Stability of Heterogeneous Beams with Three Supports—Solutions Using Integral Equations
Abstract
:1. Introduction
2. Differential Equations
Governing Equations
3. Green Function for Three-Point BVPs
3.1. Green Function for FssF Beams
3.1.1. Calculation of the Green Function if
3.1.2. Calculation of the Green Function if
3.2. Green Function for FssP Beams
3.3. Green Function for PssP Beams
4. The Stability Problem of FssF, SssP and PssP Beams with Three Supports
4.1. Solution Procedures
4.2. The Kernel for FssF Beams
4.3. The Kernel for FssP Beams
4.4. The Kernel for PssP Beams
5. Computational Results
5.1. FssF Beams
5.2. FssP Beams
5.3. PssP Beams
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Limit Cases
Appendix A.1. The Green Function for FrF Beams
Appendix A.2. The Green Function for FrP Beams
Appendix A.3. The Green Function for PrP Beams
Appendix A.4. The Kernel Function for FrF Beams
Appendix A.5. The Kernel Function for FrP Beams
Appendix A.6. The Kernel Function for PrP Beams
Appendix A.7. Characteristic Equations
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Boundary Conditions | ||
---|---|---|
(FssF beam) | (FssP beam) | (PssP beam) |
Continuity Conditions | ||
0.0000 | 2.000000 | 2.000000 | 2.000000 | 2.000000 | 2.000000 |
0.0250 | 2.000008 | 2.000013 | 2.000018 | 2.000023 | 2.000028 |
0.0500 | 2.000080 | 2.000157 | 2.000233 | 2.000310 | 2.000386 |
0.0750 | 2.000383 | 2.000761 | 2.001136 | 2.001509 | 2.001880 |
0.1000 | 2.001166 | 2.002313 | 2.003444 | 2.004561 | 2.005663 |
0.1250 | 2.002723 | 2.005379 | 2.007972 | 2.010506 | 2.012982 |
0.1500 | 2.005355 | 2.010514 | 2.015489 | 2.020289 | 2.024922 |
0.1750 | 2.009325 | 2.018181 | 2.026601 | 2.034609 | 2.042232 |
0.2000 | 2.014822 | 2.028690 | 2.041678 | 2.053850 | 2.065267 |
0.2250 | 2.021935 | 2.042170 | 2.060845 | 2.078091 | 2.094027 |
0.2500 | 2.030643 | 2.058561 | 2.083989 | 2.107147 | 2.128241 |
0.2750 | 2.040804 | 2.077623 | 2.110792 | 2.140630 | 2.167450 |
0.3000 | 2.052156 | 2.098938 | 2.140749 | 2.177991 | 2.211072 |
0.3250 | 2.064324 | 2.121912 | 2.173171 | 2.218522 | 2.258424 |
0.3500 | 2.076823 | 2.145766 | 2.207159 | 2.261337 | 2.308701 |
0.3750 | 2.089075 | 2.169522 | 2.241546 | 2.305276 | 2.360891 |
0.4000 | 2.100433 | 2.191993 | 2.274805 | 2.348745 | 2.413589 |
0.4250 | 2.110217 | 2.211797 | 2.304954 | 2.389443 | 2.464612 |
0.4500 | 2.117777 | 2.227448 | 2.329532 | 2.424019 | 2.510233 |
0.4750 | 2.122562 | 2.237538 | 2.345833 | 2.447969 | 2.543998 |
0.5000 | 2.124201 | 2.241031 | 2.351573 | 2.456659 | 2.556943 |
0.0000 | 2.000000 | 2.000000 | 2.000000 | 2.000000 | 2.000000 | 2.000000 |
