{"id":112734,"date":"2026-10-04T18:57:37","date_gmt":"2026-10-04T10:57:37","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/112734.html"},"modified":"2026-10-04T18:57:37","modified_gmt":"2026-10-04T10:57:37","slug":"%e3%80%90%e9%9b%b6%e5%9f%ba%e7%a1%80%e5%ad%a6%e6%99%ba%e8%83%bd%e4%bb%bf%e7%9c%9f-40%e3%80%91%e9%9a%8f%e6%9c%ba%e6%9c%89%e9%99%90%e5%85%83%e5%85%a5%e9%97%a8-%e6%9d%90%e6%96%99","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/112734.html","title":{"rendered":"\u3010\u96f6\u57fa\u7840\u5b66\u667a\u80fd\u4eff\u771f-40\u3011\u968f\u673a\u6709\u9650\u5143\u5165\u95e8\u2014\u2014\u6750\u6599\u4e0d\u786e\u5b9a\u6027\u5982\u4f55\u5f71\u54cd\u4f4d\u79fb\u9884\u6d4b"},"content":{"rendered":"<h2><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"941\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261004105734-6ac2311e969df.png\" width=\"1672\" \/><\/h2>\n<h3>\u8bfe\u7a0b\u6458\u8981<\/h3>\n<p>\u4e0a\u4e00\u8282\u8ba8\u8bba\u4e86\u7f51\u683c\u5212\u5206\u5e26\u6765\u7684\u6570\u503c\u8bef\u5dee&#xff0c;\u672c\u8282\u628a\u76ee\u5149\u8f6c\u5411\u6750\u6599\u672c\u8eab\u7684\u4e0d\u786e\u5b9a\u6027\u3002<span style=\"color:#1a439c\">\u6211\u4eec\u4ee5\u4e24\u5355\u5143\u62c9\u6746\u4e3a\u4f8b&#xff0c;\u8ba9\u4e24\u4e2a\u5355\u5143\u7684\u5f39\u6027\u6a21\u91cf\u968f\u673a\u53d8\u5316&#xff0c;\u91cd\u590d\u8fdb\u884c\u6709\u9650\u5143\u6c42\u89e3&#xff0c;\u89c2\u5bdf\u7aef\u90e8\u4f4d\u79fb\u7684\u5206\u5e03&#xff0c;\u5e76\u4f30\u8ba1\u4f4d\u79fb\u8d85\u8fc7\u6307\u5b9a\u9650\u503c\u7684\u6982\u7387\u3002\u8bfe\u7a0b\u540c\u65f6\u4ecb\u7ecd\u7a7a\u95f4\u76f8\u5173\u6027\u3001\u5bf9\u6570\u6b63\u6001\u5206\u5e03\u3001Monte Carlo \u7edf\u8ba1\u8bef\u5dee&#xff0c;\u4ee5\u53ca\u968f\u673a\u573a\u5411\u590d\u6742\u6709\u9650\u5143\u6a21\u578b\u6269\u5c55\u7684\u601d\u8def\u3002<\/span><\/p>\n<h3>\u4e00\u3001\u4ece\u201c\u4e00\u4e2a\u7b54\u6848\u201d\u5230\u201c\u4e00\u7ec4\u53ef\u80fd\u7684\u7b54\u6848\u201d<\/h3>\n<p>\u6b64\u524d&#xff0c;\u6211\u4eec\u7ed9\u62c9\u6746\u6307\u5b9a\u5f39\u6027\u6a21\u91cf \\\\(E&#061;210000\\\\ \\\\mathrm{MPa}\\\\)&#xff0c;\u8ba1\u7b97\u51fa\u552f\u4e00\u7684\u7aef\u90e8\u4f4d\u79fb&#xff1a;<\/p>\n<p>\\\\[ u_3&#061;\\\\frac{FL}{EA}&#061;0.476190\\\\ \\\\mathrm{mm} \\\\]<\/p>\n<p>\u4f46\u771f\u5b9e\u6750\u6599\u4e0d\u53ef\u80fd\u5904\u5904\u5b8c\u5168\u76f8\u540c\u3002\u5373\u4f7f\u662f\u540c\u4e00\u6839\u6746&#xff0c;\u4e0d\u540c\u4f4d\u7f6e\u7684\u5f39\u6027\u6a21\u91cf\u4e5f\u53ef\u80fd\u7565\u6709\u5dee\u5f02\u3002<\/p>\n<p>\u4e8e\u662f&#xff0c;\u95ee\u9898\u53d8\u6210&#xff1a;<\/p>\n<p><span