Analysis of Thick Circular Plates Undergoing Large Deflections

This investigation considers the effect of transverse shear deformation on bending of the axisymmetrically loaded isotropic and orthotropic circular and annular plates undergoing large deflection. The analysis treats the nonlinear terms of lateral displacement as fictitious loads acting on the plate. The solution of a von Kármán‐type plate is, therefore, reduced to a plane problem in elasticity and a linear plate‐bending problem. Results are presented for simply supported and clamped plates and are in good agreement with the available solutions. For plates considered in this study, the influence of shear deformation on lateral displacement becomes more significant as the orthotropic parameter increases. The linear and nonlinear solutions for orthotropic plates deviate at a low value of the maximum deflection‐to‐thickness ratio (w/h). Consequently, the extent of w/h within which the small‐deflection theory is applicable to orthotropic plates is much lower than the value of about 0.4 typically used for isotropic plates, and it depends, in general, on the degree of orthotropy. The technique employed in this study is well suited for the analysis of nonlinear plate problems.     

Detecting Multiple Open Cracks in Elastic Beams by Static Tests

This paper concerns with the identification of multiple open cracks in a beam by measurements of the damage-induced variations in the static deflection of the beam under a prescribed load condition. Each crack is simulated by an equivalent linear spring connecting the two adjacent segments of beam. Sufficient conditions on the static measurements which allow for the unique identification of the damage are presented and discussed for nonuniform beams under some ideal boundary conditions. The inverse analysis is based on an explicit expression of the crack-induced variation in the deflection of the beam under a given load distribution and it provides exact closed-form expressions of position and severity of the cracks in terms of the measured data. The theoretical results are confirmed by a comparison with static tests carried out on a steel beam with localized damages.     

Vibro-Impact Behavior of Two Orthogonal Beams

Structural interaction between two beam-like structures is a situation that occurs in piping systems, among other applications relevant to the nuclear, petroleum, biomedical, and automotive industries, for instance. This paper analytically investigates the repetitive impact dynamics of two orthogonal pinned–pinned beams subjected to base excitation at specified frequency and acceleration. The orthogonal beam configuration restricts the contact to a single point, and the contact interface is modeled by a spring. Although many approaches have been developed for multibody dynamics, the constraint and modal mapping method is efficiently applied herein to obtain the forced response through modal analysis. The vibration is described in a piecewise fashion as switching between the linear in-contact and not-in-contact states, and compatibility conditions are applied at their junctions. The development of the conjoined mode shapes and their orthogonality is derived in detail. The contact impulse is used to describe the structures’ complex interacting behavior through repetitive impact frequency-response functions. In order to determine major response factors, parameter studies are performed on contact stiffness, relative beam stiffness, contact location, modal damping, and stand-off gap.     

Transient Response of Supported Beams to Moving Forces with Sinusoidal Time Variation

Moving forces are a common loading pattern for flexible beams, found in many applications in both civil and mechanical engineering. These forces give rise to a transient response, the nature of which depends on the time variation of the amplitude of the force and its position along the beam. In addition to the possibility of numerical evaluation, closed form solutions of the beam response are beneficial for their simplicity of use, and because they allow an understanding of the system behavior. On the other hand, these prove to be rather complicated in most cases, and only a limited number of cases are available in the literature. This paper studies the simple but common case of a supported beam loaded by a force with sinusoidal time variation moving at a constant speed. Simple equations are presented for the approximated responses at and away from resonance, and their accuracy is discussed. Transient frequency response functions are also shown. Finally, as an example, the results are applied to an evaluation of the response of a beam footbridge to the action of a walker, and compared to code specifications.     

Ductile Design of Reinforced Concrete Beams Retrofitted with Fiber Reinforced Polymer Plates

For reinforced concrete beams retrofitted with fiber-reinforced polymer (FRP) plates, an analytical method is derived for determining the allowable plate area to achieve a targeted value of ductility. Nonlinear models for concrete and reinforcement are applied, and the effects of concrete confinement and spalling and of FRP plate rupture are considered. The derivation of equilibrium and compatibility equations for a rectangular cross section is presented, and the solution to the nonlinear equation for determining the allowable plate area is demonstrated with examples. Analytical results are compared with numerical and experimental data reported in the literature. Subsequently a simplified version of the method is derived, based on regression analysis, to relate the curvature ductility to the FRP plate ratio. It is noted that additional conditions need to be checked to ensure ductile performance, such as local failure of the concrete layer between tension reinforcement and FRP plate or debonding of the plate itself.     

