Questa pagina raccoglie gli esercizi svolti del corso di Teoria e progetto di costruzioni e strutture.
a = 60 mm b = 180 mm c = 360 mm
fy = ± 80 MPa γ = 2
A = ( a ⋅ b ) ⋅ 2 + [ ( h − 2 a ) ⋅ a ] = 60 ⋅ 180 ⋅ 2 + 240 ⋅ 60 = 36 000 m m 2 {\displaystyle A = (a \cdot b) \cdot 2 + [(h - 2a) \cdot a] = 60 \cdot 180 \cdot 2 + 240 \cdot 60 = 36\,000\,mm^2 }
I x x = [ 60 ⋅ 180 3 12 + 60 ⋅ 180 ⋅ ( 90 − 30 ) 2 ] ⋅ 2 + 240 ⋅ 60 3 12 = 140 400 000 m m 4 {\displaystyle I_{xx} = \left[ \frac{60 \cdot 180^3}{12} + 60 \cdot 180 \cdot (90 - 30)^2 \right] \cdot 2 + \frac{240 \cdot 60^3}{12} = 140\,400\,000\,mm^4 }
I y y = [ 180 ⋅ 60 3 12 + 60 ⋅ 180 ⋅ ( 120 + 30 ) 2 ] ⋅ 2 + 60 ⋅ 240 3 12 = 561 600 000 m m 4 {\displaystyle I_{yy} = \left[ \frac{180 \cdot 60^3}{12} + 60 \cdot 180 \cdot (120 + 30)^2 \right] \cdot 2 + \frac{60 \cdot 240^3}{12} = 561\,600\,000\,mm^4 }
I x y = 60 ⋅ 180 ⋅ [ − ( 180 2 − 60 2 ) ] ⋅ [ − ( 240 2 + 60 2 ) ] + 60 ⋅ 180 ⋅ ( 180 2 − 60 2 ) ⋅ ( 240 2 + 60 2 ) = 194 400 000 m m 4 {\displaystyle I_{xy} = 60 \cdot 180 \cdot \left[ - \left( \frac{180}{2} - \frac{60}{2} \right) \right] \cdot \left[ - \left( \frac{240}{2} + \frac{60}{2} \right) \right] + 60 \cdot 180 \cdot \left( \frac{180}{2} - \frac{60}{2} \right) \cdot \left( \frac{240}{2} + \frac{60}{2} \right) = 194\,400\,000\,mm^4 }
I ξ ξ = I x x + I y y 2 + 1 2 ( I x x − I y y ) 2 + 4 ⋅ I x y 2 = = 140 400 000 + 561 600 000 2 + 1 2 ( 140 400 000 − 561 600 000 ) 2 + 4 ⋅ 194 400 000 2 = 637 607 257 m m 4 {\displaystyle \begin{align} I_{\xi \xi} & = \frac{I_{xx} + I_{yy}}{2} + \frac{1}{2} \sqrt{(I_{xx} - I_{yy})^2 + 4 \cdot I_{xy}^2} = \\ & = \frac{140\,400\,000 + 561\,600\,000}{2} + \frac{1}{2} \sqrt{(140\,400\,000 - 561\,600\,000)^2 + 4 \cdot 194\,400\,000^2} = 637\,607\,257\,mm^4 \end{align} }
I η η = I x x + I y y 2 − 1 2 ( I x x − I y y ) 2 + 4 ⋅ I x y 2 = . . . = 64 392 742 m m 4 {\displaystyle I_{\eta \eta} = \frac{I_{xx} + I_{yy}}{2} - \frac{1}{2} \sqrt{(I_{xx} - I_{yy})^2 + 4 \cdot I_{xy}^2} = ... = 64\,392\,742\,mm^4 }
α = 1 2 arctan ( − 2 ⋅ I x y I x x − I y y ) = 21 , 35 ∘ {\displaystyle \alpha = \frac{1}{2} \arctan \left( - \frac{2 \cdot I_{xy} }{I_{xx} - I_{yy} } \right) = 21,35^\circ }
α < 45 ∘ {\displaystyle \alpha < 45^\circ } e I x x < I y y ⟶ I ξ ξ = I v v {\displaystyle I_{xx} < I_{yy} \longrightarrow I_{\xi \xi} = I_{vv} }
A H ¯ = d i s t a n z a d a A a G {\displaystyle \overline{AH} = distanza\ da\ A\ a\ G }
B H ¯ = d i s t a n z a d a B a G {\displaystyle \overline{BH} = distanza\ da\ B\ a\ G }
