Analytical Solutions to General Orthotropic Plates under Patch Loading

Orthotropic plates are widely used in bridge deck systems. However, these are not commonly treated as such within design specifications, and semianalytical solutions are not presently available for all deck types. This paper develops deflection equations for infinitely wide and simply supported thin plates considering each of the three cases of orthotropy: (1)?relatively torsionally stiff, flexurally soft; (2)?uniformly thick plate; and (3)?torsionally soft, flexurally stiff; subjected to arbitrary patch loading. These are common boundary and loading conditions encountered for bridge deck applications. The reported analytical solutions enable rapid evaluation of multiple moving patch loads to determine maximum design load effects and permit validation of numerical and finite-element methods. Application of the solutions will produce guidelines that can prescribe design demands and establish practical design simplifications for treatment of different bridge deck and slab systems in a uniform and consistent manner.     

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.     

Influence of Functionally Graded Material on Buckling of Skew Plates under Mechanical Loads

In this technical note, the critical buckling of simply supported functionally graded skew plate subjected to mechanical compressive loads is evaluated using first-order shear deformation theory in conjunction with the finite element approach. The material properties are assumed to vary in the thickness direction according to the power-law distribution in terms of volume fractions of the constituents. The effective material properties are estimated from the volume fractions and the properties of the constituents using the Mori–Tanaka homogenization method. The effects of aspect ratio, material gradient index, and skew angle on the critical buckling loads of functionally graded material plates are highlighted.     

Buckling of Simply Supported Rectangular Reissner–Mindlin Plates Subjected to Linearly Varying In-Plane Loading

This paper is concerned with the buckling analysis of simply supported rectangular Reissner–Mindlin plates subjected to linearly varying edge loads. An analytical solution is developed and the effect of load intensity variation on the critical load is investigated. The solution is verified with the commercial computer code ANSYS. It is observed that the results of the present solution are in excellent agreement with those of ANSYS. The inaccuracy of the currently cited data in some design handbooks is highlighted.     

Behavior of Reinforced Concrete Bridge Decks on Skewed Steel Superstructure under Truck Wheel Loads

This paper focuses on the behavior of skewed concrete bridge decks on steel superstructure subjected to truck wheel loads. It was initiated to meet the need for investigating the role of truck loads in observed skewed deck cracking, which may interest bridge owners and engineers. Finite-element analysis was performed for typical skewed concrete decks, verified using in?situ deck strain measurement during load testing of a bridge skewed at 49.1°. The analysis results show that service truck loads induce low strains/stresses in the decks, unlikely to initiate concrete cracking alone. Nevertheless, repeated truck wheel load application may cause cracks to become wider, longer, and more visible. The local effect of wheel load significantly contributes to the total strain/stress response, and the global effect may be negligible or significant, depending on the location. The current design approach estimates the local effect but ignores the global effect. It therefore does not model the situation satisfactorily. In addition, total strain/stress effects due to truck load increase slightly because of skew angle.     

Plastic-Buckling of Rectangular Plates under Combined Uniaxial and Shear Stresses

This paper is concerned with the plastic-buckling of rectangular plates under uniaxial compressive and shear stresses. In the prediction of the plastic-buckling stresses, we have adopted the incremental theory of plasticity for capturing the inelastic behavior, the Mindlin plate theory for the effect of transverse shear deformation, the Ramberg-Osgood stress–strain relation for the plate material, and the Ritz method for the bifurcation buckling analysis. The interaction curves of the plastic uniaxial buckling stress and the plastic shear buckling stress for thin and thick rectangular plates are presented for various aspect ratios. The effect of transverse shear deformation is examined by comparing the interaction curves obtained based on the Mindlin plate theory and the classical thin plate theory.     

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.     

Application of Orthotropic Thin Plate Theory to Filled Steel Grid Decks for Bridges

Although AASHTO LRFD specifications provide moment capacity equations as an approximate design method and recommend an orthotropic plate model as the refined method for the analysis of filled grid decks, no guidelines are provided for the determination of the flexural rigidities associated with the plate analysis. This technical note briefly reviews orthotropic thin plate theory, discusses the determination of the flexural rigidities using Huber’s assumption, and applies the theory to concrete-filled steel grid decks. The accuracy of the orthotropic plate analysis is assessed by comparing it to results of an earlier finite-element analysis.     

