ( P(k) > 0 ) ; % at least one EBC here [flags] = unpack_pt_flags (n_g, k, P(k)) ; % unpacking EBC_flag (k, 1:n_g) = flags (1:n_g) ; % populate array end % if EBC at node k end % for loop over all nodes % end get_ebc_flags function [EBC_value] = get_ebc_values (n_g, n_m, EBC_flag) EBC_value = zeros(n_m, n_g) ; % initialize to zero load msh_ebc.tmp ; % node, DOF, value (eq. number) n_c = size(msh_ebc, 1) ; % number of constraints fprintf ('Read %g EBC data with Node, DOF, Value. \n', n_c) fprintf ('(Echo of file msh_ebc.tmp) \n') disp(msh_ebc) ; % echo input for j = 1:n_c ; % loop over ebc inputs node = round (msh_ebc (j, 1)) ; % node in mesh DOF = round (msh_ebc (j, 2)) ; % DOF # at node value = msh_ebc (j, 3) ; % EBC value % Eq = n_g * (node - 1) + DOF ; % row in system matrix EBC_value (node, DOF) = value ; % insert value in array if ( EBC_flag (node, DOF) == 0 ) % check data consistency fprintf ('WARNING: EBC but no flag at node %g & DOF %g. \n', ... node, DOF) % EBC_flag (node, DOF) = 1; % try to recover from data error end % if common user error end % for each EBC EBC_count = sum (sum ( EBC_flag > 0 )) ; % check input data if ( EBC_count ~= n_c ) ; % probable user error



fprintf ('WARNING: mismatch in bc_flag count & msh_ebc.tmp') end % if user error % end get_ebc_values function [rows] = get_element_index (n_g, n_n, e_nodes) % calculate system DOF numbers of element, for gather, scatter rows = zeros (1, n_g*n_n) ; % allow for node = 0 for k = 1:n_n ; % loop over element nodes global_node = round (e_nodes (k)) ; % corresponding sys node for i = 1:n_g ; % loop over DOF at node eq_global = i + n_g * (global_node - 1) ; % sys DOF, if any eq_element = i + n_g * (k - 1) ; % el DOF number if ( eq_global > 0 ) ; % check node=0 trick rows (1, eq_element) = eq_global ; % valid DOF > 0 end % if allow for omitted nodes end % for DOF i % end local DOF loop end % for each element node % end local node loop % end get_element_index function [n_e, n_n, n_t, el_type, nodes] = get_mesh_elements () ; % input file controls (for various data generators) load msh_typ_nodes.tmp ; % el_type, connectivity list (3) n_e = size (msh_typ_nodes,1) ; % number of elements if ( n_e == 0 ) ; % data file missing error ('Error missing file msh_typ_nodes.tmp') end % if error n_n = size (msh_typ_nodes,2) - 1 ; % nodes per element fprintf ('Read %g elements with (ignored) type & %g nodes each. \n', ... n_e, n_n) fprintf ('(Echo of file msh_typ_nodes.tmp) \n') el_type = round (msh_typ_nodes(:, 1)); % el type number >= 1 n_t = max(el_type) ; % number of element types %b fprintf ('Maximum number of element types = %g. \n', n_t) nodes (1:n_e, 1:n_n) = msh_typ_nodes (1:n_e, 2:1+n_n); disp(msh_typ_nodes (:, 1:1+n_n)) % echo data % end get_mesh_elements function [n_m, n_s, P, x, y, z] = get_mesh_nodes () ; % input file controls (for various data generators) % READ MESH AND EBC_FLAG INPUT DATA % specific problem data from MODEL data files (sequential) load msh_bc_xyz.tmp ; % bc_flag, x-, y-, z-coords n_m = size (msh_bc_xyz,1) ; % number of nodal points in mesh if ( n_m == 0 ) ; % data missing ! error ('Error missing file msh_bc_xyz.tmp') end % if error n_s = size (msh_bc_xyz,2) - 1 ; % number of space dimensions fprintf ('Read %g nodes with bc_flag & %g coordinates. \n', ... n_m, n_s) fprintf ('(Echo of file msh_bc_xyz.tmp) \n') msh_bc_xyz (:, 1)= round (msh_bc_xyz (:, 1)); P = msh_bc_xyz (1:n_m, 1) ; % integer Packed BC flag x = msh_bc_xyz (1:n_m, 2) ; % extract x column y (1:n_m, 1) = 0.0 ; z (1:n_m, 1) = 0.0 ; % default to zero if (n_s > 1 ) ; % check 2D or 3D



y = msh_bc_xyz (1:n_m, 3) ; % extract y column end % if 2D or 3D if ( n_s == 3 ) ; % check 3D z = msh_bc_xyz (1:n_m, 4) ; % extract z column end % if 3D disp(msh_bc_xyz (:, 1:1+n_s)) ; % echo data % end get_mesh_nodes function list_save_beam_displacements (n_g, n_m, T) fprintf('\nCalculated Displacements \n') fprintf('\nNode, Y_displacement, Z_rotation at %g nodes \n', n_m) T_matrix = reshape (T, n_g, n_m)' ; % pretty shape % save results (displacements) to MODEL file: node_results.tmp fid = fopen('node_results.tmp', 'w') ; % open for writing for j = 1:n_m ; % node loop, save displ fprintf (fid, '%g %g \n', T_matrix (j, 1:n_g)) ; % to file fprintf ('%g %g %g \n', j, T_matrix (j, 1:n_g)) ; % to screen end % for j DOF % end list_save_beam_displacements (n_g, n_m, T) function list_save_displacements_results (n_g, n_m, T) fprintf ('\n') ; fprintf('X_disp Y_disp Z_disp at %g nodes \n', n_m) T_matrix = reshape (T, n_g, n_m)' ; % pretty shape disp (T_matrix) ; % print displacements % save results (displacements) to MODEL file: node_results.tmp fid = fopen('node_results.tmp', 'w') ; % open for writing for j = 1:n_m ; % save displacements if ( n_g == 1 ) fprintf (fid, '%g \n', T_matrix (j, 1:n_g)) ; elseif ( n_g == 2 ) fprintf (fid, '%g %g \n', T_matrix (j, 1:n_g)) ; elseif ( n_g == 3 ) fprintf (fid, '%g %g %g \n', T_matrix (j, 1:n_g)) ; elseif ( n_g == 4 ) fprintf (fid, '%g %g %g %g \n', T_matrix (j, 1:n_g)) ; elseif ( n_g == 5 ) fprintf (fid, '%g %g %g %g %g \n', T_matrix (j, 1:n_g)) ; elseif ( n_g == 6 ) fprintf (fid, '%g %g %g %g %g %g \n', T_matrix (j, 1:n_g)) ; else error ('reformat list_save_displacements_results for n_g > 6.') end % if end % for j DOF % end list_save_displacements_results (T) function [EBC_react] = recover_reactions_print_save (n_g, n_d, ... EBC_flag, EBC_row, EBC_col, T) % get EBC reaction values by using rows of S & C (before EBC) n_d = size (T, 1) ; % number of system DOF % n_c x 1 = n_c x n_d * n_d x 1 + n_c x 1 EBC_react = EBC_row * T - EBC_col ; % matrix reactions (+-) % save reactions (forces) to MODEL file: node_reaction.tmp fprintf ('\nReactions at Displacement BCs \n') fprintf ('Node, DOF, Reaction Value \n') fid = fopen('node_reaction.tmp', 'w') ; % open for writing if ( size (EBC_flag, 2) > 1 ) ; % change to vector copy flag_EBC = reshape ( EBC_flag', 1, n_d) ; % changed