B H ¯ = 360 2 ⋅ c o s 31 , 35 ∘ + ( 180 − 60 2 ) ⋅ s e n 31 , 35 ∘ = 222 , 26 m m {\displaystyle \overline{BH} = \frac{360}{2} \cdot cos \ 31{,}35^\circ + \left( 180 - \frac{60}{2} \right) \cdot sen \ 31{,}35^\circ = 222{,}26\,mm }
A H ¯ = − B H ¯ = − 222 , 26 m m {\displaystyle \overline{AH} = - \overline{BH} = - 222{,}26\,mm }
σ a d m = M I ξ ξ ⋅ B H ¯ {\displaystyle \sigma_{adm} = \frac{M}{I_{\xi \xi}} \cdot \overline{BH} }
si applica la formula inversa:
M m a x = σ a d m ⋅ I ξ ξ B H ¯ = 40 ⋅ 637 607 257 222 , 26 = 114 749 798 N ⋅ m m 2 {\displaystyle M_{max} = \frac{\sigma_{adm} \cdot I_{\xi \xi}}{\overline{BH}} = \frac{40 \cdot 637\,607\,257}{222{,}26} = 114\,749\,798\, N \cdot mm^2 }
σ B = M ⋅ B H ¯ I ξ ξ = M ⋅ 222 , 26 637 607 257 = 3 , 48 ⋅ 10 − 7 ⋅ M {\displaystyle \sigma_{B} = \frac{M \cdot \overline{BH}}{I_{\xi \xi}} = \frac{M \cdot 222{,}26}{637\,607\,257} = 3{,}48 \cdot 10^{-7} \cdot M }
σ A = M ⋅ A H ¯ I ξ ξ = M ⋅ ( − 222 , 26 ) 637 607 257 = − 3 , 48 ⋅ 10 − 7 ⋅ M {\displaystyle \sigma_{A} = \frac{M \cdot \overline{AH}}{I_{\xi \xi}} = \frac{M \cdot ( - 222{,}26)}{637\,607\,257} = - 3{,}48 \cdot 10^{-7} \cdot M }
σ m a x = M m a x ⋅ B H ¯ I ξ ξ = 40 M P a {\displaystyle \sigma_{max} = \frac{M_{max} \cdot \overline{BH}}{I_{\xi \xi}} = 40\,MPa }
![{\displaystyle A=(a\cdot b)\cdot 2+[(h-2a)\cdot a]=60\cdot 180\cdot 2+240\cdot 60=36\,000\,mm^{2}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/21537a7954dfb6ec5a8729f072b4d1e6dd5517f0)
![{\displaystyle I_{xx}=\left[{\frac {60\cdot 180^{3}}{12}}+60\cdot 180\cdot (90-30)^{2}\right]\cdot 2+{\frac {240\cdot 60^{3}}{12}}=140\,400\,000\,mm^{4}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/a3964640efb99fb87cc0e09e67be13e5bf60834f)
![{\displaystyle I_{yy}=\left[{\frac {180\cdot 60^{3}}{12}}+60\cdot 180\cdot (120+30)^{2}\right]\cdot 2+{\frac {60\cdot 240^{3}}{12}}=561\,600\,000\,mm^{4}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/3470445c035e430def8bbfa6ddc84b09a3f5e018)
![{\displaystyle I_{xy}=60\cdot 180\cdot \left[-\left({\frac {180}{2}}-{\frac {60}{2}}\right)\right]\cdot \left[-\left({\frac {240}{2}}+{\frac {60}{2}}\right)\right]+60\cdot 180\cdot \left({\frac {180}{2}}-{\frac {60}{2}}\right)\cdot \left({\frac {240}{2}}+{\frac {60}{2}}\right)=194\,400\,000\,mm^{4}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/031337ae9707b60b1ef64b0e8cd0eeed141ecab3)
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