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|
unit DGCClassSectionOfBeam;
// Calcul de caractéristiques de sections droites d'une poutre:
// - aire
// - centre de gravité
// - limites
// - moments quadratiques
// - produits de surface
// - directions principales
{$mode delphiunicode}
interface
uses
Classes, SysUtils, math,
DGCTypes;
type
{ TDGCSectionOfBeam }
TDGCSectionOfBeam = class(TList)
private
FBounds : TDGCBoundingBox;
FPerimetre : double;
FArea : double;
FCentroid : TDGCPoint2D;
FInertiaOx : double;
FInertiaOy : double;
FInertiaOxy: double;
FInertiaPolar: double;
FAnglePrincipalInertie: double;
FMainInertia_I : double;
FMainInertia_II : double;
procedure CalcPerimeter();
procedure CalcArea();
procedure CalcCentroid();
procedure ClearListe();
procedure CalcInertias();
public
function Initialiser(): boolean;
procedure Finaliser();
function GetNbVertex(): integer;
function AddVertex(const V: TDGCPoint2D): boolean; overload;
function AddVertex(const VX, VY: double): boolean; overload;
procedure CalcBounds();
function GetBounds(): TDGCBoundingBox;
function GetVertex(const Idx: integer): TDGCPoint2D;
function RemoveVertex(const Idx: integer): boolean;
procedure CalcCaracteristiques();
property Area: double read FArea;
property Perimeter: double read FPerimetre;
property Centroid: TDGCPoint2D read FCentroid;
property InertiaXGX: double read FInertiaOx;
property InertiaYGY: double read FInertiaOy;
property InertiaXY: double read FInertiaOxy;
property InertiaPolar: double read FInertiaPolar;
property MainAngleInertia: double read FAnglePrincipalInertie;
property MainInertia_I : double read FMainInertia_I;
property MainInertia_II : double read FMainInertia_II;
end;
implementation
uses
DGCDummyUnit;
// Classes utilitaires: Liste des vertex
function TDGCSectionOfBeam.GetNbVertex(): integer;
begin
Result := self.Count;
end;
procedure TDGCSectionOfBeam.ClearListe();
var
i, Nb: Integer;
begin
Nb := self.Count;
if (0 = Nb) then Exit;
for i := 0 to Nb - 1 do
begin
try
Dispose(self.Items[i]);
except
end;
end;
self.Clear();
end;
function TDGCSectionOfBeam.Initialiser(): boolean;
begin
result := false;
try
FArea := 0.00;
FPerimetre := 0.00;
FCentroid.X := 0.00;
FCentroid.Y := 0.00;
FInertiaOx := 0.00;
FInertiaOy := 0.00;
FInertiaOxy := 0.00;
FAnglePrincipalInertie := 0.00;
FMainInertia_I := 0.00;
FMainInertia_II := 0.00;
self.ClearListe();
result := true;
except
end;
end;
procedure TDGCSectionOfBeam.CalcPerimeter();
var
i, j, n: Integer;
V1, V2: TDGCPoint2D;
begin
FPerimetre := 0.00;
n := GetNbVertex();
for i := 0 to n - 1 do
begin
j := Ifthen(n = (i+1), 0, i+1);
V1 := GetVertex(i);
V2 := GetVertex(j);
FPerimetre += hypot(V2.X - V1.X, V2.Y - V1.Y);
end;
end;
procedure TDGCSectionOfBeam.CalcArea();
var
n, i, j: Integer;
V1, V2: TDGCPoint2D;
begin
FArea := 0.00;
n := GetNbVertex();
for i := 0 to n - 1 do
begin
j := Ifthen(n = (i+1), 0, i+1);
V1 := GetVertex(i);
V2 := GetVertex(j);
FArea += V1.X * V2.Y - V2.X * V1.Y;
end;
FArea:= 0.5 * Abs(FArea);
end;
//centre de gravité
procedure TDGCSectionOfBeam.CalcCentroid();
var
Q6A: Double;
n, i, j: Integer;
V1, V2: TDGCPoint2D;
begin
if (IsZero(FArea)) then exit;
Q6A := 1 / (6 * FArea);
FCentroid.X := 0.00;
FCentroid.Y := 0.00;
n := GetNbVertex();
for i := 0 to n - 1 do
begin
j := Ifthen(n = (i+1), 0, i+1);
V1 := GetVertex(i);
V2 := GetVertex(j);
FCentroid.X += (V1.X + V2.X) * (V1.X * V2.Y - V2.X * V1.Y);
FCentroid.Y += (V1.Y + V2.Y) * (V1.X * V2.Y - V2.X * V1.Y);
end;
FCentroid.X := Q6A * FCentroid.X;
