identification of boundary conditions of tapered beam-like structures using static flexibility...

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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 25 (2011) 2484–2500

0888-32

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Identification of boundary conditions of tapered beam-likestructures using static flexibility measurements

Le Wang n, Zhichun Yang

School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China

a r t i c l e i n f o

Article history:

Received 12 November 2010

Received in revised form

28 March 2011

Accepted 2 April 2011Available online 19 April 2011

Keywords:

Damage detection

Boundary condition identification

Static flexibility

Direct method

Tapered beam-like

70/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ymssp.2011.04.003

esponding author. Tel./fax: þ86 29 8846046

ail addresses: [email protected], wanglnp

a b s t r a c t

The integrity and reliability of the beam-like structures are dependent in part on their

boundary conditions, which can vary with time due to damage or aging, thus the

identification of boundary conditions might be one of the most significant aspects for

damage detection of such structures. This paper investigates a direct method for

identifying the boundary conditions of tapered beam-like structures using static

flexibility measurements. The beam is modeled by a flexible tapered beam, which is

constrained at one end by translational and rotational springs. The translational and

rotational springs are utilized to simulate the boundary conditions of the tapered beam,

and the purpose of this paper is to identify the stiffnesses of the translational and

rotational springs, i.e. translational root stiffness and rotational root stiffness. It is

theoretical proved that the static flexibility measured on the beam can be expressed as

a function of the flexural rigidity of the beam at its constrained end, translational root

stiffness and rotational root stiffness. Then, a set of linear equations for identifying the

translational and rotational root stiffnesses are formed by three or more different static

flexibility measurements. Finally, the proposed method is validated using both

simulative and experimental examples.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The integrity and safety of some structure such as cantilevers are seriously affected by the damages occurred on theirsupports (i.e. boundary conditions), thus the boundary condition identification might be one of the most significant aspectfor damage detection of such structures. However, many of the existed investigations in the field of structural damagedetection were focused on the damages occurred on the structure itself, and only a few methodologies were proposed foridentifying the boundary conditions.

In the field of boundary condition identification, two model updating methods, i.e. characteristic equation basedmethod and sensitivity analysis based method, were utilized by Pabst and Hagedorn [1]. In the characteristic equationbased method, the boundary conditions are obtained by solving the nonlinear characteristic equations, which formed bythe properties of the structure, the measured natural frequencies and the unknown boundary conditions, such as theinvestigations by Ahmadian et al. [2], Yi and Liu [3], Deng et al. [4]. In the sensitivity analysis based method, an iterativemethod is employed to obtain the boundary conditions based on the sensitivities of natural frequencies and mode shapeswith respect to changes in boundary conditions, such as the investigation by Liu and Yi [5]. Besides the model updating

ll rights reserved.

1.

[email protected] (L. Wang).

www.elsevier.com/locate/jnlabr/ymssp

dx.doi.org/10.1016/j.ymssp.2011.04.003

mailto:[email protected]

dx.doi.org/10.1016/j.ymssp.2011.04.003

Nomenclature

A measurement position matrixb vector of measured static flexibilitiesb0 width of the cross section of the beam at the

constrained endb1 width of the cross section of the beam at the

free endC coefficients determined by the boundary

conditionsD coefficients determined by the boundary

conditionsE Young’s modulus of the beamF force applied on the beamEI0 flexural rigidity of the beam at the

constrained endh0 depth of the cross section of the beam at the

constrained endh1 depth of the cross section of the beam at the

free endHstatic,hl static flexibility between the measurement

positions Lh and Ll

Hmeasstatic ‘‘measured’’ static flexibility in the simulative

examplesHtrue

static true value of the static flexibility in thesimulative examples

I(x) second moment of the area of the crosssection of the beam at position x

kr stiffness of rotational spring, i.e. rotationalroot stiffness

kt stiffness of translational spring, i.e. transla-tional root stiffness

l length of the beamLmax maximum dimensionless measurement

position

Lmin minimum dimensionless measurementposition

Lf dimensionless force measurement positionlf force measurement positionLh maximum of Lf and Ls

Ll minimum of Lf and Ls

Ls dimensionless deflection measurementposition

ls deflection measurement positionM(x) moment at the deflection measurement posi-

tion x

N number related to the truncated series, i.e.Nþ1 is the number of truncated series

p true value of the identified parametersW weighting matrixX dimensionless position along the beamx position along the beamx vector of unknown parametersY static lateral deflectionbb degree of taper in the width directionbh degree of taper in the depth directionbr dimensionless rotational root stiffnessbt dimensionless translational root stiffnessd random variable with zero mean and unit

varianceZ noise level on the measured static flexibilitykEI,hl coefficient dependent on dimensionless mea-

surement position and the tapered degreeskr,hl coefficient dependent on dimensionless mea-

surement positionm mean value of the identified parameterss standard deviation of the identified

parameters9s/m9 coefficient of variation

L. Wang, Z. Yang / Mechanical Systems and Signal Processing 25 (2011) 2484–2500 2485

methods, Waters et al. [6] proposed a method for boundary condition identification based on static stiffness measure-ments. In their method, the beam is modeled as a uniform rigid beam that is constrained by collocated equivalenttranslational and rotational springs, and the translational and rotational root stiffnesses (i.e. boundary conditions) areidentified by quasi-static stiffness measurements obtained from impact tests.

