Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation

SummaryDisplacement‐based seismic assessment of buildings containing unreinforced masonry (URM) walls requires as input, among others, estimates of the in‐plane drift capacity at the considered limit states. Current codes assess the drift capacity of URM walls by means of empirical models with most codes relating the drift capacity to the failure mode and wall slenderness. Comparisons with experimental results show that such relationships result in large scatter and usually do not provide satisfactory predictions. The objective of this paper is to determine trends in drift capacities of modern URM walls from 61 experimental tests and to investigate whether analytical models could lead to more reliable estimates of the displacement capacity than the currently used empirical models. A recently developed analytical model for the prediction of the ultimate drift capacity for both shear and flexure controlled URM walls is introduced and simplified into an equation that is suitable for code implementation. The approach follows the idea of plastic hinge models for reinforced concrete or steel structures. It explicitly considers the influence of crushing due to flexural or shear failure in URM walls and takes into account the effect of kinematic and static boundary conditions on the drift capacity. Finally, the performance of the analytical model is benchmarked against the test data and other empirical formulations. It shows that it yields significantly better estimates than empirical models in current codes. The paper concludes with an investigation of the sensitivity of the ultimate drift capacity to the wall geometry, static, and kinematic boundary conditions.

Nonlocal inflected nano-beams: A stress-driven approach of bi-Helmholtz type

The Navier-Stokes equations with the kinematic and vorticity boundary conditions on non-flat boundaries

Rapidly rotating convection-driven dynamos are investigated under different kinematic and magnetic boundary conditions using direct numerical simulations. At a fixed rotation rate, represented by the Ekman number$Ek=5\times 10^{-7}$, the thermal forcing is varied from$2$to$20$times its value at the onset of convection ($\mathcal {R}=Ra/Ra_c=2\unicode{x2013}20$, where$Ra$is the Rayleigh number), keeping the fluid properties constant ($Pr=Pr_m=1$, where$Pr$and$Pr_m$are the thermal and magnetic Prandtl numbers). The statistical behaviour of the dynamos, including the force balance, energetics, and heat transport, depends on the boundary conditions that dictate both the boundary layer and the interior dynamics. At a fixed thermal forcing ($\mathcal {R}=3$), the structure and strength of the magnetic field produced by the dynamos, especially near the walls, depend on both velocity and magnetic boundary conditions. Though the leading-order force balance in the bulk remains geostrophic, the Lorentz force becomes comparable to the Coriolis force inside the thermal boundary layer with no-slip, perfectly conducting conditions. In this case, a term signifying the work done by the Lorentz force in the turbulent kinetic energy (TKE) equation is found to have some components that extract energy from the velocity field to produce the magnetic field, while some other components extract energy from the magnetic field to produce TKE. However, with no-slip, pseudo-vacuum conditions, all the components of the work done by the Lorentz force perform unidirectional energy transfer to produce magnetic energy from the kinetic energy of the fluid to sustain dynamo action. We find heat transfer enhancement in the rotating dynamo convection, as compared with non-magnetic rotating convection, with the peak enhancement lying in the range$\mathcal {R}=3\unicode{x2013}4$. For free-slip conditions, in the absence of an Ekman layer, the dynamo action may alter the heat transport significantly by suppressing the formation of large-scale vortices. However, the highest heat transfer enhancement is found at$\mathcal {R}=3$with no-slip, perfectly conducting walls, which can be attributed to a local magnetorelaxation of the rotational constraint due to enhanced Lorentz force inside the thermal boundary layer.

On the role of higher-order conditions in distortion gradient plasticity

Object of the paper is a class of structural problems that will be abstractly defined by the following three sets of equations: the compatibility equation C(u) = e, where the operator C:u —>D is assumed linear (and so is its dual C’ that relates the internal stress a to the external actions f ∈ u’); the constitutive equation f (e) = σ, assumed to be a monotone lower semi-continuos map from D to D’. Therefore the map is obtained from a generalized potential, that turns out to be lower semi-continuos and convex. This means that the material behaviour is reversible; the external force field, supposed to derive from a functional that is assumed to be lower semi-continuos and convex. The functional defines also the (non-linear) static and kinematic boundary conditions (eventually unilateral).

The general criteria for arch bridge design, outlined in the 'static' and 'kinematic' approaches used in the 18th century, are found to agree with a modern formulation, and also allow the identification of the static and kinematic boundary conditions. Historical research should be included in the structural domain, because of such convergence of methods and results. The static approach was formulated by Coulomb, and developed by several others in France especially. The kinematic approach is mainly attributed to Mascheroni, and became an important reference point for Italian mechanical culture. The mathematical principles of both approaches are outlined. A modern analysis is given of the arch as a rigid system with unilateral constraints. Its kinematics and kinetics are presented for values of friction preventing sliding. A two-dimensional arch, consisting of several rigid voussoirs, is considered in the equilibrium configuration under its own weight. The kinematics of the arch and its elements is here governed by the impenetrability law of solid mechanics and by the limit value of dry friction. Mascheroni's mechanisms are re-analysed. An equilibrium analysis is given for the arch considered as a voussoir of increasing width.

