## Development and Applications of a Flat Triangular Element for Thin Laminated Shells

##### Abstract

Finite element analysis of laminated shells using a three-noded
flat triangular shell element is presented. The flat shell element is
obtained by combining the Discrete Kirchhoff Theory (DKT) plate bending
element and a membrane element similar to the Allman element, but
derived from the Linear Strain Triangular (LST) element.
Though this combination has been employed in the literature for linear
static analysis of laminated plates, the results presented
are not adequate to ascertain that
the element would perform well in the case of static and dynamic
analysis of general shells. The element is first thoroughly tested
for linear static analysis of laminated plates and shells and is
extended for free vibration, thermal, and geometrically
nonlinear analysis.

The major drawback of the DKT plate bending element is that the transverse displacement is not explicitly defined within the interior of the element. Hence obtaining the consistent mass matrix or the derivatives of the transverse displacement that are required for forming the geometric stiffness matrix is not straight forward. This problem is alleviated by borrowing shape functions from other similar elements or using simple displacement fields. In the present research, free vibration analysis is performed both by using a lumped mass matrix and a so called consistent mass matrix, obtained by borrowing shape functions from an existing element, in order to compare the performance of the two methods. The geometrically nonlinear analysis is performed using an updated Lagrangian formulation employing Green strain and Second Piola-Kirchhoff (PK2) stress measures. A linear displacement field is used for the transverse displacement in order to compute the derivatives of the transverse displacement that are required to compute the geometric stiffness or the initial stress matrix.

Several numerical examples are solved to demonstrate the accuracy of the formulation for both small and large rotation analysis of laminated plates and shells. The results are compared with those available in the existing literature and those obtained using the commercial finite element package ABAQUS and are found to be in good agreement. The element is employed for two main applications involving large flexible structures.

The first application is the control of thermal deformations of a spherical mirror segment, which is a segment of a multi-segmented primary mirror used in a space telescope. The feasibility of controlling the surface distortions of the mirror segment due to arbitrary thermal fields, using discrete and distributed actuators, is studied. This kind of study was required for the design of a multi-segmented primary mirror of a next generation space telescope.

The second application is the analysis of an inflatable structure, being considered by the US Army for housing vehicles and personnel. The tent structure is made up of membranes supported by arches stiffened by internal pressure. The updated Lagrangian formulation of the flat shell element has been developed primarily for the nonlinear analysis of the tent structure, since such a structure is expected to undergo large deformations and rotations under the action of environmental loads like the wind and snow loads. The wind load is modeled as a nonuniform pressure load and the snow load as lumped concentrated loads. Since the direction of the pressure load is assumed to be normal to the current configuration of the structure, it changes as the structure undergoes deformation. This is called the follower action. As a result, the pressure load is a function of the displacements and hence contributes to the tangent stiffness matrix in the case of geometrically nonlinear analysis. The thermal load also contributes to the system tangent stiffness matrix. In the case of the thermal load this contribution is similar to the initial stress matrix and hence no additional effort is required to compute this contribution. In the case of the pressure load, this contribution (called the pressure stiffness) is in general unsymmetric but can be systematically derived from the principle of virtual work. The follower effects of the pressure load have been included in the updated Lagrangian formulation of the flat shell element and have been validated using standard examples in the literature involving deformation-dependent pressure loads. The element can be used to obtain the nonlinear response of the tent structure under wind and snow loads.

The major drawback of the DKT plate bending element is that the transverse displacement is not explicitly defined within the interior of the element. Hence obtaining the consistent mass matrix or the derivatives of the transverse displacement that are required for forming the geometric stiffness matrix is not straight forward. This problem is alleviated by borrowing shape functions from other similar elements or using simple displacement fields. In the present research, free vibration analysis is performed both by using a lumped mass matrix and a so called consistent mass matrix, obtained by borrowing shape functions from an existing element, in order to compare the performance of the two methods. The geometrically nonlinear analysis is performed using an updated Lagrangian formulation employing Green strain and Second Piola-Kirchhoff (PK2) stress measures. A linear displacement field is used for the transverse displacement in order to compute the derivatives of the transverse displacement that are required to compute the geometric stiffness or the initial stress matrix.

Several numerical examples are solved to demonstrate the accuracy of the formulation for both small and large rotation analysis of laminated plates and shells. The results are compared with those available in the existing literature and those obtained using the commercial finite element package ABAQUS and are found to be in good agreement. The element is employed for two main applications involving large flexible structures.

The first application is the control of thermal deformations of a spherical mirror segment, which is a segment of a multi-segmented primary mirror used in a space telescope. The feasibility of controlling the surface distortions of the mirror segment due to arbitrary thermal fields, using discrete and distributed actuators, is studied. This kind of study was required for the design of a multi-segmented primary mirror of a next generation space telescope.

The second application is the analysis of an inflatable structure, being considered by the US Army for housing vehicles and personnel. The tent structure is made up of membranes supported by arches stiffened by internal pressure. The updated Lagrangian formulation of the flat shell element has been developed primarily for the nonlinear analysis of the tent structure, since such a structure is expected to undergo large deformations and rotations under the action of environmental loads like the wind and snow loads. The wind load is modeled as a nonuniform pressure load and the snow load as lumped concentrated loads. Since the direction of the pressure load is assumed to be normal to the current configuration of the structure, it changes as the structure undergoes deformation. This is called the follower action. As a result, the pressure load is a function of the displacements and hence contributes to the tangent stiffness matrix in the case of geometrically nonlinear analysis. The thermal load also contributes to the system tangent stiffness matrix. In the case of the thermal load this contribution is similar to the initial stress matrix and hence no additional effort is required to compute this contribution. In the case of the pressure load, this contribution (called the pressure stiffness) is in general unsymmetric but can be systematically derived from the principle of virtual work. The follower effects of the pressure load have been included in the updated Lagrangian formulation of the flat shell element and have been validated using standard examples in the literature involving deformation-dependent pressure loads. The element can be used to obtain the nonlinear response of the tent structure under wind and snow loads.

##### Collections

- Doctoral Dissertations [11291]