This paper considers a flexible tapered beam which is constrained at one end by a translational spring and a rotationalspring. By a similar approach to that used by Waters et al. [6], we deduce the theoretical formula between the staticflexibility and the boundary conditions based on the static equilibrium equation of the tapered beam. Then a directmethod is employed to obtain the boundary conditions using different combinations of the static flexibility measurements.

The layout of this paper is as follows: Section 2 describes the theory of the identification method and the principlefor choosing measurement positions; simulative and experimental examples are investigated in Sections 3 and 4,respectively; finally, conclusions are summarized in Section 5.

2. Theory of the identification method

Fig. 1 shows a tapered Euler–Bernoulli beam of rectangular cross section and length l which is constrained at one endby translational and rotational springs of stiffness kt and kr. Y(x) is the static lateral deflection, E is Young’s modulus, I(x) isthe second moment of the area of the cross section.

Suppose that the taper of the beam is such that both the width b(x) and depth h(x) of the beam vary linearly along itslength, i.e.

bðxÞ ¼ b0 1�bb

x

l

h ið1Þ

Fig. 1. Tapered Euler–Bernoulli beam with flexible boundary conditions at one end, (a) beam; (b) cross section of the constrained end; (c) cross section of

the free end.

L. Wang, Z. Yang / Mechanical Systems and Signal Processing 25 (2011) 2484–25002486

hðxÞ ¼ h0 1�bh

x

l

h ið2Þ

where bb¼1�b1/b0 and bh¼1�h1/h0 are the degrees of taper in each direction, respectively. For example bb¼bh¼0corresponds to a uniform beam, and bb¼bh¼1 corresponds to a pyramidal beam. Then, the second moment of area for asolid rectangular cross section is

IðxÞ ¼bðxÞ½dðxÞ�3

12¼ I0 1�bb

x

l

h i1�bh

x

l

h i3

ð3Þ

where I0 ¼ b0h30=12 is the second moment of area of the section at the constrained end (i.e. x¼0) of the beam.

The boundary conditions, in the presence of constraints with the translational and rotational springs of stiffness kt andkr, are given by [7,8]

EIðxÞ@2yðx,tÞ

@x2x ¼ 0

¼ kr@yðx,tÞ

@x x ¼ 0

�� ð4Þ

@

@xEIðxÞ

@2yðx,tÞ

@x2

!x ¼ 0

¼�ktyðx,tÞ9x ¼ 0

�� ð5Þ

EIðxÞ@2yðx,tÞ

@x2x ¼ l

¼ 0

�� ð6Þ

@

@xEIðxÞ

@2yðx,tÞ

@x2

!x ¼ l

¼ 0

�� ð7Þ

2.1. Static equilibrium equation

The relationship between the static lateral deflection, flexural rigidity of the beam and moment applied on the beam is

EIðxÞd2YðxÞ

dx2¼MðxÞ ð8Þ

Supposing that the force F(lf) is applied at the position lf, and based on the equilibrium of moment, the moment at thedeflection measurement position x (supposing xr lf) can be calculated by

MðxÞ ¼ Fðlf Þðlf�xÞ ð9Þ


Substituting Eqs. (3) and (9) into Eq. (8), we obtain

d2YðxÞ

dx2¼

Fðlf Þðlf�xÞ

EI0 1�bbxl

� �1�bh

xl

� �3 ð10Þ

Introducing the dimensionless length X¼x/l, Lf¼ lf/l, Eq. (10) can be rewritten as

d2YðXÞ

dX2¼

FðLf Þl3

EI0

Lf�X

ð1�bbXÞð1�bhXÞ3ð11Þ

Using the power series, one can obtain

1

1�bbX¼X1n ¼ 0

ðbbXÞn,1

1�bhX¼X1n ¼ 0

ðbhXÞn ð12Þ

Then, based on Cauchy product, one can achieve

1

ð1�bbXÞð1�bhXÞ3¼

X1n ¼ 0

XnXn

m ¼ 0

bmb b

n�mh

! ! X1n ¼ 0

Xnðnþ1Þbnh

!

¼X1n ¼ 0

Xn

k ¼ 0

XkXk

m ¼ 0

bmb b

k�mh

!Xn�kðn�kþ1Þbn�k

h

� �

¼X1n ¼ 0

XnXn

k ¼ 0

Xk

m ¼ 0

ðn�kþ1Þbmb b

n�mh

!ð13Þ

Substituting Eq. (13) into (11), one obtains

d2YðXÞ

dX2¼

FðLf Þl3

EI0

X1n ¼ 0

ðLf Xn�Xnþ1ÞXn

k ¼ 0

Xk

m ¼ 0

ðn�kþ1Þbmb b

n�mh

!ð14Þ

Then, the static lateral deflection Y(X) can be easily obtained by evaluating the above differential function

