Basic Theories and Principles of Nonlinear Beam Deformations

Basic Theories and Principles of Nonlinear Beam Deformations

  

1.1 Introduction

The minimum weight criteria in the design of aircraft and aerospace vehicles, coupled with the ever growing use of light polymer materials that can undergo large displacements without exceeding their specified elastic limit, prompted a renewed interest in the analysis of flexible structures that are subjected to static and dynamic loads. Due to the geometry of their deformation, the behavior of such structures is highly nonlinear and the solution of such prob-lems becomes very complex. The solution complexity becomes immense when flexible structural components have variable cross-sectional dimensions along their length. Such members are often used to improve strength, weight and deformation requirements, and in some cases, architects and planners are using variable cross-section members to improve the architectural aesthetics and design of the structure.

In this chapter, the well known theory of elastica is discussed, as well as the methods that are used for the solution of the elastica. In addition, the so-lution of flexible members of uniform and variable cross-section is developed in detail. This solution utilizes equivalent pseudolinear systems of constant cross-section, as well as equivalent simplified nonlinear systems of constant cross-section. This approach simplifies a great deal the solution of such com-plex problems. See, for example, Fertis [2, 3, 5, 6], Fertis and Afonta [1], and Fertis and Lee [4].

This chapter also includes, in a brief manner, important historical devel-opments on the subject and the most commonly used methods for the static and the dynamic analysis of flexible members.

1.2 Brief Historical Developments Regarding the Static and the Dynamic Analysis of Flexible members

By looking into past developments on the subject, we observe that the static analysis of flexible members was basically concentrated in the solution of simple elastica problems. By the term elastica, we mean the determination of the exact shape of the deflection curve of a flexible member. This task was carried out by using various types of analytical (closed-form) methods and techniques, as well as various kinds of numerical methods of analysis, such as the finite element method. Numerical procedures were also extensively used to carry out the complicated mathematics when analytical methods were used.

The dynamic analysis of flexible members was primarily concentrated in the computation of their free frequencies of vibration and their corresponding mode shapes. The mode shapes were, one way or another, associated with large amplitudes. In other words, since the free vibration of a flexible member is taking place with respect to its static equilibrium position, we may have large static amplitudes associated with the static equilibrium position and small vibration amplitudes that take place about the static equilibrium position of the flexible member. We may also have large vibration amplitudes that are nonlinearly connected to the static equilibrium position of the member. This gives some fair idea about the complexity of both static and dynamic problems which are related with flexible member.

A brief history of the research work associated with the static and the dynamic analysis of flexible members is discussed in this section. Since the member, in general, can be subjected to both elastic and inelastic behavior, both aspects of this problem are considered.

The deformed configuration for a uniform flexible cantilever beam loaded with a concentrated load P at its free end is shown in Fig. 1.1a. The free-body diagram of a segment of the beam of length x_o is shown in Fig. 1.1b. Note the di erence in length between the projected length x in Fig. 1.1a, or Fig. 1.1b, and the length x_o along the length of the member. The importance of such lengths, as well as the other items in the figure, are explained in detail later in this chapter and in following chapters of the book.

The basic equation that relates curvature and bending moment in its gen-eral sense was first derived by the brothers, Jacob and Johann Bernoulli, of the well-known Bernoulli family of mathematicians. In their derivation, however, the constant of proportionality was not correctly evaluated. Later on, by fol-lowing a suggestion that was made by Daniel Bernoulli, L. Euler (1707–1783) rederived the di erential equation of the deflection curve and proceeded with the solution of various problems of the elastica [7–10]. J.L. Lagrange (1736– 1813) was the next person to investigate the elastica by considering a uniform cantilever strip with a vertical concentrated load at its free end [8, 10–12]. G.A.A. Plana (1781–1864), a nephew of Lagrange, also worked on the elastica problem [13] by correcting an error that was made in Langrange’s investigation of the elastica. Max Born also investigated the elastica by using variational methods [14].

Since Bernoulli, many mathematicians, scientists, and engineers researched this subject, and many publications may be found in the literature. The methodologies used may be crudely categorized as either analytical (closed-form), or based on finite element techniques. The analytical approaches are

Fig. 1.1. (a) Large deformation of a cantilever beam of uniform cross section. (b) Free-body diagram of a beam element

based on the Euler–Bernoulli law, while in the finite element method the pur-pose is to develop a procedure that permits the solution of complex problems in a straightforward manner.

