Affine-and-minimal surface is a surface with zero mean affine curvature. By contrast to ordinary minimal surfaces consisting of saddle points, an affine-and-minimal surface can contain elliptical points. So, an elliptical paraboloid consists of all-elliptic points but this is an affine-and- minimal surface.

Fig. (4). A surface of diagonal translation.

Surfaces of diagonal translation are formed due to translation of a plane curve along a directing curve so that in the process of sliding of a generating curve along the plane stationary contour, two symmetrical points of the generatrix have the contact with the contour (Fig. 4). Such surface may be constructed on a contour of arbitrary form. The surfaces formed on a contour with symmetry, such as circle, ellipse, square, and rectangle, are of the greatest interest. Sometimes, using a method of design of surfaces of diagonal translation, one may obtain well known surfaces. For example, assuming an ellipse as the plane contour and an axis of the ellipse as the diagonal and taking a mobile generating curve in the form of a parabola, one can obtain the elliptic paraboloid (Fig. 5).

Fig. (5). A surface of diagonal translation.

Additional information on geometry of elliptic paraboloids is given in Hatiskatzi [30], where a method of construction of a line of intersection of the elliptic paraboloid with the cylinder is under consideration. Shovkoplyas [31] obtained the points of intersection of the elliptic paraboloid and the trihedral prism with the help of methods of descriptive geometry.

The motion of heavy material point on the elliptical paraboloid surface creates some exclusive periodic trajectories on its surface. It is of great interest in mathematicians. The problems of space rounding of smooth elliptic paraboloids under different angles of attack and slippage of a three-dimensional hypersonic flow of dissociating non- equilibrium air are considered in [32] where an approximate numerical method of solving the equations of a hypersonic, three-dimensional viscous shock layer is proposed.

Representations of surfaces on the plane are a typical task of the engineering geometry. It is formulated in the following way: for a given segment of the surface is required to find a flat area of an appointed form and from this form, the initial segment must be built by bending. The first step is to break the surface on the constructive modules, the borders of which will coincide with lines of seams. Next, dividing of modulus not belonging to developable surfaces into small fragments, limited by lines of horizontal sections, is used. By design, these parts can be approximated by developable surfaces with any given accuracy. However, researches have shown that the direct application of this algorithm is ineffective. Frolovskiy and Pavenko [33] offered to use additional limitations associated with the approaches of a theory of elasticity for prevention of the above-stated anomalies. An account of energy expenditures on the deformation of the starting surface is taken into consideration. In this case at the second stage, it is necessary to evaluate the energy expenditures on the deformation of the developable surfaces obtained at the first stage. As an example, a quasi-development of the elliptic paraboloid was adduced in [33].

Theocaris applies the elliptic paraboloid failure surface, constructed with the application of strength criterions, for the investigation of strength of composites, working in a three-axial stress state [34], polyurethane porous materials [35], for the evaluation of components of stress-strain state when continuous external load increases and provokes a change of material from elastic into plastic state and further till the value of the ultimate load [36]. Theocaris and Philippidis [37] affirm that geometrical parameters of the failure surface in the form of an elliptic paraboloid depend of quantity of strength anisotropy and strength properties of the materials.

It is known that the shape of a growing crystal can look like an elliptic paraboloid [38-40].

First, momentless shell theory was applied for the investigation of thin shells. Momentless theory of analysis of thin elliptic paraboloid shells was used by Fischer [45], Pavlović [46], Vlasov [47], and by other researchers [48-50]. Dulacska and Janko [48] affirm that results of momentless theory of shell in the form of an elliptic paraboloid subjected to uniform horizontal symmetrical load or inversely symmetrical load can be used for the shells, supported by boundary elements, rigid in bending and torsion. Lantukh [51] presents examples of analysis of momentless translational shells loaded by uniform load. For the solution of the problems it is necessary to use the tables of subsidiary coefficients presented in [52]. This method gives an opportunity to find firstly projections of normal forces N_xp and N_yp on the xOy plane and after that to obtain normal forces N_x, N_y in the shell. Lantukh [51] considers that the solution with the help of the tables is easier than the solution with using of a stress function. The volume of calculation cuts by half.

