Nonlinear mathematics in structural engineering

Accepted versio

Use of elastic stability analysis to explain the stress-dependent nature of soil strength

The peak and critical state strengths of sands are linearly related to the stress level, just as the frictional resistance to sliding along an interface is related to the normal force. The analogy with frictional sliding has led to the use of a ‘friction angle’ to describe the relationship between strength and stress for soils. The term ‘friction angle’ implies that the underlying mechanism is frictional resistance at the particle contacts. However, experiments and discrete element simulations indicate that the material friction angle is not simply related to the friction angle at the particle contacts. Experiments and particle-scale simulations of model sands have also revealed the presence of strong force chains, aligned with the major principal stress. Buckling of these strong force chains has been proposed as an alternative to the frictional-sliding failure mechanism. Here, using an idealized abstraction of a strong force chain, the resistance is shown to be linearly proportional to the magnitude of the lateral forces supporting the force chain. Considering a triaxial stress state, and drawing an analogy between the lateral forces and the confining pressure in a triaxial test, a linear relationship between stress level and strength is seen to emerge from the failure-by-buckling hypothesis

Postbuckling behaviour of beams with discrete nonlinear restraints

A beam with nonlinearly ‐ elastic lateral restraints attached at discrete points along its span is investigated via analytical and numerical methods. Previous results for the critical moment and the deflected shape based on an eigenvalue analysis of a similar beam with linearly ‐ elastic restraints are discussed, along with a validation of these results against an equivalent finite element model and results from numerical continuation. A beam with nonlinearly ‐ elastic restraints is then analysed with treatments for both quadratic and cubic restraint force–displacement relationships being provided. After formulation of the potential energy functionals, the governing differential equations of the system are derived via the calculus of variations and appropriate boundary conditions are applied. The equations are then solved using the numerical continuation software AUTO ‐ 07p for a standard I ‐ section beam. The variation in elastic critical buckling moment with the linear component of the restraint stiffness is tracked via a two ‐ parameter numerical continuation, allowing determination of the stiffness values at which the critical buckling modes changes qualitatively. Using these stiffness values, subsequent analyses are conducted to examine the influence of the nonlinear component of the restraint stiffness, from which post ‐ buckling equilibrium paths and deformation modes are extracted. The results of these analyses are then compared with an equivalent Rayleigh–Ritz formulation whereby the displacement components are represented by Fourier series. Equilibrium equations are derived by minimizing the potential energy functional with respect to the amplitudes of the constituent harmonics of the Fourier series. The amplitudes are solved for in the post ‐ buckling range by AUTO ‐ O7p and equilibrium paths are produced and compared to the equivalent solutions of the differential equations, with good agreement observed

Sensitivity of elastic thin-walled rectangular hollow section struts to manufacturing tolerance level imperfections.

Finite element models for elastic thin-walled rectangular hollow section (RHS) struts with pre-defined local and global geometric imperfections are developed within the commercial package ABAQUS. A unified local imperfection measurement based on equal local bending energy is proposed. The effects of imperfect cross-section profiles, imperfection wavelength in the longitudinal direction and the degree of imperfection localization on the ultimate load and equilibrium path are investigated and the most severe imperfection profiles are determined. A parametric study on the wavelength of the most severe local imperfection profile is conducted and a semi-empirical equation to approximate the corresponding wavelength is proposed. Moreover, an equation to calculate the global buckling load of thin-walled RHS struts with tolerance level doubly-symmetric cross-section local imperfections is proposed

Imperfection sensitivity of thin-walled rectangular hollow section struts susceptible to interactive buckling

A variational model describing the interactive buckling of thin-walled rectangular hollow section struts with geometric imperfections is developed based on analytical techniques. A system of nonlinear differential and integral equilibrium equations is derived and solved using numerical continuation. Imperfection sensitivity studies focus on the cases where the global and local buckling loads are close. The equilibrium behaviour of struts with varying imperfection sizes, characterized by the equilibrium paths and the progressive change in local buckling wavelength, is highlighted and compared. The numerical results reveal that struts exhibiting mode interaction are very sensitive to both local and global imperfections. The results from the variational model are verified using the finite element method in conjunction with the static Riks method and show good comparisons. A simplified method to calculate the pitchfork bifurcation load where mode interaction is triggered for struts with a global imperfection is developed for the first time. The simplified method is calibrated to predict the ultimate load for struts with tolerance level global imperfections and combined imperfections based on the parametric study, which also reveals that local and global imperfections are relatively more significant where global and local buckling are critical respectively. Finally, the ultimate load for struts with tolerance level geometric imperfections is compared with the existing Direct Strength Method (DSM). Potential dangers of making unsafe load-carrying capacity predictions by the DSM are highlighted and an improved strength equation is proposed

Sensitivity to local imperfections in inelastic thin-walled rectangular hollow section struts

Mass efficient inelastic thin-walled rectangular hollow section (RHS) struts practically always fail in a combination of local–global interactive buckling and material nonlinearity while also exhibiting high sensitivity to initial imperfections. Nonlinear finite element (FE) models for inelastic thin-walled RHS struts with pre-defined local and global geometric imperfections are developed within the commercial package Abaqus. Using a unified local imperfection measurement based on equal local bending energy, the effects of imperfect cross-section profiles, imperfection wavelength and the degree of localization in the longitudinal direction on the ultimate load and the nonlinear equilibrium path are investigated for four characteristic length struts at different material yielding stress levels. The corresponding most severe imperfection profiles are determined and are found to be qualitatively different to the linear eigenmodes in all cases. Moreover, it is found that the most severe purely periodic imperfections may be used to provide a safe approximation of the ultimate load when the corresponding amplitude is constrained to the manufacturing tolerance level. An extensive parametric study on the wavelength of the most severe periodic imperfection profile is conducted and a relationship for this is proposed in terms of the normalized local slenderness, which compares excellently with the FE results

Stability of multiple-crossarm prestressed stayed columns with additional stay systems

Prestressed stayed columns have an enhanced resistance to buckling through the effective use of crossarms and pretensioned stays when compared to conventional columns. An analytical derivation of the minimum, linear optimum and maximum initial pretension forces for configurations of prestressed stayed columns with multiple crossarms and additional stays is presented for the first time. The findings are validated through comparisons with finite element models developed in the commercial package ABAQUS. The influence of the initial pretension on the load-carrying capacity of the configurations considered is also analysed, providing insight into the actual optimum initial pretension force for the configurations accounting for the significance of geometric nonlinearities