Applied Surface Loading Research Articles

Horizontal flow aerated constructed wetlands for municipal wastewater treatment: The influence of bed depth

The influence of bed depth on the performance of aerated horizontal constructed wetlands was investigated at the pilot plant scale. Two horizontal flow subsurface constructed wetlands (HF) intensified units of different bed depth (HF1: 0.90 m and HF2: 0.55 m, 0.8 m and 0.5 m water level, respectively) were fitted with forced aeration, while a third one (HFc, 0.55 m bed depth, 0.5 m water level) was used as control and not aerated. The three HF units were operated in parallel, receiving the same municipal wastewater pre-treated in a hydrolytic up-flow sludge blanket anaerobic digester. Applied surface loading rates (SLR) ranged from 20 to 80 g biochemical oxygen demand (BOD5)/m2·d and from 3.7 to 6.7 g total nitrogen (TN)/m2·d, while it ranges from 6 to 23 g BOD5/m2·d and from 1.1 to 1.7 g TN/m2·d in the control unit. Removal of total suspended solids (TSS) and BOD5 was usually close to a 100 % in all units, whilst chemical oxygen demand (COD) removal was higher for the HF1 unit (97 % on average, range of 96–99 %) than for HF2 (92 %, 82–98 %) and HFc (94 %, 86–99 %). TN removal reached on average 33 % (16–43 %) in HFc, 37 % (10–76 %) in HF2 and 51 % (21–79 %) in HF1. High TN removal required a longer aeration time for nitrification and higher effluent recirculation ratio to enhance denitrification. The results indicate that artificial aeration and a high bed depth allows to increase the SLR by a factor of 4 in HF1 but only by a factor of 2 in HF2.

Semi-analytical solutions of static and dynamic degenerate, nondegenerate and functionally graded electro-elastic multilayered plates

In this paper mathematical modelling and semi-analytical solutions are elaborated based on the extended Stroh-like formalism for 3-D static problems of electro-elastic mono and multilayered plates. For the dynamic analysis of such structures under various excitation types, a time trigonometric series was coupled with the Stroh-like formalism. The unknown coefficient vectors are solution of the resulted eigenvalue problems and the solution is obtained in the associated basis. Mechanical and electrical forces are applied on the top and/or bottom layer and may take arbitrary forms. A flexible matrix formulation is elaborated to easily account for arbitrary mechanical and electrical surface loads and excitations. The methodological approach focuses on various aspects including the electro-elastic effects, the degeneracy of elastic isotropic materials as well as the effect of the graded material’s factor. Analytical solutions are obtained for the static behavior of degenerate and non-degenerate Elastic FGM plates, where the eigenvalues and eigenvectors are explicitly given. Mainly, semi-analytical solutions are presented for the dynamic behavior of piezoelectric sandwich plates by coupling Stroh-like formalism and trigonometric series. The presented methodological approach is applied to the dynamical analysis of sandwich plates under various excitation forms, namely sinusoidal, rectangular and sawtooth excitation pulses. The obtained numerical results demonstrate the effect of the graded factor, degeneracy and the applied surface load as well as of the excitation on mechanical and electrical responses for various laminated plates.

Full-Scale Pavement Testing of a High Polymer-Modified Asphalt Concrete Mixture

Abstract This paper presents the validation process for the structural layer coefficient determined mechanistically for high polymer (HP) asphalt mixtures (i.e., 0.54) through full-scale laboratory testing of asphalt pavements. Two instrumented full-scale pavement experiments were conducted in the PaveBox setup at the University of Nevada, Reno. A flexible pavement structure with a conventional polymer-modified asphalt layer was evaluated in Experiment No. 1. Meanwhile, a flexible pavement structure with a reduced-thickness HP asphalt concrete (AC) layer was evaluated in Experiment No. 2. Similar vertical surface deflections (except under the center of the applied surface load), higher vertical stresses at the middle of the base layer, and similar vertical stresses below the subgrade surface were observed with the reduced thickness of the HP AC layer. Moreover, better overall fatigue and rutting performance in the AC layer, slightly higher rut depths in the unbound layers, and similar total rut depths were observed by means of mechanistic analysis with the use of HP AC pavement. In general, the results and findings of this study supported the selection of 0.54.

Analytical plane-stress recovery of non-prismatic beams under partial cross-sectional loads and surface forces

High levels of strength- and stiffness-to-mass ratio can be achieved in slender structures by lengthwise tailoring of their cross-sectional areas. During use, a non-prismatic beam element can be subject to surface forces or loads acting on only a part of their cross-section. Practical examples involve tapered aircraft wings, wind turbine and helicopter rotor blades under fluid pressure and shear forces; arched beams in bridges subject to vehicular traction forces and tensile stresses in tendons of prestressed concrete. Presently, beam theories generalise the external loads to the entire cross-sectional area. However, this technique does not accurately describe surface-load boundary conditions and beams under partial cross-sectional loads. Hence, an efficient analytical plane-stress recovery methodology is introduced in the present study that generalises the external load to a specific sub-area of the cross-section of homogeneous non-prismatic beams with one plane of symmetry. As a result, the transverse stress components become piecewise distributions, i.e. non-smooth but continuous in the thickness direction. Additional novelties include the boundary equilibrium recovered considering applied surface loads and terms up to second-order derivatives of the internal forces to define the transverse stress field. Closed-form solutions for the specific case of non-prismatic beams with a rectangular cross-section loaded both on top and bottom surfaces are presented. For validation purposes, different numerical examples are modelled with results compared to solid-like finite element analyses as well with relevant analytical theories. The results show that the developed formulation predicts the stress field in non-prismatic beams under surface forces and non-uniform loads applied to a part of the cross-sectional area with goods levels of accuracy. The error associated with the proposed method escalates with the taper angle, such that a 10° taper angle could result in a 6% error at the surfaces and reduced values for interior zones, while the analytical state-of-the-art models were not able to predict the transverse stresses correctly.

Support optimization in additive manufacturing for geometric and thermo-mechanical constraints

Supports are often required to safely complete the building of complicated structures by additive manufacturing technologies. In particular, supports are used as scaffoldings to reinforce overhanging regions of the structure and/or are necessary to mitigate the thermal deformations and residual stresses created by the intense heat flux produced by the source term (typically a laser beam). However, including supports increase the fabrication cost and their removal is not an easy matter. Therefore, it is crucial to minimize their volume while maintaining their efficiency. Based on earlier works, we propose here some new optimization criteria. First, simple geometric criteria are considered like the projected area and the volume of supports required for overhangs: they are minimized by varying the structure orientation with respect to the baseplate. In addition, an accessibility criterion is suggested for the removal of supports, which can be used to forbid some parts of the structure to be supported. Second, shape and topology optimization of supports for compliance minimization is performed. The novelty comes from the applied surface loads which are coming either from pseudo gravity loads on overhanging parts or from equivalent thermal loads arising from the layer-by-layer building process. Here, only the supports are optimized, with a given non-optimizable structure, but of course many generalizations are possible, including optimizing both the structure and its supports. Our optimization algorithm relies on the level set method and shape derivatives computed by the Hadamard method. Numerical examples are given in 2-d and 3-d.

Improved Methods for Simulating Live Loads for Two-Dimensional Structural Analysis of Buried Culverts

The current AASHTO LRFD Bridge Design Specifications stipulates a “special distribution width” for two-dimensional (2D) live-load analysis of reinforced concrete (r/c) boxes and arches with less than 2 ft of soil cover. This special distribution width allows a significantly greater reduction of the applied surface load than permitted for other culvert shapes and materials. Neither the AASHTO commentary nor the underlying developmental report provide the physical reasoning for the special distribution width, or why it only applies to r/c boxes and arches. This paper provides a clear, physical understanding of the three-dimensional (3D) phenomena associated with the special distribution width and the interaction with longitudinal load spreading through soil. This is achieved with the aid of a flat-plate model, representative of the top slab of a box culvert, with a variable line-load width. The closed-form solution reveals that the physical reason is “3D stiffness effects” (3DSE), which occur when the line-load width is relatively short compared with the culvert’s longitudinal lay length. Moreover, it is shown that 3DSE disappear when the line load reaches a special width called the “critical distribution width” or Wcritical. Wcritical is dependent on the culvert’s span and length, and is a key parameter along with parameter Wmin needed to identify the limiting line-load width that evokes 3DSE. The key concepts of 3DSE, Wcritical, and Wmin are used to develop improved 2D analysis procedures using either the traditional reduced surface load approach or the more recent continuous load spreading approach.

Excitation of plane Lamb wave in plate-like structures under applied surface loading

Lamb waves play an important role in structure health monitoring (SHM) systems. The excitation of Lamb waves has been discussed for a long time with absorbing results. However, little effort has been made towards the precise characterization of Lamb wave excitation by various transducer models with mathematical foundation. In this paper, the excitation of plane Lamb waves with plane strain assumption in isotropic plate structures under applied surface loading is solved with the Hamiltonian system. The response of the Lamb modes excited by applied loading is expressed analytically. The effect of applied loading is divided into the product of two parts as the effect of direction and the effect of distribution, which can be changed by selecting different types of transducer and the corresponding transducer configurations. The direction of loading determines the corresponding displacement of each mode. The effect of applied loading on the in-plane and normal directions depends on the in-plane and normal displacements at the surface respectively. The effect of the surface loading distribution on the Lamb mode amplitudes is mainly reflected by amplitude versus frequency or wavenumber. The frequencies at which the maxima and minima of the S0 or A0 mode response occur depend on the distribution of surface loading. The numerical results of simulations conducted on an infinite aluminum plate verify the theoretical prediction of not only the direction but also the distribution of applied loading. A pure S0 or A0 mode can be excited by selecting the appropriate direction and distribution at the corresponding frequency.

Viscoelastic characterization of the primate finger pad in vivo by microstep indentation and three-dimensional finite element models for tactile sensation studies.

When we touch an object, surface loads imposed on the skin are transmitted to thousands of specialized nerve endings (mechanoreceptors) embedded within the skin. These mechanoreceptors transduce the mechanical signals imposed on them into a neural code of the incident stimuli, enabling us to feel the object. To understand the mechanisms of tactile sensation, it is critical to understand the relationship between the applied surface loads, mechanical state at the mechanoreceptor locations, and transduced neural codes. In this paper, we characterize the bulk viscoelastic properties of the primate finger pad and show its relationship to the dynamic firing rate of SA-1 mechanoreceptors. Two three-dimensional (3D) finite element viscoelastic models, a homogeneous and a multilayer model, of the primate fingertip are developed and calibrated with data from a series of force responses to micro-indentation experiments on primate finger pads. We test these models for validation by simulating indentation with a line load and comparing surface deflection with data in the literature (Srinivasan, 1989, "Surface Deflection of Primate Fingertip Under Line Load," J. Biomech., 22(4), pp. 343-349). We show that a multilayer model with an elastic epidermis and viscoelastic core predicts both the spatial and temporal biomechanical response of the primate finger pad. Finally, to show the utility of the model, ramp and hold indentation with a flat plate is simulated. The multilayer model predicts the strain energy density at a mechanoreceptor location would decay at the same rate as the average dynamic firing rate of SA-1 mechanoreceptors in response to flat plate indentation (previously observed by Srinivasan and LaMotte, 1991 "Encoding of Shape in the Responses of Cutaneous Mechanoreceptors," Information Processing in the Somatosensory System (Wenner-Gren International Symposium Series), O. Franzen and J. Westman, eds., Macmillan Press, London, UK), suggesting that the rate of adaptation of SA-1 mechanoreceptors is governed by the viscoelastic nature of its surrounding tissue.

Dynamic Adjustment of the Ocean Circulation to Self-Attraction and Loading Effects

AbstractThe oceanic response to surface loading, such as that related to atmospheric pressure, freshwater exchange, and changes in the gravity field, is essential to our understanding of sea level variability. In particular, so-called self-attraction and loading (SAL) effects caused by the redistribution of mass within the land–atmosphere–ocean system can have a measurable impact on sea level. In this study, the nature of SAL-induced variability in sea level is examined in terms of its equilibrium (static) and nonequilibrium (dynamic) components, using a general circulation model that implicitly includes the physics of SAL. The additional SAL forcing is derived by decomposing ocean mass anomalies into spherical harmonics and then applying Love numbers to infer associated crustal displacements and gravitational shifts. This implementation of SAL physics incurs only a relatively small computational cost. Effects of SAL on sea level amount to about 10% of the applied surface loading on average but depend strongly on location. The dynamic component exhibits large-scale basinwide patterns, with considerable contributions from subweekly time scales. Departures from equilibrium decrease toward longer time scales but are not totally negligible in many places. Ocean modeling studies should benefit from using a dynamical implementation of SAL as used here.