0.0250 | 2.000033 | 2.000042 | 2.000067 | 2.000100 | 2.000293 | 2.038216 |
0.0500 | 2.000462 | 2.000613 | 2.000990 | 2.001511 | 2.004361 | 2.077889 |
0.0750 | 2.002249 | 2.002979 | 2.004767 | 2.007183 | 2.019289 | 2.119074 |
0.1000 | 2.006750 | 2.008884 | 2.013983 | 2.020605 | 2.049702 | 2.161815 |
0.1250 | 2.015401 | 2.020078 | 2.030899 | 2.044229 | 2.094262 | 2.206145 |
0.1500 | 2.029394 | 2.037888 | 2.056787 | 2.078661 | 2.148228 | 2.252082 |
0.1750 | 2.049491 | 2.063007 | 2.091798 | 2.122924 | 2.207134 | 2.299619 |
0.2000 | 2.075983 | 2.095515 | 2.135248 | 2.175238 | 2.268186 | 2.348715 |
0.2250 | 2.108766 | 2.135051 | 2.186031 | 2.233702 | 2.330019 | 2.399278 |
0.2500 | 2.147465 | 2.181004 | 2.242947 | 2.296665 | 2.392061 | 2.451142 |
0.2750 | 2.191547 | 2.232672 | 2.304889 | 2.362820 | 2.454057 | 2.504040 |
0.3000 | 2.240399 | 2.289358 | 2.370903 | 2.431136 | 2.515766 | 2.557558 |
0.3250 | 2.293366 | 2.350419 | 2.440155 | 2.500707 | 2.576781 | 2.611080 |
0.3500 | 2.349744 | 2.415267 | 2.511842 | 2.570559 | 2.636388 | 2.663708 |
0.3750 | 2.408718 | 2.483339 | 2.585016 | 2.639404 | 2.693437 | 2.714177 |
0.4000 | 2.469216 | 2.554051 | 2.658291 | 2.705314 | 2.746205 | 2.760765 |
0.4250 | 2.529572 | 2.626716 | 2.729274 | 2.765312 | 2.792309 | 2.801259 |
0.4500 | 2.586620 | 2.700418 | 2.793376 | 2.814985 | 2.828768 | 2.833058 |
0.4750 | 2.633089 | 2.773688 | 2.841720 | 2.848596 | 2.852393 | 2.853522 |
0.5000 | 2.652952 | 2.833793 | 2.860604 | 2.860604 | 2.860604 | 2.860604 |
0.0000 | 1.430302 | 1.430302 | 1.430302 | 1.430302 | 1.430302 |
0.0500 | 1.430334 | 1.430377 | 1.430420 | 1.430473 | 1.430547 |
0.1000 | 1.430784 | 1.431422 | 1.432052 | 1.432831 | 1.433905 |
0.1500 | 1.432544 | 1.435455 | 1.438280 | 1.441695 | 1.446273 |
0.2000 | 1.436662 | 1.444720 | 1.452328 | 1.461255 | 1.472769 |
0.2500 | 1.443950 | 1.460773 | 1.476177 | 1.493662 | 1.515280 |
0.3000 | 1.454692 | 1.484013 | 1.510111 | 1.538821 | 1.572915 |
0.3500 | 1.468509 | 1.513634 | 1.552947 | 1.595125 | 1.643523 |
0.4000 | 1.484348 | 1.547702 | 1.602312 | 1.660025 | 1.724640 |
0.4500 | 1.500535 | 1.583160 | 1.654563 | 1.729929 | 1.813437 |
0.5000 | 1.514927 | 1.615748 | 1.704260 | 1.799175 | 1.905547 |
0.5500 | 1.525206 | 1.640163 | 1.743538 | 1.857971 | 1.991783 |
0.6000 | 1.529367 | 1.651044 | 1.762835 | 1.890806 | 2.050418 |
0.6500 | 1.526294 | 1.645008 | 1.755043 | 1.882616 | 2.046060 |
0.7000 | 1.516169 | 1.622456 | 1.720727 | 1.833762 | 1.975444 |
0.7500 | 1.500463 | 1.587353 | 1.667434 | 1.758843 | 1.871752 |
0.8000 | 1.481573 | 1.545450 | 1.604616 | 1.672479 | 1.757022 |
