Bending of multi-cross section thin/thick plate under composite transverse pressures modeled using Finite Element Method
A multi-cross section thin plate under composite transverse pressure is considered with line/area boundary conditions.
In main.m:
% Geometrical and material properties of plate
a = 1; % Length of the plate (along X-axes) [m]
b = 1; % Length of the plate (along Y-axes) [m]
E = 30e6; % Elastic Modulus [GPa --> kN/m^2]
nu = 0.3; % Poisson's ratio
t = 0.1 ; % Plate thickness [m]
I = t^3/12 ; % Moment of Inertia [m^4]nelem_x = 30;
nelem_y = 30;fcu = 30; % Concrete compressive strength [MPa]
B = b/nelem_x; % Slab width [m]
H = 180; % Slab height [mm]
C = 30; % Distance of rebars from bottom/top of the slab [mm]% In sec 3, the master layer of the plate has been defined. Now, here specify whether you want any area of the plate to have a different cross-section than the master one. In such case, the new cross-section will overwrite the existing one/ones in the area where it was specified.
reb_lay_am = 2; % Longitudinal Bars along x-direction
phisx0 = 8; % Rebars diameter on the lower edge [mm]
phix0 = 8; % Rebars diameter on the upper edge [mm]
sp_nsx0 = 0.15; % Spacing of rebars on the lower edge [mm]
sp_nx0 = 0.15; % Spacing of rebars on the upper edge [mm]
nsx0 = 1/sp_nsx0; % Amount of rebars on the lower edge per m [-]
nx0 = 1/sp_nx0; % Amount of rebars on the upper edge per m [-]
% Longitudinal Bars along y-direction
phisy0 = phisx0; % Rebars diameter on the lower edge [mm]
phiy0 = phix0; % Rebars diameter on the upper edge [mm]
nsy0 = nsx0; % Amount of rebars on the lower edge per m [-]
ny0 = nx0; % Amount of rebars on the upper edge per m [-]
sp_nsy0 = 1/nsy0; % Spacing of rebars on the lower edge [mm]
sp_ny0 = 1/ny0; % Spacing of rebars on the upper edge [mm]% Longitudinal Bars along x-direction
phisx1 = 8; % Rebars diameter on the lower edge [mm]
phix1 = 8; % Rebars diameter on the upper edge [mm]
nsx1 = 6; % Amount of rebars on the lower edge per m [-]
nx1 = 6; % Amount of rebars on the upper edge per m [-]
sp_nsx1 = 1/nsx1; % Spacing of rebars on the lower edge [mm]
sp_nx1 = 1/nx1; % Spacing of rebars on the upper edge [mm]
% Longitudinal Bars along y-direction
phisy1 = phisx1; % Rebars diameter on the lower edge [mm]
phiy1 = phix1; % Rebars diameter on the upper edge [mm]
nsy1 = nsx1; % Amount of rebars on the lower edge per m [-]
ny1 = nx1; % Amount of rebars on the upper edge per m [-]
sp_nsy1 = 1/nsy1; % Spacing of rebars on the lower edge [mm]
sp_ny1 = 1/ny1; % Spacing of rebars on the upper edge [mm]
% Define Rebar Layer Area according to inserted coordinates of nodes
% (COUNTERCLOCKWISE!)
reb1nodes = [1 1; 5 1; 5 5; 1 5];% Longitudinal Bars along x-direction
phisx1 = 8; % Rebars diameter on the lower edge [mm]
phix1 = 8; % Rebars diameter on the upper edge [mm]
nsx1 = 6; % Amount of rebars on the lower edge per m [-]
nx1 = 6; % Amount of rebars on the upper edge per m [-]
sp_nsx1 = 1/nsx1; % Spacing of rebars on the lower edge [mm]
sp_nx1 = 1/nx1; % Spacing of rebars on the upper edge [mm]
% Longitudinal Bars along y-direction
phisy1 = phisx1; % Rebars diameter on the lower edge [mm]
phiy1 = phix1; % Rebars diameter on the upper edge [mm]
nsy1 = nsx1; % Amount of rebars on the lower edge per m [-]
ny1 = nx1; % Amount of rebars on the upper edge per m [-]
sp_nsy1 = 1/nsy1; % Spacing of rebars on the lower edge [mm]
sp_ny1 = 1/ny1; % Spacing of rebars on the upper edge [mm]
% Define Rebar Layer Area according to inserted coordinates of nodes
% (COUNTERCLOCKWISE!)