Results of many investigations show that elliptic paraboloid shells have the important advantage that the bending stresses are confined to narrow zones along the boundaries and are very small. Thus the membrane (momentless) theory provides often a good approximation for the stress condition of the shell [53].

A method of double trigonometric series is one of the wide spread methods of solution of system of linear differential equations with constant curvatures (k_x = д²z/дx²; k_y = д²z/дy²). Its principal propositions as applied to shallow elliptic paraboloid shell are presented in a reference book [54]. One supposed that a shell has simply supported edges and subjected to uniform loading.

A very extensive study of elliptic paraboloid shells with elastic edge members has been carried out by H.C. Shah [55] in 1960. The solution was obtained by using double trigonometric series. But Aass Asbjӧrn [53] holds that this solution for these boundary conditions is almost impossible for practical application. Double trigonometric series can be also used for the analysis of a shell with simply supported edges and eccentrically reinforced with ribs [56]. The known system of equations for shallow orthotropic shells was supplemented with the forth order differential operator which took into account the rigidity of the ribs and their eccentricity relatively the middle shell surface. It would be interesting to investigate an influence of value and sign of the eccentricity on the accuracy of a finite result. Later on in 1984 [57], the diagrams, illustrating the change of maximum flexure and bending moment of the shell, simply supported and rigidly fixed along the contour, depending on its rise, dimensions on the plane, and the height of stiffeners, were made.

Bending of an elastic elliptic paraboloid shell rested upon ribs was studied by Zmievskiy [58]. He used a Vlasov’s shallow shell theory and obtained a system of three differential equations in displacements. This system was solved with the help of a method of the Fourier’s finite integral transformation. The shell loaded by uniform distributed load was taken for the numerical example. The numerical results obtained were compared with results obtained with the help of a finite difference method and a finite element method.

A many-waved shell consisting of elliptic paraboloids was studied by Garanin [59]. A strain state was defined by the equation:

where D is the flexural rigidity of the shell, С = Eh/R². Poison ratio was taken equal to zero. Garanin has taken into account three types of boundary conditions depending on the position of the shell in the many-waved covering. A solution was taken in a single polynomial series. A surface was given in a Cartesian coordinates. Membrane and bending theory of multi-span elliptic paraboloid shells was studied by Kashani-Sabet [60] too.

Fareed and Dawoud [61] applied a single polynomial series for analysis of a shallow translational shell rested on thin walls in two opposite edges and on an elastic foundation of Winkler’s type. Vasilenko [62] presented a solution of a problem on a stress state of an orthotropic thick shallow elliptic paraboloid shell. For a shell rectangular in plan with the edges free from normal load, a method of separation of variables was applied. So, the problem was reduced to one-dimensional boundary problem for a system of ordinary differential equations with piecewise-continuous coefficients. After that, this problem is solved by a numerical method.

Ansan [63] showed that Levy-type solutions for long cylindrical shells may be extended to elliptic paraboloid only for certain restricted boundary conditions. But Banerjee [64] proved that using of a stress function for analysis of elliptic paraboloid shells gives results not satisfying the fundamental equations for the equilibrium conditions.

Gambarova and Karini [65] analyzed the membrane and bending stresses in a uniformly loaded shell in the form of elliptic paraboloid on a rectangular plan with free supporting in the corners. They did it with the help of two absolutely different approaches. For the calculation of membrane stresses, a theory of A. Pucher for the semi-inverse problem was used. A special attention was paid to the behavior of zones siding with the corners where equilibrium conditions are satisfied at the expense of the membrane stress distribution under conditions that a deformed state is appeared from the solution of an inverse problem for a local area. A method of moire strips was used for the bending stress analysis. A picture of moire strips gave a possibility to obtain the elastic angles of turning and after that, the elastic curvatures with the help of differentiation and transversal displacements by integration.