The more widely used analytical methods include power series, com-plete and incomplete elliptic integrals, numerical procedures using the fourth order Runge–Kutta method, and the author’s method of the equivalent sys-tems which utilizes equivalent pseudolinear systems and simplified nonlinear equivalent systems.

In the power series method, the basic di erential equation is expressed with respect to x_o, i.e.

dθ	=		M		(1.1)
		E1I1f(x0)g(x0)
dx0

where f(x_o) and g(x_o) represent the variation of the moment of inertia I(x_o) and the modulus of elasticity E(x_o), respectively, with respective reference values I₁ and E₁, respectively. Note that for uniform members and linearly elastic materials we have f(x_o) = g(x_o) = 1.00. Also note that θ is the angular rotation along the deformed length of the member as shown in Fig. 1.1a.

For constant E, Eq. (1.1) is usually expressed in terms of the shear force

V_x0  as follows:

EI1	d	f(x0)	dθ	= −Vx0 cos θ	(1.2)
	dx0		dx0

or, for members of uniform I,

d²θ

EI          = −Vx  cos θ                                                        (1.3)

 

dx²₀             ⁰

In order to apply the power series method, we express θ in Eqs. (1.2) and

_{θ (x0) = θ (c) + (x0 − c) θ (c) +} ^{(x0 − c)2} _{θ (c) +} ^{(x0 − c)3} _{θ  (c) + · · ·}

2!3!

 

(1.4)

where c is any arbitrary point along the arc length of the flexible member. The di culties associated with the utilization of power series is that for

variable sti ness members subjected to multistate loadings, θ depends on both x and x_o. The coordinates x and x_o are defined as shown in Fig. 1.1.

The method of elliptic integrals so far is used for simple beams of uniform E and I that are loaded only with concentrated loads. For a uniform beam that is loaded with either a concentrated axial, or a concentrated lateral load, the governing di erential equation is of the form

d²θ

= KΓ (θ)                                                         (1.5)

 

dx²₀

where K is an arbitrary constant, and Γ(θ) is a linear combination of cos θ and sin θ. The nonlinear di erential equation given by Eq. (1.5) may be integrated by using the elliptic integral method, which requires some certain knowledge of elliptic integrals. The di culty associated with this method is that it cannot be applied to flexible members with distributed loads, or to flexible members with variable sti ness.

In the fourth order Runge–Kutta method the nonlinear di erential equa-tions are given in terms of the rotation θ, as shown by Eqs. (1.2) and (1.3). The di culty associated with this method is that for multistate loadings the expressions for the bending moment involve integral equations which are func-tions of the large deformation. In such cases, the application of the Runge– Kutta method becomes extremely di cult. However, if θ is only a function of x_o, then the method can be easily applied.

The method of the equivalent systems, which was developed initially by the author and his collaborators in order to simplify the solution of complicated linear statics and dynamics problems [5,6,15–30], was extended by the author and his students during the past fifteen years for the solution of nonlinear statics and dynamics problems [1–3, 5, 6, 31–51]. Both elastic and inelastic ranges are considered, as well as the e ects of axial compressive forces in both

5

ranges. The solution of the nonlinear problem is given in the form of equivalent pseudolinear systems, or simplified equivalent nonlinear systems, which permit very accurately and rather conveniently the solution of extremely complicated nonlinear problems. A great deal of this work is included in this text. Once the pseudolinear system is derived, linear analysis may be used to solve it because its static or dynamic response is identical, or very closely identical, to that of the original complex nonlinear problem. For very complex nonlinear problems, it was found convenient to derive first a simplified nonlinear equivalent system, and then proceed with pseudolinear analysis. Much of this work is included in this text in detail and with application to practical engineering problems.