Consistent Approach to Calculating Stresses for Fatigue Design of Welded Rib-to-Web Connections in Steel Orthotropic Bridge Decks

The current (2004) fatigue design provisions in the 3rd Ed. of the AASHTO LRFD Bridge Design Specifications identify and classify the rib-to-web (rib-to-diaphragm) connections commonly utilized in steel orthotropic bridge decks where cutouts are used. The fatigue resistance of these details has been established through full-scale laboratory testing. This paper examines how the fatigue stress range was defined and determined during the testing which established the fatigue resistance of the details. A procedure to calculate or measure stresses at the rib-to-diaphragm connection, which is consistent with the fatigue resistance published in the AASHTO LRFD Bridge Design Specifications, is presented.     

Evaluation of Load Distribution Factor by Series Solution for Orthotropic Bridge Decks

In bridge engineering, the three-dimensional behavior of a bridge system is usually reduced to the analysis of a T-beam section, loaded by an equivalent fraction of the applied live load, which is called the live load distribution factor (LDF). The LDF is defined in the both the AASHTO Standard Specifications and the LRFD Specifications primarily for concrete slabs and has inherent applicable limitations. This paper provides explicit formulas using series solutions for LDF of orthotropic bridge decks, applicable to various materials but intended for fiber-reinforced polymer (FRP) decks. The present formulation considers important parameters that represent the response characteristics of the structure that are often omitted or limited in the AASHTO Specifications. A one-term series solution is proposed based on the macroflexibility approach, in which the bridge system is simplified into two major components, deck and stringers. The governing equations for the two components are obtained separately, and the deflections and interaction forces are solved by ensuring displacement compatibility at stringer lines. The LDF is calculated as the ratio of the single stringer interaction force to the summation of total stringer interaction forces. To verify this solution, a finite-element (FE) parametric study is conducted on 66 simply supported concrete slab-on-steel girder bridges. The results from the series solution correlates well with the FE results. It is also illustrated that the series solution can be applied to predict LDF for FRP deck-on-steel girder bridges, by favorable comparisons among the analytical, FE, and testing results for a one-third-scale bridge model. The scale test specimen consists of an FRP sandwich deck attached to steel stringers by a mechanical connector. The series solution is further used to obtain multiple regression functions for the LDF in terms of nondimensional variables, which can be used for simplified design purposes.     

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.     

Dynamic Characteristics of Infinite and Finite Railways to Moving Loads

Some fundamental dynamic characteristics of a railway subjected to a harmonic or constant moving load are established and presented in this paper. The railway is modeled as an infinite or finite Timoshenko beam on viscoelastic foundation. The dynamic-stiffness matrices characterized by the complex wave numbers are employed to deal with this problem. The relationship between the forced frequency and the resonant velocity of the moving load, and the resonant frequency of the railway are especially emphasized and intensively discussed. The fundamental dynamic characteristics of a railway modeled as a Bernoulli-Euler Beam on viscoelastic foundation are also included for comparison.     

Effects of Fabrication Procedures on Fatigue Resistance of Welded Joints in Steel Orthotropic Decks

A common practice for the fabrication of steel orthotropic bridge decks in the United States is to use 80% partial joint penetration (PJP) groove welds between the closed ribs and deck plate. However, it is difficult to eliminate weld melt-through with the thin rib plates. Heat straightening after welding, sometimes combined with precambering, is used to meet the deck plate flatness requirement. To study the effects of both weld melt-through and distortion control measures on the fatigue resistance of the rib-to-deck plate welded joint, six full-scale two-span orthotropic deck specimens were subjected to laboratory testing. Specimens, 10 m long and 3 m wide with four closed ribs, were fabricated with and without weld melt-through and were heat straightened; three specimens were also precambered. To simulate the effect of repetitive truck traffic, each specimen was tested up to 8 million cycles. Test results showed that six cracks initiated from the weld toe outside the rib. Only one crack developed at the weld root inside the rib; this crack initiated from a location transitioning from the 80% PJP to 100% penetration weld. None of the cracks propagated through the deck plate thickness. Precambering was beneficial in fatigue resistance as two effectively precambered specimens did not experience cracking in the PJP welds.     

Large-Deflection Mathematical Analysis of Rectangular Plates

A large-deflection mathematical analysis of rectangular plates under uniform lateral loading is presented in this paper. The analysis is based on solving two fourth-order, second-degree, partial differential Von Kármán equations relating the lateral deflections to the applied load. This paper provides a mathematical procedure which benefits from the software and hardware computing capabilities that were unavailable when mathematical modeling was last attempted. The solution is presented in a simple form suitable for direct practical use and can be easily implemented in common spreadsheet packages. Plates with two boundary conditions, namely, simply supported edges and held edges, are considered. Comparisons are held against earlier exact and approximate solutions, including results of finite element analyses. The results show close agreement with other exact analysis methods. The solution is able to produce the same results as other exact solutions, but with a much simpler and a more practical approach.     

