Interaction between laser-welded web-core sandwich deck plate and girder under bending loads

Present paper investigates the interaction between laser-welded web-core sandwich deck plate and supporting girder under bending loads. The study is based on two linear-elastic Finite Element (FE) approaches, i.e. one using beam elements to model the girder and shell elements to model the homogenized web-core sandwich plate. With this approach the obtained FE model is considerably smaller than in the case of modeling the full, periodic, 3D geometry with shell elements. The FE solution results in stress resultants for beam and shell elements. These stress resultants do not describe accurately the periodic stress response of the sandwich plate or shear stress distribution at girder web. Therefore, the paper utilizes analytical methods to calculate these stress components from the obtained Finite Element solution. The second computational approach is based on modeling the actual 3D topology with shell elements. The two approaches are shown to be in very good agreement. The investigation shows that the effective flange width of the sandwich is different for the top and bottom face plates indicating that the interaction is different for these face plates. The present study also shows that this difference between the two faces depends strongly on the orientation of the web plates of the sandwich with respect to girder axis and the stiffness of the girder. The investigation also shows that the normal stress response in bending is dominated by the interaction between the sandwich plate and the girder, but also by the shear-induced normal stresses at the outer surface of the plate.     

Static and dynamic analysis of thin plates in bending— A novel finite element model

The static and dynamic behavior of thin flat plates in bending have been studied by means of a recently developed¹ doubly curved triangular shell element. The element's formulation is based on a modified mixed variational principle, wherein the primal variable σ (vector of shell stress resultants) and (boundary displacement vector) are required to satisfy a priori: (1) the complete shallow shell equilibrium equations, and (2) interelement C¹ displacement continuity. Several well-selected plate structures have been analyzed and the numerical results obtained indicate that the new element scheme competes most favorably with recently developed as well as with well-established elements included in commercial general-purpose finite element codes.     

Strength and stiffness requirements for intermediate ring stiffeners on discretely supported cylindrical shells

Silos in the form of a cylindrical metal shell are often supported on a ring beam which rests on discrete column supports. This support condition produces a circumferential non-uniformity in the axial membrane stresses in the silo shell. One way of reducing the non-uniformity of these stresses is to use a very stiff ring beam which partially or fully redistributes the stresses from the local support into uniform stresses in the shell. A better alternative is to use a combination of a flexible ring beam and an intermediate ring stiffener. Recent research by the authors has identified the ideal location of the intermediate ring stiffener to provide circumferentially uniform axial membrane stresses above the stiffener. To be fully effective, this intermediate ring should locally prevent both radial and circumferential displacements in the shell. This paper explores the strength and stiffness requirements for this intermediate ring stiffener. Pursuant to this goal, the cylindrical shell below the intermediate ring stiffener is analysed using the membrane theory of shells and the reactions produced by the stiffener on the shell are identified. These reactions are then applied to the intermediate ring stiffener. Vlasov's curved beam theory is used to derive closed form expressions for the variation of the stress resultants around the circumference to obtain a strength design criterion for the stiffener. A stiffness criterion is then developed by considering the ratio of the circumferential stiffness of the cylindrical shell to that of the intermediate ring stiffener. The circumferential displacements of the ring and the shell are found for the loading condition previously obtained to determine the required strength. A simple algebraic expression is developed for this intermediate ring stiffness criterion. These analytical studies are then compared with complementary finite element analyses that are used to identify a suitable value for the intermediate ring stiffness ratio for practical design.     

组合空腹梁的静力特性研究

提出了一种新的组合空腹板结构形式,它由下层交叉钢肋、上层混凝土板及连接上下层并使之共同工作的钢管或钢管混凝土剪力键组合而成。为研究该类结构的静力特性,对组合空腹板的组成单元——组合空腹梁进行了足尺模型试验,侧重研究组合空腹梁下层钢肋、上层混凝土板的内力分布规律;对组合空腹板建立了基于空间梁单元与壳单元的混合元计算模型、空间壳单元计算模型两种有限元分析方法,对试验模型进行了理论分析。将试验数据、两种有限元计算方法的计算结果进行了比较。研究结果表明,组合空腹梁的内力分布均匀,能较好地发挥材料强度,具有刚度大、整体性好的特点;对简支组合空腹梁,其下层钢肋处于拉-弯、上层混凝土板处于压-弯的受力状态,弯曲应力在总应力中占了较大的比例;研究结果同时也验证了理论分析的可靠性。     

