The Principle of Minimum Potential Energy for One-Dimensional Elements

CHAPTER

THE PRINCIPLE OF MINIMUM POTENTIAL ENERGY FOR ONE-DIMENSIONAL ELEMENTS

8

In Chapters 1–7, the element equations for springs, bars, and beams was derived using the direct equilibrium method. However, this method is not suitable for analyzing solids that are more complex. For such cases, methods based on the variational principle are more effective. Among variational principles, the principle of minimum potential energy (MPE) is the basis for most finite element method applications in mechanical and structural engineering problems. In order to understand this principle, it is initially applied on one-dimensional elements. It should be noted that the principle of MPE is only applicable to conservative (e.g., elastic) systems.

8.1 THE BASIC CONCEPT From our knowledge of physics, we know that a system is conservative if work done by its internal and external forces (i.e., forces acting on its surface, volume, or nodes) depends only on the initial and final displaced configuration. Therefore, the change of the total potential energy Π p of a conservative system is independent on the deformation history from the initial to the final configuration. According to this definition, elastic solids and structures are conservative systems. The total potential energy includes the strain energy U of the stresses or internal forces causing elastic deformation plus the potential energy W possessed by the external loads (body forces, nodal forces, surface tractions), by virtue having the capacity to do work if they move through a distance. Therefore: Πp ¼ U + W

(8.1)

According to the principle of stationary potential energy, among all possible states of an elastic system, those that satisfy the equilibrium equations yield stationary potential energy with respect to small possible displacements. If the stationary potential energy has a relative minimum, the equilibrium state of the system is stable. Taking into account the above principle, the equilibrium of a conservative system of many degrees of freedom (d1, d2, …, dn) prevails when @Π p ¼ 0, i ¼ 1, 2,…,n @di

(8.2)

The above equation is the mathematical expression of the MPE principle. Essentials of the Finite Element Method. http://dx.doi.org/10.1016/B978-0-12-802386-0.00008-6 Copyright © 2015 Elsevier Inc. All rights reserved.

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CHAPTER 8 PRINCIPLE OF MINIMUM POTENTIAL ENERGY

8.2 APPLICATION OF THE MPE PRINCIPLE ON SYSTEMS OF SPRING ELEMENTS Let us take the example of the spring of Figure 8.1. The nodal forces r1x, r2x cause nodal displacements d1x, d2x. Therefore, the deformation of the spring is ðd2x  d1x Þ. As it is already known from the physics, the elastic energy of the spring is as follows: 1 U ¼ kðd2x  d1x Þ2 2

(8.3)

In contrast, the work done by the external forces r1x, r2x is as follows: W ¼ ðr1x d1x + r2x d2x Þ

(8.4)

The forces r1x, r2x are always acting at their full value (their work is independent of the elastic behavior of the spring). Their movement through the corresponding displacements d1x, d2x is doing work in at a quantity of r1xd1x and r2xd2x, respectively, thereby losing potential of equal amount (hence the negative sign in Equation 8.4). According to Equation (8.1), the total potential of the spring demonstrated in Figure 8.1 is as follows: 1 Π p ¼ kðd2x  d1x Þ2  ðr1x d1x + r2x d2x Þ 2

(8.5)

Therefore, application of the principle of MPE (Equation 8.2) yields 8 @Π p > > < @d ¼ 0 1x

(8.6)

> > : @Π p ¼ 0 @d2x

1

2

d2x d1x

r1x

k

FIGURE 8.1 A single elastic spring subjected to axial forces on its ends.

r2x

8.3 APPLICATION OF THE MPE PRINCIPLE ON SYSTEMS OF BAR ELEMENTS

or



kðd1x  d2x Þ  r1x ¼ 0 kðd1x + d2x Þ  r2x ¼ 0

281

(8.7)

The above system of equations can now be written in a matrix form providing the spring element equation: 

r1x r2x





¼

k k k k



d1x d2x



(8.8)

