Minkowski plasticity in 3D frames: Decoupled construction of the cross‐section yield surface and efficient stress update strategy

Abstract

Summary This work makes the Minkowski sum of ellipsoids into a consolidated tool for the representation of the yield surface of arbitrarily shaped composite cross sections under axial force and biaxial bending and shows how best to use it within the incremental nonlinear analysis of three‐dimensional frames. A geometric interpretation of each term of the sum allows us to construct complex convex surfaces using a low number of ellipsoids, each of them evaluated in a robust and efficient decoupled manner. Specialized algorithms, which exploit the parameterization of the yield surface in terms of a cross‐section collapse mechanism, are proposed for an efficient stress update based on the elastic predictor–return mapping scheme on a three‐dimensional beam finite element. The derivation of the algorithmic tangent moduli for the Minkowski sum completes the elements required for an efficient path‐following nonlinear analysis. The proposed methodology is general and can be applied for the representation of any zonoid yield surface and the corresponding implicit stress integration to a variety of structural and plasticity models.

A reduced order model for nonlinear time history seismic analyzes of elasto‐plastic 3D frame structures

“…Firstly, the structure is discretized spatially by means of finite elements. [32,33] Then, the semi-discrete equations of motions are solved in time by direct integration. A wide set of different methods are reported in Andujar et al [34] to integrate step-by-step the semi-discrete equations of motion in time.…”

“…For an anisotropic plastic behavior, as occurs for reinforced concrete buildings, two different plastic collapse mechanisms are evaluated for ± P. Each mechanism is obtained as the last displacement increment Δ𝐮 𝑐 corresponding to Δ𝜆 = 0 in the incremental-iterative solution of Equation ( 27) (see Magisano et al [32] and Magisano et al [37] ), that is, when plastic collapse occurs. Other solution strategies based on direct methods for limit analysis can be also adopted as in Skordeli et al, [42] Spiliopoulos and Dais, [43] Bleyer and De Buhan, [44] El Boustani et al [45] The plastic mode is normalized as…”

An efficient isostatic mixed shell element for coarse mesh solution

“…Similarly, the flexural contributions are collected by Q f and H f , respectively. Due to the assumptions introduced above for defying the generalized stress field and by recalling Equations (19) and (20), the compatibility and compliance element matrices are evaluated through a line integration along the contour of the element as…”

“…Moreover, mixed FE are diffusely employed for the analysis of structures undergoing physical nonlinearities. [19][20][21] An interesting framework for developing mixed FE is the Trefftz method. 22 This method is based on assuming stress interpolations that a priori simultaneously satisfy both equilibrium and compatibility equations.…”

“…Recent formulations of mixed FE can be found in References 17,18. Moreover, mixed FE are diffusely employed for the analysis of structures undergoing physical nonlinearities 19‐21 …”