Higher-order beam model for stress predictions in curved beams made from anisotropic materials

Abstract

A higher-order beam model for analyzing the flexural response of curved multilayered beams with constant curvature and arbitrary constant thickness is developed. The new model is derived from the Hellinger-Reissner mixed variational statement and predicts inherently equilibrated 3D stresses from an equivalent single-layer model. As a starting assumption, the hoop stress is formulated as a series of higher-order stress resultants multiplied by Legendre polynomials. The governing equations are derived in a generalized manner such that the modeling order can be adjusted and is not defined a priori. Hence, the highest order Legendre polynomial determines the modeling order. The through-thickness shear and normal stresses are derived by integrating the generalized hoop stress in Cauchy's polar equilibrium equations. As a result, all stress fields are based on the same set of variables, thereby considerably reducing the computational effort. The three stress fields, and two displacements in the radial and hoop directions are used in the Hellinger-Reissner functional to derive a new set of stress-displacement relations. The enforcement of the classical membrane and bending equilibrium equations of curved beams in the Hellinger-Reissner functional guarantees that all interlaminar and surface traction equilibrium conditions are satisfied exactly. A validation study of a composite laminate using a high-fidelity 3D finite element model shows that the stresses are captured very accurately by the present model, but with much less computational effort than the finite element model. As a result, the developed model can provide rapid and accurate insights into the expected damage onset behavior of curved laminates.

Аналітичний Розв’язок Задачі Згину Багатошарової Симетричної Кругової Арки Під Дією Нормальної Сили У Середньому Перерізі. Повідомлення 1. Арки Великої Кривизни

“…Для кругових композитних брусів розроблено велику кількість прикладних моделей згину [7,17,16,18,20], які дають змогу враховувати неоднорідність будови, анізотропію пружних властивостей, а також піддатливість деформаціям поперечного зсуву і обтиснення. Такі моделі дозволяють враховувати різноманітні типи навантажень і закріплень, однак внаслідок прийнятих допущень поступаються в точності розв'язкам теорії пружності [8,10,12,15,19,22,11,13].…”

Computationally efficient beam elements for accurate stresses in sandwich laminates and laminated composites with delaminations

“…xz(e) satisfies all interlaminar and surface traction conditions. In fact, the mathematical consistency of enforcing the interlaminar and surface traction conditions via the integration constants and the axial equilibrium equation was demonstrated in [31,41,42] using a different assumption for the transverse shear stress.…”

Investigation of failure initiation in curved composite laminates using a higher-order beam model

“…ϕ r z R t ϕ Α ϕ Β Figure 1: Definition of a curved beam element [15]. Figure 2 shows the loads acting on this curved beam, namely prescribed shear and radial tractionsT t andP t on the top surface andT b andP b on the bottom surface, which may vary with the angular co-ordinate ϕ. Additionally, the beam may be loaded by a hoop stress σ ϕ or hoop displacement u ϕ , and/or a transverse shear stress σ rϕ or transverse displacement u r at the two ends ϕ A and ϕ B .…”

“…These stress assumptions are used within the principle of minimum complementary energy to derive a set of 2D stress-displacement relations. To guarantee interlaminar continuity of the transverse stresses, the equivalent single-layer form of Cauchy's equilibrium equations [15], namely…”

“…In a recent publication, the present authors developed a higher-order beam model from the Hellinger-Reissner mixed variational principle for stress analysis in curved multilayered beams of arbitrary thickness [15]. The model was validated using a high-fidelity 3D finite element model and the hoop stress, interlaminar shear stress and radial stress were all predicted accurately for different loading conditions and lay-ups.…”