else flag_EBC = EBC_flag ; % original vector end % if Totals = zeros (1, n_g) ; % zero input totals kount = 0 ; % initialize counter for j = 1:n_d ; % extract all EBC reactions if ( flag_EBC(j) ) ; % then EBC here % Output node_number, component_number, value, equation_number kount = kount + 1 ; % copy counter node = ceil(j/n_g) ; % node at DOF j j_g = j - (node - 1)*n_g ; % 1 <= j_g <= n_g React = EBC_react (kount, 1) ; % reaction value fprintf ( fid, '%g %g %g \n', node, j_g, React);% save fprintf ('%g %g %g \n', node, j_g, React); % print Totals (j_g) = Totals (j_g) + React ; % sum all components end % if EBC for this DOF end % for over all j-th DOF fprintf ('Totals = ') ; disp(Totals) ; % echo totals % end recover_reactions_print_save (EBC_row, EBC_col, T) function [EBC_row, EBC_col] = save_reaction_matrices (EBC_flag, S, C) n_d = size (C, 1) ; % number of system DOF EBC_count = sum (sum (EBC_flag)) ; % count EBC & reactions EBC_row = zeros(EBC_count, n_d) ; % reaction data EBC_col = zeros(EBC_count, 1) ; % reaction data if ( size (EBC_flag, 2) > 1 ) ; % change to vector copy flag_EBC = reshape ( EBC_flag', 1, n_d) ; % changed else flag_EBC = EBC_flag ; % original vector end % if kount = 0 ; % initialize counter for j = 1:n_d % System DOF loop, check for displacement BC if ( flag_EBC (j) ) ; % then EBC here % Save reaction data to be destroyed by EBC solver trick kount = kount + 1 ; % copy counter EBC_row(kount, 1:n_d) = S (j, 1:n_d) ; % copy reaction data EBC_col(kount, 1) = C (j) ; % copy reaction data end % if EBC for this DOF end % for over all j-th DOF % end sys DOF loop % end save_reaction_matrices (S, C, EBC_flag) function save_resultant_load_vectors (n_g, C) % save resultant forces to MODEL file: node_resultants.tmp n_d = size (C, 1) ; % number of system DOF fprintf ('\n') ; % Skip a line % fprintf ('Node, DOF, Resultant Force (1) or Moment (2) \n') fprintf ('Resultant input sources: \n') fprintf ('Node, DOF, Resultant input sources \n') fid = fopen('node_resultant.tmp', 'w'); % open for writing Totals = zeros (1, n_g) ; % zero input totals for j = 1:n_d ; % extract all resultants if ( C (j) ~= 0. ) ; % then source here % Output node_number, component_number, value, equation_number node = ceil(j/n_g) ; % node at DOF j j_g = j - (node - 1)*n_g ; % 1 <= j_g <= n_g value = C (j) ; % resultant value fprintf ( fid, '%g %g %g %g \n', node, j_g, value, j);% save fprintf ('%g %g %g \n', node, j_g, value); % print



Totals (j_g) = Totals (j_g) + value ; % sum all inputs end % if non-zero for this DOF end % for over all j-th DOF fprintf ('Totals = ') ; disp(Totals) ; % echo totals % end save_resultant_load_vectors (n_g, n_m, C) function [flags] = unpack_pt_flags (n_g, N, flag) % unpack n_g integer flags from the n_g digit flag at node N % integer flag contains (left to right) f_1 f_2 ... f_n_g full = flag ; % copy integer check = 0 ; % validate input for Left2Right = 1:n_g ; % loop over local DOF at k Right2Left = n_g + 1 - Left2Right ; % reverse direction work = floor (full / 10) ; % work item keep = full - work * 10 ; % work item flags (Right2Left) = keep ; % insert into array full = work ; % work item check = check + keep * 10^(Left2Right - 1) ; % validate end % for each local DOF if ( flag > check ) ; % check for likely error fprintf ('WARNING: bc flag likely reversed at node %g. \n', N) end % if likely user error % end unpack_pt_flags % ---------------------- end ------------------------

shear moment diagram quintic - rice universitypresented to show that this element gives exact...

Documents