The above equation is the same as Equation (3.7) derived by the use of the principle of direct equilibrium. Let us now apply Equation (8.2) in a system with more degrees of freedom. In order to do this, we recall the structural system of figure 3.3, which is composed of three spring elements. The total potential for this system is as follows: 1 1 1 Π p ¼ k1 ðd2x  d1x Þ2 + k2 ðd3x  d2x Þ2 + k3 ðd4x  d3x Þ2 2 2 2 ðR1x d1x + R2x d2x + R3x d3x + R4x d4x Þ

(8.9)

Taking into account the above equation, Equation (8.2) can be written: 8 @Π p =@d1x ¼ 0 > > < @Π p =@d2x ¼ 0 @Π =@d3x ¼ 0 > > : p @Π p =@d4x ¼ 0

(8.10)

After some simple algebraic operations, the above algebraic system yields 8 > > <

k1 d1x  k1 d2x  R1x ¼ 0 k1 d1x + ðk1 + k2 Þd2x  k2 d3x  R2x ¼ 0 k d + ðk2 + k3 Þd3x  k3 d4x  R3x ¼ 0 > > : 2 2x k3 d3x + k3 d4x  R4x ¼ 0

(8.11)

The above system can be written in the following matrix from:

8 9 2 38 9 d1x > R1x > 0 0 > k1 k1 > > > > < = 6 < > = k1 k1 + k2 k2 R2x 0 7 d2x 7 6 ¼4 0 k2 k2 + k3 k3 5> R > d > > > > : 3x > ; : 3x > ; R4x 0 0 k3 k3 d4x

(8.12)

Equation (8.12) is same as Equation (3.15) derived by the use of the direct equilibrium principle.

8.3 APPLICATION OF THE MPE PRINCIPLE ON SYSTEMS OF BAR ELEMENTS Following similar procedure as in Section 8.2, we assume the bar of Figure 8.2. The nodal forces r1x, r2x cause nodal displacements d1x, d2x. Therefore, the axial deformation of the bar is ΔL ¼ ðd2x  d1x Þ

(8.13)

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CHAPTER 8 PRINCIPLE OF MINIMUM POTENTIAL ENERGY

1

2

d2x d1x

r1x

E, A

r2x

FIGURE 8.2 An elastic bar subjected to axial forces on its ends.

This deformation yields strain ΔL L

(8.14)

d2x  d1x L

(8.15)

ε¼

or ε¼

According to the Hooke’s law, the corresponding stress is σ ¼ Eε

(8.16)

or σ¼E

d2x  d1x L

(8.17)

As it is already known from the mechanics of solids, the accumulated strain energy of a bar is as follows: 1 U ¼ σεV 2

(8.18)

V ¼ AL

(8.19)

where V is the volume of the bar

Taking into account Equations (8.15) and (8.17), Equation (8.18) yields U¼

EA ðd2x  d1x Þ2 2L

(8.20)

The work done by the external forces is as follows: W ¼ ðr1x d1x + r2x d2x Þ

(8.21)

8.3 APPLICATION OF THE MPE PRINCIPLE ON SYSTEMS OF BAR ELEMENTS

283

Therefore, taking into account Equations (8.20) and (8.21), the total potential energy of the bar is as follows: Πp ¼

EA ðd2x  d1x Þ2  ðr1x d1x + r2x d2x Þ 2L

(8.22)

Using the above formula, the principle of the MPE 8 @Π p > > < @d ¼ 0 1x

(8.23)

> > : @Π p ¼ 0 @d2x

yields 8 EA > > ðd1x  d2x Þ  r1x ¼ 0 < L > > : EA ðd1x + d2x Þ  r2x ¼ 0 L

(8.24)

or in a matrix form (

r1x r2x

)

" ¼

EA=L EA=L

#(

EA=L EA=L

d1x

) (8.25)

d2x

Equation (8.25) is the same as Equation (4.35) derived by the principle of direct equilibrium. Let us now apply Equation (8.2) to a system of bars with more degrees of freedom. To do this, we recall the structural system of figure 4.4, which is composed of three bar elements. The total potential of this system is: Πp ¼

1 E1 A1 1 E2 A2 1 E3 A3 ðd2x  d1x Þ2 + ðd3x  d2x Þ2 + ðd4x  d3x Þ2 2 L1 2 L2 2 L3

(8.26)

