The Design of Structural Design Norms

Abstract

This paper deals with the analysis of prismatic and nonprismatic members with axial restraints, where their material is permitted to be stressed well beyond its elastic limit. This loading on the members causes the modulus of elasticity of the material to vary along their length. The mathematical formulation of this problem, as well as its analysis, is based on the method of equivalent systems that was developed by the first author. This method permits replacement of the original nonlinear member of variable stiffness E^xI^x, with one of uniform stiffness E¹I^l, that has the same length, boundary conditions, and elastic line as the original variable stiffness member. It is proven mathematically that this equivalency exists and that the solution of the equivalent system using linear analysis yields the same results as the solution of the original nonlinear variable stiffness member. Deflections and rotations may be easily obtained using equivalent systems, and the member can be analyzed in both elastic and inelastic ranges, all the way to failure, thus permitting observation of progressive deterioration of the ability of the member to resist load, stress, and de-formation. In this manner, practical critical limits regarding these modes of behavior can be established.     

Constitutive modeling of fiber composites with a soft hyperelastic matrix

This paper presents an experimental and numerical study of unidirectional carbon fiber composites with a silicone matrix, loaded transversally to the fibers. The experiments show nonlinear behavior with significant strain softening under cyclic loading. The numerical study uses a plane-strain finite element continuum model of the composite material in which the fiber distribution is based on experimental observations and cohesive elements allow debonding to take place at the fiber/matrix interfaces. It is found that accurate estimates of the initial tangent stiffness measured in the experiments can be obtained without allowing for debonding, but this feature has to be included to capture the non-linear and strain-softening behavior.     

Apparatus for Testing Concrete under Multiaxial Compression at Elevated Temperature (mac2T)

This paper describes a new test facility for determining material mechanical properties of structural concrete. The novel facility subjects 100 mm cubic concrete specimens to true multiaxial compression (σ₁ ≠ σ₂ ≠ σ₃) up to 400 MPa at temperatures of up to 300°C. Forces are delivered through three independent loading frames equipped with servo-controlled hydraulic actuators creating uniform displacement boundary conditions via rigid platens. Specimen deformation is calculated from displacements measured to an accuracy of 10⁻⁶ m using a system of six laser interferometers. The combination of stiff loading frames, rigid platens, an accurate and reliable strain measurement system and a fast control system enables investigation of the material response in the post-peak range. The in-house developed control software allows complex multi-stage experiments involving (i) load and temperature cycling, (ii) small stress probes and (iii) arbitrary (pre-defined) loading paths. The program also enables experiments in which the values of the control parameters and the execution of the test sequences depend on the response of the specimen during the test. The capabilities of the facility are illustrated in this paper by experiments determining the effects of different heat-load regimes on the strength and stiffness of the material and tests identifying the tangent stiffness matrix of the material and the associated changes in the acoustic tensor under multiaxial compression.     

Design Sensitivity Analysis of Truss Structures with Elastoplastic Material*

ABSTRACT

A continuum-based design sensitivity analysis (DSA) method is presented for configuration (or layout) design of nonlinear structural systems with rate-independent elastoplastic material. Configuration design variables are characterized by shape and orientation changes of the structural component. A continuum-based shape DSA method that utilizes the material derivative of continuum mechanics is extended to account for effects of shape and orientation variations. The incremental analysis method, with updated Lagrangian formulation, is used to derive the design sensitivity for the nonlinear structural system.

To derive the design sensitivity, incremental energy and load forms are utilized. The first variations of energy and load forms and the static response with respect to configuration design variables are described using the material derivative. Direct differentiation is utilized to obtain the first variation of the performance measure explicitly in terms of variations of configuration design variables. With the consistent tangent stiffness matrix employed at the end of each load step to compute the design sensitivity, it is found that no iterations are necessary to compute design sensitivity. In addition, the linear design velocity is used to account for configuration design changes, with the velocity field being updated at each load step of the incremental analysis.     

A generalized approach to formulate the consistent tangent stiffness in plasticity with application to the GLD porous material model

It has been shown that the use of the consistent tangent moduli is crucial for preserving the quadratic convergence rate of the global Newton iterations in the solution of the incremental problem. In this paper, we present a general method to formulate the consistent tangent stiffness for plasticity. The robustness and efficiency of the proposed approach are examined by applying it to the isotropic material with J₂ flow plasticity and comparing the performance and the analysis results with the original implementation in the commercial finite element program ABAQUS. The proposed approach is then applied to an anisotropic porous plasticity model, the Gologanu–Leblond–Devaux model. Performance comparison between the consistent tangent stiffness and the conventional continuum tangent stiffness demonstrates significant improvement in convergence characteristics of the overall Newton iterations caused by using the consistent tangent matrix.     

