Structural Safety 33 (2011) 19–25

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Structural Safety

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Timoshenko versus Euler beam theory: Pitfalls of a deterministic approach

André Teófilo Beck a,*, Cláudio R.A. da Silva Jr. b

a Department of Structural Engineering, EESC, University of São Paulo, Brazilb Department of Mechanical Engineering, Federal University of Technology of Paraná, Brazil

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 July 2009Received in revised form 20 April 2010Accepted 26 April 2010Available online 31 May 2010

Keywords:Euler–Bernoulli beamTimoshenko beamUncertainty propagationParameterized stochastic processesMonte Carlo simulationGalerkin method

0167-4730/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.strusafe.2010.04.006

* Corresponding author. Tel.: +55 16 3373 9460; faE-mail address: atbeck@sc.usp.br (A.T. Beck).

The selection criteria for Euler–Bernoulli or Timoshenko beam theories are generally given by means ofsome deterministic rule involving beam dimensions. The Euler–Bernoulli beam theory is used to modelthe behavior of flexure-dominated (or ‘‘long”) beams. The Timoshenko theory applies for shear-domi-nated (or ‘‘short”) beams. In the mid-length range, both theories should be equivalent, and some agree-ment between them would be expected. Indeed, it is shown in the paper that, for some mid-lengthbeams, the deterministic displacement responses for the two theories agrees very well. However, the arti-cle points out that the behavior of the two beam models is radically different in terms of uncertaintypropagation. In the paper, some beam parameters are modeled as parameterized stochastic processes.The two formulations are implemented and solved via a Monte Carlo–Galerkin scheme. It is shown that,for uncertain elasticity modulus, propagation of uncertainty to the displacement response is much largerfor Timoshenko beams than for Euler–Bernoulli beams. On the other hand, propagation of the uncertaintyfor random beam height is much larger for Euler beam displacements. Hence, any reliability or risk anal-ysis becomes completely dependent on the beam theory employed. The authors believe this is not widelyacknowledged by the structural safety or stochastic mechanics communities.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

This paper presents a comparison of the Euler–Bernoulli andTimoshenko beam theories, taking into account parameter uncer-tainties and uncertainty propagation. It is widely known that theEuler–Bernoulli beam theory properly models the behavior of flex-ure-dominated (or ‘‘long”) beams. The Timoshenko theory isknown to apply for shear-dominated (or ‘‘short”) beams. In themid-length range, both theories should be equivalent, and someagreement between them would be expected.

The stochastic beam bending problem has been studied by sev-eral authors. Vanmarcke and Grigoriu [1] studied the bending ofTimoshenko beams with random shear modulus. Elishakoff et al.[2] employed the theory of mean square calculus to construct asolution to the boundary value problem of bending with stochasticbending modulus. Ghanem and Spanos [3] used the Galerkin meth-od and the Karhunem-Loeve series to represent uncertainty in thebending modulus by means of a Gaussian stochastic process. Cha-kraborty and Sarkar [4] used the Neumann series and Monte Carlosimulation to obtain statistical moments of the displacements ofcurved beams, with uncertainty in the elasticity modulus of thefoundation.

ll rights reserved.

x: +55 16 3373 9482.

In this paper, it is shown that, for some mid-length beams,deterministic displacement responses for the two beam theoriesagree very well. In this case, the theories are generally acceptedas equivalent. However, it is shown in the paper that, althoughthe theories are equivalent when compared deterministically, theirbehavior is radically different in terms of uncertainty propagation.This is shown by means of some illustrative example problems.

In Section 2, formulation of the two beam theories is presented.Representation of the uncertainty in beam parameters, via param-eterized stochastic processes, is presented in Section 3. In thenumerical examples, a Galerkin–Monte Carlo scheme is used to ob-tain the random displacement fields. The Galerkin solutions arepresented in Section 4. Section 5 shows the evaluation of firstand second order moments of the Monte Carlo solution. Twoexample problems are shown in Section 6, illustrating the large dif-ferences between the two formulations in terms of uncertaintypropagation. Section 7 discusses the effects of these differenceson reliability and risk analysis. Section 8 finishes the paper withsome conclusions.