YðXÞ ¼FðLf Þl

3

EI0

X1n ¼ 0

Lf Xnþ2

ðnþ1Þðnþ2Þ�

Xnþ3

ðnþ2Þðnþ3Þ

!Xn

k ¼ 0

Xk

m ¼ 0

ðn�kþ1Þbmb b

n�mh

!þCXþD ð15Þ

Thus, it is easily obtained

d3YðXÞ

dX3X ¼ 0

¼FðLf Þl

3ð�1þLfbbþ3LfbhÞ

EI0

�� ð16Þ

d2YðXÞ

dX2X ¼ 0

¼FðLf ÞL

3Lf

EI0

�� ð17Þ

dYðXÞ

dX X ¼ 0

¼ C

�� ð18Þ

YðXÞ X ¼ 0 ¼Dj ð19Þ

Using the dimensionless length X¼x/l, the boundary conditions in Eqs. (4) and (5) can be rewritten as

EI0

l2d2YðXÞ

dX2X ¼ 0

¼kr

l

dYðXÞ

dX

��X ¼ 0

�� ð20Þ

EI0

l3d3YðXÞ

dX3X ¼ 0

�EI0ðbbþ3bhÞ

l3d2YðXÞ

dX2X ¼ 0

¼�ktYðXÞ X ¼ 0j

�� ð21Þ

Substituting Eqs. (17) and (18) into Eq. (20), one obtains

C ¼FðLf Þl

2Lf

krð22Þ

Then, substituting Eqs. (16) and (19) into Eq. (21), one obtains

D¼FðLf Þ

ktð23Þ


Then, the static lateral deflection Y(X) in Eq. (15) can be expressed as

YðXÞ ¼FðLf Þl

3

EI0

X1n ¼ 0

Lf Xnþ2

ðnþ1Þðnþ2Þ�

Xnþ3

ðnþ2Þðnþ3Þ

!Xn

k ¼ 0

Xk

m ¼ 0

ðn�kþ1Þbmb b

n�mh

!þ

FðLf Þl2Lf

krXþ

FðLf Þ

ktð24Þ

As xr lf is assumed in the above deriving, Eq. (24) has just satisfied the situation when XrLf. When xZ lf, it is easilyobtained that

YðxÞ ¼ Yðlf ÞþY 0ðlf Þðx�lf Þ ð25Þ

Using the dimensionless length X¼x/l, Lf¼ lf/l, Eq. (25) can be expressed as

YðXÞ ¼ YðLf ÞþY 0ðLf ÞðX�Lf Þ ð26Þ

Then, based on Eqs. (24) and (26), the static lateral deflection Y(Ls) can be expressed as

YðLsÞ ¼

FðLf Þl3

EI0

X1n ¼ 0

Lf Lsnþ2

ðnþ1Þðnþ2Þ�

Lsnþ3

ðnþ2Þðnþ3Þ

!Xn

k ¼ 0

Xk

m ¼ 0

ðn�kþ1Þbmb b

n�mh

!

þFðLf ÞLf Ls

kr=l2þ

FðLf Þ

kt, 0rLsrLf

FðLf Þl3

EI0

X1n ¼ 0

LsLnþ2f

ðnþ1Þðnþ2Þ�

Lfnþ3

ðnþ2Þðnþ3Þ

!Xn

k ¼ 0

Xk

m ¼ 0

ðn�kþ1Þbmb b

n�mh

!

þFðLf ÞLf Ls

kr=l2þ

FðLf Þ

kt, Lf rLsr1

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

ð27Þ

where Ls¼ ls/l is the dimensionless deflection measurement position, 0rLsr1, ls is the deflection measurement position.Based on Eq. (27), one can easily obtain the static flexibility Hstatic,s,f at position Ls when force is applied on position Lf, i.e.

Hstatic,s,f ¼YðLsÞ

FðLf Þ¼

l3

EI0

X1n ¼ 0

Lf Lsnþ2

ðnþ1Þðnþ2Þ�

Lsnþ3

ðnþ2Þðnþ3Þ

!Xn

k ¼ 0

Xk

m ¼ 0

ðn�kþ1Þbmb b

n�mh

!

þLf Ls

kr=l2þ

1

kt, 0rLsrLf

Hstatic,f ,s, Lf rLsr1

8>>>>>>><>>>>>>>:

ð28Þ

Supposing that Lh ¼maxðLf ,LsÞ, Ll ¼minðLf ,LsÞ, Eq. (28) can be further simplified as

Hstatic,hl ¼kEI,hl

3EI0=l3þ

kr,hl

kr=l2þ

1

ktð29Þ

where Hstatic,hl indicates the static flexibility between measurement positions Lh and Ll (no matter which is the forcemeasurement position or deflection measurement position), kEI,hl is a function of dimensionless measurement positionsand the tapered degrees of the beam and kr,hl is just a function of dimensionless measurement positions, i.e.

kEI,hl ¼X1n ¼ 0

3LhLlnþ2

ðnþ1Þðnþ2Þ�

3Llnþ3

ðnþ2Þðnþ3Þ

!Xn

k ¼ 0

Xk

m ¼ 0

ðn�kþ1Þbmb b

n�mh

!ð30Þ

kr,hl ¼ LhLl ð31Þ

Based on Eq. (30), we can easily obtain the coefficient kEI,hl for some particular beam, as follows:

