bernoulli beams & trusses
DESCRIPTION
ELEMENTOS FINITOSTRANSCRIPT
![Page 1: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/1.jpg)
Basically, bars oriented in two dimensional Cartesian system.
Trusses support compressive and tensile forces only, as in bars.
Translate the local element matrices into the structural (global) coordinate system.
2D TRUSSES
![Page 2: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/2.jpg)
CONSIDER A TYPICAL 2D TRUSS IN GLOBAL X-Y PLANE
Local system:
π’β²=[π’ β² 1π’ β² 2]Global system:
π’=[π’1π’2π’3π’4 ]
![Page 3: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/3.jpg)
π’β²=[π’ β²1=π’1β cosπ+π’2β sin π π’β² 2=π’3βcosπ+π’4β sinπ ]=[cosπ sinπ 0000cosπ sin π ]β [π’1π’2π’3π’4]
(π₯1 , π¦1)
(π₯2, π¦ 2)
π
=m=
cosπ= l =π₯2β π₯1ππ
π’β²=[ππ000 0 ππ]β[π’1π’2π’3π’4 ] π’β²=πΏβπ’
ππ=β(π₯2βπ₯1)2+(π¦2βπ¦1)
2
![Page 4: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/4.jpg)
STIFFNESS MATRIX
Strain Energy:
ππ=12βπ π₯βππ₯β π΄βππ₯
π=π’ β² π‘βπΎ β²βπ’ β²Energy for the local system:
π’β²=πΏβπ’
)
π=π’ π‘β(πΏπ‘βπΎ β²βπΏ)βπ’
K
πΎ=πΈβπ΄ππ [ π0π00000 ]β[ 1 β1
β1 1 ]β[ππ0000 ππ]
Stiffness matrix for the local system:
πΎ β²=πΈβπ΄ππ
β[ 1 β1β1 1 ]
β¦
![Page 5: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/5.jpg)
πΎ=πΈβπ΄ππ [ πβπ
πβπβπ πβππ ]β[ππ0000 ππ]
πΎ=πΈβπ΄ππ
β [ π2 πβπ β π2β πβππβπ π2 βπβπβπ2
βπ2β πβπ
βπβπβπ2
π2 πβππβππ2 ]
Stiffness matrix for the global system
![Page 6: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/6.jpg)
STRESSES AT THE ELEMENT
π=πΈβπ π=πΈβπ’ β² 2βπ’ β²1
πππ=
πΈππβ [β1 1 ]β[π’ β² 1π’ β² 2]
π’β²=πΏβπ’
π=πΈππβ [βπβπππ ]β [π’1π’2π’3π’4 ]
Local system:
Global system:
![Page 7: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/7.jpg)
BERNOULLI BEAMSβ’ Beams are subject to transverse loading. including
transverse forces and moments that result in transverse deformation.
β’ They are deflection in the y direction (w), and rotation in the x-y plane with respect to the z axis.
β’ Each two-noded mean element has total of four degrees of freedrom(DOFs)
![Page 8: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/8.jpg)
INTRODUCTIONβ’ The Euler-Bernoulli beam
theory assumes that undeformed plane sections remain plane under deformation.
w= deflectionβ’ Strain are defined as:
![Page 9: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/9.jpg)
STRAIN ENERGY
Taking :
Inertia:Then strain energy:
![Page 10: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/10.jpg)
SHAPE FUNCTION CONSTRUCTION
β’ As there are four DOFs for a beam element, there should be four shape functions.
Shape functions:
For N1:
![Page 11: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/11.jpg)
SHAPE FUNCTION CONSTRUCTION For N2:
For N3:
![Page 12: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/12.jpg)
SHAPE FUNCTION CONSTRUCTIONFor N4:
The shape functions defined as:
![Page 13: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/13.jpg)
The transverse displacement is interpolated by Hermite shape functions as:
Taking :
Then:
![Page 14: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/14.jpg)
The strain energy is obtained as:
We know that:
![Page 15: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/15.jpg)
STIFFNESS MATRIX
Deriving shape functions:
![Page 16: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/16.jpg)
Each element of the matrix is integrated between [-1,1]:
![Page 17: Bernoulli Beams & Trusses](https://reader035.vdocuments.net/reader035/viewer/2022062302/577cc7a71a28aba711a191ec/html5/thumbnails/17.jpg)
The stiffness matrix is: