Implementation of rectangular members in HD and SD #2646

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Merged
andrew-platt merged 43 commits into OpenFAST:dev from luwang00:f/RectangularMembers
Apr 11, 2025
+5,808 −1,712
docs/source/user/subdyn/theory.rst
            
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    @@ -389,17 +389,17 @@ element stiffness and consistent mass matrices can be written as follows
  
        {\scriptstyle

        [k_e]= 

                \begin{bmatrix}

                             \frac{12 E J_y} {L_e^3 \left(  1+ K_{sy} \right)} & 0                                                & 0                 & 0                                                                      & \frac{6 E J_y}{L_e^2 \left(  1+ K_{sy} \right)}                        & 0                 & -\frac{12 E J_y}{L_e^3 \left(  1+ K_{sy} \right)}  & 0                                                   & 0                  & 0                                                                      & \frac{6 E J_y}{L_e^2 \left(  1+ K_{sy} \right)}                        & 0 \\

                                                                               & \frac{12 E J_x}{L_e^3 \left(  1+ K_{sx} \right)} & 0                 & -\frac{6 E J_x}{L_e^2 \left (  1+ K_{sx} \right )}                     & 0                                                                      & 0                 & 0                                                  & -\frac{12 E J_x}{L_e^3 \left (  1+ K_{sx} \right )} & 0                  & -\frac{6 E J_x}{L_e^2 \left (  1+ K_{sx} \right )}                     & 0                                                                      & 0 \\

                             \frac{12 E J_y} {L_e^3 \left(  1+ K_{sx} \right)} & 0                                                & 0                 & 0                                                                      & \frac{6 E J_y}{L_e^2 \left(  1+ K_{sx} \right)}                        & 0                 & -\frac{12 E J_y}{L_e^3 \left(  1+ K_{sx} \right)}  & 0                                                   & 0                  & 0                                                                      & \frac{6 E J_y}{L_e^2 \left(  1+ K_{sx} \right)}                        & 0 \\

                                                                               & \frac{12 E J_x}{L_e^3 \left(  1+ K_{sy} \right)} & 0                 & -\frac{6 E J_x}{L_e^2 \left (  1+ K_{sy} \right )}                     & 0                                                                      & 0                 & 0                                                  & -\frac{12 E J_x}{L_e^3 \left (  1+ K_{sy} \right )} & 0                  & -\frac{6 E J_x}{L_e^2 \left (  1+ K_{sy} \right )}                     & 0                                                                      & 0 \\

                                                                               &                                                  & \frac{E A_z}{L_e} & 0                                                                      & 0                                                                      & 0                 & 0                                                  & 0                                                   & -\frac{E A_z}{L_e} & 0                                                                      & 0                                                                      & 0 \\

                                                                               &                                                  &                   & \frac{\left(4 + K_{sx} \right) E J_x}{L_e \left (  1+ K_{sx} \right )} & 0                                                                      & 0                 & 0                                                  & \frac{6 E J_x}{L_e^2 \left (  1+ K_{sx} \right )}   & 0                  & \frac{\left( 2-K_{sx} \right) E J_x}{L_e \left (  1+ K_{sx} \right )}  & 0                                                                      & 0  \\

                                                                               &                                                  &                   &                                                                        & \frac{\left(4 + K_{sy} \right) E J_y}{L_e \left (  1+ K_{sy} \right )} & 0                 & -\frac{6 E J_y}{L_e^2 \left (  1+ K_{sy} \right )} & 0                                                   & 0                  & 0                                                                      & \frac{\left( 2-K_{sy} \right) E J_y}{L_e \left (  1+ K_{sy} \right )}  & 0  \\

                                                                               &                                                  &                   & \frac{\left(4 + K_{sy} \right) E J_x}{L_e \left (  1+ K_{sy} \right )} & 0                                                                      & 0                 & 0                                                  & \frac{6 E J_x}{L_e^2 \left (  1+ K_{sy} \right )}   & 0                  & \frac{\left( 2-K_{sy} \right) E J_x}{L_e \left (  1+ K_{sy} \right )}  & 0                                                                      & 0  \\

                                                                               &                                                  &                   &                                                                        & \frac{\left(4 + K_{sx} \right) E J_y}{L_e \left (  1+ K_{sx} \right )} & 0                 & -\frac{6 E J_y}{L_e^2 \left (  1+ K_{sx} \right )} & 0                                                   & 0                  & 0                                                                      & \frac{\left( 2-K_{sx} \right) E J_y}{L_e \left (  1+ K_{sx} \right )}  & 0  \\

                                                                               &                                                  &                   &                                                                        &                                                                        & \frac{G J_z}{L_e} & 0                                                  & 0                                                   & 0                  & 0                                                                      & 0                                                                      & -\frac{G J_z}{L_e} \\

                                                                               &                                                  &                   &                                                                        &                                                                        &                   & \frac{12 E J_y} {L_e^3 \left(  1+ K_{sy} \right)}  & 0                                                   & 0                  & 0                                                                      & -\frac{6 E J_y}{L_e^2 \left (  1+ K_{sy} \right )}                     & 0  \\

                                                                               &                                                  &                   &                                                                        &                                                                        &                   &                                                    & \frac{12 E J_x}{L_e^3 \left(  1+ K_{sx} \right)}    & 0                  & \frac{6 E J_x}{L_e^2 \left (  1+ K_{sx} \right )}                      & 0                                                                      & 0  \\

                                                                               &                                                  &                   &                                                                        &                                                                        &                   & \frac{12 E J_y} {L_e^3 \left(  1+ K_{sx} \right)}  & 0                                                   & 0                  & 0                                                                      & -\frac{6 E J_y}{L_e^2 \left (  1+ K_{sx} \right )}                     & 0  \\

                                                                               &                                                  &                   &                                                                        &                                                                        &                   &                                                    & \frac{12 E J_x}{L_e^3 \left(  1+ K_{sy} \right)}    & 0                  & \frac{6 E J_x}{L_e^2 \left (  1+ K_{sy} \right )}                      & 0                                                                      & 0  \\

                                                                               &                                                  &                   &                                                                        &                                                                        &                   &                                                    &                                                     & \frac{E A_z}{L_e}  & 0                                                                      & 0                                                                      & 0 \\

                                                                               &                                                  &                   &                                                                        &                                                                        &                   &                                                    &                                                     &                    & \frac{\left(4 + K_{sx} \right) E J_x}{L_e \left (  1+ K_{sx} \right )} & 0                                                                      & 0   \\

                                                                               &                                                  &                   &                                                                        &                                                                        &                   &                                                    &                                                     &                    &                                                                        & \frac{\left(4 + K_{sy} \right) E J_y}{L_e \left (  1+ K_{sy} \right )} & 0   \\

                                                                               &                                                  &                   &                                                                        &                                                                        &                   &                                                    &                                                     &                    & \frac{\left(4 + K_{sy} \right) E J_x}{L_e \left (  1+ K_{sy} \right )} & 0                                                                      & 0   \\

                                                                               &                                                  &                   &                                                                        &                                                                        &                   &                                                    &                                                     &                    &                                                                        & \frac{\left(4 + K_{sx} \right) E J_y}{L_e \left (  1+ K_{sx} \right )} & 0   \\

                                                                               &                                                  &                   &                                                                        &                                                                        &                   &                                                    &                                                     &                    &                                                                        &                                                                        & \frac{G J_z}{L_e} \\

                     \end{bmatrix}

         }
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Implementation of rectangular members in HD and SD #2646

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