1.
When bh¼0, bnba0 and considering 00
¼1, then

kEI,hl ¼X1n ¼ 0

bnb

3LhLlnþ2

ðnþ1Þðnþ2Þ�

3Llnþ3

ðnþ2Þðnþ3Þ

! !ð32Þ

2.
When bb¼0, bha0 and considering 00¼1, then
kEI,hl ¼X1n ¼ 0

ðnþ1Þðnþ2Þbnh

2

3LhLlnþ2

ðnþ1Þðnþ2Þ�

3Llnþ3

ðnþ2Þðnþ3Þ

! !ð33Þ

3.
When bb¼bh¼b, then
kEI,hl ¼X1n ¼ 0

ðnþ1Þðnþ2Þðnþ3Þbn

6

3LhLlnþ2

ðnþ1Þðnþ2Þ�

3Llnþ3

ðnþ2Þðnþ3Þ

! !ð34Þ


4.
When bb¼bh¼0 (i.e. uniform beam), and considering 00¼1, then
kEI,hl ¼3LhL2

l

2�

L3l

2ð35Þ

In practice, the coefficient kEI,hl cannot be determined exactly, and it can only be approximated by truncated series withNþ1 terms approximation, i.e.

k½N�EI,hl ¼XN

n ¼ 0

3LhLlnþ2

ðnþ1Þðnþ2Þ�

3Llnþ3

ðnþ2Þðnþ3Þ

!Xn

k ¼ 0

Xk

m ¼ 0

ðn�kþ1Þbmb b

n�mh

!ð36Þ

where superscript [N] indicates the Nþ1 terms’ approximation, and N can be determined by k½N�EI,hl�k½N�1�EI,hl

�� re, where e is amuch small positive number.

Supposing that we have measured several different static flexibilities Hstatic,1,Hstatic,2,y,Hstatic,n (nZ3) at differentcouples of force and deflection measurement positions ll1,lh1,ll2,lh2,y,lln,lhn. Then referring to Eq. (29), we can obtain a set oflinear equations

Ax¼ b ð37Þ

where

A¼

k½N�EI,h1l1 kr,h1l1 1

k½N�EI,h2l2 kr,h2l2 1

^ ^ ^

k½N�EI,hnln kr,hnln 1

2666664

3777775

is the measurement position matrix

b¼ fHstatic,1,Hstatic,2,. . .,Hstatic,ngT

is the vector of measured static flexibilitiesand

x¼1

3EI0=l3,

1

kr=l2,

1

kt

� �T

is the vector of unknown parameters.Supposing that the tapered degrees of the beam can be measured in advance (i.e. k½N�EI,hl is merely dependent on the

dimensionless measurement positions), the flexural rigidity at the constrained end EI0, translational root stiffness kt androtational root stiffness kr can be calculated directly by Eq. (37) using least squared method, as follows:

x¼ ðAT WAÞ�1AT Wb ð38Þ

where W is the diagonal weighting matrix for each measurement position. As the condition number of a matrix can beminimized by applying equal row weighting to the matrix [6], the weighting matrix W in this paper is determined byapplying equal row weighting to the measurement position matrix, i.e. the diagonal element of the weighting matrix iswii ¼ 1=ðmaxf9Ai19,9Ai29,9Ai39gÞ

2.

2.2. How to choose the measurement positions

When the tapered degrees of the beam can be measured in advance, the measurement position matrix is just function ofmeasurement positions. Therefore, one should choose the measurement positions in order to obtain a well conditionedmeasurement position matrix. For convenience, supposing that the n measurement positions are located by identical intervalbetween the maximum dimensionless measurement position Lmax and the minimum dimensionless measurement positionLmin, three different measurement position setups (including point static flexibility measurements and transfer static flexibilitymeasurements, see following) are analyzed, and then the better measurement position setup is selected according to thecondition number of the measurement position matrix. The three different measurement position setups are as follows:

(1)
Point static flexibility measurements
Llj ¼ Lhj ¼ LminþLmax�Lmin

ðn�1Þðj�1Þ, j¼ 1,2,. . .,n ð39Þ

(2)
Transfer static flexibility measurements with the fixed maximum measurement position
Lhj ¼ Lmax

Llj ¼ LminþLmax�Lminðn�1Þ ðj�1Þ

, j¼ 1,2,. . .,n ð40Þ

Fig. 2(c) 6


Transfer static flexibility measurements with the fixed minimum measurement position
(3)
Llj ¼ Lmin

Lhj ¼ LminþLmax�Lminðn�1Þ ðj�1Þ

, j¼ 1,2,. . .,n ð41Þ

Then, the condition numbers of the measurement position matrices can be easily analyzed for different measurementposition setups, and the better measurement position setup should be the measurement position setup with the smaller
condition number. In order to illustrate the principle for choosing the better measurement positions, two particular beams,i.e. uniform beam (bb¼bh¼0) and pyramidal beam (bb¼bh¼1), are analyzed.
2.2.1. Uniform beam

