Ejemplo 3

\( L = 40 \ \text{cm} \), \( A_{1} = 25 \ \text{cm}^{2} \), \( A_{2} = 16 \ \text{cm}^{2} \) y \( E = 2000 \ \text{Kg}/\text{cm}^{2} \)

Un elemento de dos nodos

$$ \begin{equation*} \int_{0}^{L} \mathbf{B}^{\mathrm{T}} \mathbf{D} \ \mathbf{B} \ dx \ \mathbf{u} = \int_{0}^{L} q \ \mathbf{N}^{\mathrm{T}} dx + \mathbf{F} \end{equation*} $$

área $$ \begin{equation*} A(x) = 25 - \frac{9}{40} x \end{equation*} $$

desplazamientos $$ \begin{align*} \mathbf{N} &= \begin{bmatrix} \frac{x_{2}}{x_{2} - x_{1}} - \frac{1}{x_{2} - x_{1}} x & -\frac{x_{1}}{x_{2} - x_{1}} + \frac{1}{x_{2} - x_{1}} x \end{bmatrix} \\ &= \begin{bmatrix} \frac{40}{40 - 0} - \frac{1}{40 - 0} x & -\frac{0}{40 - 0} + \frac{1}{40 - 0} x \end{bmatrix} \\ &= \begin{bmatrix} 1 - \frac{1}{40} x & \frac{1}{40} x \end{bmatrix} \end{align*} $$

deformaciones $$ \begin{equation*} \mathbf{B} = \frac{d \mathbf{N}}{d x} = \begin{bmatrix} -\frac{1}{40} & \frac{1}{40} \end{bmatrix} \end{equation*} $$

matriz constitutiva $$ \begin{equation*} \mathbf{D} = E \ A = 50000 - 450 x \end{equation*} $$

reemplazando $$ \begin{equation*} \int_{0}^{40} \begin{bmatrix} -\frac{1}{40} \\ \frac{1}{40} \end{bmatrix} \begin{bmatrix} 50000 - 450 x \end{bmatrix} \begin{bmatrix} -\frac{1}{40} & \frac{1}{40} \end{bmatrix} dx \begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix} = \begin{bmatrix} F_{1} \\ F_{2} \end{bmatrix} \end{equation*} $$

integrando $$ \begin{equation*} \begin{bmatrix} 1025 & -1025 \\ -1025 & 1025 \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix} = \begin{bmatrix} F_{1} \\ F_{2} \end{bmatrix} \end{equation*} $$

reemplazando las condiciones de contorno $$ \begin{equation*} \begin{bmatrix} 1025 & -1025 \\ -1025 & 1025 \end{bmatrix} \begin{bmatrix} 0 \\ u_{2} \end{bmatrix} = \begin{bmatrix} F_{1} \\ -300 \end{bmatrix} \end{equation*} $$

resolviendo $$ \begin{align*} F_{1} &= 300 \ \text{Kg} \\ u_{2} &= -0.293 \ \text{cm} \end{align*} $$

Desplazamientos, deformaciones y esfuerzos

desplazamientos $$ \begin{equation*} u = \mathbf{N} \ \mathbf{u} = \begin{bmatrix} 1 - \frac{1}{40} x & \frac{1}{40} x \end{bmatrix} \begin{bmatrix} 0 \\ -0.293 \end{bmatrix} = -0.007325 x \end{equation*} $$

deformación normal $$ \begin{equation*} \varepsilon = \mathbf{B} \ \mathbf{u} = \begin{bmatrix} -\frac{1}{40} & \frac{1}{40} \end{bmatrix} \begin{bmatrix} 0 \\ -0.293 \end{bmatrix} = -0.007325 \end{equation*} $$

esfuerzo normal $$ \begin{equation*} \sigma = E \varepsilon = -14.65 \ \text{Kg}/\text{cm}^{2} \end{equation*} $$

Dos elementos de dos nodos

$$ \begin{equation*} \int_{0}^{L} \mathbf{B}^{\mathrm{T}} \mathbf{D} \ \mathbf{B} \ dx \ \mathbf{u} = \int_{0}^{L} q \ \mathbf{N}^{\mathrm{T}} dx + \mathbf{F} \end{equation*} $$

Elemento 1

área $$ \begin{equation*} A(x) = 25 - \frac{9}{40} x \end{equation*} $$

desplazamientos $$ \begin{align*} \mathbf{N} &= \begin{bmatrix} \frac{x_{2}}{x_{2} - x_{1}} - \frac{1}{x_{2} - x_{1}} x & -\frac{x_{1}}{x_{2} - x_{1}} + \frac{1}{x_{2} - x_{1}} x \end{bmatrix} \\ &= \begin{bmatrix} \frac{20}{20 - 0} - \frac{1}{20 - 0} x & -\frac{0}{20 - 0} + \frac{1}{20 - 0} x \end{bmatrix} \\ &= \begin{bmatrix} 1 - \frac{1}{20} x & \frac{1}{20} x \end{bmatrix} \end{align*} $$

deformaciones $$ \begin{equation*} \mathbf{B} = \frac{d \mathbf{N}}{d x} = \begin{bmatrix} -\frac{1}{20} & \frac{1}{20} \end{bmatrix} \end{equation*} $$

matriz constitutiva $$ \begin{equation*} \mathbf{D} = E \ A = 50000 - 450 x \end{equation*} $$

reemplazando $$ \begin{equation*} \int_{0}^{20} \begin{bmatrix} -\frac{1}{20} \\ \frac{1}{20} \end{bmatrix} \begin{bmatrix} 50000 - 450 x \end{bmatrix} \begin{bmatrix} -\frac{1}{20} & \frac{1}{20} \end{bmatrix} dx \begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix} = \begin{bmatrix} F_{1} \\ F_{2} \end{bmatrix} \end{equation*} $$

integrando $$ \begin{equation*} \begin{bmatrix} 2275 & -2275 \\ -2275 & 2275 \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix} = \begin{bmatrix} F_{1} \\ F_{2} \end{bmatrix} \end{equation*} $$