Objective: Frontal crashes in which the vehicle has poor structural engagement, such as small-overlap and oblique crashes, account for a large number of fatalities. These crash modes are characterized by large intrusion and vehicle yaw rotation. Results from previous studies have shown mixed results regarding the importance and effects of these parameters. The aim of this study was to evaluate how vehicle yaw rotation, instrument panel intrusion, and the time history of the pulse angle influence chest injury outcomes.Method: This study was conducted using kinematic boundary conditions derived from physical crash tests, which were applied on a finite element simulation model of a vehicle interior including a finite element human model. By performing simulations with different levels of simplified boundary conditions and comparing the results to a simulation with a full set of boundary conditions, the influence of the simplifications was evaluated. The injury outcome measure compared between the simulations was the expected number of fractured ribs. The 3 simplifications simulated were (1) removal of vehicle yaw rotation, (2) removal of vehicle yaw rotation plus an assumption of a constant pulse angle between the x- and y-acceleration, and (3) removal of instrument panel intrusion.The kinematic boundary conditions were collected from 120 physical tests performed at the Insurance Institute of Highway Safety; 77 were small-overlap tests, and 43 were moderate overlap tests. For each test, the full set of boundary conditions plus the 3 simplifications were simulated. Thus, a total of 480 simulations were performed.Results: The yaw rotation influences occupant interaction with the frontal airbag. For the approximation without this kinematic boundary component, there was an average error in injury outcome of approximately 13% for the moderate overlap cases. Large instrument panel intrusion increases the risk of rib fracture in nearside small-overlap crashes. The mechanism underlying this increased fracture risk is a combination of increased airbag load and a more severe secondary impact to the side structure. Without the intrusion component, the injury risk was underestimated by 8% for the small-overlap crashes.Conclusion: The approximation with least error was version 2; that is, a model assuming a constant pulse angle, including instrument panel intrusion but no vehicle yaw rotation. This approximation simulates a sled test with a buck mounted at an oblique angle. The average error for this approximation was as low as 2–4%.

The contribution of shear deformations to total deflections in slender concrete beams is often ignored in design practices. The negligence of shear deformations could lead to the underestimation of total deflections in slender concrete beams, which will result in unsafe designs of slender concrete beams at service conditions. The shear deformations could be more critical in concrete beams reinforced with FRP bars than reinforced concrete beams because of the lower elastic modulus of FRP bars as comparing to steel ones. To address this problem, this paper attempts to determine shear deformations of slender concrete beams reinforced with FRP bars based on the variable angle truss model. The shear span of the slender concrete beams is divided into the truss units. The inclinations of struts in these truss units are first calculated by using the principle of virtual work. The shear deformations of the beams are then evaluated by summing all the shear deformations in the truss units. The calculated shear deformations are incorporated into ACI 440.1R-15's deflection approach (2015). Comparisons are made between the analytical and available experimental results of slender concrete beams reinforced with FRP bars with respect to the deflections at 50% and 75% of the failure loads. Better results as comparing to the existing design guidelines in calculating the deflections of the slender concrete beams reinforced with FRP bars are obtained by incorporating the proposed shear deformations.

An explicit nonlinear model for large spatial deflections of symmetric slender beams

An semi-analytical solution method is presented for the buckling analysis of plates with a reinforced cutout under both uniform and non-uniform compression load- ing. The cutout is reinforced on both sides of the plate. The solution for buckling response is achieved in two subsequent steps: pre-buckling response from in-plane stress analysis and the critical buckling load and its mode from a buckling analysis that utilizes the pre- buckling stresses. The pre-buckling stresses are ob- tained based on the principle of minimum potential energy while automatically satisfying the equilibrium equations and compatibility condition. Hence, the strain energy of the plate is directly achieved by boundary integrals only. However, there exist material and thick- ness discontinuities between the unreinforced and rein- forced regions of the plate. Unlike the pre-buckling equilibrium equations, the buckling equations, which are obtained from the Treftz criterion, are highly com- plex, and the potential functions automatically satisfy- ing these equations do not exist. Hence, the Treftz cri- terion is applied with the assumed transverse displace- ment functions utilizing the full domain integration. In both pre-buckling and buckling analyses, the form of the assumed functions (real or complex) does not neces- sarily satisfy the kinematic boundary conditions. The kinematic boundary conditions are applied by introduc- ing elastic springs along the boundaries, and by enforcing the displacement field to satisfy kinematic boundary conditions through energy minimization.

Kinematics and constitutive relations in the stress-gradient theory: interpretation by homogenization

The kinematic boundary condition results from the application of conservation of mass at the bounding surface of a fluid. For a material interface, this condition reduces to the requirement that there is no flux of fluid across the boundary. As demonstrated, this can be satisfied by fluid particles which remain on the interface. However, following Kelvin, it is highlighted that this commonly referenced scenario is a special case of a more general result. The mathematical expression of the kinematic boundary condition is more complex when there is a transfer of mass across the interface. Conservation of mass is still applicable across interfaces of this type, and the resulting condition is shown to collapse to that for a material surface when the mass flux across the interface is zero.

Zaera et al. (Int J Eng Sci 138:65–81, 2019) recently showed that the nonlocal strain gradient theory (NSGT) is not consistent when it is applied to finite solids, since all boundary conditions associated to the corresponding problems cannot be simultaneously satisfied. Given the large number of works using the NSGT being currently published in the field of generalized continuum mechanics, it is pertinent to evince the shortcomings of the application of this theory. Some authors solved the problem omitting the constitutive boundary conditions. In the current paper we show that, in this case, the equilibrium fields are not compatible with the constitutive equation of the material. Other authors solved it omitting the non-standard boundary conditions. Here we show that, in this case, the solution does not fulfil conservation of energy. In conclusion, the inconsistency of the NSGT is corroborated, and its application must be prevented in the analysis of the mechanical behaviour of nanostructures.

The solution of elastostatic and elastodynamic problems with Chebyshev spectral finite elements