Abstract

This work presents a procedure to couple shell and Generalized Beam Theory, GBT, elements. The main focus of this procedure is the possibility to model mixed beam frame structures, which the traditional shell elements are applied at the joints and GBT elements are used to model the beams/columns. Such modeling technique can use the benefits of both elements. At the joints, shell elements can easily simulate different types of geometry conditions and details, such as stiffeners and holes; meanwhile, for the beams and columns, GBT can provide high performance, accuracy and an easy modeling approach with clear results.

Highlights

  • 6.1 Type of connections to clarify the kinematic behavior in high modes . . 119 6.2 Elementary node to describe the kinematic behavior of the connection, 6.3 Kinematic mechanism of box connection; a) the stress compatibility; xi xii

  • Taking into account that the functional eq 2.62 is null for any arbitrary functions of longitudinal amplification, δ iV (x), the parentheses terms of the integral must be zero, which leads to the nonhomogeneous differential equation of equilibrium in Generalized Beam Theory (GBT): EiCM + KiCP iV,xxxx (x) − G iDMc +iDP −2μKiDμ iV,xx (x) + KiBiV (x) = iqv (s) fv (x) +iqw (s) fw (x) −iqx (s) Fx (x)

  • This study presents the variational formulation and the analysis of displacement and stress fields in GBT

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Summary

A Appendix A

Numerical example - GBT’s analysis of a generic segmented thinwalled cross-section. 2.2 Types of Distortion: a) transverse bending distortion due to the variation of transverse bending moments; b) transverse bending distortion due to the variation of transverse forces; c) transverse elongation distortion. . . 9 ix. Numerical example - GBT’s analysis of a generic segmented thinwalled cross-section. 2.2 Types of Distortion: a) transverse bending distortion due to the variation of transverse bending moments; b) transverse bending distortion due to the variation of transverse forces; c) transverse elongation distortion. 6.1 Type of connections to clarify the kinematic behavior in high modes . . 119 6.2 Elementary node to describe the kinematic behavior of the connection, 6.3 Kinematic mechanism of box connection; a) the stress compatibility; xi xii. A.5 transverse modal displacements of plate anti-symmetric modes: 5 to 9 . A.5 transverse modal displacements of plate anti-symmetric modes: 5 to 9 . 240 xiii xiv

Differences among the eigenvectors from the proposed approach and [136]235 xvii xviii
E Young’s modulus
Motivation
Literature Review and historical development of GBT
Objectives of the dissertation
Dissertation organization
Hypotheses of Generalized Beam Theory
Kinematic assumptions
Separation of variables to describe the displacement field
Variational Formulation in Generalized Beam Theory
Internal strain energy according to GBT
External load potential energy according to GBT
Equilibrium by Hamilton’s principle
Stress field in GBT: generalized internal forces
GBT’s analysis of thin-walled hollow circular cross-section
GBT’s displacement field for thin-walled hollow circular crosssection
External loads in thin-walled hollow circular cross-section according to GBT
Stress field and internal forces for thin-walled hollow circular cross-section according to GBT
Cross-section semi-discretization
Transverse Stiffness matrices
Identification and elimination of singular modes
Membrane, Plate and Membrane-Plate modes
Membrane mode and constraints
Plate modes and constraints
Mode 2, 3 and 4 - Pure longitudinal bending and Torsion
Symmetric and anti-symmetric modes
Linearization and quadratic pencil
Recover of the eigenvectors in the original coordinate system
Review of the applied solution methods in GBT’s ordinary differential equations
Shape function assortment and transverse deformation mode classification
Classification of GBT’s deformation modes concerning the ordinary differential equation
Classification of GBT’s deformation modes concerning the solution of ordinary differential equations
Shape functions based on homogeneous solutions in GBT analysis
Shape functions based on non-homogeneous solutions in GBT analysis
Non-Homogeneous solution for distortion with warping - case A with clamped-clamped boundary conditions
Simplification of the high-order derivatives and stiffness matrices for coupled modes
Non-Homogeneous solution for distortion with warping - case A with hinged-hinged boundary conditions
Non-Homogeneous solution for distortion with warping - case A with clamped-hinged boundary conditions
Numerical Examples
Cross-section analysis and load’s mode participation
Finite element solution in the longitudinal direction
Analysis of the displacement field
Analysis of the stress field
Pure GBT beam frame analysis
Internal support mechanism
Coupling of different high modes due to connections
Clamp mechanism
Connections obtained from coupling between GBT and shell elements
Definition of master and slaver degrees of freedom
Master-slave relationships based on GBT’s modes
Coupled stiffness matrix and external load vector
Numerical example of coupling GBT and shell elements
Setup of finite element and coupling matrices
Finite element solution
Analysis of displacement field
Analysis of stress field
Introduction - Brief historical development of non-linear GBT
Geometric stiffness in GBT
Principle of Virtual Works applied in non-linear GBT
Variation of longitudinal internal strain energy
Variation of transversal internal strain energy
Linear and Quadratic Initial Stresses Stiffness - Longitudinal Direction
Linear and Quadratic Initial Stresses Stiffness - Transversal Direction
Linear and Quadratic Initial Displacements Stiffness - Longitudinal Direction
Linear and Quadratic Initial Displacements Stiffness - transversal Direction
Internal forces evaluation in non-linear GBT
Properties of GBT’s coupling tensors
Development of geometric stiffness matrices in GBT
Simplification of the multiplication among several shape functions
Setup of GBT’s non-linear stiffness matrices
Numerical example of non-linear analysis in coupling GBT and shell elements
Setup of the mixed GBT-shell finite element model
Conclusion
Future research
Numerical example
Step 1 - Setup of Cross-section stiffness matrices
Step 2 - transverse elongation constrains
Step 3 - Pure elongation membrane mode
Step 4 - Membrane shear strain constraint
Step 6 - Membrane-plate modes
Step 7 - Pure longitudinal bending modes and Vlasov mode
Step 8 - Symmetric and anti-symmetric modes
Step 9 - Quadratic eigenvalue problem
A.1.10. Step 10 - eigenvectors in original coordinate system
Findings
A.1.11. Step 11 - Analysis of pure plate modes
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