05.6.03 / Structural / Cylinder model 2

Hyer thick laminated cylinder

Resolve the radial coordinate explicitly through every ply so pressure tractions, interlaminar stress, and displacement continuity are retained across a thick laminated wall.

Methodology

The Hyer route follows the generalized-plane-deformation treatment for long laminated cylinders. Each concentric orthotropic ply is transformed into the axial–hoop–radial coordinate system. The solver constructs a layer solution, enforces radial equilibrium inside each ply, applies displacement and traction continuity at every interface, and closes the system with inner and outer pressure boundary conditions.

H-01Axisymmetric strain field

ε_x = ε̄_x

ε_θ = u_r(r) / r

ε_r = du_r(r) / dr

Equation detailsExplanation · variables · model connection · reference

Defines the axial, hoop, and radial strains for an axisymmetric cylinder with generalized uniform axial extension and layerwise radial displacement.

Variablesε_xUniform generalized axial strainm/mu_rLayerwise radial displacementmrCurrent radial coordinate in the reference wallmε_θHoop strainm/mε_rRadial strainm/m

Model connectionThese kinematic fields are evaluated separately in each ply while sharing interface displacement constraints.

Theory basisHyer (1988), stress analysis of thick laminated cylinders

H-02Layer constitutive equation

σ^(k) = C̄^(k)[ε^(k) − ᾱ^(k)ΔT^(k) − β̄^(k)ΔC^(k)]

Equation detailsExplanation · variables · model connection · reference

Relates the three-dimensional stress state in each transformed orthotropic ply to mechanical strain after thermal and moisture free strains are removed.

Variablesσ^(k)Stress vector in ply kPaC̄^(k)Transformed 3D ply stiffnessPaᾱ^(k)Transformed ply CTE vector1/Kβ̄^(k)Transformed moisture-expansion vector1 or concentration basis

Model connectionThe pristine or damaged per-ply property source is resolved upstream and passed unchanged into this layerwise cylinder branch.

Theory basisCDS independent per-ply material routing

H-03Radial equilibrium

dσ_r/dr + (σ_r − σ_θ)/r = 0

dτ_rx/dr + τ_rx/r = 0

Equation detailsExplanation · variables · model connection · reference

Enforces axisymmetric equilibrium through the wall so radial and hoop stress are mechanically compatible in every ply.

Variablesσ_rRadial normal stressPaσ_θHoop normal stressPaτ_rxRadial–axial shear stressPa

Model connectionThe resulting ordinary differential system determines each layer's radial solution coefficients.

Theory basisHyer (1988), radial equilibrium formulation

H-04Interface and pressure boundary conditions

u_r^(k) = u_r^(k+1), σ_r^(k) = σ_r^(k+1), τ_rx^(k) = τ_rx^(k+1)

σ_r(r_i) = −p_i, σ_r(r_o) = −p_o

Equation detailsExplanation · variables · model connection · reference

Closes the multilayer system by preserving radial displacement and traction at ply interfaces and applying pressure tractions at the inner and outer radii.

Variablesr_iCylinder inner radiusmr_oCylinder outer radiusmp_iInternal pressurePap_oExternal pressurePa

Model connectionCDS assembles these equations and conditions into one linear coefficient system for the active laminate wall.

Theory basisHyer (1988), layer and surface conditions

Recovered results

The model recovers radial displacement and the axial, hoop, radial, and interlaminar shear stresses at requested through-wall positions. Named results provide the load and resultant histories plus the final radial-coordinate and stress-recovery table.

Selection guidance

Use shell cylinder when	Use Hyer cylinder when
Rapid sizing, membrane dominance, and large radius-to-thickness ratio	Thick walls, strong radial gradients, or interface tractions matter
Hoop/axial resultants are the primary design quantities	Layerwise radial displacement and interlaminar stress are required

Verification basis. Check single-layer isotropic limits, traction continuity, pressure boundary recovery, equilibrium, and convergence of through-wall sampling before using the layerwise solution in design decisions.

Workbench availability: released models, inputs and compatible study paths. The wider theory library includes reference formulations not available in every Workbench solve.