Euler’s Equations

S

F⁽¹⁾s · u⁽²⁾_i +



B

F⁽¹⁾_b · u⁽²⁾_i =



S

F⁽²⁾s · u⁽¹⁾_i +



B

F⁽²⁾_b · u⁽¹⁾_i . (1.130) As a special case of the above theorem, we may consider two discrete surface loads acting over points A and B of an elastic body, and suppose that no body forces are applied. When the force Fs(A) (at A) is applied, the displacements u^A_A at A and u^A_Bat B are obtained. When the force Fs(B) (at B) is applied, the displacements u^B_Aat A and u^B_Bat B are obtained. For this case (1.130) yields

Fs(A) · u^BA= Fs(B) · u^AB. (1.131) If we now adopt the notation u^B_A=αABFs(B) and u^A_B=αBAFs(A) (where the matricesαABand αBAare usually referred to as the influence coefficients), (1.131) shows thatαAB=αBA.

1.4.4 Castigliano’s Theorems

We shall now examine the case of the Theorem of Minimum Potential Energy, where body forces are absent and the surface loading Fs k(k = 1,...,K) are discrete. For such a case

V =^

B

W(u1,...,uK) −

∑

^Kk=1Fs k· uk, (1.132) where F_{s k}are surface loading and u_kare displacements of the k^th point (of the application of the force Fs k). Variation of V leads to ∑^K_k=1∂^∂V_ukδu_k= 0, while (1.123) together with the fact that the variationsδu_kare arbitrary, yield the system Fs k=_∂u^∂U

k. This result, widely known as Castigliano’s First Theorem, may be interpreted as follows: the partial derivative of the strain energy with respect to generalized displacements is equal to the corresponding generalized force.

Note that this result stands for small perturbations, and thus should be applied either to the case of relatively small loads, or to the case where the system is linear in the sense that the resulting displacements vary linearly with the loads (in such a case, U_,ukis a linear function of u_k). To develop Castigliano’s Second Theorem we write the potential energy as

V=



B

W(Fs 1,...,Fs K) −

∑

^Kk=1Fs k· uk, (1.133) and again the minimization leads to equations∑^K_k=1∂^∂V_F

s kδFs k= 0, which for arbitrary varia-tionsδFs kyield u_k=_∂F^∂U

s k. The above may put in words as:

The partial derivative of the strain energy with respect to a generalized force is equal to the corresponding generalized displacements.

The reservation mentioned above regarding the system linearity holds in this case as well.

1.5 Euler’s Equations

In this section we shall employ variational analysis (or calculus of variations) techniques that deal with minimization of functionals. In the present context, a functional is an operator that converts a set of functions to a number. A fundamental result of the calculus of variations is that the extreme values of a functional must satisfy an associated differential equation (or a set of differential equations) over the discussed domain that are generally termed Euler’s equations.

The notion “extreme values” stands for local minima, maxima or inflection points. Hence, the underlying idea is founded on the existence of a physical global quantity that remains maximal or minimal at all times, regardless of the nature (i.e. stationary or time-dependent) of the problem. Note that variational calculus is a fundamental analytical tool in many other areas of general physics and engineering, and a review of the mathematical methods associated with it may be found, e.g., in (Sagan, 1969).

Two general assumptions are typically associated with the analytical methodologies applied in calculus of variations. First, we generally assume that all functions and functionals are con-tinuous and have concon-tinuous derivatives as required. In addition, the functional values are as-sumed to be positive.

This section contains a brief survey of the techniques associated with deriving Euler’s equa-tions out of a given functional, and examples for use of these equaequa-tions in the area of elasticity.

For the sake of abbreviating, in this section, derivatives of functions of one variable, for example y(x), are denoted as ^dy_dx= y
,^d_dx²2^y= y

,^d_dx^mm^y= y^(m).

