Transverse Concentrated Load Research Articles

Critical speeds and the response of a tensioned beam on an elastic foundation to repetitive moving loads

A finite-length tensioned beam on a damped elastic foundation is acted upon by an infinite series of equally spaced and steadily moving concentrated transverse loads. The deflection response of the beam is obtained by an expansion in terms of the normal modes of vibration. Numerical results are determined for various values of the load-spacing, beam tension, foundation stiffness and damping, and for a range of load-speeds. It is found that the critical velocities for repetitive loading exist at significantly lower speeds than would be expected based upon the well-known critical speed for a single moving load. An interpretation in terms of forced vibration response is given.

Flexural-torsional stability of thin-walled composite I-section beams

An analytical study of optimal fibre direction for improving the lateral buckling strength of thin-walled composite open-section members is presented. Based on a Vlasov-type linear hypothesis, beam stiffness coefficients, which account for cross-section geometry and for the material anisotropy of the section as well as the geometrical characteristics of columns, are obtained. Uniformly distributed load, transverse concentrated load, unequal end moments, tip-loaded cantilever and columns with different types of loading, are considered. The results show that, in some cases, the web fibre angle makes a remarkable contribution to increasing the buckling load compared with the unidirectional orientation of the pultrusion process.

Matrix Cracking Effect on Delamination Growth in Composite Laminates Induced by a Spherical Indenter

An investigation was performed to study the mechanics of delamination growth in fiber-reinforced laminated composites resulting from transverse concentrated loads. The effect of matrix cracking on delamination propagation was the primary focus of the study. For simplicity's sake, only cross-ply laminates were investigated. An analytical model was proposed for determining the response of the laminates containing a matrix crack-induced delamination and subjected to a concentrated load through a spherical indenter. Two types of matrix cracks were considered in the model: a surfacebending crack and an internal shear crack. The model consists of three parts: a stress analysis, a contact analysis, and a failure analysis. A nonlinear three-dimensional finite element model based on an updated Lagrange formulation was developed for calculating stresses and strains in composites. An augmented Lagrangian method was utilized to simulate the interfacial contact condition of the embedded delamination and the indentation between the composite and the indenter. Fracture mechanics was applied to predict the delamination growth in three dimensions. In order to verify the model, results of the calculations from the analysis were compared with the existing solutions and test data available in the literature. Experiments were also conducted to further substantiate the model and the predictions from the analysis. Overall, the predictions agreed well with the existing solutions and the test data.

Response of Annular Plates to Circumferentially and Radially Moving Loads

The forced dynamic response of annular plates to circumferentially and radially moving concentrated transverse loads is investigated utilizing classical plate theory, with damping included, and solved in integral form. The boundary conditions are that the inner boundary of the plate is clamped and the outer boundary is free. An analytical expression in Fourier-Bessel series form is obtained for the forced deflection response to an arbitrarily moving concentrated load. This study includes radially moving loads and is a significant extension of the understanding of circular and annular plate dynamics. This understanding of radially moving loads is used to examine the nature of resonance conditions and corresponding critical values of the load parameters. The shapes of deflection modes of plate vibration are also presented. Damping and loading parameter sensitivities are studied in detail.

Stresses in laminated composite annular disks subjected to a concentrated transverse edge load

Polar orthotropic and rectangular orthotropic annular disks are analysed for displacements and stresses when subjected to a concentrated transverse load at the free outer edge. The inner edge of the disk is assumed to be clamped. The material properties are varied and the resulting variation in the maximum displacements and stresses is studied. In the case of rectangular orthotropic annular disks, multilayered disks are analysed. It is shown that there is a very large improvement in the factor of safety as the number of layers with different fiber orientations is increased, so much so that a multilayered rectangular orthotropic annular disk has safer stress values than a polar orthotropic annular disk with radial fibres. The finite element method with sector elements is used for the analysis.

Shearing effects on a shaft with circular surface crack under rotary bending

The shearing effects on the failure behavior of a rotating shaft with a circular surface crack subjected to transverse concentrated load and torsion are investigated by employing finite element method combined with strain energy density theory. Results of stress intensity factors and minimum strain energy density factor at an arbitrary point on the crack front are obtained for three different crack sizes. The plane strain condition has been assumed in order that the onset and direction of the mixed mode crack propagation can be determined. It is shown that the crack will distort and extend preferentially in a direction opposite to that of rotation and swing a small angle depending on the shearing stress effects.