Figs. 2–4 show the condition numbers (calculated by 2-norm) of the measurement position matrices for differentmeasurement setups for the uniform beam (Note: the condition numbers are calculated after applying equal rowweighting to the measurement position matrix in Eq. (37).) As shown in Fig. 2, with the increase of the maximummeasurement position and the decrease of the minimum measurement position, the condition number of themeasurement position matrix decreases; with the increase of the number of measurement positions, the conditionnumber of the measurement position matrix does not decrease but increases a little. It tells us that the additional staticflexibilities do not provide more useful information for improving the solution of the inverse problem. Thus, threemeasurement positions are enough for the inverse problem if both the maximum and minimum measurement positionsare ascertained. The similar curves are shown in Fig. 3, but the condition numbers of the measurement position matrices inFig. 3 are a little bigger than the condition numbers of the corresponding measurement position matrices in Fig. 2. The verybig condition numbers are occurred in Fig. 4, and it is clear that the measurement position matrices are ill conditioned.Then, it can be concluded that the better measurement position setup for the uniform beam should be the point staticflexibility measurements with a bigger maximum measurement position and a smaller minimum measurement position.However, as it is a little easier to measure the transfer static flexibilities than the point static flexibilities, meanwhile thecondition numbers of the measurement position matrices of the transfer static flexibility measurements with the fixedmaximum measurement position are nearly the same as the condition numbers of the corresponding measurement

. Condition numbers of point static flexibility measurements for uniform beam, (a) 3 measurement positions; (b) 4 measurement positions;

measurement positions; (d) 8 measurement positions.

Fig. 3. Condition numbers of transfer static flexibility measurements with the fixed maximum measurement position, (a) 3 measurement positions;

(b) 4 measurement positions; (c) 6 measurement positions; (d) 8 measurement positions.

Fig. 4. Condition numbers of transfer static flexibility measurements with the fixed minimum measurement position, (a) 3 measurement positions;

(b) 4 measurement positions; (c) 6 measurement positions; (d) 8 measurement positions.


Fig. 5. Condition numbers of point static flexibility measurements for pyramidal beam (bb¼bh¼1), (a) 3 measurement positions; (b) 4 measurement

positions; (c) 6 measurement positions; (d) 8 measurement positions.


position matrices of the point static flexibility measurements, the transfer static flexibility measurements with the fixedmaximum measurement position also can be adopted in the inverse problem.

2.2.2. Pyramidal beam

Figs. 5–7 show the condition numbers of the measurement position matrices for different measurement setups forpyramidal beam (bb¼bh¼1) (Note: the condition numbers are calculated after applying equal row weighting to themeasurement position matrix in Eq. (37).) In these figures, the truncated number N is determined by setting e¼10�15. Asthe N may be different for different measurement positions, the maximum of these N is adopted in the final calculation.

As shown in these figures, the curves are similar with the curves of the uniform beam, i.e.: (1) both the point staticflexibility measurements and transfer static flexibility measurements with the fixed maximum measurement position givebetter results, and with the increase of the maximum measurement position and the decrease of the minimum measurementposition, the condition number of the measurement position matrix decreases; (2) with the increase of the number ofmeasurement positions, the condition number of the measurement position matrix does not decrease but increases a little;(3) the condition numbers of the measurement position matrices of the transfer static flexibility measurements with thefixed maximum measurement position are a little bigger than the condition numbers of the corresponding measurementposition matrices of the point static flexibility measurements; (4) the measurement position matrix is ill conditioned whenthe transfer static flexibility measurements with the fixed maximum measurement position is adopted. However, comparingFig. 5 with 2, and Fig. 6 with 4, it is clear that the condition numbers of the measurement position matrices of the pyramidalbeam are always a little smaller than the condition numbers of the measurement position matrices of the uniform beam. Aswe known, the robustness to measurement noise of the solution of linear equations increases with the decrease of thecondition number of the coefficients matrix of the equations. Thus, it can be inferred that it is apt to obtain better results forthe pyramidal beams comparing to the uniform beams under the same measurement noise level.

3. Simulative examples

A uniform beam and a tapered beam are adopted as the simulative examples to verify the effectiveness of the proposedmethod. The common properties of the two beams are listed in Table 1, and the tapered degrees of the tapered beam arebb¼0.5 and bh¼0, respectively. In order to demonstrate the method for various boundary conditions, we introduced the

Fig. 6. Condition numbers of transfer static flexibility measurements with the fixed maximum measurement position for pyramidal beam (bb¼bh¼1),

(a) 3 measurement positions; (b) 4 measurement positions; (c) 6 measurement positions; (d) 8 measurement positions.


dimensionless translational root stiffness bt¼ktl3/(EI0) and dimensionless rotational root stiffness br¼krl/(EI0). A series of

different boundary conditions are then simulated for each beam, i.e. both bt and br are set in the range from 10�5 to 105.In the simulations, the transfer static flexibility measurements with the fixed maximum measurement position are

adopted, and three measurements positions with Lmax¼0.8 and Lmin¼0.1 are selected. The ‘‘measured’’ static flexibility issimulated as follows:

Hmeasstatic ¼Htrue

staticð1þZdÞ ð42Þ

where Hmeasstatic is the ‘‘measured’’ static flexibility, Htrue

static is the true value of the static flexibility, which is calculated byEq. (29) directly, d is a random variable with zero mean and unit variance, Z is the noise level. Apparently, the precisesolution can be obtained when the noise level Z is equal to zero. Thus, the simulations are performed for noise level Z40,and three different noise levels are considered, i.e. Z¼0.01, 0.03 and 0.05.