1.5.1 Functional Based on Functions of One Variable

We shall first calculate extreme values of integral functional, J, whose integrand, F, contains one or several functions associated with the admissible function y(x) of the C²class on the interval[x0,x1]. As an example, consider the problem

J(y) =^{ x1}

x₀ F(x,y,y
)dx → min, (1.134)

where F is a continuous function of three arguments (the problem of determining a maximum may be dispensed with F replaced by−F). The boundary values of y(x) are generally given as

y(x0) = y0, y(x1) = y1. (1.135)

The minimization of the functional J(y) leads to the Euler’s equation for its integrand, see (Sokolnikoff, 1983),

F_,y− d

dx(F_,y
) = 0, (1.136)

where, obviously,

d

dx(F,y
) = F,y
y
y

+ F,y
yy
+ F,y
x. (1.137) Equation (1.136) is a necessary condition that J(y) possess a stationary value.

A similar derivation may be carrying out for the extreme problem J(y) =

 _x1 x0

F(x,y,y
, y

,...,y^(m))dx → min, (1.138) where m≥ 1, and the admissible function y(x) belongs to the C^m+1class on the interval[x0,x1], and satisfies the boundary conditions

y(x0) = y0, y
(x0) = y0
, ... y^(m)(x0) = y0, (1.139) y(x1) = y1, y
(x1) = y1
, ... y^(m)(x1) = y1.

The minimization in this case leads to the following Euler’s equation:

F_,y− d

dx(F,y
) + d²

dx²(F,y

)−,···,+(−1)^m d^m

dx^m(F_,y(m)) = 0, (1.140)

while _dx^diiF_,y(i)are derived analogously to (1.137).

To generalize the previous case of (1.134), consider the problem defined in a different way by the functional

J(y) =^{ x1}

x₀ F(x,y1,...,yn,y1
,...,yn
)dx → min, (1.141) where n≥ 1, and F is a continuous function of 2n + 1 arguments. We suppose that the admis-sible functions yi(x) of one variable belong to the C²class on the interval[x0,x1], and that the boundary values are defined as

yi(x0) = yi0, yi(x1) = yi1, 1≤ i ≤ n. (1.142) The minimization leads to the following system of Euler’s equations:

F_,yi− d

dx(F,yi
) = 0, 1≤ i ≤ n. (1.143)

To generalize the case of (1.141) furthermore, we note that similar calculations performed on the extreme problem

J(y) =^{ x1}

x₀ F(x, y1,...,yn, y1
...,yn
,...,y^(m)₁ ,...,y^(m)n )dx → min (1.144) lead to the following system of Euler’s equations:

F_,yi− d

dx(F,yi
) + d²

dx²(F,yi

)−,···,+(−1)^m d^m dx^m(F_,y(m)

i ) = 0, 1≤ i ≤ n. (1.145) The Euler Lagrange(F, x, y(x)) and Euler Lagrange(F, x, [y1(x),...,yn(x)]) commands from the Variational Calculus package of (Maple, 2003), compute the Euler’s equations of the functionals (1.134), (1.141). The higher-order functionals (1.138), (1.144) may be reduced to these forms as well.

Remark 1.9 Consider the variational problem (1.141) when for each function, only one of the boundary conditions (1.142) is given. For example, yi(x0) = yi0, 1 ≤ i ≤ n. Clearly, the admissible functions yi(x) in this case form a larger class. The minimizing process shows that Euler’s equations given in (1.143) are obtained only if the following boundary conditions (usually called “natural”) are satisfied:

F_,yi
(x1) = 0, 1≤ i ≤ n. (1.146)

This characteristic of the variational process is one of its profound advantages in the area of elasticity, as it is capable of providing part of the boundary conditions as well.

Example 1.2 Rotating Beam.

