help_ formulas of matrix elements (material orthotropy)
TRANSCRIPT
Designations applied:
E - Young's modulus
v - Poisson's ratio
G - shear modulus
Single-sided unidirectional ribs
h - plate thickness
ha - rib height
a - rib spacing
a1 - rib width
Membrane stiffness values:
adopted as: s1 = a*h, s2 = (ha-h)*a1
D_XXXX = E*(s1+s2)/a
D_YYYY = E*h
D_XXYY = v*E*h
D_XYXY = G*h
D_XXXY = 0
D_YYXY = 0
Bending stiffness values:
adopted as: hz = h + a1/12, xv = 1 - v*v
Dyyyy= E/(12*xv)*a/((a-a1)/(h*h*h)+a1/(hz*hz*hz))
I = moment of inertia of a T-section with flange width a/xv, flangethickness h, total height ha and web thickness a1
K_XXXX = E*I/a
K_YYYY = Dyyyy
K_XXYY = v*Dyyyy
K_XYXY = (1-v)/2*Dyyyy
K_XXXY = 0
K_YYXY = 0
Shear stiffness values:
H_XX = 5/6*G*((a-a1)*h + a1*ha)/a
H_YY = 5/6*G*h
H_XY = 0
Double-sided unidirectional ribs
h - plate thickness
ha - rib height
a - rib spacing
a1 - rib width
Membrane stiffness:
adopted as: s1 = a*h, s2 = (ha-h)*a1
D_XXXX = E*(s1+s2)/a
D_YYYY = E*h
D_XXYY = v*E*h
D_XYXY = G*h
D_XXXY = 0
D_YYXY = 0
Bending stiffness:
adopted as: hz = h + a1/6, xv = 1 - v*v
Dyyyy= E/(12*xv)*a/((a-a1)/(h*h*h)+a1/(hz*hz*hz))
I = moment of inertia of a cross-shaped section with flange width a/xv,flange thickness h, total height ha and web thickness a1
K_XXXX = E*I/a;
K_YYYY = Dyyyy;
K_XXYY = v*Dyyyy;
K_XYXY = (1-v)/2*Dyyyy
K_XXXY = 0;
K_YYXY = 0;
Shear stiffness values:
H_XX = 5/6*G*((a-a1)*h + a1*ha)/a
H_YY = 5/6*G*h
H_XY = 0
Single-sided bi-directional ribs
Formulas of matrix elements (material orthotropy)
SHARELIKE (0)
h - plate thickness
ha - rib height
hb - rib height
a - rib spacing
a1 - rib width
b - rib spacing
b1 - rib width.
Membrane stiffness values:
adopted as:
s1x = a*h, s2x = (h-ha)*a1
s1y = b*h, s2y = (h-hb)*b1
D_XXXX = E*(s1x+s2x)/a
D_YYYY = E*(s1y+s2y)/b
D_XXYY = v*E
D_XYXY = G*h
D_XXXY = 0
D_YYXY = 0
Bending stiffness values:
if ha>=hb, then
a2 = a1, a1 = a-a1, h2 = ha, b2 = b1, b1 = b-b2, h1 = hb
if otherwise (hb>ha), then
a2 = b1, a1 = b-b1, h2 = hb, b2 = a1, b1 = a-b2, h1 = ha
I11 = InertiaTsec(b/xv, b2, hp, h1)
I12 = InertiaTsec(b/xv, b2+a2/6, hp+b2/12, hp+a/12)
I21 = InertiaTsec(a/xv, a2, hp, h2)
I22 = InertiaTsec(a/xv, (a2+a2+b2/6)/2, hp+b2/12, h2)
xv = 1 - v*v
GC1 = G*TorsI(b2,h1)
GC2 = G*TorsI(a2,h2)
D11 = E*a/b/(a1/I11+a2/I12)
D22 = E*b/a/(b1/I21+b2/I22)
K_XXXX = (ha>=hb) D22:D11
K_YYYY = (ha>=hb) D11:D22
K_XXYY = v*E/12.0*(h3/xv+ h1*h1*h1 *a2*b2/(a*b))
K_XYXY = G*h3/12.0 + (GC1/b+GC2/a)/4.0
K_XXXY = 0.0
K_YYXY = 0.0
xv = 1 - v*v
Shear stiffness values:
H_XX = 5/6*G*((a-a1)*h + a1*ha)/a;
H_YY = 5/6*G*h;
H_XY = 0;
Unidirectional box floor
h - floor thickness
h1 - lower plate thickness
h2 - upper plate thickness
a - rib spacing
a1 - rib width.