2. Euler and Timoshenko beam formulations

In this section, the strong and weak formulations of the prob-lems of stochastic bending of Euler–Bernoulli and Timoshenkobeams are presented. The strong form of the Euler–Bernoulli beambending problem is given by:

20 A.T. Beck, C.R.A. da Silva Jr. / Structural Safety 33 (2011) 19–25

d2

dx2 EIðx;xÞ � d2wdx2

� �¼ f ; 8ðx;xÞ 2 ð0; lÞ �X;

wð0;xÞ ¼ 0;

wðl;xÞ ¼ 0;dwdx

��ð0;xÞ ¼

dwdx

��ðl;xÞ ¼ 0; 8x 2 X;

8>>>>><>>>>>:ð1Þ

where w is the transverse displacement field, EI is the bending stiff-ness, X is a sample space and f is a load term. The strong form of theTimoshenko beam bending problem is given by:

ddx EIðx;xÞ � d/

dx

� �þ GAðx;xÞ � dw

dx � /� �

¼ 0;

ddx GAðx;xÞ � dw

dx � /� �� �

¼ �f ; 8ðx;xÞ 2 ð0; lÞ �X;

wð0;xÞ ¼ wðl;xÞ ¼ 0;

/ð0;xÞ ¼ /ðl;xÞ ¼ 0; 8x 2 X;

8>>>><>>>>: ð2Þ

where / is the angular displacement field, GA is the shear stiffness,and the remaining symbols follow Eq. (1). The angular displace-ments stochastic process in Euler–Bernoulli theory is given by thespace derivative of the transverse displacement field. Both formula-tions are given for clamped–clamped boundary conditions.

In the sequence, elasticity modulus E and beam height h will beassumed as stochastic processes. Hence, the displacement re-sponses w and / will also be stochastic processes. In order to en-sure existence and uniqueness of the solutions, the followinghypotheses are required:

H1 :

9a; �a 2 Rþ n f0g; j½a; �a�j < þ1; Pðfx 2 X : EIðx;xÞ 2 ½a; �a�;8x 2 ½0; l�gÞ ¼ 1;

9s; �s 2 Rþ n f0g; j½s; �s�j < þ1; Pðfx 2 X : GAðx;xÞ 2 ½s; �s�;8x 2 ½0; l�gÞ ¼ 1;

8>>><>>>:H2 : f 2 L2ðX;F; P; L2ð0; lÞÞ:

ð3Þ

Hypothesis H1 ensures that the elasticity modulus and beamheight are strictly positive and uniformly limited in probability[5]. Hypothesis H2 ensures that the stochastic load process has fi-nite variance. These hypotheses are necessary for application of theLax–Milgram Lemma, which ensures existence and uniqueness ofthe solution, as well as continuous dependency on the data [5,6].

The abstract variational problem associated to the strong form(Eq. (1)) of the stochastic Euler–Bernoulli beam bending problemis obtained as:

Find w 2Vsuch that :RX

R l0 EI � d2w

dx2 � d2vdx2

� �ðx;xÞdxdPðxÞ ¼

RX

R l0ðf � vÞðx;xÞdxdPðxÞ;

8v 2V:

8><>:ð4Þ

where V ¼ L2ðX;F; P;UÞ with U ¼ H20ð0; lÞ.

The abstract variational problem associated to the strong form(Eq. (2)) of the stochastic Timoshenko beam bending problem isobtained as:

Find ðw;/Þ 2W such that :RX

R l0 GA � dw

dx �/� �

�u� �

ðx;xÞdxdPðxÞ¼R

X

R l0ðf �uÞðx;xÞdxdPðxÞ;R

X

R l0 EI � d/

dx � dtdx

� �ðx;xÞdxdPðxÞ¼

RX

R l0 GA � dw

dx �/� �

�t� �

ðx;xÞdxdPðxÞ;8ðu;tÞ 2W;

8>>>>><>>>>>:ð5Þ

where W ¼ L2ðX;F; P;QÞ with Q ¼ H10ð0; lÞ � H1

0ð0; lÞ. Eq. (5) repre-sents a system of variational equations for the coupled fieldsw ¼ wðx;xÞ and / ¼ /ðx;xÞ .

Details of the formulation of stochastic Euler–Bernoulli beamsare given in [7]. For stochastic Timoshenko beams, details can befound in Ref. [8].