style=\"color:#be191c\">\u5982\u679c\u6750\u6599\u53c2\u6570\u5b58\u5728\u6ce2\u52a8&#xff0c;\u7aef\u90e8\u4f4d\u79fb\u901a\u5e38\u662f\u591a\u5c11&#xff1f;\u5b83\u6709\u591a\u5927\u53ef\u80fd\u8d85\u8fc7\u5141\u8bb8\u503c&#xff1f;<\/span><\/p>\n<p>\u8fd9\u6b63\u662f\u968f\u673a\u6709\u9650\u5143\u8981\u56de\u7b54\u7684\u95ee\u9898\u4e4b\u4e00\u3002\u5b83\u4e0d\u662f\u8ba9\u6709\u9650\u5143\u6c42\u89e3\u5668\u201c\u968f\u673a\u5730\u6c42\u89e3\u201d&#xff0c;\u800c\u662f\u4e3a\u4e0d\u786e\u5b9a\u7684\u8f93\u5165\u5efa\u7acb\u6982\u7387\u6a21\u578b&#xff0c;\u518d\u7814\u7a76\u8f93\u51fa\u5982\u4f55\u53d8\u5316\u3002<\/p>\n<p>\u4e0a\u4e00\u8282\u7684\u7f51\u683c\u8bef\u5dee\u4e0e\u672c\u8282\u7684\u6750\u6599\u4e0d\u786e\u5b9a\u6027\u4e5f\u4e0d\u8981\u6df7\u4e3a\u4e00\u8c08&#xff1a;\u524d\u8005\u6765\u81ea\u8ba1\u7b97\u8fd1\u4f3c&#xff0c;\u540e\u8005\u6765\u81ea\u6211\u4eec\u5bf9\u7269\u7406\u5bf9\u8c61\u7684\u63cf\u8ff0\u3002<\/p>\n<h3>\u4e8c\u3001\u5efa\u7acb\u4e00\u4e2a\u770b\u5f97\u6e05\u695a\u7684\u968f\u673a\u6709\u9650\u5143\u6a21\u578b<\/h3>\n<p>\u7ee7\u7eed\u4f7f\u7528\u8bfe\u7a0b\u4e2d\u7684\u4e24\u5355\u5143\u4e00\u7ef4\u62c9\u6746&#xff1a;<\/p>\n<ul>\n<li>\u6746\u957f&#xff1a;\\\\(L&#061;1000\\\\ \\\\mathrm{mm}\\\\)<\/li>\n<li>\u622a\u9762\u79ef&#xff1a;\\\\(A&#061;100\\\\ \\\\mathrm{mm^2}\\\\)<\/li>\n<li>\u53f3\u7aef\u62c9\u529b&#xff1a;\\\\(F&#061;10000\\\\ \\\\mathrm N\\\\)<\/li>\n<li>\u5de6\u7aef\u56fa\u5b9a<\/li>\n<li>\u4e24\u4e2a\u5355\u5143\u7b49\u957f&#xff1a;\\\\(L_e&#061;500\\\\ \\\\mathrm{mm}\\\\)<\/li>\n<\/ul>\n<p>\u8fd9\u6b21\u4e0d\u8981\u6c42\u4e24\u4e2a\u5355\u5143\u7684\u5f39\u6027\u6a21\u91cf\u5b8c\u5168\u76f8\u540c&#xff0c;\u800c\u662f\u5206\u522b\u8bb0\u4e3a \\\\(E_1\\\\) \u548c \\\\(E_2\\\\)&#xff1a;<\/p>\n<p>\u56fa\u5b9a\u7aef             \u5355\u5143 1                \u5355\u5143 2             \u62c9\u529b<br \/>\n u\u2081&#061;0     \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 E\u2081 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u25cf\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 E\u2082 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2192 F<br \/>\n                              \u8282\u70b9 2                     \u8282\u70b9 3 <\/p>\n<p>\u5bf9\u4e8e\u67d0\u4e00\u7ec4\u786e\u5b9a\u7684 \\\\(E_1,E_2\\\\)&#xff0c;\u5355\u5143\u521a\u5ea6\u4ecd\u6309\u719f\u6089\u7684\u516c\u5f0f\u8ba1\u7b97&#xff1a;<\/p>\n<p>\\\\[ k_1&#061;\\\\frac{E_1A}{L_e},\\\\qquad k_2&#061;\\\\frac{E_2A}{L_e} \\\\]<\/p>\n<p>\u56fa\u5b9a\u8282\u70b9 1 \u540e&#xff0c;\u5f85\u6c42\u4f4d\u79fb\u4e3a \\\\(u_2,u_3\\\\)&#xff0c;\u5bf9\u5e94\u7684\u6709\u9650\u5143\u65b9\u7a0b\u662f&#xff1a;<\/p>\n<p>\\\\[ \\\\begin{bmatrix} k_1&#043;k_2 &amp; -k_2\\\\\\\\ -k_2 &amp; k_2 \\\\end{bmatrix} \\\\begin{bmatrix} u_2\\\\\\\\u_3 \\\\end{bmatrix} &#061; \\\\begin{bmatrix} 0\\\\\\\\F \\\\end{bmatrix} \\\\]<\/p>\n<p>\u8fd9\u4e2a\u5c0f\u6a21\u578b\u8fd8\u6709\u4e00\u4e2a\u7279\u522b\u6709\u7528\u7684\u89e3\u6790\u7ed3\u679c&#xff1a;<\/p>\n<p>\\\\[ u_2&#061;\\\\frac{FL_e}{AE_1},\\\\qquad u_3&#061;\\\\frac{FL_e}{A} \\\\left(\\\\frac{1}{E_1}&#043;\\\\frac{1}{E_2}\\\\right) \\\\]<\/p>\n<p>\u56e0\u6b64&#xff0c;\u6211\u4eec\u65e2\u80fd\u7ec3\u4e60\u201c\u968f\u673a\u8f93\u5165\u2014\u6709\u9650\u5143\u6c42\u89e3\u2014\u7edf\u8ba1\u8f93\u51fa\u201d&#xff0c;\u53c8\u80fd\u7528\u89e3\u6790\u5f0f\u9010\u6b21\u68c0\u67e5\u7a0b\u5e8f\u6709\u6ca1\u6709\u7b97\u9519\u3002<\/p>\n<p>\u4f8b\u5982&#xff0c;\u82e5 \\\\(E_1&#061;210000\\\\ \\\\mathrm{MPa}\\\\)\u3001\\\\(E_2&#061;189000\\\\ \\\\mathrm{MPa}\\\\)&#xff0c;\u5219&#xff1a;<\/p>\n<p>\\\\[ u_3&#061; \\\\frac{10000\\\\times500}{100} \\\\left(\\\\frac{1}{210000}&#043;\\\\frac{1}{189000}\\\\right) \\\\approx0.502646\\\\ \\\\mathrm{mm} \\\\]<\/p>\n<p>\u7b2c\u4e8c\u4e2a\u5355\u5143\u53d8\u8f6f\u540e&#xff0c;\u7aef\u90e8\u4f4d\u79fb\u6bd4\u540d\u4e49\u503c \\\\(0.476190\\\\ \\\\mathrm{mm}\\\\) \u589e\u5927\u4e86\u3002<\/p>\n<h3>\u4e09\u3001\u600e\u6837\u63cf\u8ff0\u6750\u6599\u7684\u968f\u673a\u6027&#xff1f;<\/h3>\n<p>\u672c\u8282\u4f5c\u4e09\u4e2a\u660e\u786e\u7684\u6559\u5b66\u5047\u8bbe&#xff1a;<\/p>\n<li>\u4e24\u4e2a\u5355\u5143\u7684\u5f39\u6027\u6a21\u91cf\u5e73\u5747\u503c\u5747\u4e3a \\\\(210000\\\\ \\\\mathrm{MPa}\\\\)\u3002<\/li>\n<li>\u4e24\u8005\u7684\u53d8\u5f02\u7cfb\u6570\u5747\u4e3a \\\\(10\\\\%\\\\)\u3002\u53d8\u5f02\u7cfb\u6570\u662f\u201c\u6807\u51c6\u5dee\u00f7\u5e73\u5747\u503c\u201d\u3002<\/li>\n<li>\u4e24\u4e2a\u4f4d\u7f6e\u7684\u6750\u6599\u6027\u8d28\u5177\u6709\u6b63\u76f8\u5173\u6027&#xff1a;\u4e00\u5904\u504f\u786c\u65f6&#xff0c;\u90bb\u8fd1\u4e00\u5904\u4e5f\u66f4\u53ef\u80fd\u504f\u786c\u3002<\/li>\n<p>\u5f39\u6027\u6a21\u91cf\u5fc5\u987b\u4e3a\u6b63&#xff0c;\u56e0\u6b64\u793a\u4f8b\u91c7\u7528\u5bf9\u6570\u6b63\u6001\u5206\u5e03\u3002\u9700\u8981\u7559\u610f&#xff1a;\u5bf9\u6570\u6b63\u6001\u5206\u5e03\u53c2\u6570\u5bf9\u5e94\u7684\u662f \\\\(\\\\ln E\\\\) \u7684\u6b63\u6001\u5206\u5e03&#xff0c;\u800c\u4e0d\u662f\u76f4\u63a5\u628a \\\\(E\\\\) \u5f53\u4f5c\u6b63\u6001\u53d8\u91cf\u3002SciPy \u5bf9\u6570\u6b63\u6001\u5206\u5e03\u6587\u6863<\/p>\n<p>\u8bbe\u5f39\u6027\u6a21\u91cf\u7684\u76ee\u6807\u5e73\u5747\u503c\u4e3a \\\\(\\\\bar E\\\\)\u3001\u53d8\u5f02\u7cfb\u6570\u4e3a \\\\(c_v\\\\)&#xff0c;\u4ee3\u7801\u4e2d\u4f7f\u7528&#xff1a;<\/p>\n<p>\\\\[ \\\\sigma_{\\\\ln E}&#061;\\\\sqrt{\\\\ln(1&#043;c_v^2)},\\\\qquad \\\\mu_{\\\\ln E}&#061;\\\\ln\\\\bar E-\\\\frac{\\\\sigma_{\\\\ln E}^2}{2} \\\\]<\/p>\n<p>\u7136\u540e\u7531\u6807\u51c6\u6b63\u6001\u53d8\u91cf \\\\(Z\\\\) \u751f\u6210&#xff1a;<\/p>\n<p>\\\\[ E&#061;\\\\exp\\\\!