End Constraints Research Articles

Elasticas and buckling loads of shear deformable tapered columns with both hinged ends

This paper deals with the geometrical non-linear analyses of the buckled columns. Differential equations governing elasticas of the buckled columns are derived, in which both effects of taper type and shear deformation are included. Three kinds of tape types such as breadth, depth and square tapers are considered. Differential equations are solved numerically to obtain the elasticas and buckling loads of such columns. End constraint of both hinged ends is considered. The effects of shear deformation on the elastica of the buckled column and buckling load of column are extensively investigated. Experimental studies are presented that complement theoretical results of non-linear responses of the elasticas.

Plane Analysis of Finite Multilayered Media With Multiple Aligned Cracks—Part II: Numerical Results

In Part I of this paper, elasticity solutions were developed for finite multilayered domains, weakened by aligned cracks, that are in a state of generalized plane deformation under two types of end constraints. In Part II we address computational aspects of the developed solution methodology that must be implemented numerically, and present new fundamental results that are relevant to modern technologically important applications involving defect criticality of multilayers. The computational aspects include discussion of the various parameters that influence the accuracy with which numerical results are generated and subsequent verification by a comparison with previously reported results in the limit, as the in-plane dimensions become very large and layer anisotropy vanishes. The present solution quantifies the thus far undocumented effects of finite dimensions, crack location, and material anisotropy due to a unidirectional fiber-reinforced layer’s orientation on Mode I, II, and III stress intensity factors in composite multilayers with single and multiple interacting cracks under different loading and boundary conditions. These effects may have significant impact on defect criticality of advanced multilayered structures when cracks are in close proximity to vertical and horizontal boundaries.

Plane Analysis of Finite Multilayered Media With Multiple Aligned Cracks—Part I: Theory

Elasticity solutions are developed for finite multilayered domains weakened by aligned cracks that are in a state of generalized plane deformation under two types of end constraints. Multilayered domains consist of an arbitrary number of finite-length and finite-height isotropic, orthotropic or monoclinic layers typical of differently oriented, unidirectionally reinforced laminas arranged in any sequence in the plane in which the analysis is conducted. The solution methodology admits any number of arbitrarily distributed interacting or noninteracting cracks parallel to the horizontal bounding surfaces at specified elevations or interfaces. Based on half-range Fourier series and the local/global stiffness matrix approach, the mixed boundary-value problem is reduced to a system of coupled singular integral equations of the Cauchy type with kernels formulated in terms of the unknown displacement discontinuities. Solutions to these integral equations are obtained by representing the unknown interfacial displacement discontinuities in terms of Jacobi or Chebyshev polynomials with unknown coefficients. The application of orthogonality properties of these polynomials produces a system of algebraic equations that determines the unknown coefficients. Stress intensity factors and energy release rates are derived from dominant parts of the singular integral equations. In Part I of this paper we outline the analytical development of this technique. In Part II we verify this solution and present new fundamental results relevant to the existing and emerging technologies.

Unique local deformations of the superelastic SMA rods during stress-relaxation tests

This paper studies mechanical behavior of the superelastic shape memory alloy (SMA) rods in terms of local deformations and time via tensile loading-unloading cycles for both ends fixed end constraints. Besides the unique stress induced martensitic transformation (SIMT), SMA`s time dependent behavior when it is in mixed-phase condition upon loading and unloading, also need careful attention with a view of investigating the local deformation of the structural elements made of the same material. With this perspective, the so-called stress-relaxation tests have been performed to demonstrate and investigate the local strains-total strains relationships with time, particularly, during the forward SIMT. Some remarkable phenomena have been observed pertaining to SIMT, which are absent in traditional materials and those unique phenomena have been explained qualitatively. For example, at the stopped loading conditions the two ends (fixed end and moving end of the tensile testing machine) were in fixed positions. So that there was no axial overall deformation of the specimen but some notable increase in the axial local deformation was shown by the extensometer placed at the middle of the SMA specimen. It should be noted that this peculiar behavior termed as `inertia driven SIMT` occurs only when the loading was stopped at mixed phase condition. Besides this relaxation test for the SMA specimens, the same is performed for the mild steel (MS) specimens under similar test conditions. The MS specimens, however, show no unusual increase of local strains during the stress relaxation tests.

Mode mixity for tubular K-joints with weld toe cracks

This paper examines the mode-mixity of stress-intensity factors for surface cracks located near weld toes of the crown point at the toe of the tension brace in circular hollow section K-joints under different chord/brace end constraints. The remote loading applies a uniform stress at the end of the brace along its axis. The 3-D finite element models employ mesh-tieing between a topologically continuous, global mesh and a separate, local crack-front mesh. The linear-elastic analyses compute the mixed-mode stress-intensity factors along the crack front using an interaction-integral approach. The analyses for three different crack locations along the brace–chord intersection demonstrate that the critical location for the surface flaw lies at the crown point of the brace toe. The numerical investigation of six different boundary conditions indicate that the unbalanced loading conditions cause significant bending stresses in the chord and generate a considerably larger mode I stress-intensity factor than found for the balanced loading case. The mixed-mode stress-intensity factors indicate that the crack front experiences predominantly mode I loading, with K III → 0 near the deepest point on the front ( ϕ = π/2). The non-dimensional, mode I crack driving force, described by F I = K I / σ ¯ br π a , reaches a maximum value at the deepest point of the crack for the crack aspect ratio a/ c = 0.25 considered here. The mode-mixity angle, ψ = tan −1( K II/ K I), at ϕ = π/2 is compared for a range of practical K-joint configurations. The present study demonstrates that the mode-mixity angle ψ becomes significantly larger in the balanced loading conditions than in the unbalanced loading condition. Values of the non-dimensional stress-intensity factors ( F I = K I / σ ¯ br π a and F II = K II / σ ¯ br π a ), however, decrease for the balanced loading condition. Variations in the brace-to-chord diameter ratio ( β) and chord radius to wall thickness ratio ( γ) generate significant changes in the mode mixity. Thin-walled joints ( γ = 20) generally experience a larger crack driving force with a higher mode-mixity angle.

Design Analysis for Buckling of Tanks and Silos

When subject to combinations of internal pressure and axial compression, experience shows that the often thin cylindrical shell walls of tanks and silos are prone to a local form of buckling failure that is essentially axisymmetric and is strongly influenced by the nature of the end or intermediate support conditions. This paper will outline the basis of an analytical solution to the inherently nonlinear elastic-plastic buckling into axisymmetric modes, in which end constraints and arbitrary radial pressure loading are modeled as essentially loading imperfections on an otherwise linear, classical, critical buckling formulation. Closed form analytical expressions for the imperfect behavior, including geometric imperfections, are then used to predict either first surface or full section plasticity, allowing buckling failure to be summarized in a form that is closely analogous to the Ayrton-Perry formula for the buckling of columns. It is suggested that a suitably simplified form of this general approach to axisymmetric buckling, when combined with similarly simple methods for the buckling into nonaxisymmetric modes, could provide an important alternative to the current procedures for the buckling design of thin-walled, cylindrical, tanks and silos.

New Considerations on the Stress Field in an Off-axis Tensile Test

Two new analytical approaches are proposed for the off-axis tension test problem. Both of them are based on the Second Castigliano’s theorem. In the first one, a simple stress field that satisfies equilibrium conditions is used. In the second approach, integration constants of the stress field obtained by Pagano and Halpin are calculated directly, without the use of the displacement field. The first approach shows the points of maximum normal and shear stresses in a qualitative manner. The second approach, besides the fulfilment of equilibrium and compatibility conditions, minimizes the strain energy. Furthermore, both approaches give the same solution for bending moments and shear forces at specimen ends caused by coupling effects. After analyzing the stresses at critical points in the principal material directions, it is shown that the value of in-plane shear strength is usually underestimated. The effect of end constraints on the obtention of in-plane shear modulus is also analyzed.

Identification of the tensile force in high-tension bars using modal sensitivities

In certain field applications involving the repair or replacement of structural members, the knowledge of the tensile forces in tension members may be critical in the selection of the retrofit or rehabilitation process. To date, several methods have been presented to estimate in situ tensile forces using static and/or dynamic responses of the tension members. However, each and every one of these methods has its disadvantages as well as advantages in its procedures, level of accuracy, and equipment requirements. This paper presents a practical method to identify the tensile force in a high-tension bar using the changes in the modal sensitivities. The methodology can identify not only the tensile force but also other related parameters such as the magnitude of end constraint. The methodology utilizes the sensitivity relationship between the natural frequencies, the structural parameters, and the tensile force in the member. In this paper, the basic elements of the approach are summarized and the proposed method is validated using both numerical and experimental data. Also, the impact of model uncertainty to the performance of the parameter identification is investigated.

Strength and Failure Modes of Hoop Wound CFRP Tubes Under Compressive High Rates of Loading

A study was undertaken to investigate the response of hoop wound carbon fibre reinforced plastic (CFRP) tubes to dynamic compressive loading at strain rates in the range of 5–200/sec. An experimental rig was designed and built to test short tubular specimens under external radial pressure with minimum end constraints. The load was applied by detonating a small explosive charge inside a water filled, steel, cylindrical chamber enclosing the test specimen. For each test the external pressure and the strains, in both circumferential and longitudinal directions, were recorded on suitable digital processing equipment. Two distinct modes of failure were identified; material and structural (buckling). The mode of failure was dependent on the rate of loading and the tube diameter/thickness ratio. For 100 mm diameter tubes with diameter/thickness=40, buckling failure dominated at strain rates below 10/s. However, at higher strain rates, material failure and a considerable enhancement in burst strength was observed. For 100 mm diameter tubes, with diameter/thickness=80, a buckling mode of failure was in evidence in all the tests, irrespective of the rate of loading.

Effect of boundary constraint on the frequency response of moderately thick doubly curved cross-ply panels using mixed fourier solution functions

A hitherto unavailable analytical solution to the boundary value problem of free vibration response of shear–flexible cross-ply laminated doubly curved panels is presented. The laminated shell theory formulation is based on the first-order shear deformation theory (FSDT) including rotatory and surface-parallel inertias. The governing equations of the panel are defined by five highly coupled partial differential equations in five unknowns — three displacements, and two rotations. The assumed solution functions for the eigen/boundary-value problem are selected in terms of mixed-type double Fourier series. Extensive numerical results that are presented in this study include (1) convergence characteristics of computed natural frequencies, and (2) effects of length-to-thickness ratio, radius-to-length ratio, lamination sequence, boundary constraint and shell geometry on the normalized natural frequencies of interest. Also numerically investigated is the highly complex interaction among bending–stretching type coupling effect, membrane action due to shell curvature, surface-parallel end constraints (or lack thereof), and the effects of transverse shear deformation, rotatory inertias and surface-parallel inertias.

A protocol for characterizing the structural performance of metallic sandwich panels: application to pyramidal truss cores

All-metallic sandwich panels with truss and prismatic cores have impending application as ultra-light load bearing panels amenable to simultaneous active cooling and blast resistance. To facilitate application, a protocol for characterizing the structural performance of such panels is needed that can be used for design calculations. An approach capable of realizing this objective is presented and demonstrated for panels with pyramidal truss cores. It combines measurements, mechanism maps, finite element simulations and optimization. Mechanism maps based on beam theory are used to characterize face and core dimensions as well as to estimate minimum weight designs. Experimental measurements and finite element calculations are used to calibrate and understand the responses of the core in transverse compression, in-plane shear and stretch. Overlaying the measurements and simulations allows assessment of the mechanical properties of the core material, as affected by fabrication. The updated results are used to establish and calibrate an orthotropic constitutive law for the core. Bending tests performed on panels subject to two end constraints are used to assess the fidelity of the core constitutive law. The tests are simulated by using a stress/strain response for the faces obtained from independent tensile measurements. The comparison reveals that the loads are predicted quite accurately. However, when controlled by the core, the simulations predict a more abrupt yield point than that found experimentally.

Quasi-static axial crush response of a thin-wall, stainless steel box component

An experimental investigation was performed on the quasi-static, symmetric axial crush response mode of commercially-produced, welded AISI 304 stainless steel square box components. The first objective was to demonstrate that a specific configuration response consisting of a fold formation process and the corresponding load–displacement curve could be obtained using experimental control methods. Test specimens from a single length of tubing were examined in a progressive axial crush study, and the controls included removable grooved caps for end constraints, and shallow machined groove patterns on specimen sidewalls for collapse initiators. Consistent fold appearance and load–displacement curve shapes indicated that ten of the eleven test specimens exhibited the same symmetric, axial crush configuration response. During the first cycle of the secondary folding phase, observed differences on an average value basis were less than or equal to 3% for maximum loads, 6% for minimum loads, and 3% for energy absorption. The second objective of this investigation was to isolate the effect of alloy composition and microstructure on the axial crush configuration response. A higher concentration of carbon and smaller grain size resulted in an 18% increase in energy absorption in a secondary folding phase cycle. Overall, the results show that response can be restricted to a specific axial crush configuration response and therefore, a controlled and repeatable energy absorption process can be obtained and modifications to alloy composition and microstructure can be used to enhance energy absorption performance of the box component.

A magnetoelastic theoretical model for soft ferromagnetic shell in magnetic field

With the aid of a generalized variational method, in this paper, a theoretical model for soft ferromagnetic shells is derived to describe their magnetoelastic behavior in an applied magnetic field. Having made a quantitative comparison between the numerical predictions given by several theoretical models and the experimental results on strains of a cylindrical shell, we find that the predictions got by our model are in good agreement with the experimental data. It is also found that the Moon’s model is a special case of the model derived in this paper when the relative magnetic permeability μ r>10 4, which confirms that it is reasonable for the Moon’s model to calculate strains of the soft ferromagnetic shells. Having displayed the distribution of the equivalent magnetic force in the length of the cylindrical shell and its circumferential bending strains with different elastic end constraints, we give an explanation for the discrepancy between Moon’s analytical results and his experimental ones.

Single-walled carbon nanotubes as ultrahigh frequency nanomechanical resonators

Department of Mechanical Engineering, University of Delaware, Newark, Delaware 19716, USA~Received 16 March 2003; revised manuscript received 3 June 2003; published 28 August 2003!The high stiffness and strength, low density, and large aspect ratio of single-walled carbon nanotubes makethem good candidates as nanoresonators. Employing an atomistic modeling technique, molecular structuralmechanics, this work predicts that the fundamental frequencies of cantilevered or bridged single-walled carbonnanotubes as nanomechanical resonators could reach the level of 10 GHz–1.5 THz. The effects of tubediameter, length and end constraints on the fundamental frequency have been discerned.DOI: 10.1103/PhysRevB.68.073405 PACS number~s!: 62.25.1g, 61.46.1w, 85.35.Kt, 61.48.1c

Buckling loads of linearly tapered columns laterally restrained by multiple elastic springs

This paper discusses the numerical methods that were developed for calculating the buckling loads with the corresponding buckled shapes of linearly tapered columns laterally restrained by multiple elastic springs. In formulating the governing equation of the column, each elastic spring was modeled as a discrete Winkler foundation of the finite longitudinal length. By using this model, the differential equation governing the free vibration of the column was first derived, which was subjected to an axial compressive load, after which the differential equation governing the buckled shapes of the column was formulated using the relationship between the natural frequencies and the loads. The Runge-Kutta method was used to integrate the differential equation, and the determinant search method combined with the Regula-Falsi method was used to determine the eingenvalues, namely, the buckling loads. In the numerical examples, clamped-clamped, clamped-hinged, hinged-clamped and highed-hinged end constraints were considered. The numerical results including the buckling load parameters and the buckled shapes of the columns were presented in non-dimensional forms.

Axial Loading and Buckling Response Characteristics of Pultruded Hybrid Glass-Graphite/Epoxy Composite Beams

The effects of hybridization on the buckling characteristics of flat pultruded glass-graphite/epoxy composite beams has been investigated. Theoretical buckling loads were obtained based on Euler’s formulation. Finite element models were constructed with different beam dimensions and end constraint conditions to simulate buckling. Experiments were conducted to verify and validate the theoretical and finite element results. The experimental results were found to be consistent with the theoretical and finite element models. The buckling characteristics of composites showed that buckling strengths improve with increase in graphite fiber content. The improvements were more pronounced when glass fibers were replaced with graphite fibers located in the outer layers of the composite. The effectiveness of hybridization for improving the buckling performance of composite beams is highlighted.

