## Probabilistic system analysis – Part III

Limit theorem Chebyshev’s inequality \begin{align} {\sigma ^2} &= \int {{{\left( {x – \mu } \right)}^2}{f_X}\left( x \right)dx} \\ &\ge \int_{ – \infty }^{u – c} {{{\left( {x – \mu } \right)}^2}{f_X}\left( x \right)dx} + \int_{u + c}^\infty {{{\left( {x – \mu } \right)}^2}{f_X}\left( x \right)dx} \\ &\ge {c^2}\int_{ – \infty }^{u – c} {{f_X}\left( x…

## Probabilistic system analysis – Part II

Bernoulli process \begin{align} {\rm{P}}\left( {{\rm{sucess}}} \right) &= {\rm{P}}\left( {{X_i} = 1} \right) = p\\ {\rm{P}}\left( {{\rm{failure}}} \right) &= {\rm{P}}\left( {{X_i} = 0} \right) = 1 – p\\ {\rm{E}}\left[ {{X_t}} \right] &= p,{\mathop{\rm var}} \left( {{X_t}} \right) = p\left( {1 – p} \right) \end{align} PMF of # of arrivals (number of success $$S$$ in $$n$$ time…

## Probabilistic system analysis – Part I

Independent and disjoint Two events are independent or disjoint are two different concept. If two events are independent, there is no relation between but they can happen together. It is noted that independence can be affected by conditioning. If two events are disjoint, they has the relation that when one happens the other cannot happen…

## Differential equation – Part IV ODE system

ODE system $\left\{ \begin{array}{l} {x^{\prime}} = f\left( {x,y,t} \right)\\ {y^{\prime}} = g\left( {x,y,t} \right) \end{array} \right.,x\left( {{t_0}} \right) = {x_0},y\left( {{t_0}} \right) = {y_0}$ Linear system: $\left\{ \begin{array}{l} {x^{\prime}} = ax + by + {r_1}\left( t \right)\\ {y^{\prime}} = cx + dy + {r_2}\left( t \right) \end{array} \right.$ $$a$$, $$b$$, $$c$$, $$d$$ can be functions of $$t$$. If they…

## Transverse shear stiffness of beam in ABAQUS

Calculation of transverse shear stiffness Definition The effective transverse shear stiffness of the section of a shear flexible beam is defined in Abaqus as ${\overline K _{\alpha 3}} = f_p^\alpha {K_{\alpha 3}}$ where $${\overline K _{\alpha 3}}$$ is the section shear stiffness in the $$\alpha$$-direction;  $${K_{\alpha 3}}$$ is the actual shear stiffness of the section having…

## Beam elements in ABAQUS

Beam elements in ABAQUS make me a little confused, especially about the shear stiffness and elements for open sections. In this post, I want to make a summary of the elements in ABAQUS and give an example about the settings of beam elements. Beam element library Beam elements in a plane only have active degrees…