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When a concentrated load is introduced perpendicular to the flanges of a steel beam, this condition is referred to as Patch loading (Gozzi, 2007). This occurrence is common in many steel structures, for example at supports or during launching of bridges. Because of the usual slenderness of I-beams and other plated structures, these are sometimes reinforced with stiffeners in order to avoid buckling. Modifications, such as adding stiffeners to a beam, are done to make greater plastic deformations possible before buckling can occur; thereby increasing the resistance against failure. Transverse stiffeners are added in areas where the beam is exposed to concentrated loads (Lagerqvist, 1994). The descriptions of calculating patch loading in the Eurocode are presented for cases of double stiffeners, with the load applied in between two stiffeners with same distance to each of them, or when there is one single stiffener that is acting in line with the load. In the Eurocode there are also descriptions on how to calculate on the resistance against patch loading when there are no stiffeners added. However, the Eurocode lacks descriptions for cases when the stiffeners are misaligned. The purpose of this paper is the evaluation of the impact from transverse stiffeners to the resistance of welded I-beams, when the stiffeners are misaligned and where the length of the beam varies. Because of the complexity of such of problems it is almost impossible to find theoretical solutions (Lagerqvist & Johansson, 1996). Therefore, in this study as well as in almost all studies that aim to predict the ultimate resistances of steel beams subjected to patch loading, the results are gained empirically. The tests herein were done by FE-modeling and the results from the physical experiments done in Lagerkvist’s doctoral thesis were used for validation of the model, as conducting experiments ourselves was not economically possible. 6 The study was made in two steps. In the first step FE-models were produced under the same circumstances as the results obtained by Lagerqvist (1994). Those analyses were not part of the aim of the study; the intention for making the initial analyses was to strengthen the reliability of the results. From there, the final analyses were made with the aim in investigating the influence of stiffeners on the resistance, when these are misaligned. In this step, observations were also made with regards to the impact of the bending moment of the beam on its resistance. The initial analyses, which were made for validation of the modeling, had a satisfying correspondence to the physical experiments; hence the final analyses are assumed valid of acceptance. From observations of the results in the final analyses it is noticed that adding stiffeners is a highly preferred way of increasing the resistance for slender beams. For full utilization it is however important to have the stiffeners optimally placed, because a small deviation from this position gives an unwanted decrease in resistance.

Sandwich constructions are the most commonly used structures in aircraft and navy industries, traditionally. These structures are made up of the face sheets and the core, where the face sheets will be taking the load and is connected to other structural members, while the soft core material, will be used to absorb energy during impact like situation. Thus, sandwich constructions are mainly employed in light weight structures where the high energy absorption capability is required. Generally the face sheets will be thin, made up of either metallic or composite material with high stiffness and strength, while the core is light in weight, made up of soft material. Cores generally play very crucial role in achieving the desired properties of sandwich structures, either through geometric arrangement or material properties or both. Foams are in extensive use nowadays as core material due to the ease in manufacturing and their low cost. They are extensively used in automotive and industrial field applications as the desired foam density can be fabricated by adjusting the mixing, curing and heat sink processes. Modeling of sandwich beams play a crucial role in their design with suitable finite elements for face sheets and core, to ensure the compatibility between degrees of freedom at the interfaces. Unless the mathematical model simulates the physics of the model in terms of kinematics, boundary and loading conditions, results predicted will not be accurate. Accurate models helps in obtaining an efficient design of sandwich beams. In Structural Health Monitoring studies, the responses under the impact loading will be captured by carrying out the wave propagation analysis. The loads applied will be for a shorter duration (in the orders of micro seconds), where higher frequency modes will be excited. Wavelengths at such high frequencies are very small and hence, in such cases, very fine mesh generally is employed matching the wavelength requirement of the propagating wave. Traditional Finite element softwares takes enormous time and computational e ort to provide the solution. Various possible models and modeling aspects using the existing Finite element tools for wave propagation analysis are studied in the present work. There exists a huge demand for an accurate, efficient and rapidly convergent finite elements for the analysis of sandwich beams. E orts are made in the present work to address these issues and provide a solution to the sandwich user community. Super convergent and Spectral Finite sandwich Beam Elements with metallic or composite face sheets and soft core are developed. As a philosophy, the sandwich beam finite element is constructed with the combination of two beams representing the face sheets (top and bottom) at their neutral axis. The core effects are captured at the interface boundaries in terms of shear stress and normal transverse stress. In the case of wave propagation analysis, the equations are coupled in time domain and spatial domain and solving them directly is a difficult task. In Spectral Finite Element Method(SFEM), the displacement functions are derived by solving the transformed governing equations in the frequency domain. By transforming them and forces from time domain to frequency domain, the coupled partial differential equations will become coupled ordinary differential equations. These equations in frequency domain, can be solved exactly as they are normally ordinary differential equation with constant coefficients with frequency entering as a parameter. These solutions will be used as interpolating functions for spectral element formulation and in this respect it differs from conventional FE method wherein mostly polynomials are used as interpolating functions. In addition, SFEM solutions are expressed in terms of forward and backward moving waves for all the degrees of freedom involved in the formulations and hence, SFEM provides faster and efficient solutions for wave propagation analysis. In the present work, strong form of the governing differential equations are derived for a given system using Hamilton's principle. Super Convergent elements are developed by solving the static part of the governing differential equations exactly and hence the stiffness matrix derived is exact for point static loads. For wave propagation analysis, as the mass is not exactly represented, these elements are required in the optimal numbers for getting good results. The number of these elements required are generally much lesser than the number of elements required using traditional finite elements since the stiffness distribution is exact. Spectral elements are developed by solving the governing equations exactly in the frequency domain and hence the dynamic stiffness matrix derived is exact for the dynamic loads. Hence, one element between any two joints is enough to solve the whole system under impact loads for simple structures. Developing FE for sandwich beams is quiet challenging. Due to small thickness, the face sheets can be modeled using 1D idealization, while modeling of large core requires 2-D idealization. Hence, most finite or spectral elements requires stitching of these two idealizations into 1-D idealization, which can be accomplished in a variety of ways, some of which are highlighted in this thesis. Variety of finite and spectral finite elements are developed considering Euler and Timoshenko beam theories for modeling the sandwich beams. Simple element models are built with rigid core in both the theories. Models are also developed considering the flexible core with the variation of transverse displacements across depth of the core. This has direct influence on shear stress variation and also transverse normal stress in the core. Simple to higher order models are developed considering different variations in shear stress and transverse normal stress across depth of the core. Development of super convergent finite Euler Bernoulli beam elements Eul4d (4 dof element), Eul10d (10 dof element) are explained along with their results in Chapter 2. Development of different super convergent finite Timoshenko beam elements namely Tim4d (4 dof), Tim7d (7 dof), Tim10d (10 dof) are explained in Chapter 3. Validation of Euler Bernoulli and Timoshenko elements developed in the present work is carried out with test cases available in the open literature for displacements and free vibration frequencies are presented in Chapter 2 and Chapter 3. The results indicates that all developed elements are performing exceedingly well for static loads and free vibration. Super convergence performance for the elements developed is demonstrated with related examples. Spectral elements based on Timoshenko theory STim7d, STim6d, STim6dF are developed and the wave propagation characteristics studies are presented in Chapter 4. Euler spectral elements are derived from Timoshenko spectral elements by enforcing in finite shear rigidity, designated as SEul7d, SEul6d, SEul6dF and are presented. E orts were made in this present work to model the horizontal cracks in top or bottom face sheets using the spectral elements and the methodology is presented in Chapter 4. Wave propagation analysis using general purpose software N AST RAN and the super convergent as well as spectral elements developed in this work, are discussed in detail in Chapter 5. Modeling aspects of sandwich beam in N AST RAN using various combination of elements available and the performance of four possible models simulated were studied. Validation of all four models in N AST RAN, Super convergent Euler, Timoshenko and Spectral Timoshenko finite elements was carried out by simulating a homogenous I beam by comparing the longitudinal and transverse responses. Studies were carried out to find out the response predictions of a sandwich beam with soft core and all the predictions were compared and discussed. The responses in case of cracks in top or bottom face sheets under the longitudinal and transverse loading were studied in this chapter. In Chapter 6, Parametric studies were carried out for bringing out the sensitiveness of the important specific parameters in overall behaviour and performance of a sandwich beam, using Super convergent and Spectral elements developed. This chapter clearly brings out the various aspects of design of sandwich beam such as material selection of core, geometrical configuration of overall beam and core. Effects of shear modulus, mass density on wave propagation characteristics, effects of thick or thin cores with reference to the face sheets and dynamic effects of core are highlighted. Wave propagation characteristics studies includes the study of wave numbers, group speeds, cut off frequencies for a given configuration and identification of frequency zone of operations. The recommendations for improvement in design of sandwich beams based on the parametric studies are made at the end of chapter. The entire thesis, written in seven Chapters, presents a unified treatment of sandwich beam analysis that will be very useful for designers working in the area.

In this thesis, we investigate the propagation of some coherent and partially coherent laser beams, including a dark hollow beam (DHB), an elliptical Gaussian beam (EGB), a flat-topped beam and a twisted anisotropic Gaussian Schell-model (TAGSM) beam, through a paraxial optical system or a turbulent atmosphere. Several theoretical models are proposed to describe a DHB of circular or non-circular symmetry. Approximate analytical formulas for a DHB and a partially coherent TAGSM beam propagating through an apertured paraxial optical system are derived based on the generalized Collins formula. Analytical formulas for a DHB, an EGB, a flat-topped beam and a partially coherent TAGSM beam propagating in a turbulent atmosphere are derived in a tensor form based on the extended Huygens-Fresnel integral formula. It is found that after a long propagation distance these beams become circular Gaussian beams in a turbulent atmosphere, and this is quite different from their propagation properties in free space. The conversion of any of these beams to a circular Gaussian beam becomes quicker and the beam spot in the far field spreads more rapidly for a larger structure constant of the turbulent atmosphere, a shorter wavelength and a smaller waist size of the initial beam. Lower coherence and larger twist have a stronger effect of anti-circularization of the beam spot. Our analytical formulas provide a convenient way for studying the propagation of various laser beams through a paraxial optical system or a turbulent atmosphere. The concept of coincidence fractional Fourier transform (FRT) with an incoherent or partially coherent beam is introduced, and the optical system for its implementation is designed. The coincidence FRT is demonstrated experimentally with a partially coherent beam, and the experimental results are consistent with the theoretical results. / QC 20100831

In this thesis, hybrid stress finite element is formulated for the analysis of the isotropic, thin walled, open section beams with variable cross sections. The beam element has two nodes each having seven degrees of freedom. Assumption of stress field is sufficient to determine the element stiffness matrix. Axial, flexural and torsional effects are taken into account in the analysis. The methodology can be applied both to the tapered and the uniform beams. Throughout this study, firstly element cross-sectional properties are computed using the flow analogy of the inter-connected elements which may have different thicknesses. Then another computer program calculates the displacements and stresses at the nodes along the beam. The results obtained are compared to the results taken from literature and commercial FEM program Nastran.