Farooque A. Mirza

CONSISTENT THICK SHELL ELEMENT

Abstract Thick shell finite element with transverse shear deformation have required the use of reduced integration to provide improved results for thin plates and shells, due to the presence of spurious transverse shear strain modes. It has been found that the spurious transverse shear strain modes result from inconsistencies in the displacement fields used in the formulation of these elements. A new thick shell element has been formulated. By providing cubic polynomials for approximation of displacements, and quadratic polynomials for approximation of rotations a consistent formulation is ensured thereby eliminating the spurious modes. Rotational degrees of freedom which vary quadratically through the thickness of the element are included. This allows for a parabolic variation of the transverse shear strains and hence eliminates the need for use of the shear correction factor k as required by the Mindlin plate theory. These rotational degrees of freedom also provide cubic variations of the displacements through the thickness of the element. Thus, the normal to the middle surface is neither straight nor normal after shearing and bending, allowing for warping of the cross-section. Material non-linearities are also incorporated, along with the modified Newton-Raphson method for nonlinear analysis. Comparisons are made with the available elasticity solutions and those predicted by the eight and nine-node isoparametric shell elements. The results indicate that the consistent thick shell element provides excellent predictions of the displacements, stresses and plastic zones for both thin and thick plates and shells.

Large displacement extension of consistent shell element for static and dynamic analysis

Abstract The superior performance of the consistent shell element in the small deflection range has encouraged the authors to extend the formulation to large displacement static and dynamic analyses. The nonlinear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear extension is based on a total Lagrangian approach. A detailed derivation of the non-linear stiffness matrix and the unbalanced load vector for the consistent shell element is presented in this study. Meanwhile, a simplified method for coding the nonlinear formulation is provided by relating the components for the nonlinear B-matrices to those of the linear B-matrix. The consistent mass matrix for the shell element is also derived and then incorporated with the stiffness matrix to perform large displacement dynamic and free vibration analyses of shell structures. Newmarks method is used for time integration and the Newton-Raphson method is employed for iterating within each increment until equilibrium is achieved. Numerical testing of the nonlinear model through static and dynamic analyses of different plate and shell problems indicates excellent performance of the consistent shell element in the nonlinear range.

Consistent curved beam element

Curved beam finite elements with shear deformation have required the use of reduced integration to provide improved results for thin beams and arches due to the presence of a spurious shear strain mode. It has been found that the spurious shear strain mode results from an inconsistency in the displacement fields used in the formulation of these elements. A new curved beam element has been formulated. By providing a cubic polynomial for approximation of displacements, and a quadratic polynomial for approximation of rotations a consistent formulation is ensured thereby eliminating the spurious mode. A rotational degree of freedom which varies quadratically through the thickness of the element is included. This allows for a parabolic variation of the shear strain and hence eliminates the need for use of the shear correction factor k as required by the Timoshenko beam theory. This rotational degree of freedom also provides a cubic variation of displacements through the depth of the element. Thus, the normal to the centroidal axis is neither straight nor normal after shearing and bending allowing for warping of the cross section. Material nonlinearities are also incorporated, along with the modified Newton-Raphson method for nonlinear analysis. Comparisons are made with the available elasticity solutions and those predicted by the quadratic isoparametric beam element. The results indicate that the consistent beam element provides excellent predictions of the displacements, stresses and plastic zones for both thin and thick beams and arches.

Stability of elevated liquid-filled conical tanks under seismic loading, Part I—theory

Conical steel shells are widely used as water containments for elevated tanks. However, the current codes for design of water structures do not specify any procedure for handling the seismic design of such structures. In this paper, a numerical model is developed for studying the stability of liquid-filled conical tanks subjected to seismic loading. The model involves a previously formulated consistent shell element with geometric and material non-linearities included. A boundary element formulation is derived to obtain the hydrodynamic pressure resulting from both the horizontal and the vertical components of seismic motion acting on a conical tank which is prevented from rocking. The boundary element formulation leads to a fluid added-mass matrix which is incorporated with the shell element formulation to perform non-linear dynamic stability analysis of such tanks subjected to both horizontal and vertical components of ground motion. Although, the formulation was developed for conical vessels, it is general and can be easily modified to study the stability of any liquid-filled shell of revolution subjected to seismic loading. The accuracy of fluid added-mass formulation was verified by performing the free vibration analysis of liquid-filled cylindrical tanks and comparing the results to those available in the literature.