reb1nodes = [1 1; 5 1; 5 5; 1 5]; YieldCriterionOrder = 1; % Linearized Yield Criterion
% YieldCriterionOrder = 2; % Quadratic Yield Criterion%bc = 0; % Only external BCs
% bc = 1; % External + 1 internal BCs
bc = 2; % External + 2 internal BCs
%% External Boundary conditions (at the edges) %%
%typeextBC = 's-s-s-s' ; % Simply Supported
%typeextBC = 'c-c-c-c' ; % Fully Clamped
%typeextBC = 's-s-o-s' ; % Simply Supported at 3 out of 4 edges
%typeextBC = 'o-s-o-s' ; % Simply Supported at 2 out of 4 edges
%typeextBC = 's-o-s-o' ; % Simply Supported at 2 out of 4 edges
%typeextBC = 's-o-o-s' ; % Simply Supported at 2 out of 4 edges
typeextBC = 'o-s-o-o' ; % Simply Supported at right edge if bc == 1
% typeintBC = 's-s-s-s' ; % Simply Supported
% typeintBC = 'c-c-c-c' ; % Fully Clamped
% typeintBC = 'points'; % Point support
typeintBC = 'patchs, s-s-s-s'; % Patch support
% typeintBC = 'patchs, c-c-c-c'; % Patch clamp
%dir = 'x'; % BC along x-direction
%dir = 'y'; % BC along y-direction
%dir = 'point'; % BC in one point support
dir = 'patch'; % BC in a patch with simple support at edges
coor_x1 = 0; % Define x-coordinate of internal BC
coor_y1 = 0.967; % Define y-coordinate of internal BC
coor_x2 = 0.1; % Define x-coordinate of internal BC
coor_y2 = 1; % Define y-coordinate of internal BCif bc == 2
% 1st Internal Boundary condition
% typeintBC1 = 's-s-s-s' ; % Simply Supported
% typeintBC1 = 'c-c-c-c' ; % Fully Clamped
% typeintBC1 = 'points'; % Point support
typeintBC1 = 'patchs, s-s-s-s'; % Patch support
% typeintBC1 = 'patchs, c-c-c-c'; % Patch clamp
%dir1 = 'x'; % BC along x-direction
%dir1 = 'y'; % BC along y-direction
%dir1 = 'point'; % BC in one point support
dir1 = 'patch'; % BC in a patch with simple support at edges
% Bottom left vertex
% ROUNDED TO THREE SIGNIFICANT DIGITS!
coor1_x1 = 0; % Define x-coordinate of internal BC
coor1_y1 = 0; % Define y-coordinate of internal BC
% Top right vertex
coor1_x2 = 0.033; % Define x-coordinate of internal BC
coor1_y2 = 0.033; % Define y-coordinate of internal BC
intbcdof1 = IntBoundaryCondition(typeintBC1,NodeCoor,dir1,...
coor1_x1,coor1_y1,coor1_x2,coor1_y2,nelem_x,'b') ;
% 2nd Internal Boundary condition
% typeintBC2 = 's-s-s-s' ; % Simply Supported
% typeintBC2 = 'c-c-c-c' ; % Fully Clamped
% typeintBC2 = 'points'; % Point support
typeintBC2 = 'patchs, s-s-s-s'; % Patch support
% typeintBC2 = 'patchs, c-c-c-c'; % Patch clamp
%dir2 = 'x'; % BC along x-direction
%dir2 = 'y'; % BC along y-direction
%dir2 = 'point'; % BC in one point support
dir2 = 'patch'; % BC in a patch with simple support at edges
% Bottom left vertex
% ROUNDED TO THREE SIGNIFICANT DIGITS!
coor2_x1 = 0; % Define x-coordinate of internal BC
coor2_y1 = 0.967; % Define y-coordinate of internal BC
% Top right vertex
coor2_x2 = 0.033; % Define x-coordinate of internal BC
coor2_y2 = 1; % Define y-coordinate of internal BC
intbcdof2 = IntBoundaryCondition(typeintBC2,NodeCoor,dir2,...
coor2_x1,coor2_y1,coor2_x2,coor2_y2,nelem_x,'k') ;
legend('Slab','Internal BC1','Internal BC2');
% Merge the two Internal Boundary Conditions together
intbcdof = [intbcdof1, intbcdof2];
intbcdof = sort(intbcdof);MaxLoadSteps = 2; % Define Maximum Amount of Load Steps
UnitP = -1 ; % Define Unit Load [kN/m^2]
Pmax = 1300; % Define Maximum Load [kN/m^2]
Pstart = 0; % Starting Load [kN/m^2]
lambda = Pstart:abs((Pmax - Pstart)/(MaxLoadSteps - 1)):abs(Pmax); % Define Incrementing Load Factor [-]
lambda(1) = []; %loadType = 0; % Uniform transverse pressure
loadType = 1; % One patch load applied
%loadType = 2; % Two patch loads applied
%loadType = 3; % Three patch loads applied
%loadType = 4; % Four patch loads appliedSpecify the load location of load 1 at load1nodes, of load 2 at load2nodes etc.
%% ----------------- START ANALYSIS------------------------------------%%M. De Filippo 'Bending of multi cross-section thin/thick plate under composite transverse pressures and boundary conditions modeled using Finite Element Method', DOI: 10.13140/RG.2.2.35724.46726, Dec 2018.