Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Finite Element Basics with the Bar Element: Uniaxial Deformations—Interpolants, Stiffness Matrices and Nodal Loads Read chapter Read first chapter

2018 | OriginalPaper | Chapter

2. Truss Analysis: A Structural Mechanics Tour of the FEM—Nodal Equilibrium and Compatibility

Author : Gautam Dasgupta

Published in: Finite Element Concepts

Publisher: Springer New York

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Linear elastic behavior and small displacements and rotations (Spencer, Continuum mechanics. Longman, London, 1980) are assumed throughout. This model of linear systems allows us to obtain the nodal forces in terms of the bar stiffness matrices and nodal displacements that are the primary variables. The equilibrium equations can then be constructed in terms of displacements at nodes. The given forces and displacements are assumed to be prescribed as nodal quantities. The resulting system equation, which is in the matrix form, can be obtained in terms of nodal displacements and solved by employing the Mathematica function Solve.
A very important area of numerical analysis, i.e., solving positive definite simultaneous equation systems, will not be addressed in this textbook.
This chapter introduces Mathematica codes. To get started, a summary introduction is provided in Appendix A that should be studied before reading the current chapter.
After the first reading, Appendix B should be reviewed for additional examples and theoretical analysis.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Finite Element Basics with the Bar Element: Uniaxial Deformations—Interpolants, Stiffness Matrices and Nodal Loads
next chapter Courant’s Triangular Elements with Linear Interpolants
Footnotes
1
Details are in Sect. B.4, vide Fig. B.10.
 
2
The fonts in figures and text do not match. This issue is described in Sect. B.4.1.
 
3
‘Equating to zero the sum in the vector sense.’ Components of all forces, which are meeting at a point, in any two orthogonal directions should add up to zero.
 
4
The same principle applies to all complicated structures. Once the element stiffness matrices \(\left [k\right ]^{(i)}\) are obtained then at each node compatibility and equilibrium conditions lead to the system equations, via the system stiffness matrix \(\left [k\right ]^{(S)}\), relating all unknowns.
 
5
Roller and hinge boundary conditions are not considered in the system stiffness matrix \(\left [k\right ]^{(S)}.\) There are three zero eigenvalues, for two rigid body translations in (x, y), and one rigid body rotation on the truss plane.
 
6
Vide \(\mathfrak{n}_{i}(x,y)\) of Sect. 3.​1 for an illustration with plane problems.
 
7
There is a slight anomaly, depending on whether the displacements at ith and jth degree-of-freedom be zero or unity.
 
8
Here, a square matrix with non-zero determinant.
 
9
From Williot Mohr’s diagrams we can determine the nodal displacements from bar (positive or negative) extensions.
 
10
In Fig. 2.5, the MaTeX fonts, vide Sect. B.4.1, match the symbols in the text and figures.
 
11
It is very different from the global stiffness matrix \(\left [k\right ]^{(S)}\) of Eq. (2.1b).
 
12
In Mathematica, \(\left \{kS\right \}\) is kS; \(\left [k\right ]^{(i)}\) can be extracted as: ki = Part[kS,i] or ki = kS[[i]].
 
13
For a given distributed skin force f(x), vide Sect. 1.​5, we still approximate the displacement field to be a linear function u(x). By considering a constant bar stress, we do not satisfy equilibrium at individual sections.
 
14
Also for members, vide Eq. (2.16).
 
15
Fonts of graphics from Listing 2.1 do not match those in the text, e.g. Eq. (2.16).
 
16
Except for the incompressible case where the set of degrees-of-freedom contains isochoric shape functions and a constant pressure for each finite element.
 
17
For example, temperature distributions and the torsion in non-circular prismatic shafts.
 
18
For triangles and tetrahedra the interpolants are identical for scalar and vector problems.
 
Literature
1.
2.
Fritzson P (2004) Principles of object-oriented modeling and simulation with Modelica 2.1. Wiley, London
3.
Lubliner J, Papadopoulos P (2016) Introduction to solid mechanics, 2nd edn. Springer, New YorkMATH
4.
Ritz W (1908) Über eine neue methode zur lösung gewisser variationalprobleme der mathematischen physik. J Reine Angew Math 135:1–61MATH
5.
Spencer AJM (1980) Continuum mechanics. Longman, London (also 1990 Dover, New York)
6.
Timoshenko SP, Young DH (1965) Theory of structures, 2nd edn. McGraw-Hill, New York
Metadata
Title
Truss Analysis: A Structural Mechanics Tour of the FEM—Nodal Equilibrium and Compatibility
Author
Gautam Dasgupta
Copyright Year
2018
Publisher
Springer New York
Book
Finite Element Concepts

Print ISBN: 978-1-4939-7421-4

Electronic ISBN: 978-1-4939-7423-8

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-1-4939-7423-8_2

Premium Partners

    Image Credits