Type of Document Dissertation Author Mohan, P. Author's Email Address mohan@apollo.aoe.vt.edu URN etd-11497-15054 Title Development and Applications of a Flat Triangular Element for Thin Laminated Shells Degree PhD Department Aerospace and Ocean Engineering Advisory Committee

Advisor Name Title Kapania, Rakesh K. Committee Chair Hughes, Owen F. Committee Member Johnson, Eric R. Committee Member Nikolaidis, Efstratios Committee Member Plaut, Raymond H. Committee Member Keywords

- Finite Element Method
- Flat Shell Element
- Updated Lagrangian
- Smart Structures
- Inflatable Structures
Date of Defense 1997-11-18 Availability unrestricted AbstractFinite 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.

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