نظرية ثلاث عزوم

تُستخدم نظرية مور[3] لإستنتج الادلات الثلاث.[4]

نظرية مور.

${\displaystyle {\dfrac {PR}{RB'}}={\dfrac {SQ}{B'S}},}$

${\displaystyle {\dfrac {PR}{L1}}={\dfrac {SQ}{L2}}}$

(1)

إنحناء كمرة تحت تأثير هبوط.

${\displaystyle PR=\Delta B-\Delta A+PA'}$

(2)

${\displaystyle SQ=\Delta C-\Delta B-QC'}$

(3)

من 1,2,3 : ${\displaystyle {\dfrac {\Delta B-\Delta A+PA'}{L1}}={\dfrac {\Delta C-\Delta B-QC'}{L2}}}$

${\displaystyle {\dfrac {PA'}{L1}}+{\dfrac {QC'}{L2}}={\dfrac {\Delta A-\Delta B}{L1}}+{\dfrac {\Delta C-\Delta B}{L2}}}$

(a)

من نظرية مور الثانية:

${\displaystyle PA'=\left({\frac {1}{2}}\times {\frac {M_{1}}{E_{1}I_{1}}}\times L_{1}\right)\times L_{1}\times {\frac {1}{3}}+\left({\frac {1}{2}}\times {\frac {M_{2}}{E_{2}I_{2}}}\times L_{1}\right)\times L_{1}\times {\frac {2}{3}}+{\frac {A_{1}X_{1}}{E_{1}I_{1}}}}$

${\displaystyle QC'=\left({\frac {1}{2}}\times {\frac {M_{3}}{E_{2}I_{2}}}\times L_{2}\right)\times L_{2}\times {\frac {1}{3}}+\left({\frac {1}{2}}\times {\frac {M_{2}}{E_{2}I_{2}}}\times L_{2}\right)\times L_{2}\times {\frac {2}{3}}+{\frac {A_{2}X_{2}}{E_{2}I_{2}}}}$

تكون المعادلة النهائية:

${\displaystyle {\frac {M_{1}L_{1}}{E_{1}I_{1}}}+2M_{2}\left({\frac {L_{1}}{E_{1}I_{1}}}+{\frac {L_{2}}{E_{2}I_{2}}}\right)+{\frac {M_{3}L_{2}}{E_{2}I_{2}}}=6[{\frac {\Delta A-\Delta B}{L_{1}}}+{\frac {\Delta C-\Delta B}{L_{2}}}]-6[{\frac {A_{1}X_{1}}{E_{1}I_{1}L_{1}}}+{\frac {A_{2}X_{2}}{E_{2}I_{2}L_{2}}}]}$

منحني M / EI

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