Oznaczamy przemieszczenia i układy lokalne na prętach

Macierze sztywności dla poszczególnych elementów oraz macierz transformacji dla elementu 1:

\begin{aligned} L &= 5 \quad &x_k &= 3 \quad &x_p &= 0 \quad &y_k &= 0 \quad &y_p &= 4 \\ c &= \frac{x_k - x_p}{L} = 0.6 \\ s &= \frac{y_k - y_p}{L} = -0.8 \end{aligned} \begin{aligned} K_1 &= \begin{bmatrix} \frac{EA}{L} & 0 & 0 & -\frac{EA}{L} & 0 & 0 \\ 0 & \frac{12 EI}{L^3} & \frac{6 EI}{L^2} & 0 & -\frac{12 EI}{L^3} & \frac{6 EI}{L^2} \\ 0 & \frac{6 EI}{L^2} & \frac{4 EI}{L} & 0 & -\frac{6 EI}{L^2} & \frac{2 EI}{L} \\ -\frac{EA}{L} & 0 & 0 & \frac{EA}{L} & 0 & 0 \\ 0 & -\frac{12 EI}{L^3} & -\frac{6 EI}{L^2} & 0 & \frac{12 EI}{L^3} & -\frac{6 EI}{L^2} \\ 0 & \frac{6 EI}{L^2} & \frac{2 EI}{L} & 0 & -\frac{6 EI}{L^2} & \frac{4 EI}{L} \end{bmatrix} = \begin{bmatrix} 2000 & 0 & 0 & -2000 & 0 & 0 \\ 0 & 48 & 120 & 0 & -48 & 120 \\ 0 & 120 & 400 & 0 & -120 & 200 \\ -2000 & 0 & 0 & 2000 & 0 & 0 \\ 0 & -48 & -120 & 0 & 48 & -120 \\ 0 & 120 & 200 & 0 & -120 & 400 \end{bmatrix} \end{aligned} \begin{aligned} T_1 &= \begin{bmatrix} c & s & 0 & 0 & 0 & 0 \\ -s & c & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & c & s & 0 \\ 0 & 0 & 0 & -s & c & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 0.6 & -0.8 & 0 & 0 & 0 & 0 \\ 0.8 & 0.6 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.6 & -0.8 & 0 \\ 0 & 0 & 0 & 0.8 & 0.6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \end{aligned} \begin{aligned} K_1 &= T_1^T K_1 T_1 = \begin{bmatrix} 750.72 & -936.96 & 96 & -750.72 & 936.96 & 96 \\ -936.96 & 1297.28 & 72 & 936.96 & -1297.28 & 72 \\ 96 & 72 & 400 & -96 & -72 & 200 \\ -750.72 & 936.96 & -96 & 750.72 & -936.96 & -96 \\ 936.96 & -1297.28 & -72 & -936.96 & 1297.28 & -72 \\ 96 & 72 & 200 & -96 & -72 & 400 \end{bmatrix} \end{aligned} \begin{aligned} L &= 4 \\ K_2 &= \begin{bmatrix} \frac{EA}{L} & 0 & 0 & -\frac{EA}{L} & 0 & 0 \\ 0 & \frac{12 EI}{L^3} & \frac{6 EI}{L^2} & 0 & -\frac{12 EI}{L^3} & \frac{6 EI}{L^2} \\ 0 & \frac{6 EI}{L^2} & \frac{4 EI}{L} & 0 & -\frac{6 EI}{L^2} & \frac{2 EI}{L} \\ -\frac{EA}{L} & 0 & 0 & \frac{EA}{L} & 0 & 0 \\ 0 & -\frac{12 EI}{L^3} & -\frac{6 EI}{L^2} & 0 & \frac{12 EI}{L^3} & -\frac{6 EI}{L^2} \\ 0 & \frac{6 EI}{L^2} & \frac{2 EI}{L} & 0 & -\frac{6 EI}{L^2} & \frac{4 EI}{L} \end{bmatrix} = \begin{bmatrix} 2500 & 0 & 0 & -2500 & 0 & 0 \\ 0 & 93.75 & 187.5 & 0 & -93.75 & 187.5 \\ 0 & 187.5 & 500 & 0 & -187.5 & 250 \\ -2500 & 0 & 0 & 2500 & 0 & 0 \\ 0 & -93.75 & -187.5 & 0 & 93.75 & -187.5 \\ 0 & 187.5 & 250 & 0 & -187.5 & 500 \end{bmatrix} \end{aligned} \begin{aligned} K &= \begin{bmatrix} 750.72 & -936.96 & 96 & -750.72 & 936.96 & 96 & 0 & 0 & 0 & 0 \\ -936.96 & 1297.28 & 72 & 936.96 & -1297.28 & 72 & 0 & 0 & 0 & 0 \\ 96 & 72 & 400 & -96 & -72 & 200 & 0 & 0 & 0 & 0 \\ -750.72 & 936.96 & -96 & 3250.72 & -936.96 & -96 & -2500 & 0 & 0 & 0 \\ 936.96 & -1297.28 & -72 & -936.96 & 1391.03 & 115.5 & 0 & -93.75 & 187.5 & 0 \\ 96 & 72 & 200 & -96 & 115.5 & 900 & 0 & -187.5 & 250 & 0 \\ 0 & 0 & 0 & -2500 & 0 & 0 & 2500 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -93.75 & -187.5 & 0 & 93.75 & -187.5 & 0 \\ 0 & 0 & 0 & 0 & 187.5 & 250 & 0 & -187.5 & 500 & 0 \\ \end{bmatrix} \end{aligned}

