Shear Springs Research Articles

Tunable bandgaps in an elastic meta-plate with shape memory alloy springs

In this paper, a tunable metamaterial based on shape memory alloy springs is designed, which can achieve bandgap tuning by spring's unique shape memory effect. The variation mechanism of shear modulus and geometrical parameters of shape memory alloy springs is investigated by considering a detailed phase transition mechanism. For any specific temperatures, the energy band structure and frequency response spectrum of the metamaterial are calculated by numerical simulation. And the theoretical prediction models of bandgap boundary frequency and tuning range at different temperatures are established. The experimental test of vibration transmission of metamaterial is finally presented. The results show that (1) By varying the spring's shear modulus and height, the metamaterial exhibits excellent vibration isolation characteristics and bandgap tuning in low-frequency range of 124–226Hz. (2) The bandgap boundaries and tuning ranges can be predicted by the theoretical prediction model, which shows good agreement with both the simulation results and the experimental data. (3) By artificially designing shape memory alloy springs with larger shear modulus and wire radius, smaller helix radius and number of turns, the bandgap moves to higher frequencies. The current work can provide a reference for further engineering applications with the tunable elastic/acoustic metamaterials.

Research on out-of-plane vibration band gap calculation of two-dimensional periodic grillage structures based on spectral element method

The vibration of ship and marine structures directly impacts their structural performance and noise levels. Therefore, the vibration band gap characteristics of two-dimensional periodic grillage structures are investigated based on spectral element method in this article. Firstly, the governing motion equation of out-of-plane vibration of periodic grid structures is established in light of the Bloch theorem, and the shear spring oscillator system is introduced to simulate the first-order vertical shear vibration of bottom plate. As a result, the periodic oscillator coupling grid structure model can be proposed. Based on the periodic grid structures, the influence of material distribution on vibration band gap is obtained. Meanwhile, the dispersion relations and vibration transmission of periodic coupling model are calculated and the coupling effect of local resonance system on the band gap characteristics is analyzed. At last, a general approach for equating periodic grillage unit to numerical model is outlined and summarized on basis of the similarity of dynamic properties. The effectiveness and accuracy of the numerical method in out-of-plane vibration band gap analysis of periodic grillage structures are validated through the computation and model test.

Elastic Foundation Solution for the End Notched Flexure (ENF) Mode II Sandwich Configuration

Abstract This paper presents a closed form solution for the energy release rate of face/core debonds in the Mode II end notched flexure (ENF) sandwich configuration. The finite-length sandwich specimen is considered to have a “debonded” region and a “joined” region. In the later, the interface between the top face and the substrate (core and bottom face) is modeled by an elastic foundation, which is a uniform distribution of shear and normal springs. Based on the Timoshenko beam theory, the solution for a general asymmetric sandwich construction is derived. The energy release rate expression is derived via the J-integral. Another closed form expression for the energy release rate is derived from the energy released by a differential spring as the debond propagates. In this closed form solution there is no fitting and everything, including the foundation constants, are given in closed form. Results are produced for a range of face/core stiffness ratios and debond length/core thickness ratios, and are compared with the corresponding ones from a finite element solution. A very good agreement is observed except for small debond lengths vs specimen thickness.

Improved stiff string torque and drag prediction using a computationally efficient contact algorithm

ABSTRACT Due to the intermittent contact with the wellbore, determining torque and drag for deviated wells is difficult. Most models have ignored drill string stiffness and assumed continual contact to simplify derivation. However, the accuracy of these ‘soft-string’ models is restricted, especially at high dogleg severities. On the other hand, most ‘stiff-string’ models rely on computationally intensive approaches or continuous contact assumptions. To mitigate these issues, a computationally efficient penalty-based wellbore contact algorithm has been developed based on vector calculation, which at most requires two dot products and three arithmetic operations to determine contact locations. This algorithm is incorporated into a 3D multibody dynamics (MBD) model, which utilizes rigid drill-string segments based on the Newton-Euler formulation, connected via axial, shear, torsional, and bending springs to capture drill string flexibility. This model performs simulations faster than real-time and has been validated using surface measurements from a completed well.

