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弯曲问题的几个注解

news 来源：原创 2025/6/12 17:29:01

弯曲问题的几个注解

奇异函数

定义奇异函数为：

$\left< x-a\right>^n = \begin{cases} 0 & , x < a \\ (x-a)^n & , x > a \end{cases}$

$\left< x-a\right>^n = \begin{cases} 0 & , x < a \\ (x-a)^n =\frac{1}{(n+1)!}\frac{\text{d}(x-a)^{n+1}}{\text{d}x}& , x > a \end{cases}$

微积分关系满足如下关系：
$\frac{\text{d}(x-a)^{n+1}}{\text{d}x}=(n+1)!\left< x-a \right>^n$

$\int_{-\infty}^{x} \left< x-a\right>^n\text{d} x=\frac{1}{n+1}\left< x-a\right>^{n+1}$

例

$x = y = z = a$ ,求解挠曲线近似微分方程。

解

弯矩方程为：
$M(x)=\frac{5qa}{6}x-qa^2\left< x-a\right>^0-qa\left< x-a\right>^1-\frac{q}{2}\left< x-a\right>^2$

有挠曲线近似微分方程：

$EIw^{\prime\prime}=\frac{5qa}{6}x-qa^2\left< x-a\right>^0-qa\left< x-2a\right>^1-\frac{q}{2}\left< x-2a\right>^2$

得到：

$EIw=\frac{5qa}{36}x^3-\frac{1}{2}qa^2\left< x-2a\right>^2-\frac{1}{6}qa\left< x-2a\right>^3-\frac{q}{24}\left< x-2a\right>^4+Cx+D$

代入边界条件：

$w|_{x=0}=w|_{x=3a}=0$
得到：

$D=0，C=-\frac{37}{72}qa^3$

理想黏结的两种材料组合梁

其中性轴公式：

$h_c=\frac{1}{2}\left(\frac{E_1b_1h_1^2-E_2b_2h_2^2}{E_1b_1h_1+E_2b_2h_2}\right)$

对于等腹宽里梁：

$h_c=\frac{1}{2}\left(\frac{E_1h_1^2-E_2h_2^2}{E_1h_1+E_2h_2}\right)$

其正应力公式：

$\begin{align*} &\sigma_1=\frac{My}{I_z}\\ &\sigma_2=\frac{E_2}{E_1}\frac{My}{I_z} \end{align*}$

弹性基础梁

弹性基础梁是指支撑在弹性地基上的梁结构，地基对梁的反力与梁的位移成正比。

$\frac{\text{d}^4w}{\text{d}x^4} + k \cdot w = q(x)$

\documentclass[tikz]{standalone}
\usetikzlibrary{calc,decorations.pathmorphing,patterns,arrows.meta}\begin{document}
\begin{tikzpicture}[scale=1.8]% 定义参数\def\beamLength{6}\def\beamHeight{0.3}\def\springLength{0.5}\def\springCount{12}\def\loadHeight{0.7}      % 均布载荷高度\def\momentRadius{0.6}    % 弯矩符号半径% 绘制梁\draw[thick] (0,0) -- (\beamLength,0) -- (\beamLength,\beamHeight) -- (0,\beamHeight) -- cycle;\node[left] at (0,\beamHeight/2) {$EI$};% 绘制均布载荷（向下箭头）\draw[thick] (0,\beamHeight+\loadHeight) -- (\beamLength,\beamHeight+\loadHeight);\foreach \x in {0.5,1.5,...,\beamLength-0.5}{\draw[thick,->,>=Stealth] (\x,\beamHeight+\loadHeight) -- (\x,\beamHeight);}\node[above] at (\beamLength/2,\beamHeight+\loadHeight) {$q$};% 绘制弹簧地基\foreach \x in {1,...,\springCount}{\pgfmathsetmacro{\posX}{\x*\beamLength/(\springCount+1)}\draw[decorate,decoration={aspect=0.3, segment length=2.5pt, amplitude=2.5pt,coil}] (\posX,0) -- (\posX,-\springLength);\draw (\posX,-\springLength) -- +(-0.2,0.1) -- +(0.2,0.1);}\node[left] at (0,-\springLength/2) {$k$};% 绘制地基\fill[pattern=north east lines,pattern color=gray!30] (-0.5,-\springLength-0.2) rectangle (\beamLength+0.5,-\springLength-0.4);\draw[thick] (-0.5,-\springLength-0.2) -- (\beamLength+0.5,-\springLength-0.2);\node[below] at (\beamLength/2,-\springLength-0.3) {弹性地基};% 优化挠度曲线（增加控制点）\draw[blue,thick,dashed] plot[smooth, tension=0.7] coordinates {(0,0) (\beamLength/6,-0.08) (\beamLength/3,-0.15) (\beamLength/2,-0.22) (2*\beamLength/3,-0.15) (5*\beamLength/6,-0.08) (\beamLength,0)};\draw[blue,->,>=Latex] ($(\beamLength/2,-0.22)$) -- +(0,0.18) node[midway,right,font=\small] {挠度 $w(x)$};% 标注应力方向\draw[green!60!black,thick] (0.8*\beamLength,\beamHeight) -- (0.8*\beamLength,\beamHeight+0.15) node[above,font=\small] {拉应力 $\sigma_t$};\draw[green!60!black,thick] (0.8*\beamLength,0) -- (0.8*\beamLength,-0.15) node[below,font=\small] {压应力 $\sigma_c$};
\end{tikzpicture}
\end{document}