Frames and Conventions

An ABD stiffness is expressed in a right-handed local frame:

\[ \{\mathbf e_1,\mathbf e_2,\mathbf n\}. \]

e1 and e2 are tangent directions. n is the positive normal. Tensyl validates:

\[ \mathbf e_1 \times \mathbf e_2 = \mathbf n. \]

Positive in-plane rotation is counterclockwise about n:

\[ \mathbf e_1'=\cos\psi\,\mathbf e_1+\sin\psi\,\mathbf e_2, \qquad \mathbf e_2'=-\sin\psi\,\mathbf e_1+\cos\psi\,\mathbf e_2. \]

Rotation of ABD Stiffnesses

Generalized strains and generalized resultants use separate transforms:

\[ \boldsymbol\eta'=\mathbf T_\eta\boldsymbol\eta, \qquad \mathbf r'=\mathbf T_r\mathbf r. \]

The rotated tangent is:

\[ \mathbf C'=\mathbf T_r\mathbf C\mathbf T_\eta^{-1}. \]

This preserves power:

\[ \mathbf r'^T\boldsymbol\eta'=\mathbf r^T\boldsymbol\eta. \]

Reference Surface

The default reference surface is the mid-surface. If a stiffness is shifted by distance \(d\) along the positive normal, membrane strain at the old surface is:

\[ \boldsymbol\epsilon_\text{old} = \boldsymbol\epsilon_\text{new} - d\boldsymbol\kappa. \]

Use shift_reference_surface when superposing facesheets or moving an ABD stiffness to a different shell reference surface.

        +n
        ^
        |  positive eccentricity
        |      o stiffener centroid
--------+-------------------------- reference surface
       e1 ->       e2 completes the right-handed tangent frame

Eccentricity Inputs

Every eccentricity input follows the same sign rule:

Eccentricity is the signed distance from the reference surface to the stiffener or face centroid, measured along +n.

This applies to:

  • BeamMember.eccentricity;
  • StiffenerFamily.eccentricity;
  • stringer_eccentricity;
  • rib_eccentricity;
  • sandwich bottom_face_offset and top_face_offset.

For a cylinder whose local normal points outward, an external stringer has positive stringer_eccentricity. An internal stringer has negative eccentricity.

The eccentricity sign is not cosmetic

Flip it and you change the membrane-bending coupling block B, which gives you a physically different ABD stiffness. Tensyl raises no exception and no validation error, and the result looks entirely ordinary. Decide which way +n points before you type a sign.

Positive eccentricity adds membrane-bending coupling according to the chosen reference surface. Moving the reference surface also changes B. Equal and opposite faces in a symmetric sandwich can cancel coupling when the face stiffnesses and offsets are symmetric.

Member Angles

Member angles are measured in the local tangent frame:

  • 0 is along local e1;
  • pi/2 is along local e2;
  • positive angles follow Tensyl's positive in-plane rotation convention about n.

For the built-in Cylinder, e1 is axial and e2 is circumferential. A longitudinal stringer uses angle 0; a ring rib uses angle pi/2.

Engineering Shear

Tensyl's public strain convention uses engineering shear components gamma12, gamma13, and gamma23. Tensor shear is rejected because an unnoticed factor-of-two convention mix would corrupt the stiffnesses, energies, and resultants while leaving the matrices looking well-formed.