This example demonstrates the application of the Theorem of Minimum Potential Energy to derive Euler’s equation associated with rotating isotropic beam. In this case, there are three components that contribute to the potential energy: the strain energy, the rotational tension, and the surface loads. Let z be a coordinate along the beam axis, while we are looking for the beam axis deflection in the y direction v(z), 0 ≤ z ≤ l, see Fig. 1.13. The (bending) strain energy UBin this example will be expressed by the bending curvature v

and Young’s modulus, E (see (5.44) with 1/a33= E), namely

UB=^

BW=1 2



Bσzεz=1 2

 _l 0 EIx

v

2

, (1.147)

Figure 1.13:Rotating beam notation in Example 1.2.

where Ix=^_Ωy² is the cross-section moment (of inertia) about the x-axis. The rotational tension may be treated in two different ways. We first may consider this effect as a contributor to the strain energy. Its contribution equals to the product of the (given) tensile force, T(z), and the extension created by the bending v(z), which is written as√

1+ v
²− 1 ∼=¹₂(v
)². Hence, UT =

 _l 0

1 2T(z)

v
2

dz. (1.148)

Alternatively, the tension effect may be viewed as a transverse distributed loading of(T(z)v
)
. Since this loading is not constant but depends on the displacements (unlike the body loads in (1.124)), its potential should be therefore written as

VT= −1 2

 _l

0 (T(z)v
)
v dz. (1.149)

The given surface loads in the y direction, fy(z), do not depend on the deformation. Therefore, their potential is−^₀^l fy(z)ydz. Overall, the functional F of (1.138) may be written as

F=1 2EIx

v

2

−1

2(T(z)v
)
v− fy(z)v. (1.150) By employing the minimization results of (1.140) with F= F(z,v,v
,v

), see also P.1.13, we obtain the Euler’s equation

(EIxv

)

− (T(z)v
)
− fy(z) = 0. (1.151) For a rotation about the y-axis as shown in Fig. 1.13, the tension force is T=¹₂m(z)Ω²(l²− z²), where m(z) is the mass distribution per unit length andΩis the rotational velocity.

1.5.2 Variational Problems with Constraints

Consider a case where Lagrange multipliers should be employed within the variational prob-lem (1.134) with the isoperimeter conditions

Jk(y) =^{ x1}

x0

Fk(x,y,y
)dx = gk, 1≤ k ≤ n, (1.152) where g_kare constants and F_kare continuous functions of three arguments. The corresponding Lagrange functional has the form

JL(y) =^{ x1}

x₀ [F(x,y,y
) +

∑

ⁿk=1λkFk(x,y,y
)]dx. (1.153)

In this case we solve the variation problem JL(y) → min, by considering Lagrange multipliers, λk, as constants.

Example 1.3 Elastica.

The analysis described in this example deals with large deformations of an elastic rod.

This kind of problems are traditionally termed “Elastica”. For further reading see (Frisch-Fay, 1962), (Stronge and Yu, 1993). The results presented below are documented in P.1.14.

The bending energy of an elastic rod, which is deformed as the plane curve r(s) = [x(s), y(s)]

of length l is assumed, for simplicity, to be proportional to the integral of the squared curva-ture over the length of the curve, i.e.^₀^lκ²(s)ds, see Example 1.2. Here s ∈ [0,l] is the natural length parameter of the curve. Recall that the curvatureκ(s) of a plane curve is given as ^dθ_ds whereθ(s) is the angle between the local tangent line and the x-axis. One may therefore ask the following question: what shape will the curve take if the total turning of its tangent is given and the turning is zero andθeat the endpoints. As a constrained variational problem we write

J= F depends on two of the three variables s,θ,^dθ_ds and on the parameterλ. The Euler’s equation for the constrained problem is of the form of (1.136),

λ− 2d²θ

ds² = 0. (1.155)

Using the boundary conditions one may find the solutionθ=^λ₄s²−^λl2−4θ_4l ^es, that depends on the parameterλ, which with the constraint J1=
g gives

g= (λ

To present the above result we note that a plane curve r(s) = [x(s), y(s)] with a given curvature function κ(s) =^dθ_ds, may be reconstructed in view of

dx The resulting curve is known as Euler’s spiral (or the spiral of Cornu), and an example of it is shown in Fig. 1.14.