Wektory prawej strony:

\begin{aligned} z_2 &= \begin{bmatrix} \frac{p_x \cdot L}{2} \\ \frac{p_y \cdot L}{2} \\ \frac{p_y \cdot L^2}{12} \\ \frac{p_x \cdot L}{2} \\ \frac{p_y \cdot L}{2} \\ \frac{p_y \cdot L^2}{12} \end{bmatrix} = \begin{bmatrix} 0 \\ 48 \\ 32 \\ 0 \\ 48 \\ -32 \end{bmatrix} \end{aligned} \begin{aligned} z &= \begin{bmatrix} 0 \\ 48 \\ 32 \\ 0 \\ 48 \\ -32 \end{bmatrix}, \quad f = \begin{bmatrix} 0 \\ 0 \\ -50 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \quad d = \begin{bmatrix} 0 \\ 0 \\ 0 \\ d_4 \\ 0 \\ d_6 \end{bmatrix} \end{aligned}

Równanie MES:

\begin{aligned} K \cdot d &= z + f + r \\ \end{aligned} \begin{aligned} \begin{bmatrix} 750.72 & -936.96 & 96 & -750.72 & 936.96 & 96 & 0 & 0 \\ -936.96 & 1297.28 & 72 & 936.96 & -1297.28 & 72 & 0 & 0 \\ 96 & 72 & 400 & -96 & -72 & 200 & 0 & 0 \\ -750.72 & 936.96 & -96 & 3250.72 & -936.96 & -96 & -2500 & 0 \\ 936.96 & -1297.28 & -72 & -936.96 & 1391.03 & 115.5 & 0 & -93.75 \\ 96 & 72 & 200 & -96 & 115.5 & 900 & 0 & -187.5 \\ 0 & 0 & 0 & -2500 & 0 & 0 & 2500 & 0 \\ 0 & 0 & 0 & 0 & -93.75 & -187.5 & 0 & 93.75 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 0 \\ d_4 \\ 0 \\ d_6 \\ \end{bmatrix} &= \begin{bmatrix} 48 \\ 32 \\ 0 \\ -50 \\ 0 \\ 0 \\ \end{bmatrix} + \begin{bmatrix} r_1 \\ r_2 \\ r_3 \\ r_4 \\ r_5 \\ r_6 \\ \end{bmatrix} \end{aligned}

Wykreślamy elementy związane z zerowymi przemieszczeniami oraz obliczamy:

\begin{aligned} \begin{bmatrix} 3250.72 & -96 \\ -96 & 900 \end{bmatrix} \begin{bmatrix} d_4 \\ d_6 \end{bmatrix} &= \begin{bmatrix} 0 \\ 32 \end{bmatrix} + \begin{bmatrix} 0 \\ -50 \end{bmatrix} \end{aligned} \begin{aligned} \begin{bmatrix} -96.0 \cdot d_6 + 3250.72 \cdot d_4 \\ 900.0 \cdot d_6 - 96.0 \cdot d_4 \end{bmatrix} &= \begin{bmatrix} 0.0 \\ -18.0 \end{bmatrix} \end{aligned} \begin{aligned} d_4 &= -5.925 \cdot 10^{-4} \\ d_6 &= -2.006 \cdot 10^{-2} \end{aligned}