The Analytical Model and Horizontal Resistance of the Column-Head Dou-Gong in the Traditional Chinese Timber Structure

ABSTRACT To study the horizontal resistance of the column-head Dou-Gong (CHDG) in the traditional Chinese timber structure, the analytical model of CHDG without or with horizontal gap was proposed in this study. According to the structural form analysis of CHDG, the analytical model of CHDG considers the contact relationship between the components. CHDG has several distinct layers, of which the first layer is modeled as a rotation spring element and the second layer and above of CHDG are modeled as shear spring elements. Then, the horizontal resistance of CHDG was analyzed, and the horizontal load-displacement relationship of CHDG with horizontal gap appears triple polyline, while CHDG without horizontal gap is bilinear. Finally, the effects of parameters, including the length or width of Dadou bottom, height of Mantousun (MTS), width of horizontal gap, and vertical load, on the horizontal resistance was analyzed. The horizontal resistance of CHDG increases significantly with the length or width of the Dadou bottom, height of MTS, and vertical load increase, while the horizontal resistance of CHDG decreases significantly with width of horizontal gap increases.

Parameter identification procedure for hysteretic shear-resistant properties of beech wood dowels

To evaluate the shear-resistant behavior of wooden dowels used in Blockhaus shear walls under cyclic load, 19 specimens under ten groups of conditions were prepared and tested. The failure modes, hysteresis curves, mechanical properties, stiffness degradations, and energy dissipation capacities of the specimens were studied. The test results showed that with the increase in the number of dowels, the initial stiffness and peak load of the specimens increased greatly. The diameter of the dowels had little influence on the mechanical properties of the specimens. Furthermore, the test findings demonstrated that the pretension load between the walls greatly enhanced the initial stiffness and energy dissipation capacity of specimens. A simplified finite element model was established in Opensees. Considering the effect of material variability, the parameters of single dowel shear spring and friction spring were identified by Genetic Algorithm with modified objective function in Matlab. The identified parameters were applied to the finite element model of the multi-dowel specimens. The simulation results were in good agreement with the test results, and the validity of the numerical model and parameter identification method was verified.

A Method for Stabilizing the Vibration Amplitude of a Flip-Flow Vibrating Screen Using Piecewise Linear Springs

The flip-flow vibrating screen (FFVS) is a novel multi-body screening equipment that utilizes vibrations to classify bulk materials in the field of screening machinery. The relative amplitude of FFVSs determines the tension and ejection intensity of elastic flip-flow screen panels, which is a critical operating parameter affecting the screening performance. However, FFVSs generally suffer from large variations of relative amplitude caused by the loading of materials and the changes in shear spring stiffness (the temperature changes of the shear springs lead to their stiffness changes), which significantly reduce the screening efficiency and lifespan of FFVSs. To address this problem, this paper proposes a nonlinear stiffness-based method for stabilizing the vibration amplitude of FFVSs using piecewise linear springs. By introducing these springs between the two frames, the sensitivity of the relative amplitude to shear spring stiffness is reduced, thereby achieving the stabilization of the relative amplitude of FFVSs. In this study, the variations of the vibration amplitude of the FFVS due to the loading of materials and the changes in shear spring stiffness were first demonstrated in a reasonable operating frequency range. Then the reasonable operating frequency range and dynamics of the resultant nonlinear flip-flow vibrating screen (NFFVS) with piecewise linear springs were investigated using the harmonic balance method (HBM) and the Runge–Kutta numerical method. The operating frequency region for the NFFVS lies between the critical frequency ωcs and the frequency ωlb corresponding to the saddle-node bifurcation point. Finally, a test rig was designed to validate the theoretical predictions. Theoretical and experimental results demonstrate that piecewise linear springs can effectively stabilize the relative amplitude of the FFVS.

Assessing the significance of nonlinear rotational behavior in high damping rubber bearings for seismic performance of base‐isolated RC frame building

AbstractUtilizing high‐damping rubber bearings (HDRBs) for seismic isolation significantly enhances building seismic performance. However, previous studies have often neglected the nonlinear rotational behavior of these bearings when assessing base‐isolated systems using HDRBs. This study introduces two numerical models of the HDRBs, each comprising six springs. The first model, referred to as the BIL model, assumes linear behavior for the rotational degrees of freedom of the bearings. The second model, referred to as the BINL model, accounts for the nonlinear rotational behavior. The parameters of the bearing shear springs and the rotational springs of the two models are calibrated using experimental data. In order to investigate the influence of nonlinear rotational behavior of HDRBs on the seismic performance of base‐isolated buildings, nonlinear time history analysis is conducted on a six‐story base‐isolated RC frame building. Comparative analysis between the BIL model and the BINL model covers bearing behavior, superstructure response, and energy dissipation distribution. Results indicate a significant influence of considering nonlinear rotational behavior on column bending moments. These findings provide crucial insights for enhancing seismic design strategies for base‐isolated buildings.