We continue the discussion with the research by K.E. Bisshoppe and D.C. Drucker [52]. These two researchers used the power series method to obtain a solution for a uniform cantilever beam, which was loaded (1) by a concentrated load at its free end, and (2) by a combined load consisting of a uniformly distributed load in combination with a concentrated load at the free end of the member. J.H. Lau [53] also investigated the flexible uniform cantilever beam loaded with the combined loading, consisting of a uniformly distributed load along its span and a concentrated load at its free end, by using the power series method. He proved that superposition does not apply to large deflection theory, and he plotted some load–deflection curves for engineering applications. P. Seide [54] investigated the large deformation of an extensional simply supported beam loaded by a bending moment at its end, and he found that reasonable results are obtained by the linear theory for relatively large rotations of the loaded end.

Y. Goto et al. [55] used elliptic integrals to derive a solution for plane elas-tica with axial and shear deformations. H.H. Denman and R. Schmidt [56] solved the problem of large deflection of thin elastica rods subjected to con-centrated loads by using a Chebyshev approximation method. The finite dif-ference method was used by T.M. Wang, S.L. Lee, and O.C. Zienkiewicz [57] to investigate a uniform simply supported beam subjected (1) to a nonsym-metrical concentrated load and (2) to a uniformly distributed load over a portion of its span.

The Runge–Kutta and Gill method, in combination with Legendre Jacobi forms of elliptic integrals of the first and second kind, was used by A. Ohtsuki

[58]   to analyze a thin elastic simply supported beam under a symmetrical three-point bending. The Runge–Kutta method was also used by B.N. Rao and G.V. Rao [59] to investigate the large deflection of a cantilever beam loaded by a tip rotational load. K.T. Sundara Raja Iyengar [60] used the power series method to investigate the large deformation of a simply supported beam under the action of a combined loading consisting of a uniformly distributed load and a concentrated load at its center. At the supports he considered (1) the reactions to be vertical, and (2) the reactions to be normal to the deformed beam by including frictional forces. He did not obtain numerical results. He only developed the equations.

G. Lewis and F. Monasa [61] investigated the large deflection of a thin cantilever beam made out of nonlinear materials of the Ludwick type, and C. Truesdall [62] investigated a uniform cantilever beam loaded with a uni-formly distributed vertical load. R. Frisch-Fay in his book Flexible Bars [63] solved several elastica problems dealing with uniform cantilever beams, uni-form bars on two supports and initially curved bars of uniform cross section, under point loads. He used elliptic integrals, power series, the principle of elas-tic similarity, as well as Kirchho ’s dynamical analogy to solve such problems.

Researchers such as J.E. Boyd [64], H.J. Barton [65], F.H. Hammel, and W.B. Morton [66], A.E. Seames and H.D. Conway [67], R. Leibold [68], R. Parnes [69], and others also worked on such problems. In all the studies described above, with the exception of the research performed by the author and his collaborators, analytical approaches which include arbitrary sti ness variations and arbitrary loadings, were not treated. This is attributed to the di culties involved in solving the nonlinear di erential equations involved. These subjects, by including elastic, inelastic, and vibration analysis, as well as cyclic loadings, are treated in detail by the author, as stated earlier in this section, and much of this work is included in this text and the references at the end of the text.

Because of the di culties involved in solving the nonlinear di erential equations, most of the early investigators turned their e orts to the utilization of the finite element method to obtain solutions. However, in the utilization of the finite element method, di culties were developed, as stated earlier, re-garding the representation of rigid body motions of oriented bodies subjected to large deformations.

K.M. Hsiao and F.T. Hou [70] used the small deflection beam theory, by including the axial force, to solve for the large rotation of frame problems by assuming that the strains are small. The total sti ness matrix was formulated by superimposing the bending, geometric, and linear beam sti ness matri-ces. An incremental iterative method based on the Newton–Raphson method, combined with a constant arc length control method, was used for the solution of the nonlinear equilibrium equations.

Y. Tada and G. Lee [71] adopted nodal coordinates and direction cosines of a tangent vector regarding the deformed configuration of elastic flexible beams. The sti ness matrices were obtained by using the equations of equilib-rium and Galerkin’s method. Their method was applied to a flexible cantilever beam loaded at the free end. T.Y. Yang [72] proposed a matrix displacement formulation for the analysis of elastica problems related to beams and frames. A. Chajes [73] applied the linear and nonlinear incremental methods, as well as the direct method, to investigate the geometrically nonlinear behavior of elas-tic structures. The governing equations were derived for each method, and a procedure outline was provided regarding the plotting of the load–deflection curves. R.D. Wood and O.C. Zienkiewicz [74] used a continuum mechanics approach with a Lagrangian coordinate system and isoparametric element

1.2 The Static and the Dynamic Analysis

7

for beams, frames, arches, and axisymmetric shells. The Newton–Raphson method was used to solve the nonlinear equilibrium equations.