Dekhtyar [66] demonstrated the optimal point supporting of shells on the basis of a kinematic method of the ultimate equilibrium theory. The covering is resting on four points. The corner fragments must be made from the same shell but turned over. A boundary effect in shells resting on the several columns is examined in [49]. Interesting results as applied to the problems of analysis of plates and shells resting on point supports under action of the local loads and the point disturbances were presented by Vidyushenkov [67]. He writes that numerical methods in these cases are ineffective and that is why analytical methods continue to play an important role for research of a stress-strain state of plates and shells resting on point supports.

Many researchers used an asymptotic method of small parameter for shell analysis. A small parameter is selected depending on the structure of equations. It can have geometrical or physical sense. The presentation of a function in the form of the expansion in terms of a small parameter reduces an initial system of equations to an infinite system of the equations differing from each other only by the right parts. As a rule, a small parameter is chosen so that the first approximation has a nice calculation. In this case, all consequent approximations can be realized as more precise for the first one. First, Nash [68] has applied a method of small parameter for the analysis of an elliptic paraboloid shell. As initial equations, he has taken the equilibrium equations of K. Marguerre. But this work can be taken only as a statement paper because of its conciseness and inaccuracy.

Kozlov [69] made an analysis of a shallow shell on an elliptical plan taking as a small parameter a non-dimensional value of μ = f/a, where f is a rise of the shell, a is the semi-axis of a edge ellipse. The middle surface of the shell was taken in explicit form:

(7)

Assume then . Coefficients of the first fundamental form in the theory of surfaces were taken as A = B = 1, F = 0. A system of three differential equations of the shallow shell theory in components of an elastic displacement vector for thin shells given in Cartesian coordinates was derived in the limit of Kirchhoff’s hypotheses. After that, dimensionless coordinates ε = x/a, η = y/b were assumed. Due to the boundary conditions, parameters of an elastic displacement vector were taken as:

where μ is the small parameter, are the coefficients of decomposition. A shallow shell fixed along the elliptic contour was loaded by uniform vertical load Z = q, X = Y = 0. Kozlov [69] ascertained that it is enough to fulfill five or six approximations for practice calculations. Additionally, he showed that maximum values of displacements and bending moments for a shell examined in comparison with a plate less than 20%. The asymptotical method for the first approximation was applied for the analysis of a reinforced concrete dome of 7 m high. The support strengthened elliptic element has the spans of 45 × 63 m [70].

Srubshchik [71] has shown that the secondary edge effect phenomenon can hold when bending of elastic thin shallow shells in the form of an elliptic paraboloid under inner pressure takes place. Another approach was used by Slezinger and Barskaya [72] who investigated the influence of inhomogeneity on stress-strain state of a shallow shell with the middle elliptic paraboloid surface. The problem was considered in geometrically non-linear statement. The general equations were solved with the help of the Bubnov – Galerkin method.

The stress resultants and radial deflection of the portion of an elliptic paraboloid shell near its vertex subjected to a point load at its vertex were studied by Forsberg and Flügge [73]. The singular solutions to the homogeneous shallow-shell equations were expressed as power series.

Additional information on analytic analysis of elliptic paraboloid shells can be found in Refs [74, 75].

Modeling of entire shallow shell structures with the help of a discrete element system is one of such methods [77]. Deformations of shearing and tensile are concentrated in the units. For elastic stadia of work, stresses – deformations relations were assumed as for a plane stress state. Equations describing the behavior of such model can be obtained and solved as a finite system of linear equations. Orthographic representations of vertical displacements, inner forces and moments for a shell in the form of elliptic paraboloid on the rectangular plan are presented in a paper [78]. The authors studied two cases just when all edges of the shell are simply supported or two opposite edges are simply supported but two other edges are free. Uniformly spread loading and a load defined to sinusoidal law was considered.

A thin shallow elliptic paraboloid shell on a quadrille plan was approximated by a grid shell model by Novak [79]. A problem was solved with the help of a momentless shell theory with loading by dead load.

A method of nets became popular long before appearance of computers but contemporary computational techniques became the most important stimulator of its subsequent development and wide application of the net methods for the solution of problems of engineering practice.