Identification of Concentrated Damages in Euler-Bernoulli Beams under Static Loads

An identification procedure of concentrated damages in Euler-Bernoulli beams under static loads is presented in this work. The direct analysis problem is solved first by modeling concentrated damages as Dirac’s delta distributions in the flexural stiffness. Closed-form solutions for both statically determinate and indeterminate beams are presented in terms of damage intensities and positions. On this basis, for the inverse damage identification problem, a nonquadratic optimization procedure is proposed. The presented procedure relies on the minimization of an error function measuring the error between the analytical model response and experimental data. The procedure allows to recognize “a posteriori” some sufficient conditions for the uniqueness of the solution of the damage identification problem. The influence of the instrumental noise on the identified parameters is also explored.     

Exact Buckling Solutions For Rectangular Plates Under Intermediate and End Uniaxial Loads

This paper is concerned with the elastic buckling of rectangular plates subjected to both intermediate and end uniaxial loads. The rectangular plates have two simply supported opposite edges that are perpendicular to the in-plane load direction, while the other two plate edges can have free, simply supported, or clamped edges. The solution procedure involves the use of the Levy approach, the domain decomposition technique, and the state-space concept. The method furnishes exact stability criteria; samples of which are presented in a graphical form for plates with various boundary conditions. These results will be useful to engineers who design plates (or walls) that support intermediate floors.     

Exact Solutions for Buckling of Multispan Rectangular Plates

This paper presents exact solutions for buckling of multispan rectangular plates having two opposite edges simply supported and the other two edges being either free, simply supported, or clamped. The Levy solution procedure is employed to develop an analytical approach for buckling analysis of multispan plates. The Levy solution for each span is derived and the continuity along the interface of two spans is ensured through the implementation of the essential and natural boundary conditions at the interface. Extensive buckling factors, most of which are first-known exact solutions, are given in tabular and design chart forms for two- and three-unequal-span square plates subjected to uniaxial in-plane load in the x or y directions and biaxial in-plane load. The influence of the span ratios and plate boundary conditions on the buckling factors is discussed. Buckling factors are also obtained for two-, three-, and four-equal-span rectangular plates with various edge support conditions. The exact buckling solutions presented in this paper are of benchmark values for such plates.     

Analysis of Snap-Back Instability due to End-Plate Debonding in Strengthened Beams

The problem of end-plate debonding of the external reinforcement in strengthened concrete beams is analyzed in this paper. As experimentally observed, this mode of failure is highly brittle and poses severe limitations to the efficacy of the strengthening technique. A numerical analysis of the full-range behavior of strengthened beams in bending is herein proposed to study the stages of nucleation and propagation of interfacial cracks between the external reinforcement and the concrete substrate. This is achieved by modeling the nonlinear interface behavior according to a cohesive law accounting for Mode Mixity. The numerically obtained load versus midspan deflection curves for three- or four-point bending beams show that the process of end-plate debonding is the result of a snap-back instability, which is fully interpreted in the framework of the Catastrophe Theory. To capture the softening branch with positive slope, the interface crack-length control scheme is proposed in the numerical simulations. The results of a wide parametric study exploring the effect of the relative reinforcement length, the mechanical percentage of fiber-reinforced polymer sheets, the beam slenderness, and the ratio between Mode II and Mode I fracture energies are collected in useful diagrams. Finally, an experimental assessment of the proposed model completes the paper.     

Behavior of Pultruded Fiber-Reinforced Polymer Beams Subjected to Concentrated Loads in the Plane of the Web

Results of the behavior of pultruded fiber-reinforced polymer (FRP) I-shaped beams subjected to concentrated loads in the plane of the web are presented. Twenty beams with nominal depths from 152.4 to 304.8?mm were tested in three-point bending with a span-to-depth ratio of four. Load was applied to the top flange directly above the web—12 without bearing plates and 8 with bearing plates of varying width and thickness. All test specimens failed with a wedgelike shear failure at the upper web-flange junction. Finite-element results support experimental findings from strain gauge and digital image correlation data. Bearing plates increased beam capacity by 35% or more as a function of bearing plate width and thickness. Bearing plates increased average shear stress in the web at failure from 17.4 to 27.2?MPa—below the accepted value of in-plane shear strength (69?MPa). A design equation is presented, and predicted capacities are compared with experimental results. The average value of experimental capacity to predicted capacity is 1.12 with a standard deviation of 0.11 and coefficient of variation (COV) of 0.10 for sections up to 304.8?mm deep.     