For each model with a particular boundary conditions (i.e. bt and br couple) and a particular noise level, 100 simulationsare performed and the relative errors in the identified parameters are calculated as follows:

ep ¼9m�p9

p� 100% ð43Þ

where p indicates the true value of translational root stiffness kt, rotational root stiffness kr or flexural rigidity EI0, mindicates the estimate which is the mean value from all of the 100 simulations. The relative errors in translational rootstiffness kt, rotational root stiffness kr and flexural rigidity EI0 are denoted as ekt, ekr and eEI0

, respectively.In the simulations, we just pay attentions to the identification results in which the relative errors epr50%. Then, the

critical locus can be ascertained in the bt�br plane for ekt, ekr and eEI0, respectively. On one side of the critical locus in the

bt�br plane, the relative errors in the identified parameters are smaller than 50% and on the other side of the critical locusin the bt�br plane, the relative errors in the identified parameters are bigger than 50%.

Fig. 8 shows the critical loci of the relative errors in the identified parameters for the uniform beam with different noiselevels in the bt�br plane. In Fig. 8(a) and (b), thebt�br plane is divided as seven regions by the three critical loci: Region I,eEI0

r50%; Region II, ektr50%; Region III, ekrr50%; Region IV, eEI0r50% and ekrr50%; Region V, eEI0

r50% and ektr50%;Region VI, ektr50% and ekrr50%; Region VII, eEI0

r50%, ektr50% and ekrr50%. In Fig. 8(c), Region IV and Region VIIdisappear, and Region X in which all of ekt, ekr and eEI0

are bigger than 50% appears.As shown in Fig. 8, if the boundary conditions of a beam fall into Region I or II or III in the bt�br plane, one can only obtain

good estimate of only one parameter; if the boundary conditions of a beam fall into Region IV or V or VI, one can obtain good

Table 1Common properties of the uniform beam and tapered beam.

Young’s modulus E Poisson ratio n Density r Length l Depth b0 Height h0

71 GPa 0.3 2700 kg/m3 0.5 m 0.03 m 0.003 m

Fig. 7. Condition numbers of transfer static flexibility measurements with the fixed minimum measurement position for pyramidal beam (bb¼bh¼1),

(a) 3 measurement positions; (b) 4 measurement positions; (c) 6 measurement positions; (d) 8 measurement positions.


estimates of two parameters; if the boundary conditions of a beam fall into Region VII, one can obtain good estimates of all thethree parameters and if the boundary conditions of a beam fall into Region X, none of the parameters can be estimated well. Italso can be seen in Fig. 8, with the increase of the noise level, the critical locus of eEI0

moves towards the upper right corner of thebt�br plane, the critical locus of ekt moves towards the upper left corner of the bt�br plane and the critical locus of ekr movestowards the lower right corner of the bt�br plane. As a result, the areas of the Regions IV, V, VI and VII reduce with the increase ofthe noise level. It can be concluded that with increase of the noise level it is more difficult to obtain good estimates of at least twoparameters simultaneously.

Fig. 9 shows the critical loci of the relative errors in the identified parameters for the tapered beam with different noiselevels in the bt�br plane. The curves for the tapered beam are nearly the same as that of the uniform beam. However, theareas of the cross regions (i.e. Regions IV, V, VI and VII) in Fig. 9 are a little bigger than the corresponding areas of the crossregions in Fig. 8, especially when the noise level Z¼0.05, only Region VII disappears in Fig. 9(c). Thus, it can be concludedthat the identification results of tapered beam are a little better than the identification results of the uniform beam underthe same measurement noise level.

As shown in above simulations, the boundary condition parameters can be easily identified by Eq. (38) using the staticflexibility measurements. However, the accuracy of the identified results is depended on the region of the boundaryconditions in bt�br plane and the noise level in flexibility measurements. Unfortunately, it is impossible to know theregion of the boundary conditions and the flexibility measurement noise level in advance. Therefore, an alternativetechnique for evaluating the accuracy of the identified results is proposed as follows.

We found in our simulations that: (1) the coefficient of variation 9s/m9 of the identified parameters in the 100 simulations for aparticular case is relatively small in the region in which the relative errors in the identified parameters are smaller than 50%; (2)

Fig. 8. The critical locus of the relative errors for uniform beam with different noise level Region I, eEI0r50%; Region II, ektr50%; Region III, ekrr50%;

Region IV, eEI0r50% and eEkrr50%; Region V, eEI0

r50% and ektr50%; Region VI, ektr50% and ekrr50%; Region VII, eEI0r50%, ektr50% and ekrr50%.