Pullout resistance of granular anchors in clay for undrained condition

Granular anchors (GAs) can resist pullout/uplift forces, compression forces and also provide ground improvement. Under pullout loading, a centrally located tendon transmits the applied surface load to the base of the granular column via a base plate attachment, which compresses the column causing significant dilation of the granular material to occur, thereby forming the anchor. This paper describes a program of field testing and numerical modelling of the pullout resistance of GA installations in overconsolidated clay for the undrained (short term) condition. Pertinent modes of failure are identified for different column length to diameter (L/D) ratios. The applied pullout load is resisted in shaft capacity for short GAs or in end-bulging of the granular column for long GAs. In other words, the failure mode is dependent on the column L/D ratio. A novel modification in which the conventional flat base-plate is replaced by a suction cup was shown to significantly improve the undrained ultimate pullout capacity of short GAs.

Post-treatment of secondary wastewater treatment plant effluent using a two-stage fluidized bed bioreactor system

The aim of this study was to investigate the performance of a two-stage fluidized bed reactor (FBR) system for the post-treatment of secondary wastewater treatment plant effluents (Shahrak Gharb, Tehran, Iran). The proposed treatment scheme was evaluated using pilot-scale reactors (106-L of capacity) filled with PVC as the fluidized bed (first stage) and gravel for the filtration purpose (second stage). Aluminum sulfate (30 mg/L) and chlorine (1 mg/L) were used for the coagulation and disinfection of the effluent, respectively. To monitor the performance of the FBR system, variation of several parameters (biochemical oxygen demand (BOD5), chemical oxygen demand (COD), turbidity, total phosphorous, total coliform and fecal coliform) were monitored in the effluent wastewater samples. The results showed that the proposed system could effectively reduce BOD5 and COD below 1.95 and 4.06 mg/L, respectively. Turbidity of the effluent could be achieved below 0.75 NTU, which was lower than those reported for the disinfection purpose. The total phosphorus was reduced to 0.52 mg/L, which was near the present phosphorous standard for the prevention of eutrophication process. Depending on both microorganism concentration and applied surface loading rates (5–10 m/h), about 35 to 75% and 67 to 97% of coliform were removed without and with the chlorine addition, respectively. Findings of this study clearly confirmed the efficiency of the FBR system for the post-treatment of the secondary wastewater treatment plant effluents without any solid problem during the chlorination.

Solids hydrolysis and accumulation in a hybrid anaerobic digester-constructed wetlands system

The properties and behaviour of solids retained in a pilot plant constituted of an up-flow anaerobic sludge blanket (UASB) reactor and two constructed wetlands (CWs) were monitored over a 3-year period. The UASB (25.5 m 3) was fed with raw municipal wastewater at a flow rate of 61–112 m 3 d −1 and a volumetric loading rate (VLR) of 0.75–1.70 kg TCOD m −3 d −1. The CWs (75 m 2 each) were operated in series and received a fraction (17–20 m 3 d −1) of the UASB effluent. The applied surface loading rates (SLR) were in the range of 3800–8700 g TCOD m −2 d −1 (UASB) and 11–15 g BOD 5 m −2 d −1 (CWs). The overall system removed 95% TSS, 85% TCOD and 87% BOD 5 on average. For influent VSS, the UASB removed 72.1% and gave a hydrolysis of 63.5%, while the average surplus sludge generation was 8.7%. Over the 3-year period, TSS and VSS accumulated in the CWs at rates of 1.07 and 0.56 kg m −2 year −1, respectively. The aerobic biodegradability of the accumulated solids ranged from 23 to 92 mg O 2 g VSS −1 d −1 and increased downstream in the CWs. About 59% of the VSS that entered the CWs was removed by hydrolysis, while 24% accumulated on granular media. These low solids accumulation rates were especially remarkable considering the high COD and BOD 5 loading rates applied. The system lay-out appear to be promising in terms of preventing clogging.

