Lecture 6Energy principlesEnergy methods and variational principlesPrint version Lecture on Theory of Elasticity and Plasticity of

Dr. D. Dinev, Department of Structural Mechanics, UACEG

6.1

Contents

1 Work and energy 1

2 Strain energy 2

3 Principle of Minimum Potential Energy 3

4 Ritz method 5 6.2

1 Work and energy

Work and energy

Work• A material particle is moved from point A to point B by a force F• The infinitesimal distance along the path from A to B is a displacement du• The work dW performed by the force F is defined as

dW = F ·du

• The work done is the product of the displacement and the force in the direction of thedisplacement• The total work is

W =∫ B

AF ·du

6.3

Work and energy

Work

1

• Work = Force × Displacement6.4

Work and energy

Energy

• The energy is the capacity to do work• It is a measure of the capacity of all forces on a body to do work• Work is performed on a body trough a change in energy.

6.5

2 Strain energy

Strain energy

Internal work

• The work done by forces on an elastic solid is stored inside the body in the form of a strainenergy• Consider the uniaxial tension test where we can assume that the stresses increases slowly

from zero to σ

6.6

Strain energy

Internal work

• The strain energy stored is equal to the work done on the differential element

dU =∫

σ

0σd(

u+∂u∂x

dx)

dydz−∫

σ

0σdudydz

=σ2

2Edxdydz

6.7

Strain energy

Internal work

• The strain energy density is the strain energy per unite volume

U =dU

dxdydz=

12

σε

6.8

2

Strain energy

Internal work• The strain energy density is the shaded area under the stress-strain curve

U =12

σε

6.9

Strain energy

Internal work• General expression for the strain energy density

U =12(σxxεxx +σyyεyy +σzzεzz +2σxyεxy +2σyzεyz +2σzxεzx)

=12

σi jεi j =12

σ : ε

• The total strain energy is

Utot =12

∫V

σ : εdV

• The potential of the applied forces is

W =−[∫

Vf ·udV +

∫S

t ·udS]

6.10

3 Principle of Minimum Potential Energy

Principle of Minimum Potential Energy

TPE functional• Total potential energy of the body

ΠT PE =Utot +W =12

∫V

σ : εdV −∫

Vf ·udV −

∫S

t ·udS

6.11

Principle of Minimum Potential Energy

TPE functional• The principle states that: The body is in equilibrium if there is an admissible displacement

field u that makes the total potential energy a minimum

δΠT PE(u) = 0

where δΠT PE is a variation of the functional ΠT PE(u)• The admissible displacement field is the one that satisfy the displacement BCs.

NoteThe variational operator δ is much like the differential operator d except that it operates withrespect to the dependent variable u rather than the independent variable x

6.12

3

Principle of Minimum Potential Energy

-0.5 0.5 1.0 1.5 2.08u, u¢<

-500

500

1000

8W, U , P<

TPE functional

• Local minimum of the TPE functional6.13

Principle of Minimum Potential Energy

A gentle touch to the variational analysis

• The condition for minimum of a functional I(u) is

δ I =∂ I∂u

δu = 0

• Almost the same as the condition for a minimum of a function u(x)

du =∂u∂x

dx = 0

6.14

Principle of Minimum Potential Energy

A gentle touch to the variational analysis

• Consider a functional

I(u) =∫ b

aF[x,u(x),u′(x)

]dx

• The minimum condition is

δ I(u) =∫ b

a

(∂F∂x

δx+∂F∂u

δu+∂F∂u′

δu′)

dx = 0

• The integral can be manipulated to get the expression for the variation of u (integration byparts of the 3-rd addend)

6.15

Principle of Minimum Potential Energy

A gentle touch to the variational analysis

• The variation of the functional is

δ I(u) =∫ b

a

[∂F∂u− ∂

∂x

(∂F∂u′

)]δudx

• The non-trivial solution gives

∂F∂u− ∂

∂x

(∂F∂u′

)= 0

• The above expression is called Euler equation6.16

4

Principle of Minimum Potential Energy

A gentle touch to the variational analysis• A more general 2D case is given by

I(u,v) =∫

AF (x,y,u,v,u,x,v,x, . . . ,v,yy)

• The Euler equations are∂F∂u− ∂

∂x∂F∂u,x

− ∂

∂y∂F∂u,y

+∂ 2

∂x2∂F

∂u,xx+

∂ 2

∂x∂y∂F

∂u,xy+

∂ 2

∂y2∂F

∂u,yy= 0

∂F∂u− ∂

∂x∂F∂u,x

− ∂

∂y∂F∂u,y

+∂ 2

∂x2∂F

∂u,xx+

∂ 2

∂x∂y∂F

∂u,xy+

∂ 2

∂y2∂F

∂u,yy= 0

6.17

Principle of Minimum Potential Energy

Example• Consider a cantilever beam with length of ` and subjected to uniform load q. Using the

principle of a minimum of TPE work out the equilibrium equation6.18

4 Ritz method

Ritz method

Approximate solution• A lot of problems in elasticity there is no analytical solution of the field equations• For such cases approximate solution schemes have been developed based on the varia-

tional formulation of the problem (i.e. the principle of minimum potential energy)• The Ritz method is based on the idea of constructing a series of trial approximating func-

tions that satisfy the essential (displacement) BCs but not differential equations exactly

Note• Since the TPE functional includes the force BCs, it is require the trial solution satisfies

only the displacement BCs!6.19

Ritz method

Approximate solution• Walter Ritz (1878-1909)

6.20

5

Ritz method

Approximate solution

• The original paper6.21

Ritz method

Approximate solution

• The displacement can be expressed as

u = u0 +a1u1 +a2u2 + . . .+anun = u0 +n

∑i=1

aiui

v = v0 +b1v1 +b2v2 + . . .+bnvn = v0 +n

∑i=1

bivi

w = w0 + c1w1 + c2w2 + . . .+ cnwn = w0 +n

∑i=1

ciwi

• The terms of u0, v0 and w0 are chosen to satisfy any non-homogeneous displacement BCsand ui, vi and wi satisfy the corresponding homogeneous BCs

• These forms are not required to satisfy the stress BCs6.22

Ritz method

Approximate solution

• These trial functions are chosen from the some combinations of elementary functions(polynomials, trigonometric or hyperbolic forms)• The unknown coefficients ai, bi and ci are to be determined so as to minimize the TPE

functional of the problem• Thus we approximately satisfy the variational formulation of the problem• Using this approximation the TPE functional will be a function of these unknown coeffi-

cients

ΠT PE = ΠT PE(ai,bi,ci)

6.23

Ritz method

Approximate solution

• The minimizing condition ca be expressed as a series of

∂ΠT PE

∂ai= 0,

∂ΠT PE

∂bi= 0,

∂ΠT PE

∂ci= 0

• This set forms a system of 3n equations which gives ai, bi and ci

6

• Under suitable conditions on the choice of trial functions (completeness) the approxima-tion will improve as the number of included terms is increased• When the approximate displacement solution is obtained the strains and stresses can be

calculated from the appropriate field equations

NoteThe method is suitable to apply at problems involving one or two displacements (bars, beams,plates and shells)

6.24

Ritz method

Example

• Consider a bending of simply supported beam of length ` carrying a uniform load q6.25

Ritz method

The End

• Any questions, opinions, discussions?6.26

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