ðR1x d1x + R2x d2x + R3x d3x + R4x d4x Þ

Therefore, the principle of the MPE 8 @Π p =@d1x ¼ 0 > > > > > < @Π p =@d2x ¼ 0

(8.27)

> @Π p =@d3x ¼ 0 > > > > : @Π p =@d4x ¼ 0

using Equation (8.26) yields the following matrix equation: 8 9 R1x > > > > > > > > > < R2x > =

2

E1 A1 =L1

E1 A1 =L1

0

6 6 E1 A1 =L1 E1 A1 =L1 + E2 A2 =L2 E2 A2 =L2 6 ¼6 > > 6 0 E2 A2 =L2 R3x > E2 A2 =L2 + E3 A3 =L3 > > > 4 > > > > : ; R4x 0 0 E3 A3 =L3

38 9 > > d1x > > > > > 7> > < d2x > = 7 0 7 7 > d3x > E3 A3 =L3 7 > > > 5> > > > : > ; E3 A3 =L3 d4x 0

(8.28)

The above equation is the same as Equation (4.43) derived using the direct equilibrium principle.

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CHAPTER 8 PRINCIPLE OF MINIMUM POTENTIAL ENERGY

8.4 APPLICATION OF THE MPE PRINCIPLE ON TRUSSES Trusses are structural systems composed of bars. Since any truss member 1-2 aligned with its local coordinate axis x is often inclined with respect to the global coordinate system x-y, the change of its length should be expressed in terms of the nodal displacements d1x, d1y and d2x, d2y with respect to the system x-y. Taking into account Equations (5.3) and (5.4), the change of the length is as follows:     ΔL ¼ d2x  d1x ¼ Cd2x + Sd2y  Cd1x + Sd1y

or

ΔL ¼ ½ C S C S 

8 9 d1x > > > > > > > > > < d1y > =

(8.29)

(8.30)

> d2x > > > > > > > > : > ; d2y

Therefore, the total potential of a truss containing n-bars and k-nodes can be expressed by the following formula: Πp ¼

n X 1 Ei Ai i¼1

2 Li

ðΔLi Þ2 

k  X

Rjx djx + Rjy djy



(8.31)

j¼1

Therefore, the corresponding system of algebraic equations yielding the structure’s equation is as follows: n

@ @ @ @ @d1x @d1y @d2x @d2y

⋯

@ @ @dkx @dky

oT

Πp ¼ 0

(8.32)

8.5 APPLICATION OF THE MPE PRINCIPLE ON BEAMS We recall from Chapter 6 that the types of external loads acting on beams are: 1. transverse varying loads q; 2. transverse concentrated nodal forces riy; 3. nodal bending moments mi. The consequence of the above loads are transverse deflections u(x) and slopes ϑ(x) taking place in any point x of the beam. Therefore, for a beam element of length L, the work of the external loads is as follows: W¼

ðL 0

qðxÞuðxÞdx 

2  X i¼1

riy diy + mi ϑi



(8.33)

8.5 APPLICATION OF THE MPE PRINCIPLE ON BEAMS

285

∫ e (x)dx ϑ(x)

y

FIGURE 8.3 Deformation of a beam under bending.

where diy and ϑi are the transverse nodal displacement and the slope of the node i, respectively. Taking into account Equation (6.13), the above equation can be expressed in terms of nodal displacements W ¼

ðL

qðxÞf½N fd ggT dx  fd gT ½r T

(8.34)

0

where T ½N  ¼ ½ N1 N2 N3 N4 , fd gT ¼ f d1x ϑ1 d2x ϑ2 g, ½r T ¼ ½ r1y m1 r2y m2 

As it is already known from the mechanics of solids, during bending the slope of each cross-section of a beam (Figure 8.3) is as follows: ϑðxÞ ¼

Therefore, the geometric condition

duðxÞ dx

(8.35)

ð εðxÞdx

ϑðxÞ ¼ 

can now be written

y

(8.36)

ð duðxÞ ¼ dx

εðxÞdx y

(8.37)

The above equation yields εðxÞ ¼ y

d2 uðxÞ dx2

(8.38)

