The world-wide development of new high-speed rail lines has led to more stringent design requirements for railway bridges, mainly because high-speed trains can cause resonance in the bridge superstructure. Dynamic simulations, often utilising time-consuming finite element analysis (FEA), have become essential for avoiding such problems. Therefore, guidelines and tools to assist structural engineers in the design process are needed. Considerable effort was spent at the beginning of the project, to develop simplified models based on two-dimensional (2D) Bernoulli-Euler beam theory. First, a closed-form solution for proportionally damped multi-span beam, subjected to moving loads was derived (Paper I). The model was later used to develop design charts (Paper II) and study bridges on existing railway lines (Paper III). The model was then extended to non-proportionally damped beams (Paper IV) in order to include the effects of soil-structure interactions. Finally, the importance of the interaction between the surrounding soil and the bridge was verified by calibrating a finite element (FE) model by means of forced vibration tests of an end-frame bridge (Paper V). Recommendations on how to use the models in practical applications are discussed throughout the work. These recommendations include the effects of shear deformation, shear lag, train-bridge and soil-structure interactions, for which illustrative examples are provided. The recommendations are based on the assumption that the modes are well separated, so that the response at resonance is governed by a single mode. The results of the work show that short span bridges, often referred to as `simple´ bridges, are the most problematic with respect to dynamic effects. These systems are typically, non-proportionally damped systems that require detailed analyses to capture the `true´ behaviour. Studying this class of dynamic system showed that they tend to contain non-classical modes that are important for the structure response. For example, the bending mode is found to attain maximum damping when its undamped natural frequency is similar to that of a non-classical mode. / <p>QC 20170213</p>

The steady state response of thin-walled members subjected to harmonic forces is investigated in the present study. The governing differential equations of motion and associated boundary conditions are derived from the Hamilton variational principle. The harmonic form of the applied forces is exploited to eliminate the need to discretize the problem in the time domain, resulting in computational efficiency. The formulation is based on a generalization of the Timoshenko-Vlasov beam theory and accounts for warping effects, shear deformation effects due to bending and non-uniform warping, translational and rotary inertial effects and captures flexural-torsional coupling arising in asymmetric cross-sections. Six of the resulting seven field equations are observed to be fully coupled for asymmetric cross-sections while the equation of longitudinal motion is observed to be uncoupled. Separate closed form solutions are provided for the cases of (i) doubly symmetric cross sections, (ii) monosymmetric cross-sections, and (iii) asymmetric cross-sections. The closed-form solutions are provided for cantilever and simply-supported boundary conditions. A family of shape functions is then developed based on the exact solution of the homogeneous field equations and then used to formulate a series of super-convergent finite beam elements. The resulting two-noded beam elements are shown to successfully capture the static and dynamic responses of thin-walled members. The finite elements developed involve no special discretization errors normally encountered in other finite element formulations and provide results in excellent agreement with those based on other established finite elements with a minimal number of degrees of freedom. The formulation is also capable to predict the natural frequencies and mode-shapes of the structural members. Comparisons with non-shear deformable beam solutions demonstrate the importance of shear deformation effects within short-span members subjected to harmonic loads with higher exciting frequencies. Comparisons with shell element solution results demonstrate that distortional effects are more pronounced in cantilevers with short spans. A generalized stress extraction scheme from the finite element formulation is then developed. Also, a generalization of the analysis procedure to accommodate multiple loads with distinct exciting frequencies is established. The study is concluded with design examples which illustrate the applicability of the formulation, in conjunction with established principles of fatigue design, in determining the fatigue life of steel members subjected to multiple harmonic forces.

This work analytically solves the governing equations of the laminar film condensation from pure vapors in vertical tubes to find the condensate film thickness. The solution is then extended to turbulent flow conditions for steam. All other relevant quantities are derived from the film thickness solution. For laminar film condensation of quiescent vapors, an exact explicit solution and an approximate closed-form solution were found by using a new definition of the dimensionless film thickness, the Lambert W-function, and a logarithmic function approximation. For laminar mixed-convection film condensation with interfacial shear stress, an approximate closed-form solution was found by using a new definition of the pressure gradient. For turbulent film condensation of steam, correlations of the turbulent vapor and liquid viscosities were formed by asymptotically comparing this approximate laminar closed-form solution to a turbulent flow numerical solution. The present solutions compared very well to published numerical and experimental data.