Note: in sub-figure (c), Region IV and VII disappear, and a new Region X in which all of ekt, ekr and eEI0are bigger than 50% appears. Noise level: (a) Z¼0.01,

(b) Z¼0.03 and (c) Z¼0.05.


the coefficient of variation 9s/m9 of the identified parameters in the 100 simulation for a particular case is relatively big in theregion in which the relative errors in the identified parameters are bigger than 50%. For example, Fig. 10 shows the distributionand coefficient of variation 9s/m9of the identified parameters of the 100 simulations of a particular case for the uniform beam (thedimensionless translational root stiffness bt¼104, the dimensionless rotational root stiffness br¼10 and the noise level Z¼0.01).As the boundary conditions of the particular case fall into Region IV in Fig. 8(a), we can conclude that the accuracy of theidentified EI0 and kr is relatively high, and the accuracy of the identified kt is relatively low. Meanwhile, as shown in Fig. 10,the coefficients of variation of both EI0 and kr are relatively small, and the coefficient of variation of kt is very big. Therefore, thecoefficient of variation of the identified parameters can be utilized to evaluate the accuracy of the identified parameters, and thetechnique for evaluating the accuracy of the identified parameter should be: (1) conduct the static flexibility measurements manytimes using the same test setup; (2) identify the boundary condition parameters for each measurement using Eqs. (38) and (3)calculate the coefficients of variation of the identified EI0, kt and kr, and evaluate the accuracy of the identified parameters based

Fig. 9. The critical locus of the relative errors for tapered beam with different noise level Region I, eEI0r50%r; Region II, ektr50%; Region III, ekrr50%;

Region IV, eEI0r50% and ekrr50%; Region V, eEI0

r50% and ektr50%; Region VI, ektr50% and ekrr50%; Region VII, eEI0r50%, ektr50% and ekrr50%.

Note: in sub-figure (c), Region VII disappears. Noise level: (a) Z¼0.01, (b) Z¼0.03 and (c) Z¼0.05.


on coefficients of variation of the identified parameters, i.e. the accuracy of the identified parameter is relatively high if thecoefficient of variation of the identified parameters is relatively small, otherwise, the accuracy of the identified parameter isrelatively low.

4. Experimental validations

A uniform beam and a tapered beam were manufactured as the experimental model to illustrate the effectiveness of theproposed method, as shown in Fig. 11. The designed properties of the uniform beam and tapered beam are the same asthe corresponding beams in the simulative examples, i.e. the common properties of the two beams are listed in Table 1, thetapered degrees of the tapered beam are bb¼0.5 and bh¼0, and the designed value of the flexural rigidity at the constrainedend EI0 is 4.7925 Nm2. As show in Fig. 11, the boundary conditions for uniform beam and tapered beam are provided by auniform steel beam with fixed–fixed boundary conditions, and the designed translational root stiffness and rotational root

Fig. 10. The distributions and coefficient of variation 9s/m9 of the identified parameters of the 100 simulation of a particular boundary condition for the

uniform beam (noise level Z¼0.01, dimensionless translational root stiffness bt¼104 and dimensionless rotational root stiffness br¼10).

Fig. 11. Experimental model, (a) sketch of the geometry dimension; (b) uniform beam; (c) tapered beam.


stiffness provided by the uniform steel beam with fixed–fixed boundary conditions can be approximately calculated as

kt � 192EsIs=l3eff

kr � 4GsJs=leffð44Þ

where Es, Gs, Is, Js and leff are Young’s modulus, the shear modulus, the second moment of the area of the cross section, thetorsion constant and the effective length of steel beam, respectively. The second moment of the area of the cross section Isand the torsion constant Js can be calculated by the width bs and depth hs of the cross section of the steel beam, and leff can beapproximately indicated by the length ls of steel beam minus the width b0 at the constrained end of the uniform beam ortapered beam, as follows:

Is ¼1

12bsh

3s

Js ¼ bsh3s

1

3�0:21

hs

bs1�

h4s

12b4s

� �, hsrbs

leff � ls�b0 ð45Þ

Three different uniform steel beams are manufactured to provide three different types of boundary conditions for bothuniform beam and tapered beam, respectively. Young’s modulus Es and Poisson ratio ns of the uniform steel beams are


206 Gpa and 0.3, respectively. The designed properties of the three different uniform steel beams (i.e. three different typesof boundary conditions) in our experiment are listed in Table 2.

Similar to the simulative examples, the transfer static flexibility measurements with the fixed maximum measurementposition are adopted, and only three measurements positions with Lmax¼0.8 and Lmin¼0.1 are selected. In the experiment,the three measurement positions are denoted as measurement position 1 (corresponding to L¼0.1), measurement position2 (corresponding to L¼0.45) and measurement position 3 (corresponding to L¼0.8), and the static displacements y1, y2

and y3 corresponding to the three measurement positions are measured by laser when the beam is subjected to a 50 gweight at measurement position 3, as shown in Fig. 12. Table 3 lists the measured static displacements of the uniform andtapered beam with different types of boundary conditions, respectively.

Using the measured static displacements, Table 4 lists the measured static flexibilities of the uniform and tapered beamwith different types of boundary conditions, and Table 5 lists the identified value of the flexural rigidity at the constrainedend EI0, translational root stiffness kt and rotational root stiffness kr for different experimental models using the measuredstatic flexibilities in Table 4. As listed in Table 5, all the identified values of translational root stiffness are negative (this isunreasonable), and the identified flexural rigidity at the constrained end EI0 and rotational root stiffness kr are closed to its

Fig. 12. Test setup for the static flexibility measurement.

Table 2Designed properties of the three uniform steel beams (i.e. three different types of boundary conditions).

