Design model #122595

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Section characteristic solution

Calculate own characteristic each component of section


Component 1 - Round section, diameter 8 cm:
Area

A=π*d2 / 4=π*82 / 4=50.2 cm2

Moments of inertia

Ix=Iy=π*d4 / 64=π*84 / 64=201 cm4


Component 2 - I-section 20 :
Section width 10 cm
Section height 20 cm
Flange thickness 0.84 cm
Web thickness 0.52 cm
Area A=26.8 cm2
Moments of inertia Ix=1840 cm4, Iy=115 cm4

Component 3 - C-section 20 :
Section width 7.6 cm
Section height 20 cm
Flange thickness 0.9 cm
Web thickness 0.52 cm
Area A=23.4 cm2
Moments of inertia Ix=1520 cm4, Iy=113 cm4
Centroid location xc=2.07 cm

Common section area

A = + 50.2 + 26.8 + 23.4 = 100.4 cm2

Draw initial axes and define centroidal axes location.

Draw each shapes centroid and define their coordinates.

a1 = 0 cm; b1 = 0 cm

a2 = 8.825 cm; b2 = 10.09 cm

a3 = -10 cm; b3 = 6.07 cm

Centroid location

Xc = ΣXi*Ai / A

Yc = ΣYi*Ai / A

XC = (+X1*A1+X2*A2+X3*A3) / A = ( + 0*50.2 + 8.825*26.8 - 10*23.4) / 100.4 = 0.025 cm

YC = (+Y1*A1+Y2*A2+Y3*A3) / A = ( + 0*50.2 + 10.09*26.8 + 6.07*23.4) / 100.4 = 4.11 cm

Centroid location of each component in centroidal axes

a1 = 0-0.025 = -0.025 cm

b1 = 0-4.11 = -4.11 cm

a2 = 8.825-0.025 = 8.8 cm

b2 = 10.09-4.11 = 5.98 cm

a3 = -10-0.025 = -10 cm

b3 = 6.07-4.11 = 1.96 cm

Centroidal moments of inertia

Ix = Σ(IXown. + b2 *A) = +(201+4.112*50.2)+(1840+5.982*26.8)+(113+1.962*23.4) = 4050 cm4

Iy = Σ(IYown. + a2 *A) = +(201+0.0252*50.2)+(116+8.82*26.8)+(1520+102*23.4) = 6252 cm4

Ixy = Σ(IXYown. + a*b*A) = +(0+(-4.11)*(-0.025)*50.2)+(-30+5.98*8.8*26.8)+(0+1.96*(-10)*23.4) = 927 cm4

Angle of rotation of principal axes

tg2α=2*Ixy / (Iy-Ix)=2*927 / (6252-4050)=0.84

α = arctg(0.84) / 2 = 20.1°

Positive angle of rotation draw counterclockwise

Principal moments of inertia - moments of inertia on principal axes.

IX0 = Ix*cos2(α) + Iy*sin2(α) - Ixy*sin(2*α) =

= 4050*cos2(20.1°) + 6252*sin2(20.1°) - 2*927*sin(2*20.1°) = 3712 cm4

IY0 = Iy*cos2(α) + Ix*sin2(α) + Ixy*sin(2*α) =

= 6252*cos2(20.1°) + 4050*sin2(20.1°) + 2*927*sin(2*20.1°) = 6590 cm4

Radiuses of inertia

ix2 = IX0 / A = 3712 / 100.4 = 36.97

ix = 6.08 cm

iy2 = IY0 / A = 6590 / 100.4 = 65.64

iy = 8.1 cm




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