The USGT Method for Suspender Tensioning of Self-Anchored Suspension Bridges

Unlike earth-anchored suspension bridges, self-anchored suspension bridges (SASBs) involve a special construction stage, namely, suspender tensioning, in which the tensioning force and sequence are crucial and complicated. Against this background, an example bridge A, a SASB with a steel-concrete composite beam, is introduced in detail. Using MIDAS finite element software, a suspender tensioning scheme is formulated based on a combination method of the unstrained state method and graded tension method (the USGT method), in which a suspender is tensioned according to its unstrained length. By analyzing the bending moment change of the beam and deflection of the main cable throughout the entire construction process, a “high-to-low” suspender tensioning sequence is proposed that also involves symmetrical tensioning from the main towers to the midspan or the anchor positions. In the optimized construction process, the deviation and stress of the main towers are controlled well, thereby ensuring the safety of the main beam and main towers in the construction process.

A CONSTITUTIVE MODEL FOR BOTH LOW AND HIGH STRAIN NONLINEARITIES IN HIGHLY FILLED ELASTOMERS AND IMPLEMENTATION WITH USER-DEFINED MATERIAL SUBROUTINES IN ABAQUS

ABSTRACT Strain energy functions (SEFs) are used to model the hyperelastic behavior of rubberlike materials. In tension, the stress–strain response of these materials often exhibits three characteristics: (i) a decreasing modulus at low strains (<20%), (ii) a constant modulus at intermediate strains, and (iii) an increasing modulus at high strains (>200%). Fitting an SEF that works in each regime is challenging when multiple or nonhomogeneous deformation modes are considered. The difficulty increases with highly filled elastomers because the small strain nonlinearity increases and finite-extensibility occurs at lower strains. One can compromise by fitting an SEF to a limited range of strain, but this is not always appropriate. For example, rubber seals in oilfield packers can exhibit low global strains but high localized strains. The Davies–De–Thomas (DDT) SEF is a good candidate for modeling such materials. Additional improvements will be shown by combining concepts from the DDT and Yeoh SEFs to construct a more versatile SEF. The SEF is implemented with user-defined material subroutines in Abaqus/Standard (UHYPER) and Abaqus/Explicit (VUMAT) for a three-dimensional general strain problem, and an approach to overcome a mathematically indeterminate stress condition in the unstrained state is derived. The complete UHYPER and VUMAT subroutines are also presented.

Weakly-Emergent Strain-Dependent Properties of High Field Superconductors

All superconductors in high field magnets operating above 12 T are brittle and subjected to large strains because of the differential thermal contraction between component parts on cool-down and the large Lorentz forces produced in operation. The continuous scientific requirement for higher magnetic fields in superconducting energy-efficient magnets means we must understand and control the high sensitivity of critical current density Jc to strain ε. Here we present very detailed Jc(B, θ, T, ε) measurements on a high temperature superconductor (HTS), a (Rare−Earth)Ba2Cu3O7−δ (REBCO) coated conductor, and a low temperature superconductor (LTS), a Nb3Sn wire, that include the very widely observed inverted parabolic strain dependence for Jc(ε). The canonical explanation for the parabolic strain dependence of Jc in LTS wires attributes it to an angular average of an underlying intrinsic parabolic single crystal response. It assigns optimal superconducting critical parameters to the unstrained state which implies that Jc(ε) should reach its peak value at a single strain (ε = εpeak), independent of field B, and temperature T. However, consistent with a new analysis, the high field measurements reported here provide a clear signature for weakly-emergent behaviour, namely εpeak is markedly B, (field angle θ for the HTS) and T dependent in both materials. The strain dependence of Jc in these materials is termed weakly-emergent because it is not qualitatively similar to the strain dependence of Jc of any of their underlying component parts, but is amenable to calculation. We conclude that Jc(ε) is an emergent property in both REBCO and Nb3Sn conductors and that for the LTS Nb3Sn conductor, the emergent behaviour is not consistent with the long-standing canonical explanation for Jc(ε).