Elemento 2

área $$ \begin{equation*} A(x) = 25 - \frac{9}{40} x \end{equation*} $$

desplazamientos $$ \begin{align*} \mathbf{N} &= \begin{bmatrix} \frac{x_{2}}{x_{2} - x_{1}} - \frac{1}{x_{2} - x_{1}} x & -\frac{x_{1}}{x_{2} - x_{1}} + \frac{1}{x_{2} - x_{1}} x \end{bmatrix} \\ &= \begin{bmatrix} \frac{40}{40 - 20} - \frac{1}{40 - 20} x & -\frac{20}{40 - 20} + \frac{1}{40 - 20} x \end{bmatrix} \\ &= \begin{bmatrix} 2 - \frac{1}{20} x & -1 + \frac{1}{20} x \end{bmatrix} \end{align*} $$

deformaciones $$ \begin{equation*} \mathbf{B} = \frac{d \mathbf{N}}{d x} = \begin{bmatrix} -\frac{1}{20} & \frac{1}{20} \end{bmatrix} \end{equation*} $$

matriz constitutiva $$ \begin{equation*} \mathbf{D} = E \ A = 50000 - 450 x \end{equation*} $$

reemplazando $$ \begin{equation*} \int_{20}^{40} \begin{bmatrix} -\frac{1}{20} \\ \frac{1}{20} \end{bmatrix} \begin{bmatrix} 50000 - 450 x \end{bmatrix} \begin{bmatrix} -\frac{1}{20} & \frac{1}{20} \end{bmatrix} dx \begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix} = \begin{bmatrix} F_{1} \\ F_{2} \end{bmatrix} \end{equation*} $$

integrando $$ \begin{equation*} \begin{bmatrix} 1825 & -1825 \\ -1825 & 1825 \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix} = \begin{bmatrix} F_{1} \\ F_{2} \end{bmatrix} \end{equation*} $$

Ensamblaje y solución

ensamblando matriz global $$ \begin{equation*} \begin{bmatrix} 2275 & -2275 & 0 \\ -2275 & 2275+ 1825 & -1825 \\ 0& -1825 & 1825 \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{2} + u_{1} \\ u_{2} \end{bmatrix} = \begin{bmatrix} F_{1} \\ F_{2} + F_{1} \\ F_{2} \end{bmatrix} \end{equation*} $$

sumando $$ \begin{equation*} \begin{bmatrix} 2275 & -2275 & 0 \\ -2275 & 4100 & -1825 \\ 0& -1825 & 1825 \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{2} \\ u_{3} \end{bmatrix} = \begin{bmatrix} F_{1} \\ F_{2} \\ F_{3} \end{bmatrix} \end{equation*} $$

reemplazando condiciones de contorno $$ \begin{equation*} \begin{bmatrix} 2275 & -2275 & 0 \\ -2275 & 4100 & -1825 \\ 0& -1825 & 1825 \end{bmatrix} \begin{bmatrix} 0 \\ u_{2} \\ u_{3} \end{bmatrix} = \begin{bmatrix} F_{1} \\ 0 \\ -300 \end{bmatrix} \end{equation*} $$

resolviendo $$ \begin{align*} F_{1} &= 300 \ \text{Kg} \\ u_{2} &= -0.132 \ \text{m} \\ u_{3} &= -0.296 \ \text{m} \end{align*} $$

Desplazamientos, deformaciones y esfuerzos

Elemento 1

desplazamientos $$ \begin{equation*} u = \mathbf{N} \ \mathbf{u} = \begin{bmatrix} 1 - \frac{1}{20} x & \frac{1}{20} x \end{bmatrix} \begin{bmatrix} 0 \\ -0.132 \end{bmatrix} = -0.0066 x \end{equation*} $$

deformación normal $$ \begin{equation*} \varepsilon = \mathbf{B} \ \mathbf{u} = \begin{bmatrix} -\frac{1}{20} & \frac{1}{20} \end{bmatrix} \begin{bmatrix} 0 \\ -0.132 \end{bmatrix} = -0.0066 \end{equation*} $$

esfuerzo normal $$ \begin{equation*} \sigma = E \ \varepsilon = -13.2 \ \text{Kg}/\text{cm}^{2} \end{equation*} $$

Elemento 2

desplazamientos $$ \begin{equation*} u = \mathbf{N} \ \mathbf{u} = \begin{bmatrix} 2 - \frac{1}{20} x & -1 + \frac{1}{20} x \end{bmatrix} \begin{bmatrix} -0.132 \\ -0.296 \end{bmatrix} = 0.032 - 0.0082 x \end{equation*} $$

deformación normal $$ \begin{equation*} \varepsilon = \mathbf{B} \ \mathbf{u} = \begin{bmatrix} -\frac{1}{20} & \frac{1}{20} \end{bmatrix} \begin{bmatrix} -0.132 \\ -0.296 \end{bmatrix} = -0.0082 \end{equation*} $$

esfuerzo normal $$ \begin{equation*} \sigma = E \ \varepsilon = -16.4 \ \text{Kg}/\text{cm}^{2} \end{equation*} $$

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