\\\\left(\\\\mu_{\\\\ln E}&#043;\\\\sigma_{\\\\ln E}Z\\\\right) \\\\]<\/p>\n<p>\u4ee3\u7801\u4e2d\u7684 rho &#061; 0.60 \u7528\u6765\u63a7\u5236\u4e24\u4e2a\u5bf9\u6570\u5f39\u6027\u6a21\u91cf\u6240\u5bf9\u5e94\u7684\u6b63\u6001\u53d8\u91cf\u7684\u76f8\u5173\u6027&#xff1b;\u5b83\u4e0d\u4e25\u683c\u7b49\u4e8e\u53d8\u6362\u540e \\\\(E_1,E_2\\\\) \u7684\u76f8\u5173\u7cfb\u6570\u3002\u5bf9\u5165\u95e8\u5b9e\u9a8c&#xff0c;\u5148\u5206\u6e05\u8fd9\u4e00\u70b9\u5373\u53ef\u3002<\/p>\n<h3>\u56db\u3001\u5b8c\u6574\u53ef\u8fd0\u884c\u4ee3\u7801<\/h3>\n<p>\u8fd0\u884c\u73af\u5883\u53ea\u9700 NumPy \u548c Matplotlib\u3002\u5c06\u4e0b\u9762\u4ee3\u7801\u4fdd\u5b58\u4e3a lesson40_random_bar.py&#xff0c;\u5728\u9009\u5b9a\u7684 Python \u73af\u5883\u4e2d\u8fd0\u884c&#xff1a;<\/p>\n<p>&#034;&#034;&#034;\u7b2c\u56db\u5341\u8282&#xff1a;\u4e24\u5355\u5143\u968f\u673a\u6746\u7684\u975e\u4fb5\u5165\u5f0f Monte Carlo \u6709\u9650\u5143\u3002&#034;&#034;&#034;<\/p>\n<p>from pathlib import Path<\/p>\n<p>import matplotlib<\/p>\n<p>matplotlib.use(&#034;Agg&#034;)<br \/>\nimport matplotlib.pyplot as plt<br \/>\nimport numpy as np<\/p>\n<p># N\u3001mm\u3001MPa \u5355\u4f4d\u5236&#xff1a;1 MPa &#061; 1 N\/mm\u00b2<br \/>\nlength, area, force &#061; 1000.0, 100.0, 10000.0<br \/>\ne_mean, cv, rho &#061; 210000.0, 0.10, 0.60<br \/>\nn_samples, limit &#061; 10000, 0.55<br \/>\nelement_length &#061; length \/ 2<\/p>\n<p># \u5148\u751f\u6210\u76f8\u5173\u6807\u51c6\u6b63\u6001\u53d8\u91cf&#xff0c;\u518d\u53d8\u6362\u4e3a\u59cb\u7ec8\u4e3a\u6b63\u7684\u5f39\u6027\u6a21\u91cf\u3002<br \/>\nrng &#061; np.random.default_rng(2026)<br \/>\nz &#061; rng.standard_normal((n_samples, 2))<br \/>\nz[:, 1] &#061; rho * z[:, 0] &#043; np.sqrt(1 &#8211; rho**2) * z[:, 1]<br \/>\nsigma_log &#061; np.sqrt(np.log1p(cv**2))<br \/>\nmu_log &#061; np.log(e_mean) &#8211; 0.5 * sigma_log**2<br \/>\nmoduli &#061; np.exp(mu_log &#043; sigma_log * z)<\/p>\n<p># \u8282\u70b9 1 \u56fa\u5b9a&#xff0c;\u53ea\u6c42\u89e3\u8282\u70b9 2\u30013 \u7684\u4f4d\u79fb\u3002<br \/>\ntip_displacements &#061; np.empty(n_samples)<br \/>\nfor i, (e1, e2) in enumerate(moduli):<br \/>\n    k1, k2 &#061; e1 * area \/ element_length, e2 * area \/ element_length<br \/>\n    k_free &#061; np.array([[k1 &#043; k2, -k2], [-k2, k2]])<br \/>\n    u_free &#061; np.linalg.solve(k_free, np.array([0.0, force]))<br \/>\n    tip_displacements[i] &#061; u_free[1]<\/p>\n<p># \u6bcf\u6b21\u6709\u9650\u5143\u6c42\u89e3\u90fd\u5e94\u4e0e\u4e32\u8054\u6746\u7684\u89e3\u6790\u8868\u8fbe\u5f0f\u4e00\u81f4\u3002<br \/>\nexact &#061; force * element_length \/ area * (<br \/>\n    1 \/ moduli[:, 0] &#043; 1 \/ moduli[:, 1]<br \/>\n)<br \/>\nnp.testing.assert_allclose(<br \/>\n    tip_displacements, exact, rtol&#061;1e-12, atol&#061;1e-12<br \/>\n)<\/p>\n<p>nominal &#061; force * length \/ (area * e_mean)<br \/>\nsample_mean &#061; tip_displacements.mean()<br \/>\nsample_sd &#061; tip_displacements.std(ddof&#061;1)<br \/>\nq025, q975 &#061; np.quantile(tip_displacements, [0.025, 0.975])<br \/>\np_exceed &#061; np.mean(tip_displacements &gt; limit)<br \/>\nmc_se &#061; np.sqrt(p_exceed * (1 &#8211; p_exceed) \/ n_samples)<br \/>\ntheoretical_mean &#061; nominal * (1 &#043; cv**2)<\/p>\n<p>print(f&#034;\u6837\u672c\u6570: {n_samples}&#034;)<br \/>\nprint(f&#034;\u786e\u5b9a\u6027\u540d\u4e49\u4f4d\u79fb: {nominal:.6f} mm&#034;)<br \/>\nprint(f&#034;Monte Carlo \u5e73\u5747\u4f4d\u79fb: {sample_mean:.6f} mm&#034;)<br \/>\nprint(f&#034;\u7406\u8bba\u603b\u4f53\u5e73\u5747\u4f4d\u79fb: {theoretical_mean:.6f} mm&#034;)<br \/>\nprint(f&#034;\u4f4d\u79fb\u6837\u672c\u6807\u51c6\u5dee: {sample_sd:.6f} mm&#034;)<br \/>\nprint(f&#034;\u4f4d\u79fb 2.5%&#xff5e;97.5% \u5206\u4f4d\u533a\u95f4: [{q025:.6f}, {q975:.6f}] mm&#034;)<br \/>\nprint(f&#034;P(\u4f4d\u79fb &gt; {limit:.2f} mm) \u4f30\u8ba1: {p_exceed:.4%}&#034;)<br \/>\nprint(f&#034;\u4e0a\u8ff0\u6982\u7387\u4f30\u8ba1\u7684 Monte Carlo \u6807\u51c6\u8bef: {mc_se:.4%}&#034;)<br \/>\nprint(&#034;\u68c0\u67e5\u901a\u8fc7&#xff1a;\u6bcf\u4e2a\u6837\u672c\u7684\u6709\u9650\u5143\u4f4d\u79fb\u4e0e\u89e3\u6790\u5f0f\u4e00\u81f4\u3002&#034;)<\/p>\n<p>fig, ax &#061; plt.subplots(figsize&#061;(9, 5.2), dpi&#061;160)<br \/>\nax.hist(tip_displacements, bins&#061;55, color&#061;&#034;#257d9c&#034;, alpha&#061;0.86)<br \/>\nax.axvline(<br \/>\n    nominal, color&#061;&#034;#183b56&#034;, linestyle&#061;&#034;&#8211;&#034;, linewidth&#061;2,<br \/>\n    label&#061;f&#034;Nominal: {nominal:.3f} mm&#034;,<br \/>\n)<br \/>\nax.axvline(<br \/>\n    sample_mean, color&#061;&#034;#36a386&#034;, linewidth&#061;2,<br \/>\n    label&#061;f&#034;Sample mean: {sample_mean:.3f} mm&#034;,<br \/>\n)<br \/>\nax.axvline(<br \/>\n    limit, color&#061;&#034;#d56b32&#034;, linewidth&#061;2,<br \/>\n    label&#061;f&#034;Illustrative limit: {limit:.2f} mm&#034;,<br \/>\n)<br \/>\nax.set(<br \/>\n    xlabel&#061;&#034;Tip displacement (mm)&#034;,<br \/>\n    ylabel&#061;&#034;Number of samples&#034;,<br \/>\n    title&#061;&#034;Monte Carlo FEM: uncertain modulus, uncertain displacement&#034;,<br \/>\n)<br \/>\nax.grid(axis&#061;&#034;y&#034;, alpha&#061;0.18)<br \/>\nax.legend(frameon&#061;False)<br \/>\nfig.tight_layout()<\/p>\n<p>output &#061; Path(__file__).with_name(&#034;lesson40_displacement_distribution.png&#034;)<br \/>\nfig.savefig(output)<br \/>\nplt.close(fig)<br \/>\nprint(f&#034;\u56fe\u7247\u5df2\u4fdd\u5b58: {output}&#034;) <\/p>\n<h4>\u9884\u671f\u8f93\u51fa<\/h4>\n<p>\u4ee5\u4e0b\u662f\u4e0a\u8ff0\u4ee3\u7801\u5b9e\u9645\u8fd0\u884c\u5f97\u5230\u7684\u7ed3\u679c\u3002\u4f7f\u7528\u4e0d\u540c\u7684 NumPy \u7248\u672c\u65f6&#xff0c;\u968f\u673a\u6837\u672c\u7684\u672b\u51e0\u4f4d\u7edf\u8ba1\u503c\u53ef\u80fd\u7565\u6709\u5dee\u522b&#xff1a;<\/p>\n<p>\u6837\u672c\u6570: 10000<br \/>\n\u786e\u5b9a\u6027\u540d\u4e49\u4f4d\u79fb: 0.476190 mm<br \/>\nMonte Carlo \u5e73\u5747\u4f4d\u79fb: 0.480983 mm<br \/>\n\u7406\u8bba\u603b\u4f53\u5e73\u5747\u4f4d\u79fb: 0.480952 mm<br \/>\n\u4f4d\u79fb\u6837\u672c\u6807\u51c6\u5dee: 0.042945 mm<br \/>\n\u4f4d\u79fb 2.5%&#xff5e;97.5% \u5206\u4f4d\u533a\u95f4: [0.403046, 0.569974] mm<br \/>\nP(\u4f4d\u79fb &gt; 0.55 mm) \u4f30\u8ba1: 6.1900%<br \/>\n\u4e0a\u8ff0\u6982\u7387\u4f30\u8ba1\u7684 Monte Carlo \u6807\u51c6\u8bef: 0.2410%<br \/>\n\u68c0\u67e5\u901a\u8fc7&#xff1a;\u6bcf\u4e2a\u6837\u672c\u7684\u6709\u9650\u5143\u4f4d\u79fb\u4e0e\u89e3\u6790\u5f0f\u4e00\u81f4\u3002<br \/>\n\u56fe\u7247\u5df2\u4fdd\u5b58: &#8230;\/lesson40_displacement_distribution.png <\/p>\n<p>\u7a0b\u5e8f\u751f\u6210\u7684\u5b9e\u9645\u6570\u636e\u56fe\u5982\u4e0b&#xff1b;\u6a59\u7ebf\u53f3\u4fa7\u662f\u8d85\u8fc7\u793a\u4f8b\u4f4d\u79fb\u9650\u503c\u7684\u6837\u672c&#xff1a;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"832\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/10\/20261004105736-6ac2312075a75.png\" width=\"1440\" \/><\/p>\n<h3>\u4e94\u3001\u8bfb\u61c2\u7ed3\u679c&#xff0c;\u800c\u4e0d\u53ea\u662f\u8bfb\u51fa\u6570\u5b57<\/h3>\n<p>\u7b2c\u4e00&#xff0c;\u4e3a\u4ec0\u4e48\u5e73\u5747\u4f4d\u79fb\u6bd4\u201c\u5e73\u5747\u5f39\u6027\u6a21\u91cf\u7b97\u51fa\u7684\u4f4d\u79fb\u201d\u5927&#xff1f;<\/p>\n<p>\u540d\u4e49\u4f4d\u79fb\u662f \\\\(0.476190\\\\ \\\\mathrm{mm}\\\\)&#xff0c;\u968f\u673a\u6837\u672c\u5e73\u5747\u4f4d\u79fb\u7ea6\u4e3a \\\\(0.480983\\\\ \\\\mathrm{mm}\\\\)\u3002\u539f\u56e0\u85cf\u5728 \\\\(u_3\\\\) \u4e0e \\\\(1\/E\\\\) \u7684\u5173\u7cfb\u91cc&#xff1a;\u6750\u6599\u53d8\u8f6f\u65f6\u589e\u52a0\u7684\u4f4d\u79fb&#xff0c;\u4e0d\u80fd\u7b80\u5355\u5730\u88ab\u540c\u5e45\u5ea6\u201c\u53d8\u786c\u201d\u9020\u6210\u7684\u4f4d\u79fb\u51cf\u5c11\u62b5\u6d88\u3002<\/p>\n<p>\u5bf9\u4e8e\u672c\u4f8b\u7684\u5bf9\u6570\u6b63\u6001\u5206\u5e03&#xff0c;\u8fd8\u53ef\u63a8\u5f97\u7406\u8bba\u603b\u4f53\u5e73\u5747\u503c&#xff1a;<\/p>\n<p>\\\\[ \\\\mathbb E[u_3] &#061; \\\\frac{FL}{A\\\\bar E}(1&#043;c_v^2) &#061; 0.480952\\\\ \\\\mathrm{mm} \\\\]<\/p>\n<p>\u5b83\u4e0e\u6837\u672c\u5e73\u5747\u503c\u63a5\u8fd1&#xff0c;\u662f\u5bf9\u968f\u673a\u91c7\u6837\u548c\u7edf\u8ba1\u4ee3\u7801\u7684\u7b2c\u4e8c\u91cd\u68c0\u67e5\u3002<\/p>\n<p>\u7b2c\u4e8c&#xff0c;\u533a\u95f4\u548c\u201c\u6807\u51c6\u8bef\u201d\u56de\u7b54\u7684\u662f\u4e0d\u540c\u95ee\u9898\u3002<\/p>\n<p>\\\\([0.403046,\\\\ 