The steady state response of thin-walled members subjected to harmonic forces is investigated in the present study. The governing differential equations of motion and associated boundary conditions are derived from the Hamilton variational principle. The harmonic form of the applied forces is exploited to eliminate the need to discretize the problem in the time domain, resulting in computational efficiency. The formulation is based on a generalization of the Timoshenko-Vlasov beam theory and accounts for warping effects, shear deformation effects due to bending and non-uniform warping, translational and rotary inertial effects and captures flexural-torsional coupling arising in asymmetric cross-sections. Six of the resulting seven field equations are observed to be fully coupled for asymmetric cross-sections while the equation of longitudinal motion is observed to be uncoupled. Separate closed form solutions are provided for the cases of (i) doubly symmetric cross sections, (ii) monosymmetric cross-sections, and (iii) asymmetric cross-sections. The closed-form solutions are provided for cantilever and simply-supported boundary conditions. A family of shape functions is then developed based on the exact solution of the homogeneous field equations and then used to formulate a series of super-convergent finite beam elements. The resulting two-noded beam elements are shown to successfully capture the static and dynamic responses of thin-walled members. The finite elements developed involve no special discretization errors normally encountered in other finite element formulations and provide results in excellent agreement with those based on other established finite elements with a minimal number of degrees of freedom. The formulation is also capable to predict the natural frequencies and mode-shapes of the structural members. Comparisons with non-shear deformable beam solutions demonstrate the importance of shear deformation effects within short-span members subjected to harmonic loads with higher exciting frequencies. Comparisons with shell element solution results demonstrate that distortional effects are more pronounced in cantilevers with short spans. A generalized stress extraction scheme from the finite element formulation is then developed. Also, a generalization of the analysis procedure to accommodate multiple loads with distinct exciting frequencies is established. The study is concluded with design examples which illustrate the applicability of the formulation, in conjunction with established principles of fatigue design, in determining the fatigue life of steel members subjected to multiple harmonic forces.

This work analytically solves the governing equations of the laminar film condensation from pure vapors in vertical tubes to find the condensate film thickness. The solution is then extended to turbulent flow conditions for steam. All other relevant quantities are derived from the film thickness solution. For laminar film condensation of quiescent vapors, an exact explicit solution and an approximate closed-form solution were found by using a new definition of the dimensionless film thickness, the Lambert W-function, and a logarithmic function approximation. For laminar mixed-convection film condensation with interfacial shear stress, an approximate closed-form solution was found by using a new definition of the pressure gradient. For turbulent film condensation of steam, correlations of the turbulent vapor and liquid viscosities were formed by asymptotically comparing this approximate laminar closed-form solution to a turbulent flow numerical solution. The present solutions compared very well to published numerical and experimental data.

The steady state response of thin-walled members subjected to harmonic forces is investigated in the present study. The governing differential equations of motion and associated boundary conditions are derived from the Hamilton variational principle. The harmonic form of the applied forces is exploited to eliminate the need to discretize the problem in the time domain, resulting in computational efficiency. The formulation is based on a generalization of the Timoshenko-Vlasov beam theory and accounts for warping effects, shear deformation effects due to bending and non-uniform warping, translational and rotary inertial effects and captures flexural-torsional coupling arising in asymmetric cross-sections. Six of the resulting seven field equations are observed to be fully coupled for asymmetric cross-sections while the equation of longitudinal motion is observed to be uncoupled. Separate closed form solutions are provided for the cases of (i) doubly symmetric cross sections, (ii) monosymmetric cross-sections, and (iii) asymmetric cross-sections. The closed-form solutions are provided for cantilever and simply-supported boundary conditions. A family of shape functions is then developed based on the exact solution of the homogeneous field equations and then used to formulate a series of super-convergent finite beam elements. The resulting two-noded beam elements are shown to successfully capture the static and dynamic responses of thin-walled members. The finite elements developed involve no special discretization errors normally encountered in other finite element formulations and provide results in excellent agreement with those based on other established finite elements with a minimal number of degrees of freedom. The formulation is also capable to predict the natural frequencies and mode-shapes of the structural members. Comparisons with non-shear deformable beam solutions demonstrate the importance of shear deformation effects within short-span members subjected to harmonic loads with higher exciting frequencies. Comparisons with shell element solution results demonstrate that distortional effects are more pronounced in cantilevers with short spans. A generalized stress extraction scheme from the finite element formulation is then developed. Also, a generalization of the analysis procedure to accommodate multiple loads with distinct exciting frequencies is established. The study is concluded with design examples which illustrate the applicability of the formulation, in conjunction with established principles of fatigue design, in determining the fatigue life of steel members subjected to multiple harmonic forces.

