Basic Course in Finite element analysis (Zeng) Note Two-beam elements equation derivation (ii): Approximate solution to the flexible curve of simply supported beam

First, the "approximate" two kinds of classification

A complex function that can be approximated by a series of " Base functions " (base function) combinations, i.e. approximation of functions, has two typical methods:

1. Based on the global approximation, such as Fourier series expansion;
2. Sub- Domain -based segmentation function combination, such as finite element method.

The first function approximation is the classical Rayleigh-Ritz method in Mechanics analysis (Rayleigh-ritz), which is characterized by complex base functions, generally high-order continuous functions, usually only need to use the previous order function combination to obtain a higher approximation accuracy, such as the expansion of Fourier series.

The second function approximation is the finite element theory in modern mechanics analysis, that is, "piecewise approximation", each piecewise function is relatively simple, using linear function or two functions can be, but need more segmentation to get approximation effect, the workload is relatively large.

For more image of the two approximation methods, you can refer to a picture of Zeng teacher's book, which is the following picture

1.1 Approximate solution based on global approximation method

The core point of Rayleigh-Ritz method is to select "Test function", the test function must first satisfy the displacement boundary condition, and of course, there will be some undetermined coefficients, and then the other variables are used in this "test function" to express, through other boundary conditions or energy method (virtual work principle or minimum potential energy principle) to solve the undetermined coefficients.

1.1.1 Using virtual work principle to solve approximate solution

In simple terms, the meaning of the principle of virtual work is that if the system has a virtual displacement, then the external working virtual Force in the virtual displacement should be done within the internal virtual work.

Internal virtual

\begin{equation}
\delta U=\intop_{\varomega}\sigma_{x}\delta\varepsilon_{x}d\varomega
\end{equation}

External Force virtual

\begin{equation}
\delta W=\intop_{l}p\delta VDX
\end{equation}

Because the selected "test function" only satisfies the displacement boundary condition, the selection condition of "test function" is very broad. However, it is very important to choose a reasonable test function for different test functions to obtain the final result.

For example, for pure curved simply-supported beams, the displacement boundary condition is zero at 0 and L, and the displacement shape should be in the middle, and then gradually reduce to the end of 0. It is natural that the first thing to think about is the two-time function parabola, and the other is the trigonometric sine curve. For simplicity, the length l of the beam is taken as 1

• Test function $v1 (x) =-x^{2}+x$
• Trial function $v2 (x) =\frac{1}{4}\sin (\pi x) $

Draw the graph between the two test functions above the 0~1

It can be seen that both conform to the displacement boundary condition, and the approximate shape and the expected beam deflection curve are consistent.

When the test function adopts $c_{1}\sin\left (\FRAC{\PI x}{l}\right) $, the coefficient C1 can be obtained according to the principle of virtual work, and the final torsion curve is

\begin{equation}
\frac{4l^{4}p_{0}}{\pi^{5}\text{ei}}\sin\left (\frac{\pi x}{l}\right)
\end{equation}

When the test function uses $c_1 \left (L x-x^2\right) $, the resulting flex curve is

\begin{equation}
\frac{l^{2}p_{0}}{24\text{ei}} (Lx-x^{2})
\end{equation}

The $\frac{p_{0}}{ei}=1$ and $l=1$, respectively, draw the graph of the approximate solution of the 0~l Direct analytic solution and the two kinds of test function.

It can be seen that using $c_{1}\sin\left (\frac{\pi x}{l}\right) $ as the test function, the final to the Curve v1 (x) and the Analytic Solution V0 (x) is very close, almost no difference in, but the use of $c_1 \left (L x-x^2\ right) $ as a test function to solve the approximate solution but the difference between the analytic solution is larger, further enlarge the graph

As you can see, the direct nuances of v0 (x) and v1 (x) are seen when the graph is enlarged.

Basic Course in Finite element analysis (Zeng) Note Two-beam elements equation derivation (ii): Approximate solution to the flexible curve of simply supported beam