# Tension Structures

## The deformation gradient and the right Cauchy-Green deformation tensor for triangle element

This text presents an analytical definition of the deformation gradient and the right Cauchy-Green deformation tensor for the triangle element. The tensors and its invariants are written in terms of the nodal displacements. Download the complete text as MS-Word and PDF documents.

## Triangle element for plane stress hyperelastic FEA

This text presents a triangle element for plane stress hyperelastic Finite Element Analysis and associated computer codes for compressible or incompressible materials. The strain energy functions for Neo-Hookean and Mooney-Rivlin materials are available. Any strain energy function can be easily incorporated. The computer code generates a script file for AutoCAD and an input file for ANSYS. Download the complete text as MS-Word and PDF documents, computer source code written in Ada95 and executable code for Windows.

## Triangle element for plane strain hyperelastic FEA

This text presents a triangle element for plane strain hyperelastic Finite Element Analysis and associated computer code for compressible materials. The strain energy functions for Neo-Hookean and Mooney-Rivlin materials are available. Any strain energy function can be easily incorporated. The computer code generates a script file for AutoCAD and an input file for ANSYS. Download the complete text as MS-Word and PDF documents, computer source code written in Ada95 and executable code for Windows.

## Minimizing total potential energy to find equilibrium

The stable equilibrium configurations of a structure are associated with the local minimum points of its total potential energy. There is a fundamental difference between solving the equilibrium equation and minimizing the total potential energy. The advantages of the energy approach are: It is not necessary to derive the stiffness matrix expressions. It is not necessary to solve any system of equations. It does not matter if the structure is a mechanism, which is frequently the case of tension structures. Download the complete text as MS-Word and PDF documents.

## A simple procedure for analysis of cable network structures

This text presents a mathematical modeling of a cable finite element. It includes a total Lagrangian description using the Engineering strain definition and assumes an elastic material (linear or nonlinear). A procedure to analyze a cable network in the presence of conservative forces and small deformations is summarized. Mathematical programming techniques make the use of stiffness matrix pointless. Download the complete text as MS-Word and PDF documents, computer source code written in Ada95 and executable code for Windows. The computer code generates a script file for AutoCAD.

## A simple procedure for shape finding and analysis of fabric structures

This text presents a procedure to define the shape of a fabric structure and to analyze it in the presence of conservative forces and small deformations is summarized. The shapes are generated by loading a membrane with concentrated forces, distributed force and also by prescribing displacements. Download the complete text as MS-Word and PDF documents, computer source code written in Ada95 and executable code for Windows. The computer code generates a script file for AutoCAD.

## LIGHTS computer code

LIGHTS is a public domain 3D finite element program for the design and analysis of light structures. The program element library includes cable elements, membrane element, frame element and spring element. The computer code generates a script file for AutoCAD. A simple user's manual has been included in the documentation. Download the source code written in Ada95, examples and executable code for Windows.

## Analysis of 3D isotropic membrane structures

This text presents a mathematical modeling of an isotropic membrane finite element. It includes a total Lagrangian description with a linear elastic material, using either the Green or Engineering strain definition. Mathematical programming techniques make the use of stiffness matrix pointless. Download the complete text as MS-Word and PDF documents, computer source codes written in Ada95 and executable codes for Windows. The computer codes generate a script file for AutoCAD.

## Analysis of 3D orthotropic membrane structures

This text presents a mathematical modeling of an orthotropic membrane finite element. It includes a total Lagrangian description with a linear elastic material, using either the Green or Engineering strain definition. Mathematical programming techniques make the use of stiffness matrix pointless. Download the complete text as MS-Word and PDF documents, computer source codes written in Ada95 and executable codes for Windows. The computer codes generate a script file for AutoCAD. This is a work in progress.

## Finite element analysis for minimal shape

This text describes a mathematical model that unifies all geometrical minimal shape problems by defining geometrical finite elements. Three types of elements are defined: line, triangle and tetrahedron. The elements can be used together through the unifying concept of volume. For each element type, its corresponding isovolumetric element is also defined. The geometrical minimal shape problem is formulated as an equality constrained minimization problem. The augmented Lagrangian method is used to solve the associated unconstrained minimization problem. A quasi-Newton method is used, which avoids the evaluation of the Hessian matrix. Download the complete text as MS-Word and PDF documents, computer source codes written in Ada95 and executable codes for Windows. The computer codes generate a script file for AutoCAD.

## Form finding of tensegrity structures using finite element and mathematical programming

The major point of this text is to show that minimization of total potential energy is general rule behind the well known rule of minimizing the sum of some lengths of a truss mechanism to define a tensegrity. Moreover, the well known rule is a special case due to the usual high values of the modulus of elasticity. An innovative mathematical model is presented for form finding of tensegrity structures that is based on the finite element method and on mathematical programming. A special line element that shows constant stress for any displacement of its nodes is used to define a prestressed equilibrium configuration. The form finding is formulated as an unconstrained nonlinear programming problem, where the objective function is the total potential energy and the displacements of the nodal points are the unknowns. A connection is made with the geometrical shape minimization problem, which is defined by a constrained nonlinear programming problem. A quasi-Newton method is used, which avoids the evaluation of the tangent stiffness matrix. Download the complete text as MS-Word and PDF documents, computer source codes written in Ada95 and executable codes for Windows. The computer codes generate a script file for AutoCAD.

## A finite element for form-finding and static analysis of tensegrity structures

This text describes a mathematical model for both form finding and static analysis of tensegrity structures. A special line element that shows constant stress for any displacement of its nodes is used to define a prestressed equilibrium configuration. The form finding and static analysis are formulated as an unconstrained nonlinear programming problem, where the objective function is the total potential energy and the displacements of the nodal points are the unknowns. A quasi-Newton method is used, which avoids the evaluation of the Hessian matrix. Download the complete text as MS-Word and PDF documents, computer source codes written in Ada95 and executable codes for Windows. The computer codes generate a script file for AutoCAD.

## A strut finite element for exact isotropic hyperelastic analysis

This text describes a mathematical model of a strut finite element for isotropic hyperelastic materials. The invariants of the Right Cauchy-Green deformation tensor are written in terms of nodal displacements. The equilibrium problem is formulated as an unconstrained nonlinear programming problem, where the objective function is the total potential energy of the structure and the nodal displacements are the unknowns. The constraint for incompressibility is satisfied exactly, thereby eliminating the need for a penalty function. The results of the examples calculated by the proposed mathematical model show five significant digits in agreement when compared with commercial finite element analysis software. Download the complete text as MS-Word and PDF documents, computer source codes written in Ada95 and executable codes for Windows. The computer codes generate a script file for AutoCAD.