Beam Structures

Classical and Advanced Theories

Erasmo Carrera, Gaetano Giunta, Marco Petrolo

• Publisher: Wiley
• Year: 2011
• Print ISBN: 9780470972007
• eBook ISBN: 9781119951049
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Mechanics

Beam theories are exploited worldwide to analyze civil, mechanical, automotive, and aerospace structures. Many beam approaches have been proposed during the last centuries by eminent scientists such as Euler, Bernoulli, Navier, Timoshenko, Vlasov, etc.  Most of these models are problem dependent: they provide reliable results for a given problem, for instance a given section and cannot be applied to a different one.

Beam Structures: Classical and Advanced Theories proposes a new original unified approach to beam theory that includes practically all classical and advanced models for beams and which has become established and recognised globally as the most important contribution to the field in the last quarter of a century.

The Carrera Unified Formulation (CUF) has hierarchical properties, that is, the error can be reduced by increasing the number of the unknown variables. This formulation is extremely suitable for computer implementations and can deal with most typical engineering challenges. It overcomes the problem of classical formulae that require different formulas for tension, bending, shear and torsion; it can be applied to any beam geometries and loading conditions, reaching a high level of accuracy with low computational cost, and can tackle problems that in most cases are solved by employing plate/shell and 3D formulations.

Key features:

• compares classical and modern approaches to beam theory, including classical well-known results related to Euler-Bernoulli and Timoshenko beam theories
• pays particular attention to typical applications related to bridge structures, aircraft wings, helicopters and propeller blades
• provides a number of numerical examples including typical Aerospace and Civil Engineering problems
• proposes many benchmark assessments to help the reader implement the CUF if they wish to do so
• accompanied by a companion website hosting dedicated software MUL2 that is used to obtain the numerical solutions in the book, allowing the reader to reproduce the examples given in the book as well as to solve other problems of their own www.mul2.com

Researchers of continuum mechanics of solids and structures and structural analysts in industry will find this book extremely insightful. It will also be of great interest to graduate and postgraduate students of mechanical, civil and aerospace engineering.

1

Fundamental equations of continuous deformable bodies

This chapter recalls the geometrical and constitutive equations of continuum structural mechanics in the linear case. Symbols and reference systems that will be used throughout the book are also introduced.

1.1 Displacement, strain, and stresses

Let us consider a continuous deformable body C with volume V in a three-orthogonal Cartesian system x, y, z as shown in Figure 1.1. The following relevant variables are introduced:

Figure 1.1 Geometry and notations for a 3D deformable body.

• Displacements,
  
  , of a point P:

When the body C is subjected to mechanical and natural boundary conditions, the three continuous functions ux(x, y, z), uy(x, y, z), uz(x, y, z) give the deformed state of C. The calculation of such a deformed state remains the fundamental problem of 3D elasticity.

• Strains,
  
  
  
  
  
  These nine terms describe the deformed configuration in a three-orthogonal term of unit vectors (
  
  ) in P. The axial deformations are
  
  
  
  Shear strains, which are symmetric, are
  
  
• Stresses, σ, at a point P:
  
  
  
  Axial stresses are
  
  
  
  Shear stresses, which are symmetric, are
  
  

1.2 Equilibrium equations in terms of stress components and boundary conditions

At a generic point of the body C the following fundamental equilibrium equations hold:

(1.1)

where

,

, and

are the body forces per unit volume. On the boundary of the body C, the stress vector components at a generic point must fulfill the following conditions:

(1.2)

where nx, ny, and nz are the direction cosines, and

is the resultant of the external forces per unit area acting on the boundary of C. These are known as mechanical boundary conditions. Geometrical conditions can also be imposed stating that

(1.3)

where

,

, and

are imposed displacements on the outer surface of C.

1.3 Strain displacement relations

Strains are related to displacements by the following geometrical relations that are valid under the assumption of linearity:

(1.4)

and

are the engineering strain components which are usually denoted as

and

. In this book both symbols

and γ are used to indicate the engineering strain components.

1.4 Constitutive relations: Hooke’s law

Stress and strain comp...

Table of contents

• Cover
• Title Page
• Copyright
• About the Authors
• Preface
• Introduction
• Chapter 1: Fundamental equations of continuous deformable bodies
• Chapter 2: The Euler–Bernoulli and Timoshenko theories
• Chapter 3: A refined beam theory with in-plane stretching: the complete linear expansion case
• Chapter 4: EBBT, TBT, and CLEC in unified form
• Chapter 5: Carrera Unified Formulation and refined beam theories
• Chapter 6: The parabolic, cubic, quartic, and N-order beam theories
• Chapter 7: CUF beam FE models: programming and implementation issue guidelines
• Chapter 8: Shell capabilities of refined beam theories
• Chapter 9: Linearized elastic stability
• Chapter 10: Beams made of functionally graded materials
• Chapter 11: Multi-model beam theories via the Arlequin method
• Chapter 12: Guidelines and recommendations
• Index