Dynamic buckling of composite cylindrical panels with high-order transverse shears subjected to a transverse concentrated load

The characteristics of dynamic buckling of a geometrically non-linear cylindrical laminated composite panel subjected to a transverse concentrated step load applied at the center is studied. Attention is focused on the dynamic stability of a finite element discrete structural system. The sufficient condition for dynamic buckling, from the energy transfer consideration, is defined as the smallest load for which an unbounded motion is initiated at one generalized displacement. In other words, the dynamic buckling load associated with that generalized coordinate can be predicted by the intersecting point on the static equilibrium curve and the zero potential energy curve. This dynamic buckling criterion can also be observed by the existence of an inflection point on the generalized displacement response curve. Considering the multi-degree-of-freedom for the entire structure without damping, the dynamic buckling criterion used in a single-degree-of-freedom model gives the lower-bound dynamic buckling estimate, which will be shown in the results of the dynamic buckling analysis for a laminated composite arch. The possibility of parametric resonance due to the transverse concentrated step load on a geometrically non-linear system is discussed. Furthermore, the dynamic effects for different loading rates of the applied concentrated load are examined. Finally, the study of the damping effect, which raises the dynamic buckling load, is included.

Fundamental solutions of Mindlin plates with variable thickness for stochastic boundary elements

A procedure to develop the fundamental solutions for the boundary element method with small system stochasticity is presented herein. Analytical as well as numerical results are furnished for circilar Mindlin plates with stochastic thicknesses. A concentrated transverse point load and concentrated bending moments are considered with an arbitrary set of plate boundary conditions. The perturbation technique employed herein is correct only for the first order stochastic effects originating from the spatial variability of thickness.

Linear and non-linear failure analysis of composite laminates with transverse shear

A finite element computational procedure has been developed to find linear and non-linear (von Kármán) first-ply failure loads of composite laminates subjected to in-plane and transverse loads. The finite element model is based on first-order shear deformation theory and several phenomenological failure criteria. Linear and non-linear first-ply failure loads are computed for a uniformly distributed transverse load, a concentrated transverse load acting at the centre of the plate, and a uniformly distributed in-plane edge load for simply supported and clamped boundary conditions. It is found that the failure loads predicted by various failure criteria differ from one another by a maximum of 35% in the case of linearloads and 50% in the case on non-linear loads; the failure locations differ from each other in a random way. Furthermore, it is demonstrated that the difference between the linear and non-linear failure loads is large in the case of transversely loaded simply supported laminates and thin plates. The difference is found to be quite small in the case of in-plane (tensile) loading and thick laminates.

Initial post-buckling analysis with the spline finite-strip method

A numerical method is presented for the initial post-buckling analysis of folded plate structures. The method combines Koiter's initial post-buckling theory with the spline finite-strip method. Splines replace the often used Fourier series, in order to facilitate the description of both local non-periodic buckles which may occur under concentrated transverse loading, and of oblique buckling modes pertaining to shear. Because determination of the shape of the buckle in axial direction requires more unknowns than in the classical finite-strip method, this method can be placed mid-way between the semi-analytical finite-strip method and a full finite element method. A numerical example pertaining to a thin-walled beam loaded by a concentrated transverse force, demonstrates the interactions between two distortional buckling modes.

Stress Analysis of an Orthotropic Laminated Slab Subjected to a Transverse Load.

Stresses in a slab with three orthotropic laminated layers are analyzed with the three-dimensional theory of transversely isotropic elasticity. The slab is subjected to a transverse concentrated or distributed load at the center part on the top face. To obtain qualitative knowledge for delamination of CFRP laminates, numerical results of interlaminar stresses are given for [0°n/90°n]sym graphite/epoxy square laminates. It is shown that shear stresses on the upper interlaminar boundary are larger than those on the lower one. On each interfacial boundary, shear stresses are maxima just below the concentrated load point or the edge of the distributed load area, though these stresses are zero at the center point.

Mechanical analysis of an annular plate subject to a transverse concentrated load

The periphery deflection in an annular plate clamped along its inner edge and loaded at its outer boundary by a transverse concentrated force are studied via a series solution. The corrected expressions for the series coefficients are presented. The series convergence is accelerated via a simple technique. Approximate methods are proposed to account for the compliant clamp and shear strain effects on the plate deflections, and they are validated against simplified theoretical models.

