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…

## Differential equation – Part III Fourier series

Fourier Series To solve the ODE \({y^{\prime\prime}} + a{y^{\prime}} + by = f\left( t \right)\), in which \(f\left( t \right)\) is often the combination of \({e^t}\), \(\sin t\) and \(\cos t\), then any reasonable \(f\left( t \right)\) which is periodic, can be discontinuous but not terribly discontinuous, can be written as Fourier Series. \[f\left( t \right) = {c_0} +…

## Differential equation – Part II Second-Order ODE

Linear 2nd order ODE with constant coefficients Homogeneous: \[{y^{\prime\prime}} + A{y^{\prime}} + By = 0\] Solution: \[y = {c_1}{y_1} + {c_2}{y_2}\] in which, \({y_1}\) and \({y_2}\) are two independent solutions of the homogeneous ODE. The basic method to solve this ODE is to try \(y = {e^{rt}}\). Plug in and we can get \[{r^2}{e^{rt}} +…

## Warnings on keywords when importing input file to ABAQUS

I want to analyze a steel frame with pin connections at both sides or one side at some beams. I found the keyword *Release is very useful and convenient, so I resort to this keyword. I wrote the input file with the same syntax rules, so that it will be more convenient for me to…

## Differential equation – Part I First-Order ODE

Euler equation: \[\left\{ \begin{align} & {{x}_{n+1}}={{x}_{n}}+h \\ & {{y}_{n+1}}={{y}_{n}}+{{A}_{n}}h \\ \end{align} \right.\] \[\left\{ \begin{align} & h=\text{step}\text{size} \\ & {{A}_{n}}=f\left( {{x}_{n}},{{y}_{n}} \right) \\ \end{align} \right.\] Error of solution error = exact solution – approximate solution Method 1: smaller step Euler first-order: \(e\sim {{c}_{1}}h\), proportional to step size Method 2: find a better slop \({{A}_{n}}\) Heun’s method…

## 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…