国际期刊导读

受压或受弯开孔薄板的弹性屈曲性能 摘要：开发、论证并总结了一组闭合公式,可以估算在弯矩或压力作用下,板的单个或多个开孔对其临界弹性屈曲应力的影响。公式适用于四边简支和三边简支的板（这在设计中又称为加劲板和非加劲板）。由于有限壳单元特征值屈曲分析需要运用商业有限元程序,而这些程序又不是专为结构分析而开发的,不能很方便地用于工程设计,所以可以采用这些公式作为这种屈曲分析方法的简化替代。     

Vibration of cross-ply laminated composite plates subjected to initial in-plane stresses

Natural frequencies, modal displacements and stresses of cross-ply laminated composite plates subjected to initial in-plane stresses are analyzed by taking into account the effects of higher-order deformations and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for rectangular laminates is derived through Hamilton’s principle. Several sets of truncated approximate theories can be derived to solve the eigenvalue problems of a simply supported laminated plate. After examining the convergence properties of the lowest natural frequency, only the numerical results for M=5, which are considered to be sufficient with respect to the accuracy of solutions, are presented. Numerical results are compared with those of the published existing three-dimensional theory and FEM solutions. The modal displacement and stress distributions in the thickness direction are plotted in figures. The buckling stresses can be obtained in terms of the natural frequencies of the laminates without initial in-plane stresses.     

Hydroelastic response of a box-like floating fuel storage module modeled using non-conforming quadratic-serendipity Mindlin plate element

Presented herein are the hydroelastic responses of a floating fuel storage module under wave action. The box-like fuel storage module is modeled by an equivalent solid plate. For the analysis, a non-conforming quadratic-serendipity (NC-QS) element based on the Mindlin plate theory was developed. In this element, we make use of the reduced integration method and the superposition of non-conforming modes onto the basis function of the 8-node element in order to prevent spurious modes and shear locking phenomena. Thus the element may be applied for both thick and thin plates. The solution for the hydroelastic response involves solving the coupled plate–water equation and the water equation numerically using the hybrid finite element–boundary element (FE-BE) method. The coupled plate–water equation is derived by equating the force term in the Mindlin plate equation with the wave pressure distribution obtained from the linearized Bernoulli equation; whereas the boundary integral equation relates the displacement of the plate and water velocity potential using the free-surface Green’s function. Results of the displacement and water velocity potential are found to be more accurate when compared with existing solutions for special cases. Moreover, the stress resultants computed are significantly more accurate than previous results as spurious modes are eliminated.     

Stresses in thin spherical shells with imperfections. Part II: Influence of local imperfections

The changes in stress resultants in thin spherical shells, associated with a local imperfection introducing curvature errors in all directions, are investigated. An axisymmetric finite element model of the shell and imperfection is employed to carry out the linear elastic analysis. Parametric studies have been performed, to identify the main parameters controlling the response, for the case of internal pressure. The results are compared with those obtained in Part I for axisymmetric imperfections, and bounds for maximum elastic stress resultants are established to cover the possibility of both local and axisymmetric imperfections.     

Limit analysis of reinforced concrete cylindrical shell roofs

Two approximate yield criteria are proposed for reinforced concrete cylindrical shell element. Three lower bound solutions are developed for cylindrical shell roofs with free longitudinal edges. Two shells with different geometric parameters are analysed. Elastic analysis is also done at working load. Graphs are presented to show the variation of stress resultants at critical sections.     

The strain approach to the development of thin cylindrical shell finite elements

The method of forming thin cylindrical shell finite element displacement functions by initially developing the strain function is presented. The Kirchhoff-Love assumptions are used together with deep thin shell finite elements to develop a number of 20 degrees-of-freedom rectangular and 15 degrees-of-freedom triangular cylindrical shell elements. The elements are tested on an extended range of barrel vault roof problems and results are presented. Only in the very specific case of N_φ for the free straight-edged barrel vault using the triangular elements are the results found to be inaccurate. In all cases where the displacements and stresses can be analytically checked, the strain approach gives accurate results.     

The transition from thin plates to moderately thick plates by using finite element analysis and the shear locking problem

Two simple plate bending elements, based on Mindlin theory for analysis of both moderately thick and thin plates, are presented in this paper. These elements have either four nodes or eight nodes with 12 and 24 DOF, respectively. To illustrate the accuracy of these finite elements named as TURE12 and TURE24, several numerical examples of displacements and stresses for both thin and moderately thick plate bending problems are presented and discussed with a range of finite element meshes and thickness-to-plate length ratios. In addition, the bending and shearing behaviours of a Mindlin plate are analyzed with respect to shear locking. In order to test the shear locking, the results obtained from the Mindlin plate analysis using 4- or 8-noded elements with full, reduced, and selective reduced integration are compared with the exact classical thin plate solution.     