Uniform steel beam/boundary conditions (BC) BC 1 BC 2 BC 3

Length ls (m) 0.15 0.09 0.15

Depth bs (m) 0.01 0.01 0.01

Height hs (m) 0.001 0.001 0.0015

Translational root stiffness kt (N/m) 1.9074�104 1.5259�105 6.4375�104

Rotational root stiffness kr (Nm) 8.2488 16.4976 26.9039

Dimensionless translational root stiffness bt497.50 3980.0 1679.1

Dimensionless rotational root stiffness br0.8606 1.7212 2.8069

Table 3Measured static displacements of the uniform and tapered beam with different types of boundary conditions (the beam is subjected to a 50 g weight at

measurement position 3).

Experimental model BC 1 BC 2 BC 3

Uniform beam Tapered beam Uniform beam Tapered beam Uniform beam Tapered beam

Static displacement (mm)

y1 0.864 0.865 0.444 0.448 0.308 0.327

y2 4.75 4.80 2.71 2.75 2.05 2.15

y3 9.03 9.22 5.49 5.71 4.23 4.56

Table 4Measured static flexibilities of the uniform and tapered beam with different types of boundary conditions.



Static flexibility (m/N) Hstatic,13 0.001763 0.001765 0.000906 0.000914 0.000629 0.000667

Hstatic,23 0.009694 0.009796 0.005531 0.005612 0.004184 0.004388

Hstatic,33 0.018429 0.018816 0.011204 0.011633 0.008633 0.009306

Table 5Identified values of the flexural rigidity at the constrained end EI0, translational root stiffness kt and rotational root stiffness kr for different experimental

models using the measured static flexibilities in Table 4.



EI0 (Nm2) 6.6652 6.7788 5.1091 5.0736 5.9956 5.6009

kt (N/m) �4.1�103�4.0�103

�1.3�104�1.9�104

�1.0�104�1.8�104

kr (Nm) 10.3424 10.2959 22.5217 22.9347 30.9333 31.3968

Table 6Mean value m, standard deviation s and coefficient of variation 9s/m9of the identified EI0, kt and kr by the 100 samples for different experimental models.



EI0 m 8.0523 7.5035 5.2364 5.1548 6.1137 5.6684

s 5.5521 2.9815 0.8856 0.6918 0.9136 0.6572

9s/m9 0.6895 0.3973 0.1691 0.1342 0.1494 0.1159

kt m �7.3�103�1.7�103

�2.4�103 2.1�105 1.4�104�2.5�104

s 2.3�104 3.6�104 1.6�105 2.3�106 2.9�105 2.3�105

9s/m9 3.1351 21.4380 67.3081 10.7857 20.9354 9.2729

kr m 10.4125 10.3540 22.7521 23.1376 31.2762 31.7143

s 0.7480 0.6632 2.0873 1.9504 3.0068 2.8915

9s/m9 0.0718 0.0641 0.0917 0.0843 0.0961 0.0912

Note: the 100 samples are achieved by adding 1% artificial Gauss noise on the real measured static flexibilities.


designed values. It also can be seen from Table 5 that the identified values EI0 and kr are reasonably consistent whendifferent beam samples (i.e. uniform and tapered) are constrained by the same boundary conditions.

In order to evaluate the accuracy of the identified values of the parameters, one should conduct plenty of staticflexibility measurement. However, the huge time consumption will be involved in the plenty of static flexibilitymeasurement. An alternative method was adopted in our research to overcome the drawbacks, i.e. the plenty of staticflexibility measurement might be achieved by adding a little artificial noise to one sample of the real measured flexibilities.In this paper, we added Gauss noise (the noise level is 1%) on the real measured static flexibilities in Table 4, and obtained100 samples of static flexibilities. Then, the mean value m, standard deviation s and coefficient of variation 9s/m9 of theidentified EI0, kt and kr by the 100 samples are calculated, as listed in Table 6. The coefficients of variation 9s/m9 of theidentified EI0 and kr are relatively small, but the coefficient of variation 9s/m9 of the identified of kt is relatively big.Therefore, the identified EI0 and kr should be relatively precise but the identified kt should be relatively imprecise.

5. Conclusions

This paper investigates a direct method for identifying the boundary conditions using static flexibility measurements.The relationship between the static flexibility and boundary conditions is deduced based on the static equilibriumequation, and then a set of linear equations for identifying the boundary conditions are formed by several different staticflexibility measurements. In order to obtain robustness linear equations, the better measurement position setup isascertained by minimizing the condition number of the coefficients matrix of the linear equations, i.e. the bettermeasurement position setup should be: the point static flexibility measurements or transfer static flexibility measure-ments with the fixed maximum measurement position, the maximum measurement position should be as high aspossible, and the minimum measurement position should be as low as possible. Finally, both simulative examples and


experimental validations demonstrate that the direct method using static flexibility measurements can be utilized toidentifying the boundary conditions of the tapered beam-like structures under some particular cases.

Acknowledgment

The authors are grateful to Dr. Tim Waters for his useful suggestions in deducing some formulae, as well as Miss YanDing and Mr. Muyu Zhang for their help in conducting the experiments.

References

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identification of boundary conditions of tapered beam-like structures using static flexibility...

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