A polygonal finite element formulation for modeling nearly incompressible materials

The objective of the present paper is to develop a finite element formulation for modeling nearly incompressible materials at large strains using polygonal elements. The present finite element formulation is a simplified version of the three-field mixed formulation and, in particular, it reduces the functional of the internal potential energy by expressing the field of the average volume-change in terms of the displacement field, where the latter is discretized using the Wachspress shape functions. The reduced mixed formulation eliminates the volumetric locking in nearly incompressible materials and enhances the computational efficiency as the static condensation is circumvented. A detailed implementation of the finite element formulation is presented in this study. Also, different example problems, including eigenvalue analysis, nonlinear patch test and other benchmark problems are presented for demonstrating the accuracy and the reliability of the developed formulation for polygonal elements.

     

Geometrically nonlinear vibration of laminated composite cylindrical thin shells with non-continuous elastic boundary conditions

The geometrically nonlinear forced vibration response of non-continuous elastic-supported laminated composite thin cylindrical shells is investigated in this paper. Two kinds of non-continuous elastic supports are simulated by using artificial springs, which are point and arc constraints, respectively. By using a set of Chebyshev polynomials as the admissible displacement function, the nonlinear differential equation of motion of the shell subjected to periodic radial point loading is obtained through the Lagrange equations, in which the geometric nonlinearity is considered by using Donnell’s nonlinear shell theory. Then, these equations are solved by using the numerical method to obtain nonlinear amplitude–frequency response curves. The numerical results illustrate the effects of spring stiffness and constraint range on the nonlinear forced vibration of points-supported and arcs-supported laminated composite cylindrical shells. The results reveal that the geometric nonlinearity of the shell can be changed by adjusting the values of support stiffness and distribution areas of support, and the values of circumferential and radial stiffness have a more significant influence on amplitude–frequency response than the axial and torsional stiffness.

     

Prediction of resonantly forced vibrations for suspended footbridges

Abstract

The aim of the present paper is to investigate the performance of suspended footbridges under pedestrian loads. Indeed, several Authors have underlined the possible activation of large amplitude oscillations into suspended footbridges due to the nonlinear behavior of the hangers. In fact, the last ones act as linear elastic springs in tension and do not react in compression. Consequently, if the whole suspended footbridge or parts of it undergo large amplitude oscillations, the initial hangers’ pretension stress may become zero and the slackening may start. In these cases, the stiffness of the footbridge deck decreases drastically, and a complex dynamic response may occur. Hence, the footbridge may show unexpected vertical and torsional oscillations that the “classical” models cannot predict; these models, in fact, assume a bilateral behavior for the suspended system and, consequently, no variations of the global stiffness during the motion. Here, the response of suspended footbridges is evaluated by using a continuous model obtained by adopting the nonlinear equivalent regularization technique proposed for long span suspended bridges. The dynamic analysis, performed by means of a perturbation method, shows the possibility of the coexistence of multiple solutions, some of which are characterized by high amplitudes and by the activation of the afore-described slackening phenomenon. The response is evaluated for several values of loading, mechanical and geometrical parameters, with the main aim of highlighting the characteristics and the stability of the investigated oscillations and obtaining information and/or design indications to prevent such phenomena.

Communicated by Dumitru Caruntu.     

Stress resultants plasticity with general closest point projection

In this paper, general closest point projection algorithm is derived for the elastoplastic behavior of a cross-section of a beam finite element. For given section deformations, the section forces (stress resultants) and the section tangent stiffness matrix are obtained as the response for the cross-section. Backward Euler time integration rule is used for the solution of the nonlinear evolution equations. The solution yields the general closest projection algorithm for stress resultants plasticity model. Algorithmic consistent tangent stiffness matrix for the section is derived. Numerical verification of the algorithms in a mixed formulation beam finite element proves the accuracy and robustness of the approach in simulating nonlinear behavior.     