P.M. Varvak and L.P. Varvak [80] applied a method of nets for the analysis of a shallow elliptic paraboloid shell simply supported on a square plan. They assumed that the shell was under action of uniformly distributed load. An equation of the middle surface was taken in the following form:

Apraksina [81] carried out a comparative research of a stress-strain state of an elliptic paraboloid shell with the help of a theory of shallow and non-shallow shells given in arbitrary curvilinear coordinates. She applied FEM. The distribution of bending moments and normal forces in the principle cross-sections was studied for the diagonal section of the shell, square in a plane, with the ratio of total height to span equal to 1/5, 1/7, 1/15. A displacement vector was approximated with the help of bi-cubical polynomials with using of the node values of displacements and their derivatives in two directions.

For the analysis of elliptical parabolic domes, a shear deformable four-noded finite element based on a hybrid/mixed assumed stress is presented in a paper [20]. The element called improved Hybrid and Enhanced Shell element was developed for a general arbitrary orthogonal coordinate system. A detailed parametric study of anisotropic elliptical and parabolic shells of various configurations was carried out to investigate the effects of height ratios as well as layer lay-up schemes. A new flat quadrilateral shell element is presented in [82]. The results for uniformly distributed load pressure are obtained from both numerical and experimental work.

A triangular shallow shell rested in the corners is considered in a paper [83]. The shell edges are strengthened by the ribs. The authors show an opportunity to solve the problem by three methods. These are a method of separation of variables, a method of finite differences, and a method of finite elements. A rectangular finite element has twenty degrees of freedom. A method of separation of variables is preferred if a computer is of small capacity.

A new flat shell element was presented in [84] which can be used for practical engineering purposes.

In 1963, Pracash [85] studied a rectangular translational shell in the form of an elliptic paraboloid loaded by inner pressure. The shell edges were simply supported. Pracash reduced a governing equation to a Poison equation with a constant right part with the help of change of variables relatively the stress function. The equation obtained was solved with the help of a finite difference method. Later, Padilla and Schnobrich [86] used finite differences to treat elliptic paraboloid shells.

Novak Otakar [87] decided to study an influence of tangential displacements on the final result with the help of a finite difference method. He compared vertical displacements determined by an analysis with taking into account a hypothesis about equality of the tangential displacements to zero and without this hypothesis. It was assumed that the shell edges are rigidly fixed and the shell is subjected to a uniform distributed load. The results of analysis showed that maximum deviations of displacements, determined for an elliptic paraboloid due to an approximate analysis, constituted 15.3% in comparison with accurate results. And the mistaken increases as the shallowness increases. Deviation lies within the limits of 3.72% for a hyparshell, within the limit of 21.8% for a conoid shell. Before, he [88] derived a system of three equations in displacements for a shallow elliptic paraboloid on the square plan. Then orthographic representations of vertical displacements for two shells with the main dimensions of 18 × 18 m, the shell thickness of 6 cm and the shell rise of 2.5 m and 1 m were carried out. The load was assumed as uniformly distributed.

In 1981, Andrushkov and Rasskazov [89] obtained by a variational method a system of three equilibrium equations in displacements for a moment theory of non-shallow shells of arbitrary form defined in a Cartesian system of coordinates. The realization of the equations obtained was fulfilled for the elliptic paraboloid shells with a help of a finite difference method. They carried out the analyses of shallow and non-shallow shells. These results were compared with an analogous results determined with the help of the equations of a Vlasov’s theory of shallow shells and the moment shell theory in Cartesian coordinates written for a mixed method.

Stress-strain state of a thin shell in the form of elliptic paraboloid with taking into account plastic deformation was studied by Bozhanov and his colleagues [90]. Differential equations defining a work of a shell after elastic limit were obtained and their solutions were presented.

Some additional information for Part 2 and Part 3 of the review on a moment theory of analysis of thin shallow elliptic paraboloid shells can be found in Refs [79, 91].

