π
<-
Chat plein-écran
[^]

imagen 23


Hierarchy of files

 Downloads
 Files created online(23949)
 HP-Prime(3397)

 mViewer GX Creator Prog(735)

DownloadTélécharger


LicenceLicense : Non spécifiée / IncluseUnspecified / Included

 TéléchargerDownload

Actions



Vote :

ScreenshotAperçu


Informations

Catégorie :Category: mViewer GX Creator Prog HP-Prime
Auteur Author: luiggi
Type : Basic
Page(s) : 3
Taille Size: 299.36 Ko KB
Mis en ligne Uploaded: 16/05/2019
Uploadeur Uploader: luiggi (Profil)
Téléchargements Downloads: 1
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a2099095

Description 

Chapter 4
Analysis of 2D trusses




4.1 Introduction

This chapter deals with the static analysis of two dimensional trusses, which are
basically bars oriented in two dimensional cartesian systems. A transformation of
coordinate basis is necessary to translate the local element matrices (stiffness ma-
trix, force vector) into the structural (global) coordinate system. Trusses support
compressive and tensile forces only, as in bars. All forces are applied at the nodes.
After the presentation of the element formulation, some examples are solved by
MATLAB codes.



4.2 2D trusses

In figure 4.1 we consider a typical 2D truss in global x − y plane. The local system
of coordinates x − y  defines the local displacements u1 , u2 . The element possesses
two degrees of freedom in the local setting,

u = [u1 u2 ]
T
(4.1)

while in the global coordinate system, the element is defined by four degrees of
freedom
uT = [u1 u2 u3 u4 ] (4.2)
The relation between both local and global displacements is given by

u1 = u1 cos(θ) + u2 sin(θ) (4.3)

u2 = u3 cos(θ) + u4 sin(θ) (4.4)




A.J.M. Ferreira, MATLAB Codes for Finite Element Analysis: 51
Solids and Structures, Solid Mechanics and Its Applications 157,

c Springer Science+Business Media B.V. 2009
52 4 Analysis of 2D trusses

u4

x u2
u3


u2

θ
u1
u1

y


x


Fig. 4.1 2D truss element: local and global degrees of freedom



where θ is the angle between local axis x and global axis x, or in matrix form as

u = Lu (4.5)

being matrix L defined as  
l m0 0
L= (4.6)
0 0 l m
The l, m elements of matrix L can be defined by the nodal coordinates as
x2 − x1 y2 − y1
l= ; m= (4.7)
Le Le
being Le the length of the element,

Le = (x2 − x1 )2 + (y2 − y1 )2 (4.8)



4.3 Stiffness matrix

In the local coordinate system, the stiffness matrix of the 2D truss element is given
by the bar stiffness, as before:
 
 EA 1 −1
K = (4.9)
Le −1 1
4.5 First 2D truss problem 53

In the local coordinate system, the strain energy of this element is given by
1 T  
Ue = u Ku (4.10)
2
Replacing u = Lu in (4.10) we obtain
1 T T 
Ue = u [L K L]u (4.11)
2
It is now possible to express the global stiffness matrix as

K = LT K L (4.12)

or ⎡ ⎤
l2 lm −l2 −lm

EA ⎢ lm m2 −lm −m2 ⎥
K= ⎥ (4.13)
Le ⎣ −l2 −lm l2 lm ⎦
−lm −m2 lm m2



4.4 Stresses at the element

In the local coordinate system, the stresses are defined as σ = E. Taking into
account the definition of strain in the bar, we obtain
 
u − u1 E u1 E
σ=E 2 = [−1 1] = [−1 1]u (4.14)
Le Le u2 Le

By transformation of local to global coordinates, we obtain stresses as function of
the displacements as
E E
σ= [−1 1]Lu = [−l −m l m]u (4.15)
Le Le




4.5 First 2D truss problem

In a first 2D truss problem, illustrated in figure 4.2, we consider a downward point
force (10,000) applied at node 1. The modulus of elasticity is E = 30e6 and all
elements are supposed to have constant cross-section area A = 2. The supports
are located in nodes 2 and 4. The numbering of degrees of freedom is illustrated
in figure 4.3.

Archive contentsContenu de l'archive

Actions(s) SizeTaille FileFichier
1.90 Ko KB readme.txt
704.80 Ko KB imagen_231.hpprgm
400.66 Ko KB imagen_232.hpprgm
474.24 Ko KB imagen_233.hpprgm

Pub / Ads

Campagne de dons
Pour nous aider à financer nos déplacements sur les salons/congrès qui vous donnent du contenu exclusif

Vous aurez droit aux avantages VIP et des goodies !
45%
-
Search
-
Featured topics
Avantages VIP et goodies pour les donateurs !
Offre TI-Planet/Jarrety pour avoir la TI-83 Premium CE avec son chargeur pour 79,79€ port inclus !
Offre TI-Planet/Jarrety pour avoir la TI-Nspire CX CAS à seulement 130€ TTC port inclus!
Jailbreake ta TI-Nspire avec Ndless et profite des meilleurs jeux et applications !
1234
-
Donations / Premium
For more contests, prizes, reviews, helping us pay the server and domains...

Discover the the advantages of a donor account !
JoinRejoignez the donors and/or premium!les donateurs et/ou premium !


Partner and ad
Notre partenaire Jarrety 
-
Stats.
364 utilisateurs:
>321 invités
>38 membres
>5 robots
Record simultané (sur 6 mois):
6892 utilisateurs (le 07/06/2017)
-
Other interesting websites
Texas Instruments Education
Global | France
 (English / Français)
Banque de programmes TI
ticalc.org
 (English)
La communauté TI-82
tout82.free.fr
 (Français)