Metal ion implantation in inert polymers for strain gauge applications

Metal ion implantation in inert polymers may produce ultra-thin conducting films below the polymer surface. These subsurface films are promising structures for strain gauge applications. To this purpose, polycarbonate substrates were irradiated at room temperature with low-energy metal ions (Cu + and Ni +) and with fluences in the range between 1 × 10 16 and 1 × 10 17 ions/cm 2, in order to promote the precipitation of dispersed metal nanoparticles or the formation of a continuous thin film. The nanoparticle morphology and the microstructural properties of polymer nanocomposites were investigated by glancing-incidence X-ray diffraction and transmission electron microscopy (TEM) measurements. At lower fluences (<5 × 10 16 ions/cm 2) a spontaneous precipitation of spherical-shaped metal nanoparticles occurred below the polymer top-surface (∼50 nm), whereas at higher fluences the aggregation of metal nanoparticles produced the formation of a continuous polycrystalline nanofilm. Furthermore, a characteristic surface plasmon resonance peak was observed for nanocomposites produced at lower ion fluences, due to the presence of Cu nanoparticles. A reduced electrical resistance of the near-surface metal–polymer nanocomposite was measured. The variation of electrical conductivity as a function of the applied surface load was measured: we found a linear relationship and a very small hysteresis.

Closed-form solution for plastic zone formation around a circular tunnel in half-space obeying Mohr–Coulomb criterion

A closed-form solution for the problem of the plastic zone and stress distribution around a circular tunnel in an elastic–plastic half space, derived using bipolar coordinates, is the main scope of this paper. By assuming a uniformly applied surface loading, the whole semi-infinite space is under uniform pressure: thus the gravitational effect is neglected, while the plastic zone formation around the circular tunnel is controlled by the applied internal support pressure. The plastic behaviour of the half space is described by the Mohr–Coulomb yield criterion, and the soil is assumed to be homogeneous and isotropic with earth pressure coefficient K0 equal to unity. The critical internal pressure, where the initial yielding occurs at the tunnel wall, is derived, along with equations describing the plastic zone and plastic stresses. These equations are functions of the soil properties, which are cohesion and friction angle. This derived closed-form solution is validated through mathematical and computational analysis, and is also compared with numerical models under gravitational load, solved using the finite difference method. This innovative closed-form solution has a significant impact in practical problems by introducing a simple and effective method that allows the quick estimation of a shallow tunnel's behaviour, since it gives, in principle, the opportunity for quick and accurate calculation of the plastic zone and stress distribution around the circular tunnel. By applying different values of support pressure, the tunnel designer can easily evaluate the feasibility of different design alternatives, such as shotcrete shell and tunnel-boring machine support pressure. As a result, an efficient and innovative method of solving shallow tunnelling problems in cohesive-frictional soil is introduced.

Axisymmetric transient waves in transversely isotropic half-space

A transversely isotropic material in the sense of Green is considered. Using a series of potential functions proposed in [Eskandari-Ghadi M. A complete solution of the wave equations for transversely isotropic media. J Elasticity 2005; 81:1–19], the solutions of the transient wave equations within a half-space under surface load are obtained in the Laplace–Hankel domain for axisymmetric problems. The solutions are investigated in detail in the special case of a surface point force pulse varying with time as Heaviside function. Using Cagniard–De Hoop method, the inverse Laplace transform and inverse Hankel transform of the solutions are then obtained in the form of integrals with finite limits. For validity of the analytical results, the final formulations for surface waves are degenerated for an isotropic material and compared with the existing formulation obtained by Pekeris [The seismic surface pulse. Proc Natl Acad Sci USA 1955;41:469–80], to show that they are exactly the same. The numerical evaluations of the integrals for some transversely isotropic materials as well as an isotropic one are obtained. The present approach is then numerically verified by comparing a particular case of displacements for the surface of an isotropic half-space subjected to a point load of Heaviside function with the solutions obtained by Pekeris [The seismic surface pulse. Proc Natl Acad Sci USA 1955;41:469–80]. In addition, the wave equations for the mentioned medium are obtained on the vertical line directly under the applied surface load. The final formulations are degenerated for an isotropic material and compared with the existing formulation given in Graff [Wave motion in elastic solids. New York: Dover Publications Inc; 1975 [New Ed edition, November 1991]], to show that they are also exactly the same. Then equations are presented in graphical forms using an appropriate numerical evaluation.

