Tangent-Plane Homogenization

Tangent-plane homogenization computes a local equivalent ABD stiffness for a repeating stiffened cell that is treated as flat in the local tangent plane. The cell is assembled in the local e1-e2 plane. Surface curvature is handled later by geometry embedding and validity checks, not by the first local cell stiffness assembly.

For a cell with area \(A_\text{cell}\), a beam member contributes:

\[ \Delta\mathbf C_m = \frac{\mu_m L_m}{A_\text{cell}} \mathbf T_m^T\mathbf K_m\mathbf T_m. \]

Here:

\(\mu_m\) is multiplicity;
\(L_m\) is member length;
\(\mathbf K_m\) is the member stiffness matrix;
\(\mathbf T_m\) is the member strain map: it projects the stiffness's generalized strain onto each member's beam strain. This map is the hinge of the whole method, and it is referenced by name in the diagnostics below.

The equivalent tangent is:

\[ \mathbf C_\text{stiffness} = \mathbf C_\text{skin} + \sum_m \Delta\mathbf C_m. \]

The energy method is Tensyl's reference homogenizer because the assembled member contribution is symmetric by construction and works for graph-like canonical cells. Direct equilibrium-compatibility formulas are available for supported straight stiffener-family cases and are tested against the energy path.

This follows the equivalent-plate idea used by Nemeth for stiffened laminated plates and plate-like lattices. Tensyl treats those formulas as mechanics guidance and keeps the energy path as the executable reference.

How Geometry Enters the ABD Law

The tangent-plane homogenizer computes a local constitutive law. At a surface point, the shell or plate generalized strains and resultants are interpreted in the point's local right-handed frame:

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

The local law remains the same kind of object everywhere:

\[ \mathbf r = \mathbf C_\text{stiffness}\boldsymbol\eta. \]

Defining a barrel, dome, cone, or ellipsoid does not by itself bend the stiffener cell or insert curvature terms into the matrix above. Geometry enters through three separate mechanisms:

the surface supplies the local frame, metric, curvature, Jacobian, and positive minimum radius used to interpret and audit the law;
a stiffness field decides which ABD stiffness is present at each surface point;
a later shell, buckling, or sizing workflow uses the surface geometry to form equilibrium, loads, boundary conditions, and failure checks.

That separation is deliberate. The homogenizer answers a local constitutive question: "what resultants follow from these generalized strains in this tangent plane?" The surface answers a geometric question: "where is that tangent plane, which directions are local 1/2/n, and how curved is the midsurface?" A solver answers the global equilibrium question. Mixing those jobs would only make the numbers harder to trust.

The public stiffness-field helpers implement this separation directly:

ConstantStiffnessField returns the same canonical C8 tangent at each point and rebinds it to surface.point_at(u, v).frame. The numeric matrix is unchanged; the metadata and local frame describe where and how to read it.
HomogenizedStiffnessField calls a user-supplied cell factory at each surface point. The ABD stiffness can change pointwise if the factory changes pitch, member angle, eccentricity, section, material, or laminate with the local geometry.
ABDAtlas stores sampled linear ABD stiffnesses and interpolates the canonical C8 payload. The interpolated tangent is then bound to the target surface-point frame.

For a cylinder, e1 is axial, e2 is circumferential, and n is outward. A longitudinal stringer therefore has angle 0, and a ring rib has angle pi/2. The cylinder radius does not change the constant-field tangent, but it does set the curvature scale used by validity ratios such as \(p/R_\text{min}\) and \(h_s/R_\text{min}\).

For an ellipsoid, the same rule holds, but the frame and curvature vary over the surface. Uniform parameter spacing is not uniform physical pitch on a triaxial ellipsoid. If stiffener pitch or orientation is meant to follow physical distance, the pointwise cell factory or atlas samples must encode that choice. Tensyl will not infer a geodesic stiffener layout from the word "ellipsoid".

This is why the method is generalizable under its stated assumptions. Any smooth surface that can provide a local tangent frame and curvature scale can host the same local ABD law. The approximation is appropriate when the modeled response is scale separated from stiffener height, stiffener pitch, and local curvature, as discussed in Validity Limits. The mechanics basis follows the equivalent-plate, first-order plate/shell, differential-geometry, and homogenization sources listed in References.

