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…

Basic concepts of engineering analysis

Analysis of discrete systems The essence of a lumped-parameter mathematical model is that the state of the system can be described directly with adequate precision by the magnitudes of a finite (and usually small) number of state variables. The solution requires the following steps: System idealization: the actual system is idealized as an assemblage of…

Finite Element Procedures – Matrix, Vectors, Tensors

Special Matrix symmetric matrix identity matrix / unite matrix symmetric banded matrix the following matrix is a symmetric banded matrix of order 5 and the half-bandwidth is 2. \[{\rm{A}} = \left[ {\begin{array}{*{20}{c}} 3&2&1&0&0\\ 2&3&4&1&0\\ 1&4&5&6&1\\ 0&1&6&7&4\\ 0&0&1&4&3 \end{array}} \right]\] diagonal matrix: nonzero elements only on the diagonal of the matrix upper half of the matrix…

Examples of extracting matrices from ABAQUS

2D frame This is a plane steel frame with box sections subjected to a point load at one node figured as below. An dynamic implicit analysis step was defined and the following lines were added at the end of input file. ***STEP*MATRIX GENERATE, STIFFNESS, MASS, LOAD, Structural Damping,Viscous Damping*MATRIX OUTPUT, STIFFNESS, MASS, LOAD, Structural Damping,Viscous…

Time and Amplitude in ABAQUS

I am recently using the dynamic analysis in Abaqus, so I decided to summarize the understanding of time and amplitude in it so that I can refer to this article later if needed. Time Step Time: the time period in *STEP definition. Total Time: the sum of all the Step Time. Natural time: actual time…