Effect of Gradation on Nonlinear Stress-Dependent Behavior of a Sandy Flexible Pavement Subgrade

The development of a simple method to estimate in situ resilient modulus properties of sandy subgrade soils is described in this paper based on gradation properties, granular base to asphalt concrete pavement thickness ratios, and the applied surface loading. A total of 75 nondestructive testing (NDT) data sets were collected from conventional flexible pavements at sites consisting of sandy subgrade soils. The NDT was performed using a falling weight deflectometer at three different load levels and the NDT data from the various pavement areas were utilized to back-calculate subgrade modulus using the WESDEF program. Cores of the asphalt concrete pavements and the particle size distributions of subgrade soils at the NDT locations were obtained concurrently. The data generated such as the applied load levels, asphalt concrete and aggregate base thicknesses and their ratios, and particle size distributions were in turn related to in situ subgrade modulus properties. The effects of particle size distribution and nonlinear stress dependency of sandy subgrade modulus to the pavement responses were analyzed using the finite-element program TTI-PAVE. A horizontal to vertical modulus ratio of approximately 0.3 was assumed for the unbound aggregate base to account for its anisotropic stiffness behavior (Kim 2004). The results demonstrated that the gradation of granular soils and thicknesses of the asphalt concrete and aggregate base pavement layers influenced the subgrade modulus, which in turn has a substantial effect on the pavement response predictions that impact pavement overlay design.

LIQUEFACTION-INDUCED SETTLEMENT OF SHALLOW FOUNDATIONS AND REMEDIATION: 3D NUMERICAL SIMULATION

Calibrated Finite Element (FE) simulations are increasingly providing a reliable environment for modelling liquefaction-induced ground deformation. Effects on foundations and super-structures may be assessed and associated remediation techniques may be explored, within a unified framework. Current capabilities of such a FE framework are demonstrated via a simple 3-dimensional (3D) series of simulations. Ground liquefaction and settlement under the action of a surface load is investigated. Liquefaction hazard mitigation is explored by soil compaction and/or increase of permeability below and around the applied surface load. The potential and limitations of numerical simulation are discussed, along with future research needs and challenges towards more reliable practical application.

Numerical integration during Fourier integral analysis

Numerical integration required during Fourier integral analysis is discussed. For the case of a long and prismatic elastic medium subject to three-dimensional loads applied at the surface (e.g. live load response of buried structures), the complexity of inverse integrals depends on the relative magnitude of the load width and the distance from the load in the longitudinal direction, as well as the longitudinal spacing of the loads. The inverse integrand of the applied surface loading is more difficult to evaluate compared to those for stresses and displacements. Selection of integration schemes based on successful inversion of the applied load provides accurate solutions of stress and displacement throughout the elastic body. The use of superposition when considering complex loading cases is beneficial.

Laboratory Determination of Soil Stiffness Data for Buried Plastic Pipes

The structural behavior of a buried flexible pipe is influenced by the applied surface loading, the pipe ring stiffness, and the stiffness of the surrounding soil. The pipe stiffness is governed by national standards and can be determined by testing. The more dominant surround stiffness depends on the soil type and its placement and is consequently greatly influenced by site installation processes. There is a relative lack of soil stiffness data, and much of the available data relates to more common installation conditions in the United States (i.e., installation beneath embankments). Described in this paper is the assessment of stiffnesses of surround types used in the United Kingdom by laboratory testing of plastic pipes in a large test tank to simulate burial in a trench. Both gravel and sand surrounds were used, and either thorough or nominal compaction was applied. Static surface stresses were then applied to the buried pipes and diametral strains recorded, which are used for back-analysis of Spangler’s modulus of soil reaction ( E’). The analysis yielded significantly higher values of E’ than have previously been assumed. For a lightly compacted sand, E’ was found to be 16 MPa. However, when the same sand was thoroughly compacted, E’ was found to be 99 Mpa. For a rounded, 10-mm gravel with only nominal (i.e., light) compaction, E’ was found to be 29 MPa. Where an existing pipe is subjected to an additional surface loading, the corresponding recommend E’ values for additional diametral strain prediction are 20, 100 to 150, and 40 MPa, respectively. The results obtained are discussed in relation to currently used values of E’ and site practice.

A curved element approximation in the analysis of axi‐symmetric thin shells

AbstractThe stiffness equation is derived for curved elements of orthotropic axi‐symmetric thin shells, and equivalent applied loads are found for shells subjected to initial strains, applied surface loads and body forces. The Lure approximation of thin shells and displacement field approximation by polynomials of arbitrary degree are included in the formulae derived.