Another example in the same class deals with an elastic rod, the shape of which is expressed as x= x(s), y = y(s) while its angles at the end points (x1,y1) and (x2,y2) areθ1 andθ2, respectively. The constraints of this problem are written by the curve projections onto the x-and y- axes as, see (1.158),

 _l

0 cosθds= x2− x1, ^{ l}

0 sinθds= y2− y1. (1.160)

–0.1 –0.05 0 0.05 0.1 0.15

0.1 0.2 0.3 0.4 0.5 0.6

Figure 1.14:Euler’s spiral for l= 1,θe=³₂π,
g= 1.

y

x

Using two Lagrange multipliers, we write the (1.153) type integral JL=^{ l}

0[(dθ

ds)²+λ1cosθ+λ2sinθ]ds → min. (1.161) The resulting Euler’s equation (1.136) depends on the parametersλ1,λ2and has the form of

−2d²θ

ds² +λ2cosθ−λ1sinθ= 0. (1.162) One may now move to a new coordinate system, say
x,
y, by rotating the plane by an angle, say∆θ, so that the curve of (1.162) will be described as

d²
θ

ds² = µ cos
θ, (1.163)

where
θ=θ+∆θand µ=¹₂

λ²₁+λ²₂. This new problem is now associated with the modified boundary conditions and constraints, which are written with the transformed values(
x1,
y1), (
x2,
y2). Subsequently, the above problem should be solved under the conditions of
θ1=θ1+

∆θat(
x1,
y1) and
θ2=θ2+∆θat(
x2,
y2). The solution, namely,
θ=
θ(s,λ1,λ2), should also fulfill the modified constraints

 _l 0

cos
θds=
x2−
x1, ^{ l}

0

sin
θds=
y2−
y1, (1.164) which enables the determination ofλ1andλ2. For more details regarding the above solution, see (Oprea, 1997).

1.5.3 Functional Based on Function of Several Variables

We next consider the problem J(u) =^

ΩF(x,y,u,ux,uy) → min, (1.165) where F is a continuous function of five arguments. We suppose that the admissible functions u(x,y) belong to the C²class on the two-dimensional domainΩ, while the boundary condition is formulated in terms of a given continuous functionϕ(x,y) along the contour

u=ϕ on ∂Ω. (1.166)

The minimization leads to the Euler’s equation F_,u− ∂

∂x(F,ux) − ∂

∂y(F,uy) = 0, (1.167)

where, obviously,

∂

∂x(F,ux) = F,uxuxu_,xx+ F,uxuu_,x+ F,uxx, ∂

∂y(F,uy) = F,uyuyu_,yy+ F,uyuu_,y+ F,uyy. (1.168) More generally, if the functional F(x1,...,xn,u,ux₁,...,uxn) depends on 2n + 1 (n > 1) vari-ables including the function u(x1,...,xn) of n variables and its first derivatives, then the result-ing Euler’s equation takes the form

F_,u−

∑

ⁿi=1

∂

∂xi(F,u_xi) = 0. (1.169)

Similar to (1.165) case, when higher (second-order) derivatives are included the extreme problem becomes

J(u) =^

GF(x,y,u,ux,uy,uxx,uxy,uyy) → min, (1.170) where the admissible functions u(x,y) belong to the C³class on the domainΩ, and take speci-fied continuous values on the boundary. These calculations lead to the following Euler’s equa-tion

F_,u− ∂

∂x(F,ux) − ∂

∂y(F,uy) + ∂²

∂x²(F,uxx) + ∂²

∂x∂y(F,uxy) + ∂²

∂y²(F,uyy) = 0. (1.171) We shall not treat here explicitly the general Euler’s equation, where the functional F depends on the function u(x1,...,xn) (of n variables) and its higher derivatives (of order ≤ m). Such derivation is implemented in P.1.13, which was used to create the following examples.

Example 1.4 Variational Problem Related to Poisson’s Equation.

Of a particular interest is the special class of Euler’s equations that emerge from the func-tional

F= a44(ux)²− 2a45uxuy+ a55(uy)²+ 2u f (x,y), (1.172) where f is a given function, and the real numbers ai j, i, j = 4,5 satisfy a44 > 0, a55 >

0, a44a₅₅−a²₄₅> 0. In this particular case of (1.165), the minimizing process yields the Euler’s equation known as the generalized Poisson’s equation inΩ:

a₄₄u_,xx− 2a45u_,xy+ a55u_,yy= f (x,y). (1.173) In the isotropic case, where a44= a55, a45= 0, the differential operator of (1.173) is propor-tional to the classic Laplacian∇⁽²⁾=_∂x^∂22+_∂y^∂22 (which is a scalar square of the “nabla” operator

∇= {_∂x^∂,_∂y^∂}), see also (Sokolnikoff, 1983). More general versions of this operator appear in Chapter 3.

Example 1.5 Variational Problem Related to the Biharmonic problem.

One may examine the variational problem of the functional

F= (uxx+ uyy)²− 2 f (x,y)u, (1.174) where f is a given function, and the admissible functions, u(x,y), belong to the C⁴class on the two-dimensional domain,Ω, and are subjected to the boundary conditions on the contour,∂Ω, given by

u=ϕ, d

dnu= h on ∂Ω, (1.175)

where _{d n}^d is a normal derivative and ϕ(x,y), h(x,y) are given functions on∂Ω. In this case of (1.170), we find that the minimization process yields the Euler’s equation known as the fundamental form of the biharmonic equation inΩ:

∇⁽⁴⁾u= f , (1.176)

where∇⁽⁴⁾=_∂x^∂44+ 2_∂x^∂2⁴∂y²+_∂x^∂44 is the biharmonic operator, see (Sokolnikoff, 1983). Replac-ing the functional of (1.174), for example, by

F= c0(uxx)²+ c1uxxuxy+ c2uxxuyy+ c3uxyuyy+ c4(uyy)²+ 2 f (x,y)u, (1.177) where ciare real numbers satisfying certain “positiveness” conditions, one obtains (1.176) with a more general version of the biharmonic operator∇⁽⁴⁾, see P.1.13,

c0

∂⁴

∂x⁴+ c1

∂⁴

∂x³∂y+ c2

∂⁴

∂x²∂y²+ c3

∂⁴

∂x∂y³+ c4

∂⁴

∂x⁴. (1.178)

This operator is extensively treated in Chapter 3.

Example 1.6 Dynamic Torsion of a Beam.

Consider an isotropic beam of circular cross-section, the axis of which is stretched along the z-axis and is acted upon by a torsional moment distribution mz(z,t). The beam is characterized by a torsional rigidity, D(z), and a polar moment of inertia, Ip(z). The twist angle is denoted φ(z,t). In this case V and T may be simplified to (see also Remark 7.5)

V=1 2

 _l

0 D(z)(φ,z)²dz−^{ l}

0

m_z(z,t)φdz, T=1 2

 _l 0

Ip(z)(φ,t)²dz, (1.179) and therefore, according to (1.125)

J(φ) =^{ t2}

t1

(T −V)dt. (1.180)

We then arrive at a functional F(z,t,φ,φz,φt) defined in (1.165) that takes the form F=1

2Ip(z)(φt)²−1

2D(z)(φz)²+ mz(z,t)φ. (1.181) Subsequently, (1.167) yields with the aid of (1.168) the governing Euler’s equation, which may be titled as the “equation of motion” in this case, see P.1.13,

[D(z)φ_,z],z− Ip(z)φ_,tt+ mz(z,t) = 0. (1.182)

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