The first results on the determination of critical pressure for a rigidly fixed thin shallow shell in the form of an elliptic paraboloid with the plane elliptic contour were derived by Pogorelov [94] who assumed that a bulge area was not small. With the help of a geometrical method he obtained the formula for the critical pressure:

(8)

where Е, ν, δ are correspondently Young’s modulus, Poison ratio, the thickness of the shell; R₁, R₂ are principal radii of curvatures of the elliptic paraboloid in the center of buckling. It should be noted that a formula (8) coincides with a classical formula for a critical pressure for a closed spherical shell with radius of Babenko [95] declined a priory assumption about the form of a shell bulge and obtained the lower appraisal of the critical outer pressure:

(9)

also with the help of a geometric method. This critical pressure is attained in the case of a small area of bulge localized close by the ε- environment of one of elliptic paraboloid points placed outside the ε- environment of the rigidly fixed elliptic edge.

An approximate formula:

for the determination of critical pressure p_cr for an elliptical paraboloid shell on the rectangular plan is presented in a book of Kollar and Dulacska [96]. An algorithm of numerical determination of critical surface load for a shallow shell on a rectangular plan with the spans of 20 by 10 m is presented in [97]. Nagy transformed a system of geometrically nonlinear equations into the form convenient for numerical integration by an iteration method with the application of a finite difference method. Stability of an orthotropic elliptic paraboloid with a fixed contour subjected to outer pressure may be examined by the Koiter’s method as it is shown in [98]. The equilibrium equations obtained by a variational method are solved by a Bubnov – Galerkin method. As an example, a layered shell was studied and the influence of the shell structure and the shell rise was evaluated. Babenko [99] derived the expressions for precritical momentless forces and the relations arising from the conditions of rigid fixation of a plane edge of a non-shallow shell subjected to uniform outer pressure. These relations gave a possibility to calculate uniformly the meanings of critical pressure taking into consideration the Babenko’s stability criterion of shells with the second order middle surfaces. B.I. Babenko illustrated his method by a numerical calculation of an elliptic paraboloid shell.

Qiang, Wei Che, and Huajin [100] simulate dynamic response of single-layer elliptical paraboloid latticed shells under impact based on rate-dependent isotropic hardening material model, master-slave contact point searching method and penalty function method. The effect of all kinds of parameters (such as span, rise-to-span ratio, and cross-sectional areas of members) is analyzed on the structural dynamic buckling. The results show that the displacements of characteristic nodes stain energy and total energy of the structure increase all of a sudden. The results of the stability analysis presented in [101] would be useful for the design and application of single-layer latticed shells with semi-rigid joints because the joints in the most spatial structures are semi-rigid. Parametric analysis of latticed shells was carried out using ANSYS with the help of finite element models. The behavior of single-layer latticed shells with semi-rigid joints is highly nonlinear and that is why the influence of the rigidity of a joint on the critical load of a single-layer dome has been studied by Huihuan Ma and his collegues [101].

Zingoni and Balden [102] report the results of a numerical study undertaken on the buckling behavior of lightly stiffened elliptic paraboloidal steel panels intended for use as long-span shuttering for lightweight concrete bridge decks, walkways and floors. A linear buckling analysis was carried out using the FEM program ABAQUS, to determine the lowest buckling loads (first mode solution) for the pressure loading and the gravity loading. The numerical comparisons show that for the lightly stiffened shallow elliptic paraboloidal steel panels in question, the linear buckling analysis predicts the first buckling load reasonably well for low aspect ratios (b/a = 0.2 and b/a = 0.4).

The buckling analyses for linear buckling, elastic buckling, and elastic-plastic buckling were performed in a paper [103]. The results of buckling loads were formulated based on two procedures for practical design use. One is explicit formula for elastic buckling loads using a knock down factor and the other one is an implicit expression for buckling loads interpreted as a column buckling strength in terms of generalized slenderness ratio. An efficient method for evaluating elastic-plastic buckling loads of two-way elliptic paraboloid lattice domes of single layer under vertical loads was proposed in [104]. The comparison of the evaluated buckling loads with those obtained from FEM elasto-plastic buckling analysis proves that the procedure is valid and efficient enough to be applied in design practice.

