パイプの断面二次モーメント

memo.md

Wolfram Language 14.1.0 Engine for Microsoft Windows (64-bit) での実行結果

Integrate[y^2 * 2 * Sqrt[r^2-y^2],{y,-r,r}, Assumptions -> r>0] // TeXForm

$$ \frac{\pi r^4}{4} $$

Integrate[y^2 * 2 * Sqrt[r^2-y^2],{y,0,r}, Assumptions -> r>0] // TeXForm

$$ \frac{\pi r^4}{8} $$

Integrate[y^2 * 2 * (Sqrt[r^2-y^2] - Sqrt[(r-t)^2-y^2]),{y,0,r-t}, Assumptions -> r>0 && t>0 && t<r] // TeXForm

$$ \frac{1}{4} \left(r^4 \tan ^{-1}\left(\frac{r-t}{\sqrt{t (2 r-t)}}\right)+(r-t) \sqrt{t (2 r-t)} \left(r^2-4 r t+2 t^2\right)\right)-\frac{1}{8} \pi (r-t)^4 $$

FullSimplify[Integrate[y^2 * 2 * Sqrt[r^2-y^2],{y,r-t,r}, Assumptions -> r>0 && t>0 && t<r]] // TeXForm

$$ \frac{1}{8} \left(-2 r^4 \tan ^{-1}\left(\frac{r-t}{\sqrt{t (2 r-t)}}\right)+\pi r^4-2 (r-t) \sqrt{t (2 r-t)} \left(r^2-4 r t+2 t^2\right)\right) $$

 FullSimplify[Integrate[(y+h)^2 * 2 * (Sqrt[r^2-y^2] - Sqrt[(r-t)^2-y^2]),{y,0,r-t}, Assumptions -> h>0 && r>0 && t>0 && t<r]] // TeXForm

$$ \frac{1}{24} \left(6 r^2 \left(4 h^2+r^2\right) \tan ^{-1}\left(\frac{r-t}{\sqrt{t (2 r-t)}}\right)+12 h^2 (r-t) \left(\pi (t-r)+2 \sqrt{t (2 r-t)}\right)+32 h t \left(3 r^2-3 r t-2 r \sqrt{t (2 r-t)}+t \left(\sqrt{-t (t-2 r)}+t\right)\right)+3 (r-t) \left(2 \sqrt{t (2 r-t)} \left(r^2-4 r t+2 t^2\right)-\pi (r-t)^3\right)\right) $$

FullSimplify[Integrate[(y+h)^2 * 2 * Sqrt[r^2-y^2],{y,r-t,r}, Assumptions -> h>0 && r>0 && t>0 && t<r]] // TeXForm

$$ \frac{1}{24} \left(12 h^2 \left(\pi r^2+2 \sqrt{t (2 r-t)} (t-r)\right)-6 r^2 \left(4 h^2+r^2\right) \tan ^{-1}\left(\frac{r-t}{\sqrt{t (2 r-t)}}\right)+32 h (t (2 r-t))^{3/2}+3 \pi r^4+12 \sqrt{t^7 (2 r-t)}-6 r \left(6 \sqrt{t^5 (2 r-t)}+r \sqrt{t (2 r-t)} (r-5 t)\right)\right) $$

JIS G 3466 で規定される角形鋼管の断面二次モーメント

SquarePipeMomentOfInertia[A_,B_,r_,t_]:= 4 * Integrate[(y+(A/2-r))^2 * (Sqrt[r^2-y^2] - Sqrt[(r-t)^2-y^2]) , {y,0,r-t} , Assumptions -> A>0 && h>0 && r>0 && t>0 && t<r] 
                                       + 4 * Integrate[(y+(A/2-r))^2 *  Sqrt[r^2-y^2]                      , {y,r-t,r} , Assumptions -> A>0 && h>0 && r>0 && t>0 && t<r]
                                       + ( 2*t * (A-2*r)^3 )/12 + ( (B-2*r) * A^3 )/12 - ( (B-2*r) * (A-2*t)^3 )/12

SquarePipeMomentOfInertia[A,B,r,t] // TeXForm

$$ \frac{1}{12} A^3 (B-2 r)+\frac{1}{12} \left(6 r^2 \left(A^2-4 A r+5 r^2\right) \tan ^{-1}\left(\frac{r-t}{\sqrt{t (2 r-t)}}\right)+3 A^2 (r-t) \left(\pi (t-r)+2 \sqrt{t (2 r-t)}\right)+4 A \left(-6 r^2 \left(\sqrt{-t (t-2 r)}-2 t\right)+4 \left(\sqrt{-t^5 (t-2 r)}+t^3\right)+3 \pi r (r-t)^2-2 r t \left(\sqrt{-t (t-2 r)}+6 t\right)\right)-3 \pi (r-t)^2 \left(5 r^2-2 r t+t^2\right)+2 r \left(3 r^2 \left(5 \sqrt{t (2 r-t)}-16 t\right)+2 \left(\sqrt{t^5 (2 r-t)}-8 t^3\right)+r t \left(5 \sqrt{t (2 r-t)}+48 t\right)\right)-12 \sqrt{t^7 (2 r-t)}\right)+\frac{1}{12} \left(3 A^2 \left(\pi r^2+2 \sqrt{t (2 r-t)} (t-r)\right)-6 r^2 \left(A^2-4 A r+5 r^2\right) \tan ^{-1}\left(\frac{r-t}{\sqrt{t (2 r-t)}}\right)-4 A \left(3 \pi r^3+4 \sqrt{t^5 (2 r-t)}-2 r \sqrt{t (2 r-t)} (3 r+t)\right)+15 \pi r^4+12 \sqrt{t^7 (2 r-t)}-2 r \left(2 \sqrt{t^5 (2 r-t)}+5 r \sqrt{t (2 r-t)} (3 r+t)\right)\right)-\frac{1}{12} (A-2 t)^3 (B-2 r)+\frac{1}{6} t (A-2 r)^3 $$

Simplify[SquarePipeMomentOfInertia[A,B,r,t]] // TeXForm

$$ -\frac{1}{12} t \left(-2 A^3+A^2 (-6 B-6 (\pi -4) r+3 \pi t)+4 A \left(t (3 B-4 t)+6 (\pi -3) r^2-3 (\pi -2) r t\right)+t^2 (3 \pi t-8 B)-4 (9 \pi -28) r^3+6 (5 \pi -16) r^2 t-12 (\pi -4) r t^2\right) $$

FullSimplify[SquarePipeMomentOfInertia[A,B,r,t]] // TeXForm

$$ \frac{1}{12} t \left(2 A^3-3 t \left(\pi A^2+4 A (B-(\pi -2) r)+2 (5 \pi -16) r^2\right)+6 A^2 (B+(\pi -4) r)+4 t^2 (4 A+2 B+3 (\pi -4) r)-24 (\pi -3) A r^2+4 (9 \pi -28) r^3-3 \pi t^3\right) $$