Monitoring of Steel Railway Floor Beams Prestressed by Steel Plates

Steel cables and tendons are commonly used in reinforcing steel beams as well as in concrete beams. However, the structural detail of cable anchors in steel beams tends to be complicated, and the effect on reducing live load stresses is not significant because of the relatively small stiffness of cables and tendons. On the contrary, by using high strength steel plates instead of cables and tendons, structural detail of the anchor area becomes simpler, and live load stresses as well as dead load stresses can be reduced in steel beams because of the relatively large stiffness of steel plates. In this study, the steel plate prestressing method is applied to beam specimens and intermediate floor beams of a steel railway through girder bridge. The behavior of the reinforcing steel plates and reinforced steel beams is monitored during prestressing and live loading, in order to assess the effects of prestressing and reinforcement. The study confirmed that these effects are beneficial to the performance of steel railway floor beams.     

Closed-Form Solutions for Vibrations of Nearly Vertical Strings and Beams by Means of Asymptotic Methods

Top tensioned strings and beams are often used in civil and marine applications. Typically these members have constant cross sections, and a pronounced, usually linear, tension variation, due to the effects of gravity. In this paper simple, approximate formulas for the natural frequency of such strings are derived, based on asymptotic techniques, while for the tensioned beam case approximate closed-form results are developed by the Wentzel–Kramers–Brillouin method. Both derivations are shown in reasonable detail. While similar work is known for a beam with varying axial tension this is believed to be the first time that a single analytic expression is developed for the full length of the beam. A simple example in which the bottom tension is only 9% of the top tension is analyzed for cases with and without bending stiffness, and the solutions have been compared to the exact solution for the string case and to the results from three finite-element programs for the beam case. The accuracy was found to be very good, even in this situation, in which the tension variation is large.     

Theoretical Study on Short-Term and Long-Term Deflections of Fiber Reinforced Polymer Prestressed Concrete Beams

This paper presents the methods for predicting the short-term and time-dependent deflections of fully or partially prestressed concrete beams with fiber reinforced polymer (FRP) tendons under sustained bending moment and axial force. The age-adjusted effective modulus method is used to model the creep behavior in the concrete and the relaxation in the FRP prestressing tendons. A tension-stiffening model is proposed to evaluate the stiffness of the section after cracking. The analytical values are compared to the test results and it is found that the analytical values are in good agreement with the experimental results.     

Dynamic Response of Plates on Elastic Foundation to Moving Loads

A procedure incorporating the finite strip method and a spring system has been developed and applied to treat the dynamic response of plate structure resting on an elastic foundation to moving loads. The response to a single moving concentrated load is first investigated and then the effects of velocity, elastic foundation stiffness, moving path, and distance between multiple moving loads are studied. The response under a moving harmonic load with constant velocity is finally treated and the effect of the load frequency is investigated. Results indicate that the foundation stiffness and the velocity and frequency of the moving load have significant effects on the dynamic response of the plate and on resonant velocities. Some of these findings might find use in practical applications.     

Correction Factors in Series Solutions for One- and Two-Dimensional Boundary Value Problems

A Fourier cum polynomial series solution with correction factors is presented herein for differential equations with variable coefficients. The differential equations correspond to a wide range of boundary value problems. The correction factors included herein are: (1) modified Lanczos correction; (2) Bessel J; and (3) loading correction factor. These correction factors are introduced in terms of Fourier and polynomial series. The main purpose of using correction factors through a set of series is to improve convergence of the proposed solution, using the first two terms of the series. For the loading correction factor, a Fourier series expansion coupled with orthogonality conditions leads to evaluating undetermined Fourier coefficients of arbitrarily applied loads using concepts of summation equations. Representative boundary value problems are provided to demonstrate the efficiency and accuracy of the first two terms of the proposed solution with correction factors.     