Robustness of Corner-Supported Modular Steel Buildings with Core Walls

This paper studies the dynamic response of corner-supported modular steel buildings with a core wall system, under progressive collapse scenarios, associated with corner module removals. Since using secondary systems such as concrete core in mid- to high-rise buildings is currently unavoidable, understanding their impact on load transfer between modules during collapse scenarios becomes essential. The designated four-, eight-, and twelve-story buildings were modelled using the macro-model-based finite element method in Abaqus. In addition, three different locations are considered for the concrete shear core within the building plan, leading to nine various case scenarios. Each vertical and horizontal inter-module connection was modelled by one axial and two shear springs with predefined nonlinear force-displacement behavior. The local and global buckling, which plays an essential role in the building’s stability, was considered to obtain accurate results. Finally, parametric studies on the building response were carried out, including the intra-module connection rigidity and inter-module connection stiffness. The results demonstrated that the core wall could maintain the robustness of a modular steel building through two mechanisms dependent on its location within the plan. In addition, preventing plastic hinges from forming in beams could be introduced as an anti-collapse mechanism in the corner module removal scenarios.

A CPT-based multi-spring model for lateral monopile analysis under SLS conditions in sand

Monopiles are the most common Offshore Wind Turbine (OWT) foundations due to their simplicity in design, fabrication, and installation. However, large new-generation turbines have led to significant changes in monopile dimensions, necessitating extensive finite element analyses and ground investigations to meet design requirements. While Cone Penetration Test (CPT)-based p-y methods can analyse slender pile lateral behaviour, they often miss additional resistance mechanisms relevant to rigid monopiles. This paper introduces CPT-informed resistance mechanisms for monopiles to incorporate additional lateral resistances beyond p-y modelling capabilities. Distributed moment–rotation (m-θ) springs are defined by repurposing CPT-based axial capacity estimation methods for piles; and pile tip shear and moment springs are informed by approximating a residual bearing stress post-installation using local CPT qc values. The performance of the multi-spring model is appraised against data reported from monotonic pile pushover tests conducted at two sand sites. Results show that the multi-spring model is capable of predicting pile head deflections reasonably well within serviceability deflection limits against the reported test data, but ultimate failure loads cannot be predicted using the proposed model. A clear sensitivity in pile response to local variations in CPT qc is demonstrated.

Flexural vibration of elastic composite beams using a third-order deformation kinematics

The present study introduces a one-dimensional finite element (FE) model that utilizes third-order deformation kinematics to analyze the flexural vibration of two-layered composite beams with partial shear slip. The proposed model incorporates a three-node straight beam element, utilizing the field-consistent approach and full numerical integration of the stiffness matrix. This approach is adopted to effectively address stress oscillation and shear locking issues, ensuring accurate and reliable results. The analysis focuses on the third-order deformation kinematics of longitudinal displacement within the beam depth, examining each of the two layers individually. The interface between the two material layers is joined using deformable shear connectors, which are represented as dispersed shear springs over the length of the beam. This study focuses on conducting a flexural vibration analysis of two-layered composite beams. The analysis includes the examination of both free vibration and forced vibration responses while considering or omitting the influence of damping under various loading conditions. The validity of the proposed FE model is assessed by the utilization of a two-dimensional model developed within the ABAQUS software. Additionally, the suggested model is evaluated through the process of verification by comparing it with previously published results.

