Alcan Structural Engineering

VARIATIONAL and RAYLEIGHRITZ METHODS

Adapted from (Reich 1993)

5.1 Multifield Variational Principles

1 A Multifield variational principle is one that has more than one master field (or state variable), that is more than one unknown field is subject to independent variations.

2 In linear elastostatics, we can have displacement, u, strains e, or stress a as potential candidates for master fields. Hence seven combinations are possible, (Felippa 2000), Table 5.1.

3 In this course, we shall focus on only the Total Potential Energy, and the Hu-Washizu varia-tional principles.

u	e	er	Name
Single Field
Y	Y	Total Complementary Potential Energy No name
Two Fields
Y		Y	Hellinger-Reissner
Y	Y		de Veubeke
	Y	Y	No name
Three Fields
Y	Y	Y	Hu-Washizu

Table 5.1: Functionals in Linear Elasticity

Table 5.1: Functionals in Linear Elasticity

5.2 Total Potential Energy Principle 5.2.1 Static; Euler

4 The expression for the total potential energy (TPE) is given by def n = u-We

Tit def

2 Jn

the functional for the general form of the potential energy variational principle is obtained

5 A variational statement is obtained by taking the first variation of n and setting this scalar quantity equal to zero.

6 The variational statement for the general form of the potential energy functional (i.e. Equation 5.2) is m= SeTDedQ - SeTDeodQ + SeT^odQ - SuTbdQ - ¿uTtdr =0 (5.3)

Tt which is the Principle of Virtual Work.

7 We rewrite the strain-displacement relations in terms of a linear differential operator L (Eq. 3.41)

where L is a linear differential operator and u is the displacement vector defined in Eq. 3.42:

dx 0

dy U

dy dx dz 0

dz 0

8 Since the differential operator L is linear, the variation of the strains 5e can be expressed in terms of the variation of the displacements ¿u

We consider two forms of the variational statement n n n n n

1. In terms of strains: which is more suitable for the derivation of the corresponding Euler equations,

ön = J ö(Lu)TDedQ	- [ ¿(Lu)TDe0dQ + [ ö{Lu)Ta0dÜ
n	J n J n
	su
- f öuTbdQ -	[ ¿uTtdr = 0 (5.7)
n	ht
v-v—
SWe

2. In terms of displacements: (using Equation 5.7) which is more suitable for the subsequent discretization.

/ ö(Lu)T D(Lu)dQ - ö(Lu)T DeodQ+ / ö(Lu)T aodQ 'n Jn Jn

10 To obtain the Euler equations for the general form of the potential energy variational principle the volume integrals defining the virtual strain energy SU in Equation 5.7 must be integrated by parts in order to convert the variation of the strains S(Lu) into a variation of the displacements Su.

11 Integration by parts of these integrals using Green's theorem (Eq. 1.46)

[ R^-dü = - [ ^SdÜ + / RSnTäT In dx Jn dx Jr yields

[ ö(Lu)T^odQ = iöuTG^odr - I öu1 L1 ^dQ Jn Jr Jn

where G is a transformation matrix containing the direction cosines for a unit normal vector such that the surface tractions t are defined as t = G^ and the surface integrals are over the entire surface of the body r.

12 Substituting Equation 5.10-a into Equation 5.7, the variational statement becomes m = - ! 5uT {Lt [D(e - eo) + o"o] +b}dQ Jn s---'

Figure 5.1: Tonti Diagram for the Total Potential Energy, (Cervenka, J. 1994)

13 Since ¿u is arbitrary the expressions in the integrands within the braces must both be equal to zero for ¿n to be equal to zero. Recognizing that the stress-strain relationship appears in both the volume and surface integrals, the Euler equations are

(BE): Equilibrium LTa + b = 0 on Q (5.12) (NBC): Natural B.C. Go* - t = 0 onI\ (5.13)

where the first Euler equation is the equilibrium equation and the second Euler equation defines the natural boundary conditions. The natural boundary conditions are defined on rt rather than r because both the applied surface tractions t and the matrix-vector product Ga are identically zero outside rt.

14 In general, only certain forms of differential equations are Euler equations of a variational functional.

15 For the elastostatic problem, it is possible to start from the Euler equations, and then derive the total potential energy functional by performing the operations just presented in reverse order.

16 The Tonti diagram for the TPE is shown in Fig. 5.2. In this diagram, strong connections are shown by solid lines, weak connections by spring-like symbols, boxes with solid lines denote the primary unknown field, variables inside dashed boxes are internally derived fields, and shaded boxes indicate prescribed fields.

5.2.2 Dynamic; Euler/Lagrange

17 In dynamics, we define the Lagrangian function as the difference between the kinetic and the potential energies

and Hamilton's principle states that The motion of a particle acted on by conservative forces between two arbitrary instants of time t\ and t2 is such that the line integral over the Lagrangian function is an extremum for the path motion.

is an extremum.

18 If the path can be defined in terms of generalized coordinates qi (i = 1,2,3), then it can be shown that

t1 ft2

dqi dt \dqiJ

19 If all qi are linearly independent (i.e. no constraints among them), the variations ôqi are independent of t, except Sqi = 0 at t1 and t2. Therefore, the coefficients of ôq1, ôq2, and ôq3 vanish separately yielding the Lagrangian equations of motion, also known as the Euler-Lagrange equations.

Hamilton's principle can be expressed as:

where K is the kinetic energy, n the potential energy, and Wnc the work done by nonconservative forces acting on the system (including damping).

It should be noted that TPE is a special case of Hamilton's principle in the absence of kinetic energy.

The Kinetic energy is given by pdu d u

A conservative force is one for which the sum of the potential and kinetic energies is conserved).

■ Example 5-1: Hamilton's Principle

For a uniform cross section bar of length L, cross sectional area A, Young's modulus E, and mass density p fixed at one end, and connected to a rigid support at the other by a spring with stiffness k

0 0