Large deflections of point loaded cantilevers with nonlinear behaviour

Thin beams, being flexible, form a curve with large deflections when subjected to sufficiently large transverse loads. Therefore, geometrical nonlinearity occurs, and the problem must be formulated in terms of the nonlinear theory of bending. In this paper, the beam is constructed from nonlinear elastic material, and subjected to several transverse concentrated loads. Due to the large deflection of the beam, the exact expression of the curvature of the deflected shape is used in the Bernoulli-Euler relationship. Therefore, this leads to a second order nonlinear differential equation for the transverse deflection. The solution of this equation is obtained by using the fourth-order Runge-Kutta method, and the arc length is evaluated using Simpson's Rule. The results obtained from this procedure are compared with previously published results for thin beams of linear elastic materials in order to verify the theory and the method of analysis.

Investigation of the moment distribution in a bent circular plate by speckle-shearing interferometry

This paper describes the application of a simple speckle-shearing interferometric method for the determination of moment distribution in a clamped circular plate under a concentrated transverse load at its center. The present system consists of an image-shearing camera using a Fresnel biprism of small angle. Reduction of spurious speckle- noise and enhancement in the clarity of the fringe was achieved by using a white light source at the Fourier filtering stage. Experimental values have been compared with the existing analytical solution. Fairly close agreement between the experimental and theoretical values shows the accuracy and reliability of the method.

Dynamic Fracture of Beam under Transverse Load

The dynamic fracture response of a long beam of brittle elastic material subjected to a transverse time-dependent concentrated load is investigated. If the magnitude of this load is increased to a critical value, a crack will propagate from the tensile side of that cross section. An analysis is presented which relates the transverse force, the bending moment, the axial force, and the crack length at the fracturing section. Consequently, the crack length and other parameters of interest can be determined for any given force. The analysis is applied to several specific examples and typical results are presented. If the value of the elastic modulus is appropriately modified, the results also apply for plane strain fracture of a plate subjected to a line load.

A two-dimensional theory for piezoelectric layers used in electro-mechanical transducers—II: Applications

The approximate 2-D theory derived in Part I is applied here to the time harmonic problems of plane-strain and axisymmetry. Also, the transient axisymmetric solution is obtained for a circular layer that is suddenly released from a transverse concentrated central load on one face and an opposing ring load along the outer edge of the other face. For this problem the output voltage is computed as a function of time when the crystal is an element in a known electrical circuit.

Linear and nonlinear stability analysis of cylindrical shells

The paper describes the application of a curved isoparametric shell element to large displacement analyses including instability phenomena. A total Lagrangian formulation has been adopted using the standard incremental/iterative solution procedure. The linear stability analyses usually performed for the initial position were repeated at several advanced fundamental states on the nonlinear prebuckling path. Thus a current estimate of the final failure load is given. The method has been applied to several perfect and imperfect cylindrical shells under uniform pressure or wind load. Finally the example of a cylindrical panel under one concentrated transverse load is discussed.

OPTIMAL DESIGN OF BEAM-COLUMNS SUBJECTED TO CONCENTRATED MOMENTS

The paper is concerned with the problem of minimizing the transverse deflection at the free end of a cantilevered, elastic beam-column of given mass. The beam-column is acted upon simultaneously by an axial compression, a transverse concentrated load and a concentrated bending moment at the free end. The concentrated bending moment may be regarded as the reaction on the beam-column from the rest of the rigid frame of which it is an integral part. Solutions are presented for beam-columns of sandwich and solid cross-section. It is shown that optimization leads to a substantial decrease in the free end deflection compared to that of a prismatic member.

Optimal design of multi-purpose tie-beams

The paper presents a solution of the problem of minimizing the maximum deflection of a simply-supported beam under a transverse concentrated load. The volume (mass) of the beam is given, as is the maximum longitudinal elongation if the beam were to act as a tie. An optimal design for two requirements (beam and tie action) not only unifies the design procedure of mass-produced structural-mechanical elements, but also provides a practically acceptable design, in the sense that the resulting shape does not vanish at any point along its length, a drawback of many optimal designs for a single requirement. However, it is shown that, for cross sections of solid construction, as opposed to those of sandwich construction, the constraint on longitudinal elongation is weaker than on the (finite) maximum stress. The effectiveness of the optimal design is judged by comparing it with a prismatic member of the same volume.

The Nonlinear Behavior of Elastic Slender Straight Beams Undergoing Small Strains and Moderate Rotations

The nonlinear behavior of slender and initially straight beams is investigated. A set of equilibrium equations for a beam undergoing moderate rotations and small strains is derived. The derivation includes several assumptions which are clearly stated and explained. These equations are used to investigate the deformations of a cantilevered beam which is loaded by a concentrated transverse load at the free end. The numerical results are in very good agreement with the experimental results.