FCentroid.Y := Q6A * FCentroid.Y;
end;
// calcul des moments quadratiques
procedure TDGCSectionOfBeam.CalcInertias();
var
V1, V2: TDGCPoint2D;
WU: Double;
n, i, j: Integer;
Q1, Q2 : double;
begin
if (IsZero(FArea)) then exit;
FInertiaOx := 0.00;
FInertiaOy := 0.00;
FInertiaOxy := 0.00;
n := GetNbVertex();
for i := 0 to n - 1 do
begin
j := Ifthen(n = (i+1), 0, i+1);
V1 := GetVertex(i);
V2 := GetVertex(j);
WU := (V1.X * V2.Y - V2.X * V1.Y);
FInertiaOy += WU * (V1.X ** 2 + V1.X * V2.X + V2.X ** 2);
FInertiaOx += WU * (V1.Y ** 2 + V1.Y * V2.Y + V2.Y ** 2);
FInertiaOxy += WU * (V1.X * V2.Y +
2 * V1.X * V1.Y +
2 * V2.X * V2.Y +
V2.X * V1.Y);
end;
// moments IxOx et IyOy par rapport au repère général
FInertiaOx := (FInertiaOx / 12); //- FArea * sqr(FCentroid.Y);
FInertiaOy := (FInertiaOy / 12); //- FArea * sqr(FCentroid.X);
// produit de surface
FInertiaOxy := (FInertiaOxy / 24);
// transport des moments vers le centroide
// IxOx = IxGx + S.Yg ** 2, avec S: aire de la section et Yg ordonnée du CDG
// IxGx = IxOx + S.Xg ** 2, avec S: aire de la section et Xg abscisse du CDG
// IxGy = IxOy + S.Xg.Yg
FInertiaOx := FInertiaOx - FArea * FCentroid.Y ** 2;
FInertiaOy := FInertiaOy - FArea * FCentroid.X ** 2;
FInertiaOxy := FInertiaOxy - FArea * (FCentroid.X * FCentroid.Y);
// moment quadratique polaire: Io = IxGx + IyGy
FInertiaPolar := FInertiaOx + FInertiaOy;
// Moments d'inertie principaux, axes principaux
// cf cours https://campus.mines-douai.fr/pluginfile.php/9832/mod_resource/content/0/Chapitre4_webLatex/co/C3_MomentsPrincipaux.html
WU := FInertiaOy - FInertiaOx;
FAnglePrincipalInertie := ifthen (IsZero(WU), 0.00, 0.50 * ArcTan(2 * FInertiaOxy / WU));
Q1 := 0.50 * (FInertiaOx + FInertiaOy);
Q2 := sqrt((0.50 * (FInertiaOx - FInertiaOy)) ** 2 + FInertiaOxy ** 2);
FMainInertia_I := Q1 - Q2;
FMainInertia_II := Q1 + Q2;
end;
procedure TDGCSectionOfBeam.CalcCaracteristiques();
begin
CalcBounds();
CalcPerimeter();
CalcArea();
CalcCentroid();
CalcInertias();
end;
procedure TDGCSectionOfBeam.Finaliser();
begin
self.ClearListe();
end;
function TDGCSectionOfBeam.AddVertex(const V: TDGCPoint2D): boolean; overload;
var
pS : ^TDGCPoint2D;
begin
result := false;
try
New(pS);
pS^ := V;
self.Add(pS);
Result := True;
except
end;
end;
function TDGCSectionOfBeam.AddVertex(const VX, VY: double): boolean; overload;
var
EWE: TDGCPoint2D;
begin
EWE.X := VX; EWE.Y := VY;
Result := AddVertex(EWE);
end;
procedure TDGCSectionOfBeam.CalcBounds();
const INFINITEE = 1E30;
var
i, Nb: Integer;
EWE: TDGCPoint2D;
begin
FBounds.X1 := INFINITEE;
FBounds.Y1 := INFINITEE;
FBounds.X2 := -INFINITEE;
FBounds.Y2 := -INFINITEE;
Nb := GetNbVertex();
if (0 = Nb) then exit;
for i := 0 to Nb - 1 do
begin
EWE := getVertex(i);
if (EWE.X < FBounds.X1) then FBounds.X1 := EWE.X;
if (EWE.Y < FBounds.Y1) then FBounds.Y1 := EWE.Y;
if (EWE.X > FBounds.X2) then FBounds.X2 := EWE.X;
if (EWE.Y > FBounds.Y2) then FBounds.Y2 := EWE.Y;
end;
end;
function TDGCSectionOfBeam.GetBounds(): TDGCBoundingBox;
begin
result := FBounds;
end;
function TDGCSectionOfBeam.GetVertex(const Idx: integer): TDGCPoint2D;
var
pS : ^TDGCPoint2D;
begin
pS := self.Items[Idx];
Result := pS^;
end;
function TDGCSectionOfBeam.RemoveVertex(const Idx: integer): boolean;
begin
Result := False;
try
Dispose(self.Items[Idx]);
finally
self.Delete(Idx);
Result := True;
end;
end;
end. |
TDGCSectionOfBeam : une classe pour le calcul des caractéristiques d'une section droite de poutre
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