CrossMember class
This class defines a member of a continuous beam for the Cross Process. Euler-Bernoulli flexural behavior is assumed (Navier beam theory).
See definition of CrossSolver class.
Contents
Author
Luiz Fernando Martha
History
@version 1.01
Initial version 1.00: August 2022
Initially prepared for the course CIV 2801 - Fundamentos de Computação Gráfica, 2022, second term, Department of Civil Engineering, PUC-Rio.
Version 1.01: September 2022
Created constant properties tol_discrim and tol_rootpos; Created properties displv, bendmom, spanmom_pos and spanmom. Created private methods fixedEndUniformLoadDispl and displShapeFcnVector. Created public methods internalDispl, bendingMoment and momentRoots.
Class definition
classdef CrossMember < handle
See documentation on handle super-class.
Public attributes
properties (SetAccess = public, GetAccess = public)
EI = 0; % flexural stiffness
len = 0; % member length
q = 0; % vertical distributed load (top-down positive)
ml = 0; % left end moment
mr = 0; % right end moment
k = 0; % rotational stiffness coefficient
displv = []; % vector of member transversal displacements
bendmom = []; % vector of member bending moments
spanmom_pos = 0; % position of maximum span moment
spanmom = 0; % maximum span moment
end
Class (constant) properties
properties (Constant)
tol_discrim = 1e-7; % tolerance for discriminant of quadratic
% equation
tol_rootpos = 1e-2; % tolerance for root position proximity to
% member extremities
end
Constructor method
methods
%------------------------------------------------------------------
function memb = CrossMember(EI,len,q)
if (nargin > 0)
memb.EI = EI;
memb.len = len;
memb.q = q;
memb.ml = 0;
memb.mr = 0;
memb.k = 0;
memb.displv = [];
memb.bendmom = [];
memb.spanmom_pos = 0;
memb.spanmom = 0;
end
end
end
Private methods
methods (Access = private)
%------------------------------------------------------------------
% Computes element transversal fixed-end displacement at given
% cross-section positions (given in a array) for an applied
% uniformely distributed load.
% Output:
% dvI: array of fixed-end transversal displacements
% Input arguments:
% contin_ini: rotation continuity at initial node
% (0 -> hinge, 1 -> continuous)
% contin_end: rotation continuity at end node
% (0 -> hinge, 1 -> continuous)
% x: array of element cross-section positions in local x-axis
function dvI = fixedEndUniformLoadDispl(memb,contin_ini,contin_end,x)
% Basic element length properties
L = memb.len;
L2 = L * L;
L3 = L2 * L;
% Calculate transversal displacements
%%%%%%% COMPLETE HERE - CROSSMEMBER: 01 %%%%%%%
end
%------------------------------------------------------------------
% Evaluates member flexural displacement shape functions in local
% at given cross-section positions (given in a array).
% Flexural shape functions give the transversal displacement of an
% element subjected to unit nodal rotations.
% Output:
% Nv: a (2 x np) matrix with flexural displacement shape functions:
% ini -> initial node
% end -> end node
% Nv = [ N_ini;
% N_end ]
% Input arguments:
% contin_ini: rotation continuity at initial node
% (0 -> hinge, 1 -> continuous)
% contin_end: rotation continuity at end node
% (0 -> hinge, 1 -> continuous)
% x: array of element cross-section positions in local x-axis
function Nv = displShapeFcnVector(memb,contin_ini,contin_end,x)
% Get number of points
np = size(x,2);
% Basic element length properties
L = memb.len;
L2 = L * L;
% Compute shape function values based on end rotation
% continuity conditions
%%%%%%% COMPLETE HERE - CROSSMEMBER: 02 %%%%%%%
Nv = [ Nv2;
Nv4 ];
end
end
Public methods
methods
%------------------------------------------------------------------
% Computes member transversal internal displacements vector at
% given cross-section positions (given in a array) from nodal
% rotations. Store them in corresponding member property (displv).
% Output:
% maxv: maximum absolute transversal displacement value.
% Input arguments:
% contin_ini: rotation continuity at initial node
% (0 -> hinge, 1 -> continuous)
% contin_end: rotation continuity at end node
% (0 -> hinge, 1 -> continuous)
% rot_ini: rotation of initial node
% rot_end: rotation of end node
% x: array of element cross-section positions in local x-axis
function maxv = internalDispl(memb,contin_ini,contin_end,rot_ini,rot_end,x)
% Get internal displacements from local fixed-end applyed load
dvI = memb.fixedEndUniformLoadDispl(contin_ini,contin_end,x);
% Evaluate element shape function matrix at given cross-section
% positions
Nv = memb.displShapeFcnVector(contin_ini,contin_end,x);
% Assemble a vector of member end node rotations
membrot = [rot_ini rot_end];
% Compute internal displacements from global analysis (from
% nodal rotations)
dvII = membrot * Nv;
% Assemble total element transversal displacements vector
memb.displv = dvI + dvII;
% Compute maximum absolute displacement value
maxv = max([abs(min(memb.displv)) abs(max(memb.displv))]);
end
%------------------------------------------------------------------
% Computes member bending moment vector at given cross-section
% positions (given in a array) from nodal rotations. Store them
% in corresponding member property (bendmom).
% Computes maximum span moment and its section position along
% member and store them in the corresponding member properties
% (spanmom and spanmom_pos). Then maximum span moment position
% corresponds to the cross-section in which the internal shear
% force is null.
% Input arguments:
% mom_ini: bending moment of initial node
% mom_end: bending moment of end node
% x: array of element cross-section positions in local x-axis
% Output:
% maxm: maximum absolute bending moment value
function maxm = bendingMoment(memb,mom_ini,mom_end,x)
%%%%%%% COMPLETE HERE - CROSSMEMBER: 03 %%%%%%%
% Compute maximum bending moment value
maxm = max([abs(min(memb.bendmom)) abs(max(memb.bendmom))]);
end
%------------------------------------------------------------------
% Computes roots of member quadratic bending moment diagram.
% Returns number of roots and root positions.
% If a root is outside member span or close to a member
% extrimity, it is not considered.
% Input arguments:
% mom_ini: bending moment of initial node
% mom_end: bending moment of end node
% Output:
% nroots: number of roots of quadratic bending moment diagram
% xroots: array of roots positions in local x-axis
function [nroots,xroots] = momentRoots(memb,mom_ini,mom_end)
nroots = 0;
xroots = zeros(2);
%%%%%%% COMPLETE HERE - CROSSMEMBER: 04 %%%%%%%
end
%------------------------------------------------------------------
% Cleans data structure of a CrossMember object.
function memb = clean(memb)
memb.EI = 0;
memb.len = 0;
memb.q = 0;
memb.ml = 0;
memb.mr = 0;
memb.k = 0;
memb.displv = [];
memb.bendmom = [];
memb.spanmom_pos = 0;
memb.spanmom = 0;
end
end
end