Transient Interaction of a Circular Plate and a Fluid Medium

The transient dynamic response of an elastic circular plate subjected to a suddenly applied pressure is determined for several edge boundary conditions. The plate boundary is attached to a semi-infinite, radially rigid tube which is filled with an acoustic fluid, and pressure is applied to the in-vacuo side of the plate. The transient solution is determined by using a technique in which the plate is subjected to a periodic pressure function constructed of appropriately signed and time-shifted Heaviside step functions, and by relying on a physical mechanism which returns the plate and fluid near the plate to an unstrained state of rest between pulses. The plate response is presented for a number of radius-to-thickness ratios and edge boundary conditions when interacting with water. Comparisons are also made with solutions obtained using a plane wave approximation to the fluid field.

The effects on rubber elasticity of the addition and scission of cross-links under strain

Edwards’s equilibrium theory of rubber elasticity is used to study the effect on the network elasticity of the consecutive addition and removal of cross-links under different strains. The treatment is compared with those of Flory, Scanlan and others based on classical rubber elasticity theory. For a composite network made by first introducing ( v 1 + v 0 1 ) links in an isotropic state, then adding v 2 at deformation λ, and finally removing v 0 1 of the original group, the strain-dependent free energy at some subsequent deformation ξ (relative to the initial unstrained state) is shown under certain conditions to be F (ξ) = ½ kT [( v 1 + ф v 2 ) Ʃ i ξ 2 i + ( v 2 - ф v 2 ) Ʃ i (ξ i /λ i ) 2 ], where ф = ф{ v 1 , v 0 1 , v 2 ). A similar equation has been obtained by Flory. When v 0 1 = 0, ф = 0, confirming the familar ‘two -network’ theory for this case. The ‘memory’ effects which occur when v 0 1 is non-zero are discussed.

Large amplitude waves in bounded media I. Reflexion and transmission of large amplitude shockless pulses at an interface

Techniques which can be used to analyse the interaction of large amplitude elastic waves in a bounded medium are described. Although presented in the context of uniaxial stretching deformations in an elastic string or bar, these techniques can be used to analyse the behaviour of any system whose response is described by the nonlinear one-dimensional wave equation. In this first paper the bounded medium is contained between two parallel planes which separate it from other similar media. These are of semi-infinite extent along the axis of propagation which is normal to the interfaces. The paper is in two parts. In the first part the reflexion and transmission of an incident pulse when it arrives at an interface with a semi-infinite medium is described and the ideas of nonlinear impedance, reflexion coefficient and transmission coefficient are introduced. The results are quite general: no special forms for the stress-strain relations of either elastic materials is assumed. The results for a single interface are used to analyse the decay of a pulse as it moves back and forth between two interfaces. This decay occurs because at each contact with the interface energy is radiated across the interface to the surrounding medium. The algorithm s obtained have simple graphical interpretations. The general theory is used to discuss the decay of a pulse in a layer of saturated soil which is bounded from above by sea water and from below by rock. This pulse is triggered by a seismic disturbance deep inside the rock. The theory is also used to analyse the decay in the oscillation which occurs in a shock tube w hen a diaphragm separating air at high pressure from air at atmospheric pressure is ruptured. The bound gas is contained between the closed end of the tube and the contact discontinuity which is generated w hen the diaphragm bursts. In the second part of the paper a more detailed account is given of what happens when a pulse is partially reflected and partially transmitted at an interface. This is achieved by noting that the responses of m any elastic materials can be correlated, both qualitatively and quantitatively, by a family of stress-strain laws for which the governing nonlinear equations for this problem can be solved exactly. These laws are sufficiently general to locally curve fit any prescribed stress-strain law to an error 0 ([stra in ]4) in some vicinity of the unstrained state. They can also be used to fit the response of a polytropic gas during isentropic flow to within an error of 1 % as the density changes by a factor of ten! The reflexion of a large amplitude pulse from rigid and perfectly free interfaces is given special emphasis as is the reflexion from an interface with a Hookean material.