3. Uncertainty representation

In most engineering problems, complete statistical informationabout uncertainties is not available. Sometimes, the first and sec-ond moments are the only information available. The probabilitydistribution function is defined based on experience orheuristically.

In order to apply Galerkin’s method, an explicit representationof the uncertainty is necessary. In this paper, uncertain parametersare modeled as parameterized stochastic processes. These are de-fined as a linear combination of deterministic functions and ran-dom variables [9]:

jðx;xÞ ¼XN

i¼1

giðxÞniðxÞ; ð6Þ

where fgigNi¼1 are deterministic functions and fnigN

i¼1 are randomvariables.

4. Galerkin method

The Galerkin method and direct Monte Carlo simulation areused in this paper to obtain sample realizations of the beams ran-dom displacements, from samples of the beams randomparameters.

Approximated solutions for the qth realization of the transversedisplacement random process, for the Euler–Bernoulli beam, aregiven by:

wqmðx;xqÞ ¼

Xm

i¼1

wiquiðxÞ; ð7Þ

where fwiqgmi¼1 are coefficients to be determined and fuig

mi¼1 are

interpolating functions for the qth realization. Observing that

C20ð0; lÞ

H20ð0;lÞ ¼ H2

0ð0; lÞ, and considering a complete orthonormal set

U ¼ fuig1i¼1 of U [10], such that span½U�U ¼ U

� �. Since approxi-

mated numerical solutions are derived in this paper, the solutionspace has finite dimensions. This implies truncation of the completeorthonormal set U, which results in Um ¼ fuig

mi¼1 and

Um ¼ span½Um�. The approximated variational problem associatedto the Euler–Bernoulli beam is obtained by inserting Eq. (7) in Eq.(4):

For the qth realization; find fwiqgmi¼1 2 Rm such that :Pm

i¼1

R l0 EIðx;xqÞ � d2ui

dx2 �d2uj

dx2

� �ðxÞdx

wiq ¼

R l0 f ðx;xqÞ �ujðxÞdx;

8wj 2W:

8>>><>>>:ð8Þ

This problem can also be written in matrix form:

Find uq 2 Rn such thatKquq ¼ Fq;

�ð9Þ

where Kq 2MmðRÞ, with elements given by:

Kq ¼ ½kqij�m�m; kq

ij ¼Z l

0EIðx;xqÞ �

d2ui

dx2 �d2uj

dx2

!ðxÞdx: ð10Þ

The loading term is given by,

Fq ¼ ff qi g

mi¼1; f q

i ¼Z l

0f ðx;xqÞ �uiðxÞdx: ð11Þ

For the qth realization of Timoshenko beam displacements, approx-imated Galerkin solutions are obtained as:

A.T. Beck, C.R.A. da Silva Jr. / Structural Safety 33 (2011) 19–25 21

wqmðx;xÞ ¼

Pmi¼1

wiqwiðxÞ;

/qmðx;xÞ ¼

Pmi¼1

/iqwiðxÞ;

8>>><>>>: ð12Þ

where fðwiq;/iqÞgmi¼1 are coefficients to be determined and fwig

mi¼1

are interpolating functions. Let Q ¼ spanfwigmi¼1 be a set generated

by truncation of a complete orthonormal set W ¼ fwig1i¼1 in Q, with

wi 2 C0ð0; lÞ \ C1ð0; lÞ; 8i 2 N. Replacing Eq. (12) in Eq. (5), one ar-rives at the approximated variational problem for the Timoshenkobeam:

For the qth realization; find fðwiq;/iqÞgmi¼1 2 R2�m such that;

Pmi¼1

R l0 EIðx;xqÞ � dwi

dx � wj

� �ðxÞdx

h iwiq

�R l

0 GAðx;xqÞ � ðwi � wjÞðxÞdxh i

/iq

8><>:9>=>; ¼ R l

0 f ðx;xqÞ � wjðxÞdx;

Pmi¼1

R l0 EIðx;xqÞ � dwi

dx �dwj

dx

� �ðxÞ þ GAðx;xqÞ � ðwi � wjÞðxÞ

h idx

n o/iq

¼Pmi¼1

R l0 GAðx;xqÞ � dwi

dx � wj

� �ðxÞdx

h iwiq; 8wj 2 Qm:

8>>>>>>>>>>>>><>>>>>>>>>>>>>:ð13Þ

The approximated variational problem consists in finding thecoefficients of the linear combination expressed in Eq. (13). Usinga vector–matrix representation, the system of linear algebraicequations defined in Eq. (13) is written as:

For the qth realization; find ðwq;/qÞ 2 R2�m such that :

Aqwq þ Bq/q ¼ Fq;

Cqwq ¼ Dq/q;

8><>: ð14Þ

where Aq;Bq;Cq;Dq 2MmðRÞ. Elements of these matrices are given

by:

Aq ¼ ½aqij�m�m; aq

ij ¼R l

0 EIðx;xqÞ � dwidx �wj

� �ðxÞdx;

Bq ¼ ½bqij�m�m; bq

ij ¼�R l

0 GAðx;xqÞ � ðwi �wjÞðxÞdx;

Cq ¼ ½cqij�m�m; cq

ij ¼R l

0 GAðx;xqÞ � dwidx �wj

� �ðxÞdx;

Dq ¼ ½dqij�m�m; dq

ij ¼R l

0 EIðx;xqÞ � dwidx �

dwj

dx

� �ðxÞ þGAðx;xqÞ � ðwi �wjÞðxÞ

h idx:

8>>>>>>><>>>>>>>:ð15Þ

The loading term is given by Eq. (11). Solution of the linear sys-tem in Eq. (14) is obtained as:

/q ¼ ðAqCq�1Dq þ BqÞ�1Fq;

wq ¼ Cq�1DqðAqCq�1

Dq þ BqÞ�1Fq:

(ð16Þ

It is important to note that conversion of the continuous prob-lem (Eq. (5)) to the discretized form (Eq. (13)) results in de-cou-pling of the displacement fields w and /, following Eq. (16).

5. Statistical moments and reliability problem

In the following, Monte Carlo simulation is used to study thepropagation of uncertainty through the Timoshenko and Euler–Bernoulli bending models. In order to compare the solutions, it isinteresting to focus on some statistics of the results.

Estimates for expected value and variance of random variableswðxÞ ¼ wðx;xÞ and /ðxÞ ¼ /ðx;xÞ, for a fixed point x 2 ½0; l�, are ob-tained from the set of displacement fields samples fwðx;xiÞgN

i¼1

and f/ðx;xiÞgNi¼1:

l̂wðxÞ ¼ 1N

� �PNi¼1

wðx;xiÞ;

r̂2wðxÞ¼ 1

N�1

� �PNi¼1

wðx;xiÞ � l̂wðxÞ

h i2;

8>>><>>>: ^l̂/ðxÞ ¼ 1

N

� �PNi¼1

/ðx;xiÞ;

r̂2/ðxÞ¼ 1

N�1

� �PNi¼1

/ðx;xiÞ � l̂/ðxÞ

h i2:

8>>><>>>:ð17Þ

In order to study the effects of differences in uncertainty prop-agation in reliability or risk analysis, a simple reliability problem isdefined. An admissible displacement, at mid-spam, is defined aswADM ¼ � l

200, where ‘‘l” is the beam length. The associated proba-bility of failure is given by:

Pf ¼ PðBÞ; ð18Þ

where P stands for probability and B ¼ x 2 Xjwð l2 ;xÞP � l

200

� .

This can be estimated from the same set of simulated displace-ments, by:

bPf ¼1N

� �XN

i¼1

1BðxiÞ; ð19Þ

where 1B : X! f0;1g with:

1BðxÞ ¼1; x 2 B;

0; x R B;

�ð20Þ

is the characteristic function of set B.

6. Numerical examples

In this section, two numerical examples are presented. In thefirst example, the elasticity modulus is considered a random field.In the second example, the height of the beam’s cross-section israndom. In both cases, uncertainty is modeled by parameterizedstochastic processes. In both examples, the beam is clamped atboth ends, the span (l) equals one meter, the cross-section is rect-angular with b ¼ 1

30 m and h ¼ 125 m and the beam is subject to an

uniform distributed load of f ðxÞ ¼ 100 kPa=m; 8x 2 ½0; l�.Fig. 1 shows the exact, deterministic transverse (left) and angu-

lar (right) displacement responses, obtained via Euler–Bernoulliand Timoshenko beam theories. These results are obtained forthe mean values of the parameters to be considered random inthe following. It is observed that the two theories yield very closeresults, with transverse mid-spam displacements agreeing within97%. From a deterministic point of view, the two theories couldbe considered equivalent, for this beam.