A theoretical study is conducted for the lateral torsional buckling of wooden beam-deck assemblies consisting of twin beams braced by tongue-and-groove decking at the top. Two models are developed, each with a series of analytical and numerical solutions formulated. The first model targets twin-beam-deck assemblies where deck boards and other components are detailed to provide full continuous lateral restraint while the second model is built for situations where the beams are allowed to sway laterally and the relative lateral movement between the beams is partially restrained by the deck boards. In the first model, focus is on wind uplift while in the second model, both gravity and uplift loading scenarios are investigated. In the first model, an energy method is adopted and the principle of stationary potential energy is evoked to formulate closed-form solutions, energy-based solutions and a finite element solution. The validity of the present solutions is verified against a finite element based ABAQUS model. Similarly, a family of solutions is developed under the sway model and verified against the ABAQUS. Parametric studies are conducted for both models to examine the effects of various variables on the buckling capacity. A comparative investigation on the behavioral difference between the two models under ABAQUS is also presented. Overall, the restraining effects of deck boards bracing either on the beam compression or tension side is observed to have a significant influence on the lateral torsional buckling capacity of the twin-beam-deck assemblies.

The present dissertation contributes to advancing methods of determining the elastic and inelastic buckling resistance of compressive members with single angle and back-to-back double angle cross-sections with end conditions representative of those commonly used in steel construction. The first contribution develops an elastic buckling solution for members with asymmetric sections, such as unequal-leg angle members, connected to gusset plates at both ends and subjected to pure compression. In this case, the gusset plate connections at the member ends provide a fixity restraint to the member within the plane of the gusset and nearly a pin restraint in a plane normal to the gusset. Since both directions do not coincide with the principal directions of the member, the classical flexural-torsional buckling solutions provided in standards become inapplicable. In this context, a variational principle is formulated based on non-principal directions and then used to derive the governing differential equations and associated boundary conditions for the problem. The coupled equations are then solved analytically subject to the boundary conditions, and the characteristic equations are recovered and solved for the flexural-torsional buckling load of the member. The validity of the solutions derived is assessed against 3D shell elastic eigen-value buckling models based on ABAQUS for benchmark cases and the solution is shown to accurately predict the elastic buckling load and mode shapes. The effect of non-principal end restraints on the buckling load of compression members is then investigated for members with angle and zed cross-sections in a parametric study. It is observed that when a member end is fixed about a non-principal direction and pinned about the orthogonal direction, the flexural-torsional buckling load of the member is significantly influenced by the angle of inclination between the fixity axis and the minor principal axis. The second contribution aims to obtain the inelastic buckling resistance for single angle compression members with end gusset plate connections by taking into consideration the effects of material and geometric nonlinearity, initial out-of-straightness, residual stresses, and load eccentricity induced by the offset of the member centroidal axis from the end gusset plate connection. Towards this goal, a series of 3D shell models based on ABAQUS are developed and validated through comparisons against experimental results by others and then used to generate a database of compressive capacities for over 900 eccentrically loaded angle members with various geometrical dimensions and load eccentricities. The database is then used to investigate the effect of slenderness ratio, leg width ratio, connected leg width-to-thickness ratio and gusset plate-to-angle thickness ratio on the compressive resistance of the members, assess the accuracy of solutions available in present design standards, and develop improved design expressions for the compressive resistance for the members. The third contribution develops solutions for predicting the elastic buckling resistance of back-to-back double angle assemblies with end gusset plates and intermediate interconnectors subjected to compressive loads. Towards this goal, two novel models are developed. (1) A thin-walled finite element buckling solution is formulated and implemented into a MATLAB code. The formulation treats each angle member as a line of 1D thin-walled beam elements where then both angle members are connected at intermediate points along the span at the locations of interconnectors. The formulation is equipped with a multi-point constraint feature to enforce the kinematic constraints at the interconnector locations and at both extremities of the member. The model captures the tendency of both angles to open relative to one another in between interconnectors while undergoing flexural-torsional buckling. (2) An analytical buckling solution is developed for the limiting case where enough interconnectors are provided between members to force the two angles to essentially behave as a monolithic entity. The resistance predicted by the former model was then shown to asymptotically approach that predicted by the later model as the number of interconnectors is increased. The validity of the finite element model is assessed against 3D shell models based on ABAQUS and published experimental results, and then used to assess the validity of present design rules based on the effective slenderness concept. The present models are then used to carry out a parametric study of 1250 runs while varying the member slenderness ratio, leg width ratio, connected leg width-to-thickness ratio, and angle spacing-to-thickness ratio. The database of results generated is used to develop a simple expression to characterize the elastic buckling load/stress of the assembly. The possible integration of the new expression with present design provisions in standards to predict the inelastic buckling resistance of the member is illustrated through a design example.