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CHAPTER 8 PRINCIPLE OF MINIMUM POTENTIAL ENERGY

Taking into account Equation (6.13), Equation (8.38) can correlate the strain of each point x of the beam to the nodal displacements

fεðxÞg ¼ y

8 9 d1y > > > > > > > > > < ϑ1 > =

d2 ½ N1 N2 N3 N4  > dx2 d2y > > > > > > > > : > ; ϑ2

(8.39)

where N1, N2, N3, and N4 are shape functions given in Equations (6.14)–(6.17). After some simple algebraic operations, Equation (8.39) yields fεðxÞg ¼ y½Bfd g

(8.40)

fd g ¼ f d1x ϑ1 d2x ϑ2 gT

(8.41)

where

and ½B ¼

1 ð12x  6LÞ ð6xL  4L2 Þ ð12x + 6LÞ ð6xL  2LÞ 3 L

(8.42)

Since the stress-strain relationship is given by Hooke’s law σ ðxÞ ¼ EεðxÞ, taking into account Equation (8.40), it can be written fσ ðxÞg ¼ Ey½Bfd g

(8.43)

Therefore, the strain energy U of a beam under bending can be correlated to the nodal displacements using Equations (8.40) and (8.43): ð U¼

1 fσ gT fεgdV V2

(8.44)

where V is the volume of the beam. Since dV ¼ dAdx, the above equation can be written U¼

1 2

ð ð fσ gT fεgdAdx

(8.45)

A L

Taking into account Equations (8.40) and (8.43), Equation (8.45) yields E U¼ 2

ð ð f½Bfd ggT ½Bfd gy2 dAdx

(8.46)

A L

Taking into account the following property of the linear algebra f½Bfd ggT ¼ fd gT ½BT

Equation (8.46) can now be written U¼

E 2

(8.47)

ð ð fd gT ½BT ½Bfd gy2 dAdx A L

(8.48)

8.5 APPLICATION OF THE MPE PRINCIPLE ON BEAMS

287

Since the quantity ð I¼

y2 dA

(8.49)

A

represents the moment of inertia, Equation (8.48) yields U¼

EI 2

ð fd gT ½BT ½Bfdgdx

(8.50)

L

Therefore, taking into account Equations (8.34) and (8.50), the total potential energy for a beam element takes now the form: Πp ¼

EI 2

ðL

fd gT ½BT ½Bfdgdx 

0

ðL

qðxÞfd gT ½N T dx  fd gT fr g

(8.51)

0

Applying the MPE principle to the above equation, the following formula can be obtained @Π p ¼0 @ fdg

(8.52)

yielding

ðL  ðL T EI ½B ½Bdx fd g  ½N T qðxÞdx  fr g ¼ 0 0

(8.53)

0

Let ff g ¼

ðL

½N T qðxÞdx + fr g

(8.54)

0

Then, Equation (8.53) takes the form:

ðL  T EI ½B ½Bdx fd g ¼ f f g

(8.55)

0

Comparing the above equation with Equation (6.32), it can be concluded that the stiffness matrix of the beam element is as follows:

ðL  ½k ¼ EI ½BT ½Bdx

(8.56)

0

Taking into account Equation (8.42), the above equation yields 2

12

6L 12

6L

3

7 6 6 6L 4L2 6L 2L2 7 7 EI 6 7 ½k  ¼ 3 6 7 L 6 6 12 6L 12 6L 7 5 4 2 2 6L 2L 6L 4L

(8.57)

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CHAPTER 8 PRINCIPLE OF MINIMUM POTENTIAL ENERGY

REFERENCES [1] Wunderlich W, Pilkey WD. Mechanics of structures, variational and computational methods. Boca Raton: CRC Press; 2003. [2] Logan DL. A first course in the finite element method. Boston MA: Cengage Learning; 2012. [3] Fish J, Belytschko T. A first course in the finite elements. New York: Wiley; 2007. [4] Bhatti MA. Fundamental finite element analysis and applications. Hoboken: John Wiley & Sons; 2005. [5] Hartmann F, Katz C. Structural analysis with finite elements. Heidelberg: Springer; 2007.