0.8500 | 1.462350 | 1.502853 | 1.540969 | 1.585485 | 1.642410 |
0.9000 | 1.445718 | 1.465619 | 1.484801 | 1.507815 | 1.538331 |
0.9500 | 1.434358 | 1.439708 | 1.444994 | 1.451512 | 1.460475 |
0.9750 | 1.431330 | 1.432696 | 1.434058 | 1.435753 | 1.438115 |
0.9800 | 1.430961 | 1.431838 | 1.432713 | 1.433804 | 1.435326 |
0.9900 | 1.430467 | 1.430687 | 1.430907 | 1.431182 | 1.431566 |
0.9975 | 1.430312 | 1.430326 | 1.430340 | 1.430357 | 1.430381 |
0.0000 | 1.430302 | 1.430302 | 1.430302 | 1.430302 | 1.43030 |
0.0500 | 1.430674 | 1.430833 | 1.431353 | 1.432366 | 1.43519 |
0.1000 | 1.435700 | 1.437868 | 1.444538 | 1.455748 | 1.47845 |
0.1500 | 1.453609 | 1.461967 | 1.484671 | 1.514784 | 1.55600 |
0.2000 | 1.490195 | 1.508606 | 1.551816 | 1.596797 | 1.64233 |
0.2500 | 1.546037 | 1.576049 | 1.637247 | 1.689125 | 1.73202 |
0.3000 | 1.618623 | 1.659962 | 1.734342 | 1.787809 | 1.82642 |
0.3500 | 1.705053 | 1.756927 | 1.840447 | 1.892859 | 1.92721 |
0.4000 | 1.803191 | 1.865196 | 1.955048 | 2.004906 | 2.03510 |
0.4500 | 1.911581 | 1.984124 | 2.077841 | 2.123440 | 2.14901 |
0.5000 | 2.028652 | 2.113167 | 2.206504 | 2.244631 | 2.26416 |
0.5500 | 2.150917 | 2.250231 | 2.332179 | 2.357130 | 2.36848 |
0.6000 | 2.266076 | 2.386936 | 2.429884 | 2.437159 | 2.44005 |
0.6500 | 2.293059 | 2.452225 | 2.457446 | 2.457895 | 2.45806 |
0.7000 | 2.172303 | 2.316725 | 2.402885 | 2.418421 | 2.42438 |
0.7500 | 2.026359 | 2.154562 | 2.296332 | 2.339839 | 2.35800 |
0.8000 | 1.876349 | 1.986250 | 2.157401 | 2.236818 | 2.27423 |
0.8500 | 1.726880 | 1.812434 | 1.985291 | 2.106331 | 2.17705 |
0.9000 | 1.586370 | 1.639609 | 1.773783 | 1.919260 | 2.04977 |
0.9500 | 1.475409 | 1.493344 | 1.547757 | 1.635968 | 1.79607 |
0.9750 | 1.442131 | 1.447093 | 1.463185 | 1.493419 | 1.57070 |
0.9800 | 1.437921 | 1.441141 | 1.451681 | 1.471908 | 1.52642 |
0.9900 | 1.432224 | 1.433044 | 1.435766 | 1.441151 | 1.45684 |
0.9975 | 1.430422 | 1.430474 | 1.430646 | 1.430990 | 1.43201 |
= 50,000 | = 500,000 | |||
0.0000 | 1.430302 | 1.430302 | 1.430302 | 1.430302 |
0.0500 | 1.439304 | 1.467133 | 1.483508 | 1.486263 |
0.1000 | 1.498693 | 1.539600 | 1.546449 | 1.547261 |
0.1500 | 1.579860 | 1.609894 | 1.613513 | 1.613924 |
0.2000 | 1.662822 | 1.684347 | 1.686682 | 1.686943 |
0.2500 | 1.748836 | 1.765158 | 1.766854 | 1.767044 |
0.3000 | 1.840403 | 1.853428 | 1.854753 | 1.854900 |
0.3500 | 1.939021 | 1.949752 | 1.950830 | 1.950950 |
0.4000 | 2.045077 | 2.053978 | 2.054864 | 2.054962 |
0.4500 | 2.157153 | 2.164302 | 2.165007 | 2.165085 |
0.5000 | 2.270140 | 2.275295 | 2.275799 | 2.275855 |
0.5500 | 2.371795 | 2.374598 | 2.374870 | 2.374900 |
0.6000 | 2.440857 | 2.441527 | 2.441591 | 2.441598 |
0.6500 | 2.458107 | 2.458144 | 2.458148 | 2.458148 |
0.7000 | 2.426022 | 2.427372 | 2.427501 | 2.427515 |
0.7500 | 2.363057 | 2.367247 | 2.367649 | 2.367693 |
0.8000 | 2.284880 | 2.293734 | 2.294583 | 2.294677 |
0.8500 | 2.198218 | 2.215921 | 2.217618 | 2.217805 |
0.9000 | 2.096723 | 2.137438 | 2.141358 | 2.141792 |
0.9500 | 1.909498 | 2.051914 | 2.067177 | 2.068868 |
0.9750 | 1.665896 | 1.969510 | 2.027310 | 2.033900 |
0.9800 | 1.600374 | 1.929960 | 2.016812 | 2.027035 |
0.9900 | 1.481535 | 1.736193 | 1.865164 | 2.013432 |
0.9975 | 1.433728 | 1.463150 | 1.659478 | 2.003187 |
0.0050 | 1.000038 | 1.000089 | 1.000139 | 1.000203 | 1.000291 |
0.0250 | 1.000946 | 1.002203 | 1.003454 | 1.005009 | 1.007171 |
0.0500 | 1.003744 | 1.008664 | 1.013501 | 1.019434 | 1.027537 |
0.0750 | 1.008283 | 1.018999 | 1.029356 | 1.041818 | 1.058417 |
0.1000 | 1.014401 | 1.032695 | 1.050019 | 1.070410 | 1.096801 |
0.1250 | 1.021899 | 1.049184 | 1.074473 | 1.103551 | 1.140070 |
0.1500 | 1.030560 | 1.067907 | 1.101790 | 1.139857 | 1.186254 |
0.1750 | 1.040153 | 1.088335 | 1.131176 | 1.178259 | 1.234011 |
0.2000 | 1.050443 | 1.109986 | 1.161977 | 1.217974 | 1.282501 |
0.2250 | 1.061192 | 1.132414 | 1.193647 | 1.258436 | 1.331245 |
0.2500 | 1.072159 | 1.155206 | 1.225725 | 1.299237 | 1.380002 |
0.2750 | 1.083106 | 1.177960 | 1.257790 | 1.340055 | 1.428676 |
0.3000 | 1.093790 | 1.200274 | 1.289427 | 1.380600 | 1.477247 |
0.3250 | 1.103973 | 1.221726 | 1.320183 | 1.420549 | 1.525707 |
0.3500 | 1.113416 | 1.241869 | 1.349530 | 1.459481 | 1.574002 |
0.3750 | 1.121889 | 1.260218 | 1.376823 | 1.496781 | 1.621959 |
0.4000 | 1.129173 | 1.276257 | 1.401272 | 1.531525 | 1.669148 |
0.4250 | 1.135072 | 1.289461 | 1.421934 | 1.562324 | 1.714591 |
0.4500 | 1.139416 | 1.299330 | 1.437770 | 1.587206 | 1.756046 |
0.4750 | 1.142077 | 1.305442 | 1.447778 | 1.603709 | 1.788296 |
0.5000 | 1.142973 | 1.307513 | 1.451208 | 1.609538 | 1.801390 |
0.0050 | 1.000380 | 1.000632 | 1.001263 | 1.002520 | 1.006254 |
0.0250 | 1.009316 | 1.015351 | 1.029852 | 1.056549 | 1.121698 |
0.0500 | 1.035409 | 1.056708 | 1.103187 | 1.174045 | 1.290759 |
0.0750 | 1.074091 | 1.114340 | 1.191978 | 1.286865 | 1.396462 |
0.1000 | 1.120924 | 1.179364 | 1.278177 | 1.375881 | 1.463389 |
0.1250 | 1.172332 | 1.245932 | 1.355211 | 1.445158 | 1.513286 |
0.1500 | 1.225865 | 1.310985 | 1.422771 | 1.501934 | 1.555706 |
0.1750 | 1.280037 | 1.373333 | 1.482749 | 1.551418 | 1.594801 |
0.2000 | 1.334066 | 1.432842 | 1.537309 | 1.596787 | 1.632513 |
