Straight Beams Rested on Nonlinear Elastic Foundations – Part 2 (Numerical Solutions, Results and Evaluation)

2014 ◽  
Vol 684 ◽  
pp. 21-29 ◽  
Author(s):  
Šárka Michenková ◽  
Karel Frydrýšek ◽  
Marek Nikodým
Keyword(s):  
Difference Method ◽  
Numerical Solutions ◽  
Nonlinear Problems ◽  
Central Difference ◽  
Practical Applications ◽  
Reaction Forces ◽  
Central Difference Method ◽  
Linear And Nonlinear ◽  
Nonlinear Elastic ◽  
Nonlinear Solutions

This paper presents numerical solutions of straight plane beam structures rested on an elastic (Winkler's) foundation. It is a continuation of our previous work (see Part 1 of this article) focused on practical applications and solutions including nonlinearities in the foundation (i.e. bilateral linear, bilateral linear + cubic, bilateral linear + cubic + quintic approximations and unilateral approximation for dependencies of reaction forces on deflection in the foundation). For solutions of nonlinear problems of mechanics (i.e. differential 4th-order equations), the Finite Difference Method (i.e. the Central Difference Method) is applied in combination with the Newton (Newton–Raphson) Method. Finally, in one example, linear and nonlinear approaches are solved, evaluated and compared. In some cases, there are evident major differences between the linear and nonlinear solutions.

Straight Beams Rested on Nonlinear Elastic Foundations – Part 1 (Theory, Experiments, Numerical Approach)

2014 ◽  
Vol 684 ◽  
pp. 11-20 ◽  
Author(s):  
Karel Frydrýšek ◽  
Šárka Michenková ◽  
Marek Nikodým
Keyword(s):  
Difference Method ◽  
Least Squares Method ◽  
Reaction Force ◽  
Nonlinear Problems ◽  
Numerical Approach ◽  
Nonlinear Dependence ◽  
Central Difference ◽  
Central Difference Method ◽  
Nonlinear Elastic ◽  
Numerical Approaches

This paper presents theory, experiments and numerical approaches suitable for the solution of straight plane beams rested on an elastic (Winkler's) foundation, including nonlinearities. The nonlinear dependence of the reaction force on displacement in the foundation (i.e. the experimental data) can be described via bilateral linear or bilateral linear + cubic or bilateral linear + cubic + quintic approximations, or by unilateral approximation (i.e. by using the Least Squares Method). These applications lead to linear or nonlinear differential 4th-order equations. For solutions of nonlinear problems of mechanics, the Finite Difference Method (i.e. the Central Difference Method) and boundary conditions are applied. The solution and its evaluation is performed in second part of this article.

scholarly journals Frequency-Dependent FDTD Algorithm Using Newmark’s Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Bing Wei ◽  
Le Cao ◽  
Fei Wang ◽  
Qian Yang
Keyword(s):  
Differential Equation ◽  
Electric Field ◽  
Field Intensity ◽  
Time Domain ◽  
Difference Method ◽  
Electric Field Intensity ◽  
Polarization Vector ◽  
Solution Stability ◽  
Central Difference ◽  
Central Difference Method

According to the characteristics of the polarizability in frequency domain of three common models of dispersive media, the relation between the polarization vector and electric field intensity is converted into a time domain differential equation of second order with the polarization vector by using the conversion from frequency to time domain. Newmarkβγdifference method is employed to solve this equation. The electric field intensity to polarizability recursion is derived, and the electric flux to electric field intensity recursion is obtained by constitutive relation. Then FDTD iterative computation in time domain of electric and magnetic field components in dispersive medium is completed. By analyzing the solution stability of the above differential equation using central difference method, it is proved that this method has more advantages in the selection of time step. Theoretical analyses and numerical results demonstrate that this method is a general algorithm and it has advantages of higher accuracy and stability over the algorithms based on central difference method.

Numerical Method for Earthquake-Induced Pounding between Girder and Shear Key. I: Theory

2010 ◽  
Vol 34-35 ◽  
pp. 1402-1405
Author(s):  
Wei He
Keyword(s):  
Seismic Response ◽  
Difference Method ◽  
Equations Of Motion ◽  
Earthquake Ground Motion ◽  
Analysis Model ◽  
Central Difference ◽  
Kelvin Model ◽  
Shear Key ◽  
Analytical Perspective ◽  
Central Difference Method

Earthquake ground motion can induce out-of-phase vibrations between girders and shear keys, which can result in impact or pounding. The paper investigated pounding between girder and shear key from an analytical perspective. By introducing the initial gap in the analysis model, the elastomer stiffness played a role in the transverse vibration as well. A simplified model of bridge transverse seismic response considering girder-shear key pounding was developed. The equations of motion of the bridge response to transverse ground excitation were assembled and solved using the central difference method. Pounding was simulated using a contact force-based model—Kelvin model. Thus, the girder-shear key pounding effects and bridge transverse seismic response can be obtained by using a step-by-step direct integration the central difference method with the appropriate parameters. The proposed method is very useful in the seismic design of bridge.

Computational Methods Used in the Determination of Loading Rate: Experimental and Clinical Implications

1999 ◽  
Vol 15 (4) ◽  
pp. 404-417 ◽  
Author(s):  
C. Mark Woodard ◽  
Margaret K. James ◽  
Stephen P. Messier
Keyword(s):  
Difference Method ◽  
Ground Reaction Force ◽  
Loading Rate ◽  
Reaction Force ◽  
Vertical Force ◽  
Vertical Ground Reaction Force ◽  
Central Difference ◽  
Symptomatic Knee Osteoarthritis ◽  
Central Difference Method ◽  
Vertical Ground

Our purpose was to compare methods of calculating loading rate to the first peak vertical ground reaction force during walking and provide a rationale for the selection of a loading rate algorithm in the analysis of gait in clinical and research environments. Using vertical ground reaction force data collected from 15 older adults with symptomatic knee osteoarthritis and 15 healthy controls, we: (a) calculated loading rate as the first peak vertical force divided by the time from touchdown until the first peak; (b) calculated loading rate as the slope of the least squares regression line using vertical force and time as the dependent and independent variables, respectively; (c) calculated loading rate over discrete intervals using the Central Difference method; and (d) calculated loading rate using vertical force and lime data representing 20% and 90% of the first peak vertical force. The largest loading rate, which may be of greatest clinical importance, occurred when loading rates were calculated using the fewest number of data points. The Central Difference method appeared to maximize our ability to detect differences between healthy and pathologic cohorts. Finally, there was a strong correlation between methods, suggesting that all four methods are acceptable. However, if maximizing the chances of detecting differences between groups is of primary importance, the Central Difference method appears superior.

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