Strength analysis of concrete-filled thin-walled steel box columns

An elasto-plastic finite displacement analysis of concrete-filled thin-walled steel stub-columns of box shape is presented. In the analysis, a hardening-softening model is used to describe rationally the elasto-plastic behavior of concrete. A contact element for the interface combined with a bilinear constrained shell element for the plate and a cubic element for the concrete is employed. Both initial geometrical imperfections and residual stresses are also considered in the plate elements. Analytical results are then compared with previous experimental results. In addition, a parmetric study is conducted to investigate the effects of various parameters such as the plate aspect ratio, the plate width-thickness ratio and the concrete strength on the column strength. Finally, a design formula for concrete-filled stub-columns in compression is proposed.     

Buckling of rings in column-supported bins and tanks

Theories for the out-of-plane buckling of rings under uniform circumferential compression are well established. However these theories are not applicable to rings in column-supported bins where the circumferential stress in the ring varies significantly over the cross-section and around the circumference.

This paper deals with the out-of-plane buckling of annular plate rings in column-supported bins and tanks. The stress distributions in such rings are first examined using a finite element shell analysis. A closed-form solution for the buckling of rings under non-uniform circumferential stresses is then derived. Numerical results from the closed-form solution are compared with those from a finite element shell buckling analysis, and close agreement is found. The significant effect of stress non-uniformity on the buckling predictions is demonstrated. Finally, simplified equations are given which are suitable for structural design purposes, and which closely model the predictions of the more rigorous solution.     

A linearly-independent higher-order extended numerical manifold method and its application to multiple crack growth simulation

The numerical manifold method (NMM) can be viewed as an inherent continuous-discontinuous numerical method, which is based on two cover systems including mathematical and physical covers. Higher-order NMM that adopts higher-order polynomials as its local approximations generally shows higher precision than zero-order NMM whose local approximations are constants. Therefore, higher-order NMM will be an excellent choice for crack propagation problem which requires higher stress accuracy. In addition, it is crucial to improve the stress accuracy around the crack tip for determining the direction of crack growth according to the maximum circumferential stress criterion in fracture mechanics. Thus, some other enriched local approximations are introduced to model the stress singularity at the crack tip. Generally, higher-order NMM, especially first-order NMM wherein local approximations are first-order polynomials, has the linear dependence problems as other partition of unit (PUM) based numerical methods does. To overcome this problem, an extended NMM is developed based on a new local approximation derived from the triangular plate element in the finite element method (FEM), which has no linear dependence issue. Meanwhile, the stresses at the nodes of mathematical mesh (the nodal stresses in FEM) are continuous and the degrees of freedom defined on the physical patches are physically meaningful. Next, the extended NMM is employed to solve multiple crack propagation problems. It shows that the fracture mechanics requirement and mechanical equilibrium can be satisfied by the trial-and-error method and the adjustment of the load multiplier in the process of crack propagation. Four numerical examples are illustrated to verify the feasibility of the proposed extended NMM. The numerical examples indicate that the crack growths simulated by the extended NMM are in good accordance with the reference solutions. Thus the effectiveness and correctness of the developed NMM have been validated.     

Effect of unequal settlement of foundations on the stress resultants of hyperbolic cooling towers and the unequal settlement tolerance limit

Using the finite element solution of hyperbolic cooling towers supported by a column system, based on the curved axisymmetric finite element, the effects of unequal settlement of foundations on the stress resultants of tower shells are investigated. This partial edge effects spreads from the bottom of the shell up to 1/10−1/5th of the tower height. By way of an illustrative numerical example of a cooling tower being constructed, the tolerance limit of unequal settlement is given. It is proved that the equal settlement of this cooling tower from full-scale measurements satisfies that limit.     

Nonlinear response of shell structures: effects of plasticity modelling and large rotations

Many structural applications require nonlinear finite element analyses in order to assess response and capacity. Plastic deformations may be accounted for by means of thickness integration or stress resultants. The stress resultant model employed herein is based on Ilyushins' linear yield criterion for thin shells. The corners present with this criterion are circumvented by means of a simplification, hence, there is no need for multi-surface stress resultant updates. A backward Euler difference is employed in the stress resultant update, and a consistent tangent is used in the Newton–Raphson iterations on the global equilibrium. Limit points are traversed by means of an orthogonal trajectory method. The response of compression dominated shells with imperfections typically corresponds to limit point behaviour. For stress resultant plasticity, the nonlinear transition from initial yield to full plasticity in shell bending is missed. Hence, the efficiency obtained by eliminating thickness integration is countered by some inaccuracy in the response simulation. This is investigated by means of comparison with finite element simulations employing integration through thickness (with linear or nonlinear hardening). Both steel and aluminium alloys are considered. In collapse response of slender structures, the straining of the material may be moderate, but the motion may be governed by large rigid body translations and rotations. A way of accounting for this by means of the co-rotated approach is presented. Triangular high-performance facet shell elements are employed. By example computations, the importance of nonlinear geometry contributions is illustrated.     

