Repeated flattening and constraint-imposing operations in thin plates can leave mechanically distinct stress histories even when the final geometry appears identical. This makes history equivalence and reduction central issues in sequence-dependent plate mechanics. In this study, a finite-element stress fingerprint framework is presented to manage path-dependent stress histories in repeatedly loaded thin plates. A distortion-flattening cycle is defined by applying stress-free Zernike distortion to an anisotropic thin plate and then forcibly flattening it. The resulting top-surface von Mises stress field is represented as a field-level fingerprint that depends on the imposed mode and orientation. In the baseline formulation, exchanged operation orders produce final stress differences only at the numerical-error level. Histories containing the same set of operations can therefore be represented by a single sorted sequence and stored as one representative entry in an offline stress-fingerprint database. The framework is further extended by introducing stress feedback, where the previous stress state modifies the subsequent distortion response. This generates a unique distribution of pair similarities across all Zernike mode pairs. In this regard, a dissimilarity tolerance is introduced to identify commutable pairs and prune the sequence tree. It shows that inconsistencies between states can be managed at a low level by rearranging the given scenario to increase the commutativity of adjacent pairs. The proposed finite-element framework provides engineering procedures for steering repetitive loading sequences toward more predictable stress-history responses.
Conventional nodal concentrated-hinge models prescribe hinge locations before analysis, which is restrictive when critical stations change after internal-force redistribution in web-tapered steel frames. This paper presents a self-locating enriched beam–column element for fixed-mesh monotonic multi-hinge analysis of web-tapered steel frames. Within a standard two-node host, a node-vanishing rotational enrichment is formed by subtracting the cubic Hermite interpolant from a unit-rotation ramp. Its amplitude equals the localized hinge rotation and is removed from the global system by static condensation, leaving the host connectivity unchanged. At each converged control state, the algorithm searches eligible end and interior stations by comparing bending demand with position-dependent capacity. It inserts the selected hinge at zero amplitude, re-equilibrates the unchanged control state, and repeats the search on the redistributed frame. Tapered-cantilever tests show independent convergence of the enriched and conforming-split models toward the exact continuum response of the adopted beam–hinge problem. Further tests assess coarse-mesh self-location, descending response, and differences from independent distributed-plasticity fiber predictions. A single-bay study demonstrates movement of the first hinge into a member interior as the load pattern changes. In an eight-bay frame, the re-equilibrated analysis follows a nine-event sequence containing multiple interior hinges, while an intact-elastic companion remains subcritical at the final-event load. The coarse enriched model reproduces the event sequence and closely matches the global response and aggregate hinge rotations of independently locating fine conforming models while retaining a much smaller global system.
A strong-form meshfree method, called Taylor series-based meshfree (TSMF) method, is developed, which is efficient and robust while solving elliptic partial differential equations that exhibit sharp local field variable gradients over complex three-dimensional geometries. No background mesh is required in TSMF method, either for differentiation or integration, thus rendering it purely a node-based approach. The unique field variable value and its derivatives, at any interior domain node, are enforced by Taylor series expansion in a least square sense resulting in TSMF method shape functions. The TSMF method consists of derivative-term shape functions as well, thus no post-processing is required for flux computation. Only one variable (and its associated parameters), controlling local cloud size, needs to be assigned, compared with several tunable algebraic parameters used in various other contemporary meshfree methods. TSMF method local cloud topology selection strategy, guided by Taylor radius of convergence concept (specially for the problems having singularity in the field variable), is introduced to systematically handle the local cloud topologies, particularly in regions featuring sharp gradients or geometric singularities.Several validation problems are solved by TSMF method using both uniform and random distributions of field nodes. A progressive decrease in the global error is observed in the three-dimensional complex geometry problems involving the local high gradients of field variable distribution. It is observed, based on several numerical experiments, that the flux convergence rate is almost comparable in value with the corresponding field variable convergence rate. A close to quadratic flux convergence rate is thus obtained by TSMF method when the corresponding field variable convergence rate is quadratic, marking an improvement over the standard classical mesh-based methods where the flux convergence typically degrades.
This study presents an anisotropic strain-gradient elasticity framework in which direction-dependent microstructural interactions arise from a geometrically defined nonlocal domain. The resulting characteristic-length tensor comes from the second moments of a three-dimensional ellipsoidal averaging domain, giving it a direct geometric meaning similar to a fabric tensor. A positive-definite strain-gradient energy is discretized using smooth, continuously differentiable Hermite finite elements in one, two, and three dimensions, together with a corrected consistent mass formulation. For an axial characteristic length of 0.20 m and transverse lengths of 0.02 m, rotating the microstructural orientation produces a monotonic response. Static displacement increases by about 4%, while the fundamental frequency decreases from 186.3 Hz to 173.1 Hz. The normalized frequency ranges from 1.053 to 0.979 relative to a volume-equivalent isotropic reference of 176.8 Hz. The axial length dominates the transverse contribution, producing a 5.8% stiffness increase compared with only 0.5% in the transverse-dominant case. A Directional Stiffening Index, measured against a volume-equivalent isotropic reference, is predominantly positive, ranging from 0% to about 30% across the anisotropy ratios and orientations studied. Only a few high-angle configurations show small negative values. These trends allow the stiffness and vibration frequency of anisotropic microstructured materials to be tailored through controlled variation of characteristic length and orientation.
A mixed finite element formulation, derived consistently from the Gâteaux differential of the governing functional, is presented for the bending and free-vibration analysis of laminated composite shells. Five geometries – the flat plate and the cylindrical, spherical, hyperbolic paraboloid and elliptic paraboloid shells – and five kinematic models – the first-order theory together with the parabolic, exponential, hyperbolic and trigonometric higher-order theories – are treated within a single element by retaining the thickness function as a free parameter. In contrast to displacement-based formulations, the force and moment resultants, including their higher-order counterparts, are obtained directly as primary nodal unknowns, with the same accuracy as the displacements and without any post-processing differentiation. The in-plane and transverse shear stresses are subsequently recovered from these resultants: the former through a fully coupled inversion of the constitutive relations, the latter through thickness integration of the three-dimensional equilibrium equations, which restores interlaminar continuity and the traction-free surface conditions. Non-dimensional deflections, natural frequencies, stress resultants and stresses are validated against three-dimensional elasticity solutions and previously published higher-order and first-order results, showing excellent agreement throughout.