Membrane stiffness values:
adopted as:
s1 = a*h1, s2 = a*h2, s0 = (h-h1-h2)*a1
D_XXXX = E*(s1+s2+s0)/a
D_YYYY = E*(h1+h2)
D_XXYY = v*E*(h1+h2)
D_XYXY = G*(h1+h2)
D_XXXY = 0
D_YYXY = 0
Bending stiffness values:
adopted as:
hz1 = h1 + a1/12, hz2 = h2 + a1/12, xv = 1 - v*v
Dyyyy= a/( (a-a1)/InertiaIsec(1/xv,0,h,h1,h2) +a1/InertiaIsec(1/xv,0,h,hz1,hz2)
Iy = InertiaIsec(a/xv,a1,h,h1,h2)
K_XXXX = E*Iy/a
K_YYYY = Dyyyy
K_XXYY = v*Dyyyy
K_XYXY = (1-v)/2*Dyyyy
K_XXXY = 0
K_YYXY = 0
Shear stiffness values:
H_XX = 5/6*G*((a-a1)*(h1+h2) + a1*ha)/a
H_YY = 5/6*G*(h1+h2)
H_XY = 0
Bi-directional box floor
h - floor thickness
h1 - lower plate thickness
h2 - upper plate thickness
a - rib spacing
a1 - rib width
b - rib spacing
b1 - rib width
Membrane stiffness values:
adopted as:
s1 = a*h1, s2 = a*h2, s0x = (h-h1-h2)*a1, s0y = (h-h1-h2)*b1
D_XXXX = E*(s1+s2+s0x)/a
D_YYYY = E*(s1+s2+s0y)/b
D_XXYY = v*E*(h1+h2)
D_XYXY = G*(h1+h2)
D_XXXY = 0.0
D_YYXY = 0.0
Bending stiffness values:
adopted as:
a2 = a1, a1 = a-a2, double b2 = b1, b1 = b-b2, xv = 1 - v*v
GC1 = G*TorsI(a2,h-h1-h2)
GC2 = G*TorsI(b2,h-h1-h2)
Dpl = InertiaIsec(1/xv,0,h,h1,h2)
I11 = InertiaIsec(b/xv, b2, h, h1, h2)
I12 = InertiaIsec(b/xv, b2+a2/6.0, h, h1+a2/12.0, h2+a2/12.0)
I21 = InertiaIsec(a/xv, a2, h, h1, h2)
I22 = InertiaIsec(a/xv, a2+b2/6.0, h, h1+b2/12.0, h2+b2/12.0)
D11 = E*a/b/(a1/I11+a2/I12)
D22 = E*b/a/(b1/I21+b2/I22)
K_XXXX = D22
K_YYYY = D11
K_XXYY = v*E*(Dpl+h3/12.0*a2*b2/(a*b))
K_XYXY = (1.0-v)/2.0*E*Dpl+(GC1/b+GC2/a)/4.0
K_XXXY = 0.0
K_YYXY = 0.0
Shear stiffness values:
H_XX = 5.0/6.0*G*((a-a1)*(h1+h2) + a1*h)/a
H_YY = 5.0/6.0*G*((b-b1)*(h1+h2) + b1*h)/b
H_XY = 0.0
Grillage
h - rib height
a - rib spacing
a1 - rib width
b - rib spacing
b1 - rib width
Membrane stiffness values:
D_XXXX = E*h*a1/a
D_YYYY = E*h*b1/b
D_XXYY = 0.0
D_XYXY = G*h*(1.-(a-a1)*(b-b1)/(a*b))
D_XXXY = 0.0
D_YYXY = 0.0
Bending stiffness values:
a2 = a1; a1 = a-a2; b2 = b1; b1 = b-b2
D=E*h3/12, a23=a2*a2*a2, b23=b2*b2*b2
K_XXXX = D * a2/a
K_YYYY = D * b2/b
K_XXYY = v*D * (a2*b2)/(a*b)
K_XYXY = D*(1-v*v*a2*b2/(a*b))/(a*b*(a/a23+b/b23))
K_XXXY = 0.0
K_YYXY = 0.0
Shear stiffness values:
H_XX = 5.0/6.0*G*h*a1/a
H_YY = 5.0/6.0*G*h*b1/b
H_XY = 0.0
Slab on trapezoidal plate
h - plate thickness
h1 - steel plate height
a - steel plate rib spacing
a1 - bottom rib width
a2 - top rib width