Two-dimensional modelling of laminated piezoelectric composites: analysis and numerical results

We propose a new approach to laminated piezoelectric plates based on a refinement of the electric potential as function of the thickness coordinate of the laminate and accounting for shear effects. Moreover, the variation of the electric potential as function of the thickness coordinate is modelled for each layer of the laminate. The equation for the laminated piezoelectric plate are then obtained by using a variational formulation involving mechanical surface loads or prescribed electric potential on the top and bottom faces of the plate. In addition to the equations for the generalized stress resultants (due to the shear effects), the equation of the electric charge conservation is also deduced for the 2D model.Particular attention is devoted to the single piezoelectric plate and bimorph structure and the through-thickness distribution of the displacements, electric potential as well as stresses are given for different kinds of electromechanical loads. The results thus obtained are compared to those provided by a finite element method performed for the full 3D model. A good agreement is observed for plates made of layers of PZT-4 piezoelectric material. The comparison ascertains the effectiveness of the present 2D approach to piezoelectric laminates.     

Thermal analysis of FRP chimneys using consistent laminated shell element

Due to their high corrosion and chemical resistance, fiber reinforced plastic (FRP) materials are increasingly being used in the construction of industrial chimneys. The design of a chimney is governed by wind loads as well as thermal loads resulting from the differences among the ambient, the operating and the curing temperatures. This study involves an investigation for the thermal stresses induced in angle-ply laminated FRP chimneys, using an in-house developed laminated shell element model. The finite element model is verified by performing thermal analysis of a number of plate and shell problems and comparing the results to those available in the literature. An extensive parametric study is then conducted using the shell element model to identify the parameters which significantly affect thermal stresses induced in FRP chimneys.The study indicates that the thermal stresses are only affected by the inclination of the lamina plies, the percentage of fibers content and the through thickness temperature distribution. Analyses also show that localized cracks in the direction perpendicular to the fibers are expected to occur due to the thermal loads. Finally, thermal stress values that can be used in the design of FRP chimneys, when cracking is considered, are presented as function of the through thickness temperature distributions.     

Beam and shell element model for advanced analysis of steel structural members

Plastic zone method of advanced analysis, which uses shell elements to model the entire structure, is the most accurate method available to predict the ultimate strength and behavior of steel frames. The disadvantage of such full shell plastic zone models is that it is computationally expensive and hence its use is limited to small structures. Beam elements in commercial finite element packages can model residual stress and capture spread of plasticity, but cannot model local buckling of plates that the member is made up of, which leads to unloading and failure in steel frames. A hybrid model using shell elements only in the regions vulnerable to elastic or inelastic local buckling and beam elements in other locations could overcome this limitation of full beam element model. The issues in using this hybrid model are, knowing a priori the location and length of the shell element region and connecting the beam and shell regions without any artificial stress concentrations or incompatible displacements. In this study, in addition to addressing these issues, the hybrid model is systematically evaluated by studying its performance in structural elements. It is seen that the hybrid model strength predictions has an average error of only 0.91% but requires on an average 83% less computational time when compared to the full shell plastic zone models.     

Beam and shell element model for advanced analysis of steel structural members

Plastic zone method of advanced analysis, which uses shell elements to model the entire structure, is the most accurate method available to predict the ultimate strength and behavior of steel frames. The disadvantage of such full shell plastic zone models is that it is computationally expensive and hence its use is limited to small structures. Beam elements in commercial finite element packages can model residual stress and capture spread of plasticity, but cannot model local buckling of plates that the member is made up of, which leads to unloading and failure in steel frames. A hybrid model using shell elements only in the regions vulnerable to elastic or inelastic local buckling and beam elements in other locations could overcome this limitation of full beam element model. The issues in using this hybrid model are, knowing a priori the location and length of the shell element region and connecting the beam and shell regions without any artificial stress concentrations or incompatible displacements. In this study, in addition to addressing these issues, the hybrid model is systematically evaluated by studying its performance in structural elements. It is seen that the hybrid model strength predictions has an average error of only 0.91% but requires on an average 83% less computational time when compared to the full shell plastic zone models.