Starting with tensor calculus and the variational form of the Hamiltonian functional, a generalized theory is formulated for doubly curved thin shells. The formulation avoids geometric approximations commonly adopted in other formulations. The theory is then specialized for cylindrical and toroidal shells as special cases, both of interest in the modeling of straight and elbow segments of pipeline systems. Since the treatment avoids geometric approximations, the cylindrical shell theory is believed to be more accurate than others reported in the literature. By adopting a set of consistent geometric approximations, the present theory is shown to revert to the well known Flugge shell theory. Another set of consistent geometric approximations is shown to lead to the Donnell-Mushtari-Vlasov (DMV) theory. A general closed form solution of the theory is developed for cylinders under general harmonic loads. The solution is then used to formulate a family of exact shape functions which are subsequently used to formulate a super-convergent finite element. The formulation efficiently and accurately captures ovalization, warping, radial expansion, and other shell behavioural modes under general static or harmonic forces either in-phase or out-of-phase. Comparisons with shell solutions available in Abaqus demonstrate the validity of the formulation and the accuracy of its predictions. The generalized thin shell theory is then specialized for toroidal shells. Consistent sets of approximations lead to three simplified theories for toroidal shells. The first set of approximations has lead to a theory comparable to that of Sanders while the second set of approximation has lead to a theory nearly identical to the DMV theory for toroidal shells. A closed form solution is then obtained for the governing equation. Exact shape functions are then developed and subsequently used to formulate a finite element. Comparisons with Abaqus solutions show the validity of the formulation for short elbow segments under a variety of loading conditions. Because of their efficiency, the finite elements developed are particularly suited for the analysis of long pipeline systems.

Starting with tensor calculus and the variational form of the Hamiltonian functional, a generalized theory is formulated for doubly curved thin shells. The formulation avoids geometric approximations commonly adopted in other formulations. The theory is then specialized for cylindrical and toroidal shells as special cases, both of interest in the modeling of straight and elbow segments of pipeline systems. Since the treatment avoids geometric approximations, the cylindrical shell theory is believed to be more accurate than others reported in the literature. By adopting a set of consistent geometric approximations, the present theory is shown to revert to the well known Flugge shell theory. Another set of consistent geometric approximations is shown to lead to the Donnell-Mushtari-Vlasov (DMV) theory. A general closed form solution of the theory is developed for cylinders under general harmonic loads. The solution is then used to formulate a family of exact shape functions which are subsequently used to formulate a super-convergent finite element. The formulation efficiently and accurately captures ovalization, warping, radial expansion, and other shell behavioural modes under general static or harmonic forces either in-phase or out-of-phase. Comparisons with shell solutions available in Abaqus demonstrate the validity of the formulation and the accuracy of its predictions. The generalized thin shell theory is then specialized for toroidal shells. Consistent sets of approximations lead to three simplified theories for toroidal shells. The first set of approximations has lead to a theory comparable to that of Sanders while the second set of approximation has lead to a theory nearly identical to the DMV theory for toroidal shells. A closed form solution is then obtained for the governing equation. Exact shape functions are then developed and subsequently used to formulate a finite element. Comparisons with Abaqus solutions show the validity of the formulation for short elbow segments under a variety of loading conditions. Because of their efficiency, the finite elements developed are particularly suited for the analysis of long pipeline systems.