0.569974]\\\\ \\\\mathrm{mm}\\\\) \u662f\u6a21\u62df\u4f4d\u79fb\u7684 2.5%&#xff5e;97.5% \u5206\u4f4d\u533a\u95f4&#xff0c;\u63cf\u8ff0\u5728\u65e2\u5b9a\u6750\u6599\u6982\u7387\u6a21\u578b\u4e0b&#xff0c;\u4e0d\u540c\u6750\u6599\u6837\u672c\u53ef\u80fd\u4ea7\u751f\u600e\u6837\u7684\u4f4d\u79fb\u3002\u5b83\u4e0d\u662f\u201c\u5e73\u5747\u4f4d\u79fb\u768495%\u7f6e\u4fe1\u533a\u95f4\u201d\u3002<\/p>\n<p>0.2410% \u5219\u662f\u8d85\u9650\u6982\u7387\u4f30\u8ba1\u7684 Monte Carlo \u6807\u51c6\u8bef&#xff0c;\u63cf\u8ff0\u53ea\u62bd\u53d6 10000 \u6b21\u65f6&#xff0c;\u6240\u62a5 6.19% \u6709\u591a\u5927\u7684\u62bd\u6837\u6ce2\u52a8\u3002\u8fd9\u91cc\u7684 0.2410% \u5e94\u8bfb\u4f5c\u7ea6 0.241 \u4e2a\u767e\u5206\u70b9\u3002<\/p>\n<p>\u7b2c\u4e09&#xff0c;\u4e0d\u80fd\u628a\u793a\u4f8b\u9650\u503c\u5f53\u4f5c\u5de5\u7a0b\u5224\u636e\u3002<\/p>\n<p>\\\\(0.55\\\\ \\\\mathrm{mm}\\\\) \u662f\u4e3a\u4e86\u7ec3\u4e60\u6982\u7387\u8ba1\u7b97\u800c\u8bbe\u5b9a\u7684\u6f14\u793a\u9650\u503c\u3002\u5b9e\u9645\u5de5\u7a0b\u7684\u5141\u8bb8\u4f4d\u79fb\u5fc5\u987b\u7531\u5177\u4f53\u7ed3\u6784\u3001\u529f\u80fd\u8981\u6c42\u548c\u9002\u7528\u89c4\u8303\u786e\u5b9a\u30026.19% \u4e5f\u53ea\u5728\u672c\u8282\u5047\u5b9a\u7684\u6750\u6599\u5206\u5e03\u3001\u53d8\u5f02\u7cfb\u6570\u53ca\u76f8\u5173\u6027\u6210\u7acb\u65f6\u624d\u6709\u610f\u4e49\u3002<\/p>\n<p>\u8fd8\u6709\u4e00\u4e2a\u5bb9\u6613\u5ffd\u7565\u7684\u73b0\u8c61&#xff1a;\u5728\u8fd9\u4e2a\u7b49\u622a\u9762\u3001\u53f3\u7aef\u7ed9\u5b9a\u62c9\u529b\u7684\u4e00\u7ef4\u6746\u4e2d&#xff0c;\u4e24\u5355\u5143\u7684\u8f74\u5411\u5e94\u529b\u5747\u4e3a \\\\(F\/A&#061;100\\\\ \\\\mathrm{MPa}\\\\)\u3002\u968f\u673a\u5f39\u6027\u6a21\u91cf\u6539\u53d8\u7684\u662f\u5e94\u53d8\u548c\u4f4d\u79fb&#xff0c;\u4e0d\u4f1a\u4f7f\u672c\u4f8b\u7684\u8f74\u5411\u5e94\u529b\u4e5f\u968f\u673a\u53d8\u5316\u3002\u6362\u6210\u4f4d\u79fb\u63a7\u5236\u3001\u53d8\u622a\u9762\u6216\u66f4\u590d\u6742\u7684\u7ed3\u6784\u540e&#xff0c;\u60c5\u51b5\u53ef\u80fd\u4e0d\u540c\u3002<\/p>\n<h3>\u516d\u3001\u4ece\u4e24\u4e2a\u968f\u673a\u6570\u8d70\u5411\u968f\u673a\u573a<\/h3>\n<p>\u672c\u8282\u5b9e\u9645\u4e0a\u628a\u6750\u6599\u5212\u6210\u4e24\u6bb5&#xff1a;\u6bcf\u6bb5\u4f7f\u7528\u4e00\u4e2a\u968f\u673a\u5f39\u6027\u6a21\u91cf\u3002\u8fd9\u662f\u4e00\u4e2a\u5f88\u7c97\u7684\u3001\u5206\u6bb5\u5e38\u6570\u7684\u7a7a\u95f4\u968f\u673a\u6a21\u578b\u3002\u771f\u5b9e\u6750\u6599\u53ef\u80fd\u6cbf\u4f4d\u7f6e \\\\(x\\\\) \u8fde\u7eed\u53d8\u5316&#xff0c;\u6b64\u65f6\u8981\u63cf\u8ff0 \\\\(E(x,\\\\omega)\\\\)&#xff0c;\u8fd8\u8981\u56de\u7b54\u201c\u76f8\u8ddd\u591a\u8fdc\u7684\u4e24\u5904\u6750\u6599\u6027\u8d28\u4ecd\u7136\u76f8\u4f3c\u201d\u2014\u2014\u8fd9\u4fbf\u6d89\u53ca\u7a7a\u95f4\u76f8\u5173\u957f\u5ea6\u3002<\/p>\n<p>\u4e00\u79cd\u5e38\u89c1\u7684\u968f\u673a\u573a\u8868\u793a\u65b9\u6cd5\u662f Karhunen\u2013Lo\u00e8ve&#xff08;KL&#xff09;\u5c55\u5f00\u3002\u4f8b\u5982&#xff0c;\u4e3a\u4fdd\u6301\u5f39\u6027\u6a21\u91cf\u4e3a\u6b63&#xff0c;\u53ef\u4ee5\u5148\u8868\u793a\u5176\u5bf9\u6570\u573a&#xff1a;<\/p>\n<p>\\\\[ \\\\ln E(x,\\\\omega) \\\\approx m(x)&#043; \\\\sum_{j&#061;1}^{M} \\\\sqrt{\\\\lambda_j}\\\\,\\\\phi_j(x)\\\\,\\\\xi_j(\\\\omega) \\\\]<\/p>\n<p>\u5176\u4e2d&#xff0c;\\\\(\\\\phi_j(x)\\\\) \u63cf\u8ff0\u7a7a\u95f4\u53d8\u5316\u6a21\u5f0f&#xff0c;\\\\(\\\\lambda_j\\\\) \u8868\u793a\u76f8\u5e94\u6a21\u5f0f\u7684\u6743\u91cd&#xff0c;\\\\(\\\\xi_j\\\\) \u662f\u968f\u673a\u7cfb\u6570\u3002\u4fdd\u7559\u6709\u9650\u9879\u540e&#xff0c;\u5c31\u80fd\u7528\u6709\u9650\u4e2a\u968f\u673a\u53d8\u91cf\u8fd1\u4f3c\u8fde\u7eed\u968f\u673a\u573a\u3002\u672c\u8282\u4ee3\u7801\u5c1a\u672a\u5b9e\u73b0 KL \u5c55\u5f00&#xff1b;\u4e24\u4e2a\u5355\u5143\u5404\u53d6\u4e00\u4e2a\u6a21\u91cf&#xff0c;\u662f\u5e2e\u52a9\u6211\u4eec\u5148\u638c\u63e1\u6574\u4e2a\u8ba1\u7b97\u94fe\u6761\u7684\u8d77\u70b9\u3002\u968f\u673a\u4ecb\u8d28\u6709\u9650\u5143\u4e0e KL \u5c55\u5f00\u7684\u539f\u59cb\u7814\u7a76<\/p>\n<h3>\u4e03\u3001\u672c\u8282\u7ec3\u4e60<\/h3>\n<li>\u628a cv \u6539\u4e3a 0\u3002\u4f60\u9884\u8ba1\u76f4\u65b9\u56fe\u4f1a\u53d8\u6210\u4ec0\u4e48\u6837&#xff1f;\u7aef\u90e8\u4f4d\u79fb\u5e94\u56de\u5230\u591a\u5c11&#xff1f;<\/li>\n<li>\u4fdd\u6301\u5176\u4ed6\u53c2\u6570\u4e0d\u53d8&#xff0c;\u628a rho \u6539\u4e3a 0\u3002\u6bd4\u8f83\u4e24\u6b21\u4f4d\u79fb\u6807\u51c6\u5dee&#xff0c;\u601d\u8003&#xff1a;\u4e24\u4e2a\u5355\u5143\u201c\u540c\u65f6\u504f\u8f6f\u201d\u7684\u673a\u4f1a\u6539\u53d8\u4e86\u5417&#xff1f;<\/li>\n<li>\u5c06 n_samples \u4f9d\u6b21\u8bbe\u4e3a 1000\u300110000\u300150000\u3002\u89c2\u5bdf\u8d85\u9650\u6982\u7387\u4f30\u8ba1\u548c Monte Carlo \u6807\u51c6\u8bef\u5982\u4f55\u53d8\u5316\u3002\u6837\u672c\u66f4\u591a\u4f1a\u51cf\u5c0f\u62bd\u6837\u8bef\u5dee&#xff0c;\u4f46\u4e0d\u4f1a\u81ea\u52a8\u4fee\u6b63\u9519\u8bef\u7684\u6750\u6599\u6982\u7387\u6a21\u578b\u3002<\/li>\n<p>\u672c\u8282\u5b8c\u6210\u4e86\u201c<span style=\"color:#1c7331\">\u968f\u673a\u6750\u6599\u53c2\u6570 \u2192 \u91cd\u590d\u6709\u9650\u5143\u6c42\u89e3 \u2192 \u4f4d\u79fb\u5206\u5e03 \u2192 \u8d85\u9650\u6982\u7387\u201d\u7684\u6700\u5c0f\u95ed\u73af<\/span>\u3002\u63a5\u4e0b\u6765\u53ef\u4ee5\u628a\u8fd9\u79cd\u65b9\u6cd5\u79fb\u690d\u5230\u4e8c\u7ef4\u6216\u4e09\u7ef4\u7f51\u683c&#xff1a;<span 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