Stąd możemy obliczyć reakcje:

\begin{aligned} r &= \begin{bmatrix} 750.72 & -936.96 & 96 & -750.72 & 936.96 & 96 & 0 & 0 \\ -936.96 & 1297.28 & 72 & 936.96 & -1297.28 & 72 & 0 & 0 \\ 96 & 72 & 400 & -96 & -72 & 200 & 0 & 0 \\ -750.72 & 936.96 & -96 & 3250.72 & -936.96 & -96 & -2500 & 0 \\ 936.96 & -1297.28 & -72 & -936.96 & 1391.03 & 115.5 & 0 & -93.75 \\ 96 & 72 & 200 & -96 & 115.5 & 900 & 0 & -187.5 \\ 0 & 0 & 0 & -2500 & 0 & 0 & 2500 & 0 \\ 0 & 0 & 0 & 0 & -93.75 & -187.5 & 0 & 93.75 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 0 \\ d_4 \\ 0 \\ d_6 \end{bmatrix} - \begin{bmatrix} 0 \\ 48 \\ 32 \\ 0 \\ -50 \\ 0 \end{bmatrix} = \begin{bmatrix} -1.48 \\ -2 \\ -3.96 \\ -49.76 \\ 0 \\ 1.48 \\ -44.24 \\ 26.99 \end{bmatrix} \end{aligned}

Reakcje na rysunku

Sprawdzenie

\begin{aligned} \Sigma X &= 0 \quad -1.48 + 1.48 = 0 \\ \Sigma Y &= 0 \quad 2 + 49.76 + 44.24 - 24 \cdot 4 = 0 \\ \Sigma M_A &= 0 \quad -3.96 - 49.76 \cdot 3 - 50 + 24 \cdot 4 \cdot 5 - 44.24 \cdot 7 + 1.48 \cdot 4 + 26.99 = 0 \end{aligned}

Dodatkowo - obliczenia sił wewnętrznych

\begin{aligned} f_1 &= \begin{bmatrix} 2000 & 0 & 0 & -2000 & 0 & 0 \\ 0 & 48 & 120 & 0 & -48 & 120 \\ 0 & 120 & 400 & 0 & -120 & 200 \\ -2000 & 0 & 0 & 2000 & 0 & 0 \\ 0 & -48 & -120 & 0 & 48 & -120 \\ 0 & 120 & 200 & 0 & -120 & 400 \end{bmatrix} \begin{bmatrix} 0.6 & -0.8 & 0 & 0 & 0 & 0 \\ 0.8 & 0.6 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.6 & -0.8 & 0 \\ 0 & 0 & 0 & 0.8 & 0.6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 0 \\ d_4 \\ 0 \\ d_6 \end{bmatrix} = \begin{bmatrix} 0.71 \\ -2.38 \\ -3.96 \\ -0.71 \\ 2.38 \\ -7.97 \end{bmatrix} \end{aligned} \begin{aligned} f_2 &= \begin{bmatrix} 2500 & 0 & 0 & -2500 & 0 & 0 \\ 0 & 93.75 & 187.5 & 0 & -93.75 & 187.5 \\ 0 & 187.5 & 500 & 0 & -187.5 & 250 \\ -2500 & 0 & 0 & 2500 & 0 & 0 \\ 0 & -93.75 & -187.5 & 0 & 93.75 & -187.5 \\ 0 & 187.5 & 250 & 0 & -187.5 & 500 \end{bmatrix} \begin{bmatrix} d_4 \\ d_6 \end{bmatrix} - \begin{bmatrix} 0 \\ 48 \\ 32 \\ 0 \\ -50 \\ 0 \end{bmatrix} = \begin{bmatrix} -1.48 \\ -51.76 \\ -42.03 \\ 1.48 \\ -44.24 \\ 26.99 \end{bmatrix} \end{aligned} \begin{aligned} x_{\text{max}} &= \frac{51.76}{24} = 2.157 \\ M_{\text{max}} &= -42.03 + \frac{1}{2} \cdot 51.76 \cdot 2.157 = 13.793 \end{aligned}

Siły wewnętrzne na rysunku:

MES

Metody numeryczne - MES - Przykład 3

Przykład 3