Free vibrations of tapered beams laterally restrained by elastic springs

This paper discusses the development of numerical methods for calculating the natural frequencies of tapered beams that are laterally restrained by elastic springs. In formulating the governing equation of the beam, each elastic spring is modeled as a discrete Winkler foundation of the finite longitudinal length, and the effect of axial load is included. By using this model, the differential equation governing the free vibration of the beam is derived, which is solved numerically. The Runge-Kutta method is used to integrate the differential equation, and the determinant search method combined with the Regula-Falsi method is used to determine the eingenvalues, namely, the natural frequencies. In the numerical examples, clamped-clamped, clamped-hinged, hinged-clamped and hinged-hinged end constraints are considered. The numerical results, including the frequency parameters and mode shapes of free vibrations are presented in non-dimensional forms.

NOVEL METHODS FOR CALCULATING NATURAL FREQUENCIES AND BUCKLING LOADS OF COLUMNS WITH INTERMEDIATE MULTIPLE ELASTIC SPRINGS

Numerical methods for calculating both the natural frequencies and buckling loads of columns with intermediate multiple elastic springs are developed. In formulating the governing equations of the column, each elastic spring is modeled as a discrete Winkler foundation of the finite longitudinal length. By using this model, the differential equations governing both the free vibration and buckled shapes of the column are derived, which are solved numerically. The Runge–Kutta method is used to integrate the differential equations, and the determinant search method combined with Regula–Falsi method is used to determine the eingenvalues, namely, the natural frequencies and buckling loads. In the numerical examples, fixed–fixed, fixed-hinged, hinged-fixed and hinged–hinged end constraints are considered. The numerical results including the frequency parameters, mode shapes of free vibrations and buckling loads are presented in non-dimensional forms.

Lateral Resistance of Full-Scale Pile Cap with Gravel Backfill

A static lateral load test was performed on a full-scale 3×3 pile group driven in saturated low-plasticity silts and clays. The steel pipe piles were attached to a concrete pile cap which created a “fixed-head” end constraint. A gravel backfill was compacted in place on the backside of the cap. Lateral resistance was therefore provided by pile–soil–pile interaction, as well as base friction and passive pressure on the cap. In this case, passive resistance contributed about 40% of the total resistance. The log–spiral method provided the best agreement with measured resistance. Estimates of passive pressure computed using the Rankine method significantly underestimated the resistance while the Coulomb method overestimated resistance. The cap movement required to fully mobilize passive resistance in the gravel backfill was about 6% of the cap height. This is somewhat larger than reported in other studies likely due to the underlying clay layer. The p-multipliers developed for the free-head pile group provided reasonable estimates of the pile–soil–pile resistance for the fixed-head pile group once gaps adjacent to the pile were considered.

Load carrying capacity of composite slabs with various end constraints

In many cases, the load carrying capacity of composite slabs depends on the shear-bond resistance at the sheet-concrete interface. At the ultimate state, the tension forces in the hogging region of a continuous composite slab are mainly transferred by the negative bending reinforcement and the shear-bond resistance in the region do not significantly influence the load carrying capacity of the slab. To identify the shear-bond action in composite slabs, seven simply supported one-span composite slabs and two continuous composite slabs were tested. Different end restraints had been used in the simply supported slabs. The slabs with end anchorage of steel shear connectors were found to bear a higher shear-bond strength than that of slabs without end anchorage. The shear-bond strength was calibrated based on a linear regression of the test results of the one-span composite slabs with end anchorage. The prediction of the shear-bond resistance was also found in close agreement with the vertical shear force at the onset of the initial shear-bond slip in the two-span continuous composite slabs. It is suggested that the shear-bond slip model be reasonable to predict the shear-bond resistance of a continuous composite slab. However, the shear span of the continuous composite slabs must be related to the sagging region, which could be derived on an elastic analysis base, or simply taken as 0.8 L for the side span and 0.6 L for the interior span. At the onset of the initial shear-bond slip, the mean ratios of the vertical shear force to shear-bond resistance (VeVu) are 1.065 for the one-span slabs and 1.165 for the two-span continuous composite slabs, which are on the safe side. Because of the shear-bond failure at the sheet-concrete interface, composite slabs would not be capable of developing full plastic moments in the spans.