Analytical Model for Geometric and Material Nonlinear Analysis of Externally Prestressed Beams

The analysis of beams prestressed by external slipping tendons involves various difficulties related to the coupling between the local strain of the tendons and the global deformation of the beam. The structural behavior of the beam–tendon system at collapse is ruled both by the nonlinearity of materials and by geometric nonlinear effects. Recent scientific papers have shown the relevance of the geometric effects in evaluating the failure load of externally prestressed beams by considering the tendon eccentricity variation. The change of eccentricity is however only one of the geometric nonlinear effects. In this work the writers present a complete geometric and mechanical nonlinear analytical model based on the theory of small strains and moderate rotations deduced from the finite deformation theory.     

Nonlinear Zigzag Theory for Buckling of Hybrid Piezoelectric Rectangular Beams under Electrothermomechanical Loads

A new efficient electromechanically coupled geometrically nonlinear (of von Karman type) zigzag theory is developed for buckling analysis of hybrid piezoelectric beams, under electrothermomechanical loads. The thermal and potential fields are approximated as piecewise linear in sublayers. The deflection is approximated as piecewise quadratic to explicitly account for the transverse normal strain due to thermal and electric fields. The longitudinal displacement is approximated as a combination of third order global variation and a layerwise linear variation. The shear continuity conditions at the layer interfaces and the shear traction-free conditions at the top and bottom are used to formulate the theory in terms of three primary displacement variables. The governing coupled nonlinear field equations and boundary conditions are derived using a variational principle. Analytical solutions for buckling of symmetrically laminated simply supported beams under electrothermal loads are obtained for comparing the results with the available exact two-dimensional (2D) piezothermoelasticity solution. The comparison establishes that the present results are in excellent agreement with the 2D solution which neglects the prebuckling transverse strain effect.     

Damage-Coupled Progressive Failure and Yield Line Analyses of Metal Plates

Continuum damage mechanics based progressive failure analysis of an aluminum alloy AL2024-T3 plate has been carried out. Isotropic continuum damage mechanics model proposed by Chandrakanth and Pandey in 1995 has been implemented in a nonlinear finite element computational scheme based on damage-coupled and damage-uncoupled elastoplastic constitutive relationship. In order to model the progressive growth of damage and plasticity from extreme fibers toward the neutral axis, discrete layered approach has been adopted in the formulation using Ahmed’s degenerate isoparametric shell element, which accounts for shear deformation. A critical damage criteria is used for determining the onset and propagation of failure in the plate. Damage-coupled and damage-uncoupled analyses have been carried out on rectangular and triangular plates of aluminum alloy Al2024-T3. Yield line patterns have been generated using extensive nonlinear progressive failure analysis and comparison with conventional yield line analysis has been made. It is envisioned that employing the methodology presented herein, yield line pattern generation for structural components with complex shapes can be obtained, which would significantly assist engineers in analysis and design of structures.     

Strength and Ductility of Concrete Beams Reinforced with Carbon Fiber-Reinforced Polymer Plates and Steel

Seven concrete beams reinforced internally with varying amounts of steel and externally with precured carbon fiber-reinforced polymer (FRP) plates applied after the concrete had cracked under service loads were tested under four-point bending. Strains measured along the beam depth allowed computation of the beam curvature in the constant moment region. Results show that FRP is very effective for flexural strengthening. As the amount of steel increases, the additional strength provided by the carbon FRP plates decreases. Compared to a beam reinforced heavily with steel only, beams reinforced with both steel and carbon have adequate deformation capacity, in spite of their brittle mode of failure. Clamping or wrapping of the ends of the precured FRP plate enhances the capacity of adhesively bonded FRP anchorage. Design equations for anchorage, allowable stress, ductility, and amount of reinforcement are discussed.     

Closed-Form Solutions for Buckling of Long Anisotropic Plates with Various Boundary Conditions under Axial Compression

Closed-form solutions for buckling of long plates with flexural/twist anisotropy with the short edges simply supported and with the longitudinal edges simply supported, clamped, or elastically restrained in rotation under axial compression are presented. An energy method (Rayleigh–Ritz) is employed to obtain the critical buckling loads. The critical buckling loads are expressed in terms of minimum nondimensional buckling coefficients and stiffness parameters. The new closed-form solutions show an excellent agreement when compared to existing solutions and finite-element analysis. Due to their simplicity and accuracy, the new closed-form solutions can be confidently used as an alternative to computationally expensive structural analysis to assess buckling in the preliminary design phase of composite structures.