Orthotropic Plate Model for Estimating Deflections in Filled Grid Decks

Concrete filled grid bridge decks exhibit orthogonal elastic properties and significant two-way bending action enabling orthotropic plate theory to determine structural response for these elements. Current American Association of State Highway and Transportation Officials load and resistance factor design (LRFD) specifications employ an orthotropic plate model to predict live load moment in concrete filled grid bridge decks but provide no guidance for computing displacement, a potentially important serviceability consideration. This paper presents equations to approximate the maximum deflection in concrete filled grid bridge decks based on orthotropic plate theory, multiple patch loads, LRFD design truck and tandem load cases, the influence of multiple spans, and the two most common deck orientations.     

Tilting Analysis of Rectangular Elastic Layers Bonded between Rigid Plates

A theoretical approach to determine the tilting stiffness of a rectangular elastic layer bonded between two rigid plates is presented. On the basis of two kinematics assumptions, the governing equation for the mean pressure is derived from the equilibrium equations. Using the approximate shear boundary condition, the mean pressure is solved and the tilting stiffness of the bonded rectangular layer is then established in an explicit single-series form. Whereas the finite element method can be applied to calculate the stiffness, the series solution provides a convenient way for parametric studies. Through the obtained pressure expressions, the horizontal displacements are derived from the corresponding equilibrium equations, from which the shear traction on the bonding surface can be found. The error of using the approximate shear boundary condition is negligible for the tilting stiffness, but becomes significant for the horizontal displacements and bonding shear stresses near the edges of the rectangular layers.     

Analytical and Numerical Models of Postbuckling of Orthotropic Symmetric Plates

Two analytical perturbation methods which give approximate solutions of postbuckling behavior of orthotropic simply supported plates are considered in this work: the method of Chandra and Raju and the method of Shen and Zhang. The reproduction of the algebraic developments of these methods by the Mathematica symbolic manipulator program has revealed that there are errors in the formulas included in the original paper by Chandra and Raju. After a revision and correction of these errors, the analytical results of both methods for a set of 23 orthotropic plates are compared, an excellent agreement being found for a wide range of values of geometrical and mechanical parameters in which many actual plates lie. A numerical simulation performed on a reduced sample of six plates using finite-element code ABAQUS has validated analytical results. The present work is intended as a first step in the investigation of the possibility of using reliable analytical formulas in the design of composite plates.     

Buckling of Laminated Composite Rectangular Plates

The present study estimates the critical/buckling loads of laminated composite rectangular plates under in-plane uniaxial and biaxial loadings. The formulation is based on the first-order shear deformation theory and von-Karman-type nonlinearity. Chebyshev series is used for spatial discretisation and quadratic extrapolation is used for linearization. An incremental iterative approach is used for estimation of the critical load. Different combinations of simply supported, clamped and free boundary conditions are considered. The effects of plate aspect ratio, lamination scheme, number of layers and material properties on the critical loads are studied.     

Vibration of Pressure Loaded Nonlinear Rectangular Plates with Cross Stiffener

The resonant frequency response of large static pressure loaded, nonlinear rectangular plates with a cross stiffener have been investigated theoretically. The nonlinear Berger equation was solved by applying the finite-difference method. Replacing the partial differential equation governing the small amplitude vibration of static pressure loaded plates and the boundary conditions by the finite-difference equations approximately, the simultaneous, homogeneous, and algebraic equations are obtained. Under the condition that the determinant of coefficient matrix must be equal to zero, the resonant frequencies are determined. The numerical procedure is simpler than the procedures based on the von Kármán theory, and reasonable results are obtained.     

Thermomechanical Postbuckling of Cross-Ply Laminated Rectangular Plates

Postbuckling analysis is presented for shear deformable cross-ply laminated composite rectangular plates subjected to the combination of in-plane edge compressive mechanical loading and thermal loads due to a linearly varying temperature across the thickness. The formulation is based on the first-order shear deformation theory and von-Karman-type nonlinearity. The analysis uses a quadratic extrapolation technique for linearization and Chebyshev polynomials for spatial discretization. An incremental iterative approach is employed to estimate the critical load. The boundary conditions consisting of clamped, simply supported, free edge, and their combinations are considered. The effects of the thinness ratio, aspect ratio, lamination scheme, the number of layers, and the modulus ratio on the critical load/limit load and postbuckling behavior are studied.