Some considerable research work was performed on nonlinear vibration of beams. D.G. Fertis [2, 3, 5] and D.G. Fertis and A. Afonta [39, 40] applied the method of the equivalent systems to determine the free vibration of variable sti ness flexible members. D.G. Fertis [2, 3], and D.G. Fertis and C.T. Lee [38, 41, 48] developed a method to be used for the nonlinear vibration and instabilities of elastically supported beams with axial restraints. They have also provided solutions for the inelastic response of variable sti ness members subjected to cyclic loadings. D.G. Fertis [49, 51] used equivalent systems to determine the inelastic vibrations of prismatic and nonprismatic members as well as the free vibration of flexible members.

S. Wionowsky-Krieger [75] was the first one to analyze the nonlinear free vibration of hinged beams with an axial force. G. Prathap [76] worked on the nonlinear vibration of beams with variable axial restraints, and G. Prathap and T.K. Varadan [77] worked on the large amplitude vibration of tapered clamped beams. They used the actual nonlinear equilibrium equations and the exact nonlinear expression for the curvature. C. Mei and K. Decha-Umphai

[78]  developed a finite element approach in order to evaluate the geometric nonlinearities of large amplitude free- and forced-beam vibrations. C. Mei [79], D.A. Evensen [80], and other researchers worked on nonlinear vibrations of beams.

Analytical research work regarding the inelastic behavior of flexible struc-tures is very limited. D.G. Fertis [2, 3, 49] and D.G. Fertis and C.T. Lee [2–4, 47, 49] did considerable research work on the inelastic analysis of flexible bars using simplified nonlinear equivalent systems, and they have studied the general inelastic behavior of both prismatic and nonprismatic members. G. Prathap and T.K. Varadan [81] examined the inelastic large deformation of a uniform cantilever beam of rectangular cross section with a concentrated load at its free end. The material of the beam was assumed to obey the stress– strain law of the Ramberg–Osgood type. C.C. Lo and S.D. Gupta [82] also worked on the same problem, but they used a logarithmic function of strains for the regions where the material was stressed beyond its elastic limit.

F. Monasa [83] considered the e ect of material nonlinearity on the re-sponse of a thin cantilever bar with its material represented by a logarithmic stress–strain function. Also J.G. Lewis and F. Monasa investigated the large deflection of thin uniform cantilever beams of inelastic material loaded with a concentrated load at the free end. Again the stress–strain law of the material was represented by Ludwick relation.

In the space age we are living today, much more research and development is needed on these subjects in order to meet the needs of our present and future high technology developments. The need to solve practical nonlinear problems is rapidly growing. Our structural needs are becoming more and more nonlinear. I hope that the work in this text would be of help.

The problem of inelastic vibration received considerable attention by many researchers and practicing engineers. Bleich [86], and Bleich and Salvadory [87], proposed an approach based on normal modes for the inelastic analysis of beams under transient and impulsive loads. This approach is theoretically sound, but it can be applied only to situations where the number of possible plastic hinges is determined beforehand, and where the number of load rever-sals is negligible. Baron et al. [88], and Berge and da Deppo [89], solved the required equation of motion by using methods that are based on numerical in-tegration. This, however, involved concentrated kink angles which are used to correct for the amount by which the deflection of the member surpasses the ac-tual elastic–plastic point. The methodology is simple, but the actual problem may become very complicated because multiple correction angles and several hinges may appear simultaneously. Lee and Symonds [90], have proposed the method of rigid plastic approximation for the deflection of beams, which is valid only for a single possible yield with no reversals. Toridis and Wen [91], used lumped mass and flexibility models to determine the response of beams.