Nonlinear modal coupling in a T-shaped piezoelectric resonator induced by stiffness hardening effect

The nonlinear modal coupling in a T-shaped piezoelectric resonator, when the former two natural frequencies are away from 1:2, is studied. Experimentally sweeping up the exciting frequency shows that the horizontal beam exhibits a nonlinear hardening behavior. The first primary resonance of the vertical beam, owing to modal coupling, exhibits an abrupt amplitude increase, namely the Hopf bifurcation. The frequency comb phenomenon induced by modal coupling is measured experimentally. A Duffing-Mathieu coupled model is theoretically introduced to derive the conditions of the modal coupling and frequency comb phenomenon. The results demonstrate that the modal coupling results from nonlinear stiffness hardening and is strictly dependent on the loading range and sweeping form of the driving voltage and the frequency of the piezoelectric patches.

     

预应力混凝土平面杆系结构的有限元方法

建立了基于有限元方法的考虑材料和几何非线性的任意截面预应力混凝土平面杆系结构的数值分析模型,可用于模拟预应力混凝土大跨度梁、单向偏压细长柱等的非线性全过程结构响应。引入修正的Rodriguez截面模型确定截面切线刚度,其中混凝土的贡献通过截面边界顶点定义的梯形单元来实现;在此基础上利用传统的平面非线性杆单元导出了标准有限元公式。通过两个算例验证了该模型的可靠性和适用性。     

A new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function

This paper proposes a new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function. Firstly, a Wronskian condition involving a free function is introduced for the equation. Secondly, by solving the Wronskian condition, some exact solutions are presented. Thirdly, the dynamical behaviors are analyzed by choosing specific functions in the Wronskian condition. In addition, some exact solutions are graphically illustrated by using Mathematica symbolic computations. The dynamical behaviors include stationary y-breather, line-soliton resonance, line-soliton-like phenomenon, parabola–soliton interaction, cubic–parabola–soliton resonance, kink behavior, and singular waves. These results not only illustrate the merits of the proposed method in deriving new exact solutions but also novel dynamical behaviors in the (3 + 1)-dimensional nonlinear evolution equation.

     

Yield criteria for quasibrittle and frictional materials: A generalization to surfaces with corners

Convexity of a yield function (or phase-transformation function) and its relations to convexity of the corresponding yield surface (or phase-transformation surface) is essential to the invention, definition and comparison with experiments of new yield (or phase-transformation) criteria. This issue was previously addressed only under the hypothesis of smoothness of the surface, but yield surfaces with corners (for instance, the Hill, Tresca or Coulomb–Mohr yield criteria) are known to be of fundamental importance in plasticity theory. The generalization of a proposition relating convexity of the function and the corresponding surface to nonsmooth yield and phase-transformation surfaces is provided in this paper, together with the (necessary to the proof) extension of a theorem on nonsmooth elastic potential functions. While the former of these generalizations is crucial for yield and phase-transformation functions, the latter may find applications for potential energy functions describing phase-transforming materials, or materials with discontinuous locking in tension, or contact of a body with a discrete elastic/frictional support.     

无额外自由度广义有限元非线性分析

将无额外自由度的广义有限元法由线弹性分析扩展到弹塑性大变形分析.局部强化函数的构建依赖于已有节点,不引入额外自由度,避免了线性相关性问题.在更新拉格朗日框架下,通过控制方程弱形式的线性化推导得到了节点内力的率形式,并分为材料和几何两部分.考虑超弹性和亚弹-塑性两种材料模型,采用Newton-Raphson迭代求解,给出...     

刚接与铰接混合连接杆系结构的几何非线性分析

本文提出用子结构原理解决具有刚接与铰接混合连接空间杆系结构的几何非线性分析,实现其非线性稳定性分析的载荷-位移全过程跟踪。该法无须单独推导刚接、铰接以及一端刚接一端铰接单元的弹性刚度矩阵和几何刚度矩阵,而可以直接由空间梁单元退化得到,而且可以将平面问题与空间问题、刚接与铰接混合连接体系进行统一处理,算例表明,本文方法对于杆系结构的统一和整体分析是有效的。     

Nonlinear low-velocity impact on damped and matrix-cracked hybrid laminated beams containing carbon nanotube reinforced composite layers

This paper investigates the low-velocity impact response of a shear deformable laminated beam which contains both carbon nanotube reinforced composite (CNTRC) layers and carbon fiber reinforced composite (CFRC) layers. The effect of matrix cracks is considered, and a refined self-consistent model is selected to describe the degraded stiffness caused by the damage. The beam including damping effects rests on a two-parameter elastic foundation in thermal environments. Based on a higher-order shear deformation theory and von Kármán nonlinear strain–displacement relationships, the motion equations of the beam and impactor are established and solved by means of a two-step perturbation approach. The material properties of both CFRC layers and CNTRC layers are assumed to be temperature-dependent. To assess engineering application of this hybrid structure, two conditions for outer CNTRC layers and outer CFRC layers are compared. Besides, the effects of the crack density, volume fraction of carbon nanotube, temperature variation, the foundation stiffness and damping on the nonlinear low-velocity impact behavior of hybrid laminated beams are also discussed in detail.

