Kinematic fidelity and instability transitions of functionally graded shallow arches with elastic rotational restraints

Tekin Atacan A., Mollamahmutoğlu Ç., Sariçobanoğlu Yalçinkaya K. A.

Journal of the Brazilian Society of Mechanical Sciences and Engineering, cilt.48, ss.1-37, 2026 (Hakemli Dergi)

Özet

Nonlinear equilibrium-path transitions of functionally graded (FG) circular shallow arches with elastic rotational end restraints under full-span uniformly distributed vertical dead loading are investigated. Starting from finite-deformation fiber kinematics and finite-strain measures, three small-material-strain Euler–Bernoulli models are derived: an exact-rotation (ER) model retaining exact centerline geometry and curvature, a metric-consistent von Kármán (VK) model, and a shallow von Kármán (SVK) model. A nine-field mixed field formulation is discretized by the generalized differential quadrature method and solved using bordered cylindrical arc-length continuation. Model fidelity is assessed against two-dimensional finite-deformation, finite-strain ABAQUS simulations for 50 cases using load- and displacement-reversal landmarks. ER closely reproduces the ABAQUS equilibrium paths. VK remains below a 15% mean combined landmark-error threshold for most cases, whereas SVK preserves response topology but exceeds this threshold in parts of the high-rise/high-slenderness region. The occasional closer agreement of SVK with ER, relative to VK, is explained by cancellation between higher-order compatibility terms. Within the investigated range and along primary paths traced without branch switching, identical restraints yield only nonlinear bending or snap-through, whereas snap-back and looping occur with non-identical restraints. Restraint asymmetry is therefore enabling but insufficient; geometry and graded stiffness also govern the transition. For (L/t = 20) and dimensionless left/right rotational flexibilities (α₀ = 1, α₁ = 2) the snap-through-to-looping transition is bracketed by (f/L = 0.175–0.176) for a homogeneous arch and (0.161–0.162) for an FG arch with ceramic-to-metal modulus ratio (E* = E_c/E_m = 1.5509) and material-gradient index (p = 1). The results establish applicability limits for reduced models and extend looping from concentrated or partial loading to full-span vertical loading.