Numerical modelling of energy piles under combined loading

The growing use of fossil fuels and other non-renewable energy sources has made climate change a critical global issue. In order to counter this threat, several countries are engaged in an ecological transition, and are looking for technologies using renewable energy sources. In this context, energy geostructures, such as thermo-active (or energy) piles, have been developed, consisting in fixing heat exchanger pipes to the reinforcement cages of foundation piles to extract/inject the heat from/into the ground with the purpose of meeting the building heating and cooling demands. Their specificity is their dual function: structural support and energy exchanger.
 In the case of energy piles, two aspects can be critical and should be considered in their design. The first is the nature of the cyclic thermal loading, which can affect the mechanical response of the energy pile. In fact, during temperature variation along the pile, stresses change and pile head movements are induced (Figure 1), due to the thermal dilatancy/contraction of the pile and the behaviour of the soil-pile interface [1, 2, 3, 4, 5, 6]. Consequently, cyclic thermal loading can induce a deterioration of the shear stresses at the soil-pile interface and hence a deterioration of the pile's bearing capacity [7]. The second aspect concerns the adaptation of design under combined lateral and axial loads. Indeed, the co-existence of a lateral loading can affect the axial response of the pile and vice-versa [8]. These configurations are the most favourable for installing heat exchanger pipes since, mechanically, they require reinforcement cages all along the pile height. Studies on energy piles have mainly investigated their behaviour under axial loading. Energy piles under lateral loading have hardly been considered [9].
 The aim of this paper is to present a practical calculation tool for modelling energy piles that takes into account the combined loading on the pile response. An original 1D finite element approach is developed for engineering practice, taking into account the rheology of the problem. The pile is discretised in beam finite elements with three degrees of freedom at each node (vertical displacement, horizontal displacement and rotation). The soil is modelled by surface shear and normal springs. This tool is based on the solution of the equilibrium equation of the global system by an iterative plastic correction procedure. This correction is based on the yield criterion defined in the code.
 The main strength of this approach lies in its capability to consider a 3D failure envelope for an energy pile, capturing its behaviour under combined axial, lateral and cyclic thermal loading. It can clearly represent the critical effects of these loads by adopting an appropriate behaviour law for the soil-pile interface. In addition to being practical and easy to use, this tool has the advantage of reducing calculation time compared to more complex 3D numerical methods, especially in the case of cyclic thermal loading.

Grain-Level Numerical Simulations for the Effective Elasticity of Weakly Cemented Sandstones.

Weakly cemented sandstones are characteristic of loose-bonding contacts, large porosities, and high-clay contents. This study presents a discrete element method (DEM)-based numerical study for the effective elasticity of such rocks that mainly depends on the mechanical behavior of intergranular contact regions. The DEM scheme employs a set of normal and shear springs to phenomenologically describe the mechanical behavior of intergranular finite-sized cements defined by three morphological parameters: cement thickness, bonding radius, and grain radius. Applications to two digital models established in terms of contact-bonding and distant-bonding modes, respectively, where spherical quartz grains are randomly packed together with adding cements under the specified confining pressure, are compared with the theoretical predictions by the contact-bonding and distant-bonding cement theories, which demonstrates a good agreement generally for small contact widths, small contact thicknesses, and large-magnitude moduli, especially for the effective shear modulus. Applications to a series of artificial sandstone samples made in terms of different proportions of quartz grains and clays (a mixture of epoxy and kaolinite) under loose compaction for weak cementation demonstrate a good agreement with ultrasonic measurements. Numerical investigations for the micromechanical characteristics (differential stress fields, force chains, and fabric tensors) of artificial samples subject to applied axial strains demonstrate that the strong mechanical behavior of weakly cemented sandstones tends to appear inside the cohesive aggregates of stiff grains because of their relatively large sizes with loose compaction.

DNN-MLVEM: A Data-Driven Macromodel for RC Shear Walls Based on Deep Neural Networks

This study proposes the DNN-MVLEM, a novel macromodel for the non-linear analysis of RC shear walls based on deep neural networks (DNN); while most RC shear wall macromodeling techniques follow a deterministic approach to find the right configuration and properties of the system, in this study, an alternative data-driven strategy is proposed instead. The proposed DNN-MVLEM is composed of four vertical beam-column elements and one horizontal shear spring. The beam-column elements implement the fiber section formulation with standard non-linear uniaxial material models for concrete and steel, while the horizontal shear spring uses a multi-linear force–displacement relationship. Additionally, three calibration factors are introduced to improve the performance of the macromodel. The data-driven component of the proposed strategy consists of a large DNN that is trained to predict the force–displacement curve of the shear spring and the three calibration factors. The training data is created using a parametric microscopic FEM model based on the multi-layer shell element formulation and a genetic algorithm (GA) that optimizes the response of the macromodel to match the behavior of the microscopic FEM model. The DNN-MVLEM is tested in two types of examples, first as a stand-alone model and then as part of a two-bay multi-story frame structure. The results show that the DNN-MVLEM is capable of reproducing the results obtained with the microscopic FEM model up to 100 times faster and with an estimated error lower than 5%.

Improvement of size effect simulation based on an energy-balanced exponential softening bond model and fracture energy regularization

Although the Mode I fracture of rock demonstrates a noticeable size effect, current contact models in the discrete element method cannot simulate the size effect very well. An exponential softening bond model is proposed to address this issue based on an exponential damage law and an elliptic yield criterion. Its basic formulas are established on energy balance to ensure that the total fracture energy of the normal and shear springs always equals the fracture energy dissipated by the resultant contact force. The simulations of uniaxial tensile and three-point bending tests were conducted to demonstrate the performances of the proposed model on the fracture simulations. The particle size dependency is successfully eliminated by fracture energy regularization, in which the force–displacement relationship is adjusted based on fracture energy and particle size. The simulation results show that the exponential softening bond model effectively simulates the whole load–displacement curves of experimental tests. It also demonstrates a perfect energy balance among external work, internal energy, and fracture energy. Besides, compared with the other two bond models, the exponential softening bond model is more precise in representing the fracture toughness, the fracture process zone, and the size effect of the Mode I fracture of limestone.

Analysis of Shear Stagger Deformation of Existing Shield Tunnel below Induced by Quasirectangular Shield Tunneling

To use urban underground space more efficiently is a research hotspot of urban underground development. Compared with conventional circular tunnels, quasirectangular tunnels have the characteristics of larger space utilization and higher economic benefits, gradually used in urban rail engineering in recent years. But the ground disturbance induced by the quasirectangular tunneling needs further researches. This paper derives the calculation formula of the additional stress of existing shield tunnel below caused by the quasirectangular shield tunneling by mirror image method and Mindlin solution. The existing shield tunnel is simplified as an elastic foundation short beam connected by tensile springs and shear springs. This paper establishes the energy-deformation coupling equation by principle of minimum potential energy so that the disturbance induced by quasirectangular tunneling to the existing shield tunnel below can be analyzed. The FEM is established with the actual engineering as the background. It is by comparing the numerical simulation results and field monitoring data that the theoretical prediction formula is verified. The research results show that the theoretical calculation results consist highly with numerical simulation results and field monitoring data, verifying practicality of formulas in this paper. Calculation results of shear stagger model are more accurate, which can be also used to analyze the staggered and splayed amount between segments. As the tunnel construction proceeds, uplift deformation, segment stagger, shear force, and segment splay of existing shield tunnel gradually increase, which are symmetrical about the central axis of new tunnel. After the shield machine passes through the central axis of the existing shield tunnel for 20 meters, stable disturbance to the existing shield tunnel occurs. Stagger amount and shear force between segments reach the minimum at the position with maximum uplift deformation of existing shield tunnel and reach the maximum at inflection point of the uplift deformation curve.

Cohesive-zone based fracture mechanics model of an edge delamination in bimaterial beam under mixed-mode bending test

The problem of an edge-split bilayer beam consisted of bonded dissimilar materials under mixed-mode bending (MMB) is investigated by using a structural mechanics procedure. In this analytical procedure, the bilayer beam is modeled as two Timoshenko beams joined by using interface tensile and shear springs, and the fracture mechanics parameters including the strain energy release rate and phase angle are estimated from the spring responses. Numerical examples of bilayer beams consisted of isotropic or orthotropic bimaterials and with symmetric or asymmetric dimensions are presented and compared to continuum based finite-element solutions with either cohesive-zone or perfectly-bonded interface models. It is shown that the strain energy release rates obtained from these models are in good agreement. The phase angles obtained from the analytical model and the cohesive-zone-interface based continuum model are also consistent with each other, and agree to the crack-tip deformation. On the other hand, the phase angles obtained from the oscillating crack-tip fields related to the perfectly-bonded interface do not match to the crack-tip deformation as well as the cohesive-zone interface models. Consequently, it is recommended to characterize the mode-mixity dependent interfacial adhesion and fatigue responses by using the fracture mechanics parameters obtained by using the analytical solution for the MMB fracture problem.

