bernoulli beams & trusses
DESCRIPTION
ELEMENTOS FINITOSTRANSCRIPT
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
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 ]
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
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¦
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
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:
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)
INTRODUCTIONβ’ The Euler-Bernoulli beam
theory assumes that undeformed plane sections remain plane under deformation.
w= deflectionβ’ Strain are defined as:
STRAIN ENERGY
Taking :
Inertia:Then strain energy:
SHAPE FUNCTION CONSTRUCTION
β’ As there are four DOFs for a beam element, there should be four shape functions.
Shape functions:
For N1:
SHAPE FUNCTION CONSTRUCTION For N2:
For N3:
SHAPE FUNCTION CONSTRUCTIONFor N4:
The shape functions defined as:
The transverse displacement is interpolated by Hermite shape functions as:
Taking :
Then:
The strain energy is obtained as:
We know that:
STIFFNESS MATRIX
Deriving shape functions:
Each element of the matrix is integrated between [-1,1]:
The stiffness matrix is: