Figure 1: \(5a \times 5a\)

Figure 2: \(6a \times 6a\)

Center of gravity

\[ A_1 := 6a \cdot 6a = 36a^2 \] \[ A_2 := 5a \cdot 5a = 25a^2 \] \[ z_1 := 3a \qquad z_2 := 3{,}5a \] \[ z_C := \frac{z_1 \cdot A_1 - z_2 \cdot A_2}{A_1 - A_2} = 1{,}86a \] \[ y_C := z_C = 1{,}86a \]

Moments of inertia

\[ I_{yy} = \frac{(6a)^4}{12} - \frac{(5a)^4}{12} = 55{,}92\,a^4 \]

\[ I_{zg} = \frac{(6a)^4}{12} + A_1 \cdot \left( \left( \frac{1}{2} \cdot 6a - 1{,}86a \right) \cdot \sqrt{2} \right)^2 - \frac{(5a)^4}{12} - A_2 \cdot \left( \left( (6a - 1{,}86a - 2{,}5a) \cdot \sqrt{2} \right)^2 \right) = 15{,}01\,a^4 \]

\[ M_{yg} = M_y \cdot \frac{\sqrt{2}}{2} \qquad M_{zg} = -M_y \cdot \frac{\sqrt{2}}{2} \]

Normal stresses

\[ \sigma = \frac{M_{yg}}{I_{yg}} \cdot z_g - \frac{M_{zg}}{I_{zg}} \cdot y_g \]

Point K

\[ z_g = \frac{1}{2} \cdot 6a \cdot \sqrt{2} = 3 \cdot \sqrt{2} \cdot a \] \[ y_g = \frac{1}{2} \cdot 6a \cdot \sqrt{2} - 1{,}86a \cdot \sqrt{2} = 1{,}612 \cdot a \]

\[ M_y = 24 \cdot 10^3 \,\text{Nm} \]

\[ \sigma_K = \frac{M_y \cdot \frac{\sqrt{2}}{2}}{55{,}92 \cdot a^4} \cdot (3 \cdot \sqrt{2} \cdot a) - \frac{(-M_y \cdot \frac{\sqrt{2}}{2})}{15{,}01 \cdot a^4} \cdot (1{,}612 \cdot a) = \frac{3110{,}108}{a^3} \]

Strength condition

\[ \frac{3110{,}108}{a^3} \leq 200 \cdot 10^6 \quad \Rightarrow \quad a \geq \sqrt[3]{\frac{3110{,}108}{200 \cdot 10^6}} = 0{,}02496\,\text{m} \]

\[ a \geq 24{,}96\,\text{mm} \quad \Rightarrow \quad \text{przyjmuję: } a = 25\,\text{mm} \]

Bending

Strength of materials - Bending - Example 9

Example 9