Discrete and continuous models of linear elasticity: history and connections

This paper tracks the development of lattice models that aim to describe linear elasticity of solids and the field equations of which converge asymptotically toward those of isotropic continua, thus showing the connection between discrete and continuum. In 1759, Lagrange used lattice strings/rod dynamics to show the link between the mixed differential-difference equation of a one-dimensional (1D) lattice and the partial differential equation of the associated continuum. A consistent three-dimensional (3D) generalization of this model was given much later: Poincaré and Voigt reconciled the molecular and the continuum approaches at the end of the nineteenth century, but only in 1912 Born and von Kármán presented the mixed differential-difference equations of discrete isotropic elasticity. Their model is a 3D generalization of Lagrange’s 1D lattice and considers longitudinal, diagonal and shear elastic springs among particles, so the associated continuum is characterized by three elastic constants. Born and von Kármán proved that the lattice equations converge to Navier’s partial differential ones asymptotically, thus being a formulation of continuous elasticity in terms of spatial finite differences, as for Lagrange’s 1D lattice. Neglecting shear springs in Born–Kármán’s lattice equals to Navier’s assumption of pure central forces among molecules: in the limit, the lattice behaves as a one-parameter isotropic solid (“rari-constant” theory: equal Lamé parameters, or, equivalently, Poisson’s ratio $$\upsilon =1/4$$ ). Hrennikoff and McHenry revisited the lattice approach with pure central interactions using a plane truss; the equivalent Born–Kármán’s lattice in plane stress in the limit tends to a continuum with Poisson’s ratio $$\upsilon = 1/3$$ . Contrary to McHenry–Hrennikoff’s truss, Born–Kármán’s lattice leads to a “free” Poisson’s ratio bounded by its “limit’ bound ( $$\upsilon =1/4$$ for plane strain or 3D elasticity; $$\upsilon =1/3$$ for plane stress elasticity). Unfortunately, Born–Kármán’s lattice model does not comply with rotational invariance principle, for non-central forces. The consistent generalization of Lagrange’s lattice in 3D was achieved only by Gazis et al. considering an elastic energy that depends on changes in both lengths and angles of the lattice. An alternative consistent three-parameter elastic lattice is the Hrennikoff’s, with additional structure in the cell. We also discuss the capability of nonlocal continuous models to bridge the gap between continuum isotropic elasticity at low frequencies and lattice anisotropic elasticity at high frequencies.

Stick model for response prediction of as-built and retrofitted infilled RC frames

The Stick model is a Multi-degree of Freedom system consisting of a series of story-level lumped masses connected in series by non-linear shear springs suitably calibrated to simulate the interstory behavior. The Stick model has been recently introduced to predict the seismic performances of existing buildings in terms of engineering demand parameters, such as interstory drift ratios and peak floor accelerations, both at the building and at the large-scale level. This model has been demonstrated to predict the damage concentration and distribution with sufficient accuracy still preserving computational time with respect to refined finite element models. Previous works proposed a calibrated model to represent the behavior of existing gravity load-designed infilled building typologies. Recent advancements suggest the adoption of simplified procedures to generate interstory shear spring backbones allowing the extension of the model to different building typologies also accounting for possible retrofit interventions.In this paper, a simplified procedure is proposed to generate interstory shear spring backbones. The procedure allows generating interstory backbones accounting for typical features characterizing existing buildings, explicitly simulating typical failure modes of non-conforming RC elements, infill-induced brittle failures, and introducing the effect of possible retrofit interventions to increase both the building structural capacity and the stiffness.

Analytical review of design aspects of helical spring

The curvature correction factor is an important parameter for evaluating the stress of helical springs. A novel modelled curvature correction factor (KG) was proposed in terms of the shear correction factor (KS) and spring index (C). This factor (KC) falls between the Wahl and Bergstrasser correction factors, considering the stresses due to shear, torsional, and curvature effects. The ratio of the actual deflection to estimated deflection and the ratio of the actual stiffness to estimated stiffness are mirror images of each other. In addition, when C continued to decrease, the curvature correction factor increased sharply. In practical design, the actual deviation is greater than the evaluated deflection, and the evaluated deflection may need to be increased by a certain percentage to obtain the actual deflection. Therefore, a general equation was developed for this purpose. The ratios of the actual and evaluated deflections and the actual and evaluated stiffnesses were mirror images of each other. For suitable assumed data, the torsional shear stress, maximum shear stress, and modelled curvature stress were almost straight and parallel to each other.