ティモシェンコ梁たわみ公式

片持ち等分布†

境界値問題†

\( q=EIv''''_{(z)} \) \( EIv''''_{(z)}=q \) \( EIv'''_{(z)}=qz+A \) \( EIv''_{(z)}=\frac{qz^2}{2}+Az+B \) \( EIv'_{(z)}=\frac{qz^3}{6}+\frac{Az^2}{2}+Bz+C \) \( EIv_{(z)}=\frac{qz^4}{24}+\frac{Az^3}{6}+\frac{Bz^2}{2}+Cz+D \) \( M_{(z)}=-EI(v''_{(z)}+\frac{q}{kGA}) \) \( S_{(z)}=-EIv'''_{(z)} \) \( \theta_{(z)}=\frac{M'_{(z)}}{kGA}-v'_{(z)} \) \( v_{(0)}=0 \) \( D=0 \) \( S_{(\ell)}=0=-EIv'''_{(\ell)} \) \( EIv'''_{(z)}=q\ell+A \) \( -q\ell-A=0 \) \( A=-q\ell \) \( M_{(\ell)}=0=-EI(v''_{(\ell)}-\frac{q}{kGA}) \) \( EIv''_{(\ell)}=\frac{q\ell^2}{2}-q\ell^2+B \) \( 0=-\frac{q\ell^2}{2}+q\ell^2-B+\frac{qEI}{kGA} \) \( B=\frac{q\ell^2}{2}+\frac{qEI}{kGA} \) \( \theta_{(0)}=\frac{M'_{(0)}}{kGA}-v'_{(0)} \) \( M'_{(0)}=q\ell \) \( -v'_{(0)}=-\frac{C}{EI} \) \( \theta_{(0)}=0=\frac{q\ell}{kGA}-\frac{C}{EI} \) \( C=\frac{q\ell EI}{kGA} \) \( EIv_{(z)}=\frac{qz^4}{24}-\frac{q\ell z^3}{6}+\frac{q\ell^2 z^2}{4}-\frac{qEIz^2}{2kGA}+\frac{q\ell EIz}{kGA} \) \( v_{(z)}=\frac{q}{24EI}(z^4-4\ell z^3+6\ell^2 z^2)+\frac{q}{2kGA}(2\ell z-z^2) \) \( v_{(\ell)}=\frac{q}{24EI}(\ell^4-4\ell^4+6\ell^4)+\frac{q\ell^2}{2kGA} \) \( v_{(\ell)}=\frac{q\ell^4}{8EI}+\frac{q\ell^2}{2kGA} \)

単純梁等分布†

境界値問題†

\( q=EIv''''_{(z)} \) \( EIv''''_{(z)}=q \) \( EIv'''_{(z)}=qz+A \) \( EIv''_{(z)}=\frac{qz^2}{2}+Az+B \) \( EIv'_{(z)}=\frac{qz^3}{6}+\frac{Az^2}{2}+Bz+C \) \( EIv_{(z)}=\frac{qz^4}{24}+\frac{Az^3}{6}+\frac{Bz^2}{2}+Cz+D \) \( M_{(z)}=-EI(v''_{(z)}+\frac{q}{kGA}) \) \( S_{(z)}=-EIv'''_{(z)} \) \( \theta_{(z)}=\frac{M'_{(z)}}{kGA}-v'_{(z)} \) \( v_{(0)}=0 \) \( D=0 \) \( M_{(0)}=0 \) \( EIv''_{(0)}=B \) \( -EIv''_{(0)}=-B \) \( M_{(0)}=-B-\frac{qEI}{kGA} \) \( B=-\frac{qEI}{kGA} \) \( S_{(\frac{\ell}{2})}=0 \) \( EIv'''_{(\frac{\ell}{2})}=\frac{q\ell}{2}+A \) \( 0=-\frac{q\ell}{2}-A \) \( A=-\frac{q\ell}{2} \) \( v_{(\ell)}=0 \) \( EIv_{(z)}=\frac{qz^4}{24}-\frac{q\ell z^3}{12}-\frac{qEIz^2}{2kGA}+\frac{q\ell^3 z}{24}+\frac{qEI\ell z}{2kGA} \) \( v_{(z)}=\frac{q}{24EI}(z^4-2\ell z^3+q\ell^3 z)+\frac{q\ell^2}{2kGA}(-z^2+\ell z) \) \( v_{(\frac{\ell}{2})}=\frac{q\ell^4}{24EI}(\frac{1}{16}-\frac{1}{4}+\frac{1}{2})+\frac{q\ell^2}{2kGA}(-\frac{1}{4}+\frac{1}{2}) \) \( v_{(\frac{\ell}{2})}=\frac{5q\ell^4}{384EI}+\frac{q\ell^2}{8kGA} \)

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