     

Nonlinear dynamic behavior analysis of an elastically restrained double-beam connected through a mass-spring system that is nonlinear

Some complex engineering structures can be modeled as multiple beams connected through coupling elements. When the coupling element is elastic, it can be simplified as a mass-spring system. The existing studies mainly concentrated on the double-beam coupled through elastic connectors, where the connector is simplified as the equivalent linear stiffness element or linear mass-spring system. Furthermore, many researches ignore rotational boundary restraints in analyzing dynamic behavior of the double-beam connected through elastic connectors, limiting their engineering generality. Considering the above limitations, this study attempts to employ the cubic nonlinear stiffness in the coupling mass-spring system and study the potential application of the mass-spring system that is nonlinear on the vibration control of the double-beam system. Using the variational method and the generalized Hamiltonian method build the corresponding system’s governing functions. Applying the Galerkin truncation method (GTM) obtains the dynamic behavior of the double-beam connected through a mass-spring system that is nonlinear. According to this study, the change of the mass-spring system that is nonlinear significantly influences the dynamic behavior of the double-beam system, where the complex dynamic behavior occurs under certain parameters of the mass-spring system that is nonlinear. Suitable parameters of the mass-spring system that is nonlinear are good at the vibration suppression at the boundary of the vibration system. Furthermore, the mass-spring system that is nonlinear can change the characteristics of the double-beam system’s kinetic energy transfer. For the vibration model established in this work, a quasi-periodic vibration state can be regarded as a sign of the occurrence of the targeted energy transfer of the double-beam connected through a mass-spring system that is nonlinear.

     

Geometric Nonlinearity

The nonlinear behavior or structures has attracted considerable attention in the last decade. This paper is specifically concerned with three-dimensional skeletal structures wherein nonlinearities are attributable to change of geometry. The frameworks that are considered possess particular relevance to roofing systems.

The outline theoretical analysis incorporates: (a) effects of finite joint deformation; (b) change of member lateral stiffness; (c) change of member length due to bowing. Cable systems are discussed simply as a particular case of the more general solution.

Theoretical and experimental results are compared for a relatively complex rigid jointed structure.

An example of a shallow dome is presented in which the salient design considerations that must be made for this form of structure are clearly demonstrated.     

Dynamic analysis of a novel wide-tunable microbeam resonator with a sliding free-of-charge electrode

In this paper, a MEMS-based resonator with a novel effective stiffness tunability is presented. The performance of the proposed resonator is based on the transversal vibration of the two porous cantilever microbeams with a rectangular microplate at the end of the structure. The microplate as a free-of-charge slider electrode is in contact with two other fixed substrate electrodes via the thin layer of dielectric material. Applying a constant DC voltage to the two fixed electrodes leads to the movement of free electrons in the slider and eventually to the formation of two series capacitors. As a result, the slider meets a nonlinear electrostatic force proportional to the square of the applied DC voltage. It will act as a nonlinear spring with a tunable stiffness during the oscillation of the resonator. The coupled nonlinear equations governing the longitudinal and transversal vibration of the resonator are extracted in the presence of the nonlinear voltage-sliding spring. Its steady-state solution is obtained based on a physically based learning method that makes it possible to obtain frequency response for the first harmony as well as for the higher harmonies and to predict primary and secondary resonances in different harmonies of the response. The effect of the applied tuning DC voltage, the geometrical parameters of the resonator, and the cantilever's porosity on the dynamic response of the resonator are investigated. It is shown that the tuning stiffness of this voltage-sliding spring provides a highly effective solution to realize an extreme tunable range. In the end, a modified tunable structure is introduced in which the folded beams are replaced with common ones. The modified resonator by making the nonlinear behavior of the resonator least can improve its performance significantly.

     

The increment stiffness matrix and total quantum stiffness in nonlinear analyses

In this paper, the expressions of both increment stiffness matrix and total quantum stiffness matrix in nonlinear analyses are derived in detail, and their relationship is discussed in mathematical meaning. The results given in our paper will be of great importance to the analyses of nonlinear numerical and nonlinear stability in finite element methods.