Inputs

skin is an ABDStiffness for the unstiffened skin or laminate.
BeamSection supplies centroidal beam stiffness products. Those products may be entered directly or produced from isotropic thin-wall section geometry.
BeamMember supplies member length, angle, eccentricity, and multiplicity inside a finite canonical cell.
StiffenerFamily supplies angle, spacing, eccentricity, and multiplicity for the direct equilibrium-compatibility path.
CanonicalUnitCell.area is the tangent-plane area represented by one repeated cell.

Beam Section Quantities

BeamSection stores centroidal member-local stiffnesses:

Quantity	Meaning	Common units, US customary
`EA`	axial stiffness	`lbf`
`EIy`	bending stiffness about member-local `y`	`lbf*in^2`
`EIz`	bending stiffness about member-local `z`	`lbf*in^2`
`GJ`	torsional stiffness	`lbf*in^2`
`kGAy`	in-plane shear stiffness	`lbf`
`kGAz`	transverse shear stiffness	`lbf`

Omitted shear stiffnesses contribute zero in the current homogenizer and are recorded as assumptions in the result.

BeamSection still asks for stiffness products (EA, EIy, EIz, GJ, kGAy, kGAz) because the homogenizer consumes centroidal beam stiffnesses. The thin-wall section helpers are an upstream calculation layer; they do not change the member strain map.

Thin-Wall Section Geometry

The geometry helpers represent an isotropic stiffener as rectangular wall segments in the member-local (y, z) section plane. The X axis runs along the stiffener, z follows the wall normal used for member eccentricity, and y is the in-plane transverse section axis. Section constructors measure centroid_z from their own z = 0 construction datum; member eccentricity is still measured from the wall reference surface. If those datums differ, shift the centroid coordinate before building the cell.

For each segment, Tensyl sums the rectangular area contribution and then shifts to the section centroid. The resulting geometric properties are:

\[ A = \int_A dA,\qquad I_y = \int_A (z-z_c)^2\,dA, \]

\[ I_z = \int_A (y-y_c)^2\,dA,\qquad I_{yz} = \int_A (y-y_c)(z-z_c)\,dA. \]

The segment endpoints are midline coordinates. This is the intended thin-wall modeling convention; local corner buildup, weld radii, and exact flange/web overlap are outside this geometry helper.

For an isotropic material, the generated BeamSection uses:

\[ EA = E A,\qquad EI_y = E I_y,\qquad EI_z = E I_z,\qquad EI_{yz} = E I_{yz}. \]

The torsion value uses the open-section St Venant thin-wall approximation:

\[ J_{\mathrm{sv}} \approx \sum_i \frac{l_i t_i^3}{3},\qquad GJ = G\,J_{\mathrm{sv}}. \]

This is intentionally modest. It does not include closed-cell Bredt torsion, restrained warping, local flange/web stress recovery, crippling, or joint details. If those effects matter, compute the section properties externally and pass a BeamSection directly. There is no shame in outsourcing a problem to the tool that actually solves it.

There is no universal torsion constant for a stiffener drawing. Use the GJ that belongs to the member idealization and boundary condition in the equivalent wall. A freely warping blade, tee, angle, or channel should usually use an open-section St Venant value. A tube, closed hat, or box should use a closed-cell torsional stiffness only when the closed shear-flow path is truly part of the member model; if the skin is already modeled as the plate skin, do not count the same skin again inside the stiffener J. If joints, end constraints, or neighboring structure restrain warping, use a section-analysis or finite-element torsional stiffness for that restrained condition.

Transverse shear stiffnesses are optional. If shear_correction_y or shear_correction_z is supplied, Tensyl computes kGAy or kGAz as \(\kappa G A\). If a correction is omitted, the corresponding shear stiffness is left as None, and the homogenization result records the same omitted-shear assumption used for hand-entered BeamSection values.

Diagnostics

The homogenizer returns HomogenizationResult, not just an ABD stiffness. The result records:

symmetry, positive-semidefinite status, rank, member count, and cell area;
assumptions attached to the member strain map and section inputs;
a ValidityReport with scale-separation ratios and warning codes.

Two methods agreeing is necessary, not sufficient

Energy-vs-direct agreement is a good sign, but it is not proof. Both paths share the same member strain map, so they can agree and still be wrong together. For high-consequence use, you still need independent literature, test, or finite-element evidence.

Limits

The first homogenizer is a tangent-plane model. It does not model local joints, fasteners, stiffener crippling, curved stiffener geodesics, or full shell equilibrium. Use geometry validity ratios such as h_over_R, p_over_R, and p_over_L_response to decide whether the local flat-cell assumption is reasonable for the intended response mode.