In all the models developed in the above references, the precise location of the point of reversal of loading is very essential. A hysteretic model where the location of the loading reversal point is not required and where the reversal is automatically accounted for, was first suggested by Bonc [92] for a spring-mass system, and it was later extended by Wen [93] and by Iyender and Dash [94]. In recent years Sues et al. [95] have provided a solution for a single degree of freedom model for degrading inelastic model. This work was later extended by Fertis [2, 3] and Fertis and Lee [38], and they developed a model that adequately describes the dynamic structural response of variable and uniform sti ness members subjected to dynamic cyclic loadings. In their work, the material of the member can be stressed well beyond its elastic limit, thus causing the modulus E to vary along the length of the member. The derived di erential equations take into consideration the restoring force behavior of such members by using appropriate hysteretic restoring force models.

The above discussion, is not intended to provide a complete historical treatment of the subject, and the author wishes to apologize for any uninten-tional omission of the work of other investigators. It provides, however, some insight regarding the state of the art and how the ideas regarding these very important subjects have been initiated.

1.3 The Euler–Bernoulli Law of Linear and Nonlinear Deformations for Structural Members

From what we know today, the first public work regarding the large deforma-tion of flexible members was given by L. Euler (1707–1783) in 1744, and it was continued in the appendix of his well known book Des Curvis Elastics [7]. According to Euler, when a member is subjected to bending, we cannot neglect the slope of the deflection curve in the expression of the curvature unless the deflections are small. Euler has published about 75 substantial volumes, he was a dominant figure during the 18th century, and his contributions to both pure and applied mathematics made him worthy of inclusion in the short list of giants of mathematics – Archimedes (287–212 bc), I. Newton (1642–1727), and C. Gauss (1777–1855).

We should point out, however, that the development of this theory took place in the 18th century, and the credits for this work should be given to Jacob Bernoulli (1654–1705), his younger brother Johann Bernoulli (1667– 1748), and Leonhard Euler (1707–1783). Both Bernoulli brothers have con-tributed heavily in the mathematical sciences and related areas. They also worked on the mathematical treatment of the Greek problems of isochrone, brahistochrone, isoperimetric figures, and geodesies, which led to the devel-opment of the new calculus known as the calculus of variations. Jacob also introduced the word integral in suggesting the name calculus integrals. G.W. Leibniz (1646–1716) used the name calculus summatorius for the inverse of the calculus differentialis.

The Euler–Bernoulli law states that the bending moment M is proportional to the change in the curvature produced by the action of the load. This law

may be written mathematically as follows:
	1		=	dθ	=		M	(1.6)
	r					EI
				dx0

where r is the radius of curvature, θ is the slope at any point x_o, where x_o is measured along the arc length of the member as shown in Fig. 1.1a, E is the modulus of elasticity, and I is the cross-sectional moment of inertia.

Figure 1.1a depicts the large deformation configuration of a uniform flexi-ble cantilever beam, and Fig. 1.1b illustrates the free-body diagram of a seg-ment of the beam of length x_o. Note the di erence in length size between x and x_o in Fig. 1.1b. For small deformations we usually assume that x = x_o. For small deformations we can also assume that L = L_o in Fig. 1.1a, because under this condition the horizontal displacement ∆ of the free end B of the cantilever beam would be small.

In rectangular x, y coordinates, Eq. (1.6) may be written as

1

=

y

= −

M

(1.7)

r

1 + (y )2  3/2

EI

where

y  =

dy

(1.8)

dx

y  =

d2y

(1.9)

dx2

and y is the vertical deflection at any x. We also know that

y = tan θ  or  θ = tan−1 y

(1.10)

Equation (1.7) is a second order nonlinear di erential equation, and its exact solution is very di cult to obtain. This equation shows that the deflections are no longer a linear function of the bending moment, or of the load, which means that the principle of superposition does not apply. The consequence is that every case that involves large deformations must be solved separately, since combinations of load types already solved cannot be superimposed. The consequences become more immense when the sti ness EI of the flexible mem-ber varies along the length of the member. We discuss this point of view in greater detail, with examples, later in this chapter.

When the deformation of the member is considered to be small, y in Eq. (1.7) is small compared to 1, and it is usually neglected. On this basis, Eq. (1.7) is transformed into a second order linear di erential equation of the form

1	= y  = −	M	(1.11)
r		EI

The great majority of practical applications are associated with small de-formations and, consequently, reasonable results may be obtained by using Eq. (1.11). For example, if y = 0.1 in Eq. (1.7), then the denominator of this equation becomes

1 + (0.1)2  3/2

= 0.985

(1.12)

which suggests that we have an error of only 1.52% if Eq. (1.11) is used.

