有限元教案_第三章(杆系单元2)

2 Principle of minimum potential energy and Rayleigh-Ritz method
Let {δ } = [u1 u2 ... un ]Tbe the structure displacement vector that must be determined. Potential Πp Is a function of the ui. Symbolically,
Since
d Πp
2
ku = f
du
2
=k>0
So that means the minimum potential energy gives the equilibrium state of a system
Chapter 3 FEM in One-Dimensional Problem
2 Principle of minimum potential energy and Rayleigh-Ritz method
d d u (E A ) +f = 0 d x d x
F x
f-force per unit length; F-concentrated force on the end E-elastic modulus; A- cross-sectional area L- non-stretched length of the bar; u(x)- displacement ( u(x) is a function of x)
Chapter 3 FEM in One-Dimensional Problem
2 Principle of minimum potential energy and Rayleigh-Ritz method
Example 3.3: Potential energy of a spring
f
x
Fig. 3.3 linear spring with axial load
Chapter 3 FEM in One-Dimensional Problem
2 Principle of minimum potential energy and Rayleigh-Ritz method
Principle of minimum potential energy Among all admissible displacements of a conservation system, those that satisfy the equations of equilibrium make the potential energy Πp minimum with respect to small variations of displacement.
保守力系统，在所有许可位移场（ 对于保守力系统，在所有许可位移场（满足边 界条件和协调要求）中，对应于平衡状态的位 界条件和协调要求） 总势能取最小值。（ 。（总势能为最小值 移场使得总势能取最小值。（总势能为最小值
的位移满足静力平衡条件） 的位移满足静力平衡条件）
Chapter 3 FEM in One-Dimensional Problem
Π p = Π p (u1 u2 ... un )
Applying the principle of minimum potential energy , we obtain variation of the functional where
δΠ p =
Π p u1
δΠ p = 0
δu1 + ... +
Π p ui
=0
for
i = 1,...n
f
1 2 Π p = ku fu 2
If only displacements along the x axis are allowed
δΠ p = ( ku f )δu = 0
dΠ p = ku f = 0
du Which gives the equilibrium equation
δu variation of admissible displacement u(x),
displacement boundary condition
satisfying the
δu(0) = 0
d dδu u δΠp = [E fδu]d Fδu(L) x A 0 d d x x L d d u d u = [ (E A ) + f ]δud + (E x A F)δu(L) 0 d x d x d L x
Potential energy of system elastic potential energy concentrated force distributed force
Chapter 3 FEM in One-Dimensional Problem
2 Principle of minimum potential energy and Rayleigh-Ritz method
δΠ p =
Π p u1
δu1 + ... +
Π p un
δun = 0
For any δui ≠ 0 only
Π p ui
=0
for
i = 1,...n
Π p ui
=0
for
i = 1,...n
These are n equations to be solved for the n values of displacements ui that define the static equilibrium configuration.
How to obtain the minimum of the Πp
1 2 Π p = ku fu 2
?
δΠ p = 0
δΠ p =
Π p u1
δu1 + ... +
Π p u n
δu n
Chapter 3 FEM in One-Dimensional Problem
2 Principle of minimum potential energy and Rayleigh-Ritz method
Chapter 3 FEM in One-Dimensional Problem
2 Principle of minimum potential energy and Rayleigh-Ritz method
Introduction
1
2
3
4
Three- dimensional body
Chapter 3 FEM in One-Dimensional Problem
2 Principle of minimum potential energy and Rayleigh-Ritz method
Introduction
Functional(泛函 ), in terminology, a functional is an integral expression that implicitly contains the governing differential equations.
du = EA δu dx 0 = EA
∫0
L
d du ( EA )δudx dx dx
du δu( L ) dx L
∫0
L
d du (EA )δudx dx dx
Chapter 3 FEM in One-Dimensional Problem
2 Principle of minimum potential energy and Rayleigh-Ritz method
Let u be the displacement of the spring under the load f, Its potential energy Πp has two parts: Π p = U + (1)Strain energy of the spring
1 2 U = ku 2 (dU = Fdx)
Chapter 3 FEM in One-Dimensional Problem
2 Princiபைடு நூலகம்le of minimum potential energy and Rayleigh-Ritz method
(2)Potential energy of applied load
f
= fu
So
It does work, thereby losing potential of equal amount.
Chapter 3 FEM in One-Dimensional Problem
2 Principle of minimum potential energy and Rayleigh-Ritz method
σ = Eε
f = Aσ
L
du ε= dx
f 0 L F x
1 d 2 u Πp (u) = ∫ [ E ( ) fu]d F (L) x u A 0 2 d x
Chapter 3 FEM in One-Dimensional Problem
2 Principle of minimum potential energy and Rayleigh-Ritz method
Introduction
In the preceding discussion we employed the direct approach of assembling the system-governing equilibrium equations. However, it can only solve some relative simple engineering problems. For a continuum, governing equilibrium equations for state variables can in many analyses also be obtained using potential energy.