Graphite-epoxy multilayered elliptic paraboloid shell panels having rectangular planforms are widely used in various fields of aerospace because of their light weight, high stiffness, and strength. The thermomechanical postbuckling response of these shell panels was obtained by Maloy K. Singha et al [105] within the framework of FEM. Numerical results presented in this paper show the effect of temperature dependence of material properties on limit loads of shallow panels.

Bearing capacity of the elliptic paraboloid grid shell with semi-rigid joints is analyzed in [106, 107]. The influence of section of steel bars, rise-span ratio and initial imperfection on the bearing capacity is also investigated in these papers. The nonlinear buckling behavior of an elliptic paraboloid suspended dome structure of span 110m×80m is investigated in [108, 109] by introducing geometric nonlinearity, initial geometric imperfection, material elastic-plasticity and half-span distribution of live loads. The study shows that the coefficient of stable bearing capacity usually is not minimal when the initial geometric imperfection configuration is taken as the first order buckling mode.

Additional information on stability of elliptic paraboloid shells can be taken in [110-112].

Ballesteros [113] gave the information on the collapse of an elliptical paraboloid shell during the process of removing the formwork. The structure showed clear imperfections in the geometry and had construction defects.

The results of experimental control of formulas (8) and (9) are presented in [114]. The elliptical paraboloid shells were made by pressing of the shallow spherical shells in specially manufactured contrivances. The shells underwent the deformations and took the form of a elliptic paraboloid with principal radii of curvatures R₁, R₂. The moment stresses appearing in the process of bending are very small and that is why they do not have an influence on value of critical load . The investigations showed that a formula (8) gives a critical pressure for weakened conditions of a rigidly fixed edge but a formula (9) answers the rigid fixing of a plane elliptic edge. The loss of stability of an elliptic paraboloid shell begins after appearance of a small hollow at the top of the shell as in a case with spherical shells. An installation for the investigation of after-critical deformations of shallow elliptic paraboloid shells rigidly fixed along an elliptical edge of the shell subjected to outer pressure is offered by Babenko, Koshelev, and Avedyan [115, 116] who tested a paraboloid of revolution. The shells for the testing have been made by a method of deposition of copper on shell by evaporation in vacuum. Loading by outer pressure was realized with the help of a method of pumping out of air from a space under the shell. The aftercritical unstable branches of pressure-deflection dependences in the shell top and values of the low critical loads determined in the experiments were compared with results of asymptotic solution of the proper shell theory equations.

In his report, R.E. Row [117] presented a condensed account of the activities of the Department “Association on Cement and Concrete” (UK) and described the test of a one-tenth scale model of the elliptic paraboloid shell (1.45×1.45m, a thickness is 0.64 cm). This model copied a part of covering of a garage in Lincoln designed by Dr. Hajnal Konyi. This test was very important with a point of view of determination of possibilities of the shell models and elaboration of methods of loading and taking down of readings.

A shallow shell model of roofing of the Smithfield Poultry Market, London, UK, was used for the determination of characteristics of local loss of stability [118, 119]. The shell measures 68.6×38.12 m and its total height is 9.1 m. A model was made in a scale of 1:12 (5.22×3.25 m).

G. Franz [120] describes testing of shell models with a thickness of 3 mm made of several layers of glass fibers and epoxy resin. The shells were tested without thickening of the edges with resting on immovable supports (α_k = 0.138), with thickening of the edges with resting on immovable supports (α_k = 0.128), and with resting on the edges with opportunity of lateral displacement (α_k = 0.32). All of the models were loaded by uniformly distributed load. The models were with the dimensions of 1200 by 600 mm in plan. The test results showed that the flexure coefficients of α_k being a factor of a formula for the determination of a critical load:

Coincide nearly with the theoretical coefficients of Karman and Mushtary – Surkin and also with experimental values of Schmidt. Here, A is an area of the shell surface, the shells have the edge beams and rest on the supports as membranes. The application of models for researching works is a useful addition for better understanding of the behavior of thin-walled structures.

Natural vibrations of a shallow elliptic paraboloid on an elliptic plan were considered by Kozlov [122] in 1972. A surface equation was taken in the explicit form (7). A principle of Ostrogradskiy – Hamilton was used for the writing down of a functional for the subsequent application of a Ritz’s method. A problem on dynamic behavior of shallow shell on Winkler bedding under vertical seismic forces was under consideration in [123]. Fan Jiashen used equations of Vlasov’s technical shell theory for solution of which an apparatus of a theory of potential and Fourier’s trigonometric series were applied. A method presented may be applied for the investigation of behavior of shell under seismic load.

An interesting problem is examined in a paper [124], where sound pressure caused by vibrations of a square steel shell in the form of an elliptic paraboloid (a = b = 1 m)

is assessed. The shell thickness is 0.002 m. A shallow shell possesses of inner viscous-elastic damping. The damping coefficient ε is equal to 0.001. The Voigt-Kelvin model of damping was assumed. The inertial components provoked the motion in the tangential directions of the shell middle surface are neglected in the model. Vertical vibrations of a shell are defined by normal displacements:

where S_mn(t) are unknown functions of time and w_mn(x,y) are the shape function for the (m,n)-th mode. Kozien and Niziol [124] consider that shallow shells are commonly used as constructional elements of heavy duty machines. The results of analysis show an important influence of the radii of curvatures of a viscoelastic shallow shell on dispersion of the pressure in a chosen control point in the acoustic surrounding medium. Natural vibrations of a body in the form of an elliptic paraboloid placed in a limited volume of liquid and vibrations of a level of this liquid are studied by Hisashi Hukuda [125] with the application of the quasi-Lame equation.

The Galerkin method coupled with integral transform method was applied by Yung-Tze Chen [126] for determining the natural frequencies of cracked shallow elliptic paraboloid shells simply supported on four edges.

Dynamic analysis of shell structures can be solved efficiently by finite element method. Different computational models for laminated composites were proposed by researches. Modeling the shells of double curvature, Sahu and Datta [127] applied a finite element taking into account transversal deformations and inertia of rotation. A used method spreads a theory of dynamic transversal deformation with the application of the first approximation of Sandler to a problem of laminated composite doubly curved shells. After that, this problem was reduced to the theories of Love and Donnel by the use of tracers. Non-dimensional natural frequencies:

(a/b = 1; a/h = 100; E₁₁ = E₂₂ = 25; G₂₃ = 0,2E₂₂; G₁₂ = G₁₃ = 0,5E₂₂; ν₁₂ = 0,25) were determined for free supported shells in the form of sphere, elliptic and hyperbolic paraboloids for research of the effect of geometry of the shell and a quantity of layers of the composed shell. The obtained results give a good coincidence with analogues results presented in [128] where FEM was used also.

Free vibration problem of stiffened shells with different size and position of the cutouts was examined in [129] where natural frequency and mode shapes were studied. An eight-noded curved quadratic isoparametric finite element was used. The finite element model proposed in [129] can successfully analyze vibration problems of stiffened composite shells with cutout. It was noticed that for clamped shells the maximum fundamental frequency always occurs along the diagonal of the shell. Sarmila Sahoo [130] insists that the information available for vibration behavior of all common forms of both anticlastic and synclastic shells with cutouts is far from complete.

The Kriging algorithm [131] was developed in [132] to quantify uncertainties in natural frequencies of cantilever composite shallow doubly curved shells and its efficacy was compares with direct Monte Carlo simulation technique. An eight noded isoparametric quadratic element with five degrees of freedom at each node was considered in finite element formulation. Four layered graphite-epoxy symmetric angle-ply and cross-ply laminated composite cantilever shallow elliptic paraboloid shell with thickness of 5 mm was considered and it was observed that Kriging model can handle the large number of input parameters. The results obtained by employing Kriging models are found accurate with the results found by using direct Monte Carlo simulation.

Any information on experimental investigations of natural and forced vibrations of elliptic paraboloid shells was not found out in scientific-and-technical sources.