1.4 Integration of the Euler–Bernoulli Nonlinear Differential Equation

Figure 1.2 depicts the large deformation configuration of a tapered flexible cantilever beam loaded with a concentrated vertical load P at its free end. In this figure, y is the vertical deflection of the member at any x, and θ is its rotation at any x. We also have the relations

y =			dy		(1.13)
			dx
d2y
y  =					(1.14)
		dx
and
y = tan θ  or			θ = tan−1 y		(1.15)

In rectangular x, y coordinates, the Euler–Bernoulli law for large defor-mation produced by bending may be written as [2, 3] (see also Eq. (1.7):

We integrate Eq. (1.17) by making changes in the variables. We let y = p and then y = p . Thus, from Eq. (1.16), we obtain

	p		= λ (x)		(1.18)
[1 + p2]3/2
where			Mx
	λ (x) =				(1.19)
		ExIx
Now we rewrite Eq. (1.18) as follows:
	dp/dx		= λ (x)		(1.20)
[1 + p2]3/2

By multiplying both sides of Eq. (1.20) by dx and integrating once, we find

	dp
		=	λ (x) dx	(1.21)
	[1 + p2]3/2
We can integrate Eq. (1.21) by making the following substitutions:
	p = tan θ			(1.22)
	dp = sec2 θ dθ			(1.23)

By using the beam element shown in Fig. 1.2b and applying the Pythagorean theorem, we find

(ds)2 = (dx)2 + (dy)2

or

ds =  (dx)2 + (dy)2  1/2

(1.24)

ds

=

dy

2

1/2

1/2

1 +

=  1 + (tan θ)2

dx

dx

(1.25)

Thus,

=

1 + p2  1/2

cos θ =

dx

=

1

(1.26)

ds

[1 + p2]1/2

and from Eq. (1.22), we find

sin θ = p cos θ =

p

(1.27)

[1 + p2]1/2

  

By substituting Eqs. (1.22) and (1.23) into Eq. (1.21) and also making use of Eqs. (1.26) and (1.27), we find

		sec2 θ dθ
		1 + sin2		θ
		2				3/2	=   λ (x) dx	(1.28)
			cos	θ

or, by performing trigonometric manipulations, Eq. (1.28) reduces to the fol-lowing equation:

cos θ dθ =   λ (x) dx

(1.29)

Integration of Eq. (1.29), yields

sin θ = ϕ (x) + C

(1.30)

where the function ϕ(x) represents the integration of λ(x).

Equation (1.30) may be rewritten in terms of p and y by using Eq. (1.27). Thus,

		p	= ϕ (x) + C		(1.31)
	[1 + p2]1/2
		y		= ϕ (x) + C	(1.32)
	1 + (y )2  1/2

where C is the constant of integration which can be determined from the boundary conditions of the given problem. If we will solve Eq. (1.32) for y (x), we obtain the following equation:

		ϕ (x) + C
y (x) =			(1.33)

1 − [ϕ (x) + C]²

Integration of Eq. (1.33) yields the large deflection y(x) of the member. Thus,

y (x) =	x		ϕ (η) + C	dη	(1.34)

⁰            1 − [ϕ (η) + C]²

This shows that when M_x/E_xI_x is known and it is integrable, then the Euler–Bernoulli equation may be solved directly for y (x) as illustrated in the solution of many flexible beam problems in [2, 3]. In the same references, utilization of pseudolinear equivalent systems is made, which simplify a great deal the solution of such problems. A numerical integration may be also used for Eq. (1.34), or Eq. (1.16), by using the Simpson’s rule discussed in the following section of this text.