Effect of Embedding Medium Viscoelasticity on the Transient Response to Plane Waves of Arbitrarily Thick Circular Cylinders

The transient response of a circular cylinder of arbitrary thickness, embedded in a viscoelastic medium and impinged upon by plane waves, is obtained. Dilatational and shear incident waves are considered. The solution is valid within the scope of the linear theories of elasticity and viscoelasticity. The method of solution, which circumvents the difficulties encountered in the customary transform approaches, relies upon (a) the construction of a train of incident pulses from steady-state components, where each pulse represents the time history of the transient stress in the incident wave and (b) the existence of a physical mechanism that, between pulses, restores the disturbed particles of the cylinder and the surrounding medium to an unstrained state of rest. The influence on the major stress and displacement response of variations in the viscoelastic properties of the medium is investigated for the case of incident step waves. The results for the viscoelastic media are compared with corresponding results for the limiting cases of elastic-unrelaxed and elastic-relaxed media.

Transient Response of a Circular Cylinder of Arbitrary Thickness, in an Elastic Medium, to a Plane Dilatational Wave

The response of a hollow cylinder of arbitrary thickness, embedded in an elastic medium, to a transient plane pressure wave is presented. The solution is valid within the scope of the linear theory of elasticity. The technique for obtaining the solutions relies upon (a) the construction of a train of incident pulses from steady-state components, where each pulse represents the time history of the transient stress in the incident wave, and (b) the existence of a physical mechanism which, between pulses, restores the disturbed particles of the cylinder and the surrounding medium to an unstrained state of rest. The validity of the technique is demonstrated by (a) comparisons with published data for limiting cases and (b) results obtained for a broad range of values of cylinder and surrounding medium parameters. The influence on the cylinder response of liner thickness and cylinder-medium impedance mismatch, when the incident wave is a step pulse, is investigated.

Rupture of Rubber. V. Cut Growth in Natural Rubber Vulcanizates

It has been noted in Part I of this series (referred to hereafter as I), that if a nicked specimen of a natural rubber vulcanizate is slowly stretched, tearing occurs at the tip for quite small applied forces. In the initial stages, this tearing continues only as long as the deformation of the specimen is being increased, and virtually ceases if the deformation is held constant. This tearing is essentially time independent, and is termed “static” cut growth. If, however, the deformation is continued until the cut has grown by a few hundredths of a millimeter the growth becomes time dependent and catastrophic tearing takes place, the cut suddenly increasing in length by perhaps a millimeter or so. If a nicked specimen is alternately stretched and relaxed to the unstrained state, the cut gradually grows even though the applied force is less than that required to produce catastrophic tearing. This phenomenon is termed “dynamic” cut growth. This behavior can be compared to that of gum GR-S vulcanizates described in Part III, where static cut growth of the above type does not occur, a dead load on a test piece producing a more or less steady rate of cut growth. In the present paper, measurements on natural rubber gum vulcanizates only are described, and the numerical results expressed in terms of the theory developed in previous papers (Parts I, II and III). It has been shown in I and II that the tear behavior of differently shaped test pieces cut from thin sheets of thickness t may be correlated by means of the concept of the energy for tearing. This is defined as the value of T[=(1/t)(∂W/∂c)l] at the instant of tear, and is denoted by Tc. In the definition of T, is the total elastic energy stored in the test piece, c the length of the cut, and the subscript l indicates that the differentiation is to be carried out at constant displacement of those parts of the boundary that are not force-free. It was also shown that a convenient and direct method of obtaining Tc is by the use of the “simple extension” tear test piece described in I and shown in Figure 1, and this has been used for most of the experiments. Under most conditions, T for this test piece is nearly independent of the cut length, width of the test piece, and modulus of the rubber; T is very nearly equal to 2F/twhere F is the force applied to the arms. In the cases where the use of the above approximate relation between T and F introduces an appreciable error, the exact theory given in I was used.

The Centre of Shear of Aerofoil Sections

Consider a cantilever beam of uniform cross section whose generators are parallel to the z-axis and whose lateral surface is free from surface tractions. The line of centroids of the cross sections in the unstrained state is taken as the z-axis, and the x- and y-axes are the principal axes of the cross section at the centroid of the fixed end z = 0.The other end of the beam (z = l) is subject to forces which reduce to a single force with components (Wx, Wv, 0), transverse to the z-axis, acting through the load point L of this end section (see Fig. 1). The co-ordinates of L are taken as (p, q, l).