Stability of elevated liquid-filled conical tanks under seismic loading, Part II-Applications

In this paper, the numerical model developed in the previous paper is used to study the seismic performance of elevated liquid-filled steel conical tanks. A number of conical tanks which are classified as tall or broad tanks according to the ratio of the tank radius to its height are considered. The consistent shell element is used to model the tank surfaces, while the coupled boundary-shell element formulation is employed to obtain the fluid added-mass which simulates the dynamic pressure resulting from a seismic motion. Linear springs are used to model the supporting towers. The natural frequencies of the liquid-filled tanks due to both horizontal and vertical excitations are evaluated. This is followed by a non-linear dynamic analysis, using an appropriately scaled real input ground motion, and which includes the effect of both geometric and material non-linearities. Thin-walled structures of this kind may exhibit inelastic behaviour and a tendency to develop localized buckles, thus diminishing stiffness. The consequence could lead to overall instability of the structure. In general, time-history analyses indicate that liquid-filled conical tanks, often possessing apparently adequate safety factors under hydrostatic loading, are shown to be very sensitive to seismic loading when ground motion frequencies contain those of the fundamental frequencies of the vessels themselves.

Nonlinear analysis of reinforced concrete slabs by a discrete finite element approach

Abstract A rational numerical model is described for the nonlinear analysis of reinforced concrete slabs subjected to monotonic loading. The model is based on a discrete finite element representation of a concrete slab and its embedded reinforcing steel bars, which permits the interface behaviour such as bond-slip and dowel-action to be explicitly simulated as a local phenomenon. This material nonlinearities for both concrete and steel are considered and the smeared cracking approach is used. The nonlinear finite element equations are solved by an incremental-iterative procedure together with a constant stiffness matrix approach. The validity of the proposed model is demonstrated through a numerical example for which the solutions obtained are compared with the available experimental results.

Thermal stress analysis due to welding processes by the finite element method

Abstract A finite element model has been developed to predict the residual stresses generated in weldments during fabrication. The thermal history of the weld piece is computed using a three-dimensional heat flow model which serves as input for computation of stresses. A micro-structural model, based on the Avrami equation, and the grain growth law have also been employed to predict the grain growth due to welding. A coupled thermal elastic visco-plastic formulation including the micro-structural changes has been developed to predict the overall deformations and residual stresses caused by a welding thermal cycle. The model is applied to an austenitic type of steel, namely AISI 308, and the predictions are in good agreement with experimental results reported in the literature. The longitudinal stresses are found to be as high as the yield stress, and the transverse stresses are found to be almost half of the longitudinal stresses.

Modelling of double chord rectangular hollow section T-joints by finite element method

Abstract A linear elastic T-joint comprised of double chord RHS has been modelled by treating the mated flanges as thin plates supported by coupled linear springs thus simulating the action of the side walls and connecting bottom flanges. A rigid rectangular inclusion is presumed for the branch member. Two loading conditions are analyzed—branch member axial force and branch member bending moment. The finite element formulation that is proposed incorporates rectangular plate and edge boundary spring elements. The model is then used to determine the punching shear and rotational stiffnesses of both double chord T-joints and single chord T-joints, thus demonstrating its versatility. The numerical values obtained are in good agreement with the experimental results available in the literature.

A simplified welding arc model by the finite element method

Abstract A welding arc model is proposed to determine the thermal cycle for various materials and for different weld types. In this paper we discuss the iterative procedure employed for non-linear heat transfer analysis using the finite element method, and in particular the boundary conditions employed to determine the heat generated in the HAZ and its dissipation during welding. The model only focuses on the grain growth zone with the objective to compute the thermal regime that controls the microstructural changes that occur during welding. Both solidified weld and liquid-solid transition zones are excluded from the model. The heat generated by the arc is applied to the vertical face of the welded edge and is indifferent to the size of the HAZ. The analytical results obtained for low carbon steel and for austenitic stainless steel, and for GMA and GTA weld types, correlate well with the observed experimental data reported in the literature.

Finite element modeling of transient heat transfer and microstructural evolution in welds: Part II. Modeling of grain growth in austenitic stainless steels

During welding, structures are subjected to localized heating and cooling cycles, as described in Part I.[1] A mathematical model is proposed to determine the metallurgical changes that occur in austenitic stainless steel due to the welding thermal cycle. The proposed kinetic model computes the austenite grain growth as a function of time and temperature. It is based on a Zener pinning grain growth model. The results obtained indicate that the model is in good agreement with the experimental data reported in the literature. Furthermore, it was observed that rewriting the kinetic constant in the grain growth equation as a function of the peak temperature led to improved results for the majority of the cases examined.