1.5 Simpson’s One-Third Rule

Simpson’s one-third rule is one of the most commonly used numerical method to approximate integration. It is used primarily for cases where exact inte-gration is very di cult or impossible to obtain. Consider, for example, the integral

b

δ =	f (x) dx	(1.35)
	a

between the limits a and b. If we divide the integral between the lim-its x=a and x=b into n equal parts, where n is an even number, and if y₀, y₁, y₂, . . . , y_n−1, y_n are the ordinates of the curve y = f(x), as shown in Fig. 1.3, then, according to Simpson’s one-third rule we have

a	b	f (x) dx = 3		(y0 + 4y1 + 2y2 + 4y3 + · · · + 2yn−2 + 4yn−1 + yn)
			λ

where

_{λ  =} b − a n

 

Simpson’s rule provides reasonably accurate results for practical applications. Let it be assumed that it is required to determine the value δ of the integral

L

δ =	x2dx	(1.38)
	0

We divide the length L into 10 equal segments, yielding λ = 0.1L. By applying Simpson’s rule given by Eq. (1.36), and noting that y = f(x) = x², we find

			1.5 Simpson’s One-Third Rule	15
	1L
δ =	0.		(1) (0)2 + (4) (0.1)2 + (2) (0.2)2 + (4) (0.3)2 + (2) (0.4)2
	3			L2
+ (4) (0.5)2 + (2) (0.6)2 + (4) (0.7)2 + (2) (0.8)2 + (4) (0.9)2 + (1) (1)2

₌    L³ 3

 

Note that for λ = 0.1L, the values of f(x) are y_o = (0)², y₁ = (0.1L)², y₂ = (0.2L)², and so on. In this case, the exact value of the integral is ob-tained.

As a second example, let it be assumed that it is required to find the value

δ of the integral

L

δ =	x3dx	(1.39)
	0

Again, we subdivide the length L into 10 equal segments, yielding λ = 0.1L. In this case, the Simpson’s one-third rule yields

δ = ^0.1L (1) (0)3 + (4) (0.1)3 + (2) (0.2)3 + (4) (0.3)2 + (2) (0.4)3 3

 

+ (4) (0.5)³ + (2) (0.6)³ + (4) (0.7)³ + (2) (0.8)³ + (4) (0.9)³ + (1) (1)³  L³

₌   0.75L⁴  ₌ L⁴

34

 

The exact value of the integral is obtained again in this case.

More complicated integrals may be also evaluated in a similar manner, as shown later in this text. For example, let it be assumed that it is required to determine the length L of a flexible bar given by the integral

L =	0	840			1 + (y )2  1/2	dx		(1.40)
where					G (x)
y (x) =								(1.41)
				1 − [G (x)]2  1/2
and
G (x) = 1.111 (10)−6 x2 − 0.783922								(1.42)

Equation (1.40) is an extremely important equation in nonlinear mechanics for the analysis of flexible bars subjected to large deformations [2,3]. It relates the length L of the bar with the slope y at points along its deformed shape.

For illustration purposes, we use here n=10, and from Eq. (1.37) we obtain

λ  = ^{840 − 0} = 84 10

 

From Eq. (1.40), we note that

	f (x) =		1 + (y )2  1/2		(1.43)
The  values  of	f(x)  at  x	=  0, 84, 168, . . ., 840		are	designated  as
yo, y1, y2, . . ., y10,	respectively,	and they are obtained		by	using Eq. (1.43)

in conjunction with Eqs. (1.42) and (1.41). For example, for x=0, we have

G (0)

= −0.783922

y (0)

=

−0.783922

=

−

1.262641

1 − (−0.783922)2

y0 = f (0) =

1 + (−1.262641)2 = 1.610671

For x=84 in., we have

G (84) = 1.111 (10)6 (84)2 − 0.783922 = 0.776083

y (84) =

−0.776083

=

−

1.230645

1 − (−0.776083)2

y1 = f (84) =

1 + (−1.230645)2 = 1.585713

In a similar manner, the remaining points y₂, y₃, . . ., y₁₀, can be deter-mined. On this basis, Eq. (1.36) yields

L = ⁸⁴₃ [1.610671 + (4) (1.585713) + (2) (1.518561) + (4) (1.426963)

 

+   (2) (1.328753) + (4) (1.236242) + (2) (1.156021) + (4) (1.090986) + (2) (1.042370) + (4) (1.011280) + 1]

=     ⁸⁴₃ (38.106817) = 1, 067 in.

 

It should be realized that the value obtained for L is an approximate one, but better accuracy can be obtained by using larger values for the parameter n in Eq. (1.37). For practical applications, however, the design engineer usually has a fair idea about their accuracy requirements, and satisfactory and safe designs can be obtained by using approximate solutions.

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