0.2250 | 1.387633 | 1.489886 | 1.588239 | 1.639912 | 1.669810 |
0.2500 | 1.440701 | 1.545030 | 1.636849 | 1.681877 | 1.707165 |
0.2750 | 1.493394 | 1.598853 | 1.684026 | 1.723276 | 1.744759 |
0.3000 | 1.545916 | 1.651863 | 1.730301 | 1.764355 | 1.782554 |
0.3250 | 1.598500 | 1.704442 | 1.775890 | 1.805074 | 1.820303 |
0.3500 | 1.651374 | 1.756798 | 1.820677 | 1.845096 | 1.857524 |
0.3750 | 1.704736 | 1.808878 | 1.864149 | 1.883732 | 1.893431 |
0.4000 | 1.758730 | 1.860203 | 1.905266 | 1.919853 | 1.926867 |
0.4250 | 1.813415 | 1.909478 | 1.942270 | 1.951800 | 1.956240 |
0.4500 | 1.868675 | 1.953769 | 1.972551 | 1.977382 | 1.979566 |
0.4750 | 1.923800 | 1.987051 | 1.992816 | 1.994141 | 1.994726 |
0.5000 | 1.966030 | 2.000000 | 2.000000 | 2.000000 | 2.000000 |
= 50,000 | = 500,000 | |||
0.0050 | 1.012356 | 1.101002 | 1.337454 | 1.432062 |
0.0250 | 1.196323 | 1.408628 | 1.449777 | 1.454511 |
0.0500 | 1.367709 | 1.467128 | 1.478710 | 1.479979 |
0.0750 | 1.447720 | 1.500308 | 1.505839 | 1.506440 |
0.1000 | 1.497758 | 1.530319 | 1.533634 | 1.533994 |
0.1250 | 1.537763 | 1.560180 | 1.562432 | 1.562676 |
0.1500 | 1.574108 | 1.590676 | 1.592328 | 1.592507 |
0.1750 | 1.609223 | 1.622078 | 1.623355 | 1.623493 |
0.2000 | 1.644166 | 1.654485 | 1.655507 | 1.655618 |
0.2250 | 1.679429 | 1.687905 | 1.688743 | 1.688833 |
0.2500 | 1.715212 | 1.722274 | 1.722971 | 1.723046 |
0.2750 | 1.751529 | 1.757450 | 1.758033 | 1.758096 |
0.3000 | 1.788237 | 1.793189 | 1.793676 | 1.793729 |
0.3250 | 1.825015 | 1.829106 | 1.829507 | 1.829551 |
0.3500 | 1.861331 | 1.864623 | 1.864946 | 1.864981 |
0.3750 | 1.896371 | 1.898902 | 1.899150 | 1.899176 |
0.4000 | 1.928968 | 1.930770 | 1.930946 | 1.930964 |
0.4250 | 1.957555 | 1.958677 | 1.958786 | 1.958798 |
0.4500 | 1.980206 | 1.980750 | 1.980803 | 1.980809 |
0.4750 | 1.994896 | 1.995041 | 1.995055 | 1.995056 |
0.5000 | 2.000000 | 2.000000 | 2.000000 | 2.000000 |
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Kiss, L.; Messaoudi, A.; Szeidl, G. Stability of Heterogeneous Beams with Three Supports—Solutions Using Integral Equations. Appl. Mech. 2023, 4, 254-286. https://doi.org/10.3390/applmech4010015
Kiss L, Messaoudi A, Szeidl G. Stability of Heterogeneous Beams with Three Supports—Solutions Using Integral Equations. Applied Mechanics. 2023; 4(1):254-286. https://doi.org/10.3390/applmech4010015
Chicago/Turabian StyleKiss, László, Abderrazek Messaoudi, and György Szeidl. 2023. "Stability of Heterogeneous Beams with Three Supports—Solutions Using Integral Equations" Applied Mechanics 4, no. 1: 254-286. https://doi.org/10.3390/applmech4010015