6.1. Random elasticity modulus

In this example, the elasticity modulus is modeled as a param-eterized stochastic process:

Eðx;xÞ ¼ lE þffiffiffi3p� rE n1ðxÞ cos

xl

� �þ n2ðxÞ sin

xl

� �h i; ð21Þ

where lE is the mean value, rE is the standard deviation and fn1; n2gare uniform orthogonal random variables. Numerical solutions areobtained for rE ¼ ð 1

10Þ � lE.Results obtained via Monte Carlo simulation are shown in

Figs. 2–5. Fig. 2 shows the envelope (largest and smallest values)among the 15,000 samples obtained, for transverse (left) and angu-lar (right) beam displacements. Fig. 3 shows the mean values, andFig. 4 shows the variance of both displacement fields, obtained forthe two beam theories. Fig. 5 shows the cumulative distributionfunction, obtained via simulation, of the displacement fields.

Results presented in Fig. 1 suggest that the Euler–Bernoulli andTimoshenko beam theories are equivalent for this problem. Now,Figs. 2–5 make very clear that the two theories are completely dif-ferent in terms of uncertainty propagation. It is observed that the

Fig. 1. Exact deterministic solutions for transverse displacements (left) and angular displacements (right).

Fig. 2. Envelope of samples for transverse (left) and angular (right) beam displacements.

Fig. 3. Mean value of transverse (left) and angular (right) beam displacements.

22 A.T. Beck, C.R.A. da Silva Jr. / Structural Safety 33 (2011) 19–25

uncertainty in elasticity modulus propagates much more throughthe Timoshenko model than through the Euler–Bernoulli beammodel. The explanation for this behavior can be drawn from a com-parison of Eqs. (1) and (2). The uncertainty in elasticity modulusalso represents uncertainty in the stiffness modulus G, throughthe relation:

E ¼ 2Gð1þ tÞ: ð22Þ

where t is the Poisson coefficient. The two uncertainty terms affectthe coupled system of Timoshenko beam equations.

The two sets of Monte Carlo realizations, obtained for the Eulerand Timoshenko beam displacements, can be written as:

Ew ¼ wxiðxÞ 2 Rjwxi

ðxÞ ¼ wðx;xiÞ; ðx;xiÞ 2 ½0; l� � fxigNi¼1;

n00w00 solution of Eq:ð1Þ:g;

Tw ¼ wxiðxÞ 2 Rjwxi

ðxÞ ¼ wðx;xiÞ; ðx;xiÞ 2 ½0; l� � fxigNi¼1;

n00w00 solution of Eq:ð2Þ:g:

8>>>>>><>>>>>>:ð23Þ

Fig. 4. Variance of transverse (left) and angular (right) beam displacements.

Fig. 5. Cumulative distribution functions of transverse beam displacements.

A.T. Beck, C.R.A. da Silva Jr. / Structural Safety 33 (2011) 19–25 23

It is observed in Fig. 2 (left) that Ew � Tw. Hence, there are real-izations of the Timoshenko beam displacements which are notcontained in the set of realizations of Euler displacements. Resultspresented in Fig. 1 show no hint of this behavior.

6.2. Random cross-section height

In this example, the beam cross-section height is modeled as aparameterized random process:

hðx;xÞ ¼ lh þffiffiffi3p� rh n1ðxÞ cos

xl

� �þ n2ðxÞ sin

xl

� �h i; ð24Þ

where lh is the mean value, rh ¼ 110

� �� lh is the standard deviation

and fn1; n2g are uniform, independent random variables.Results obtained via Monte Carlo simulation are shown in

Figs. 6–9. Fig. 6 shows the envelope (largest and smallest values)among the 15,000 samples obtained, for transverse (left) and angu-lar (right) beam displacements. Fig. 7 shows the mean values, andFig. 8 shows the variance of both displacement fields, obtained forthe two beam theories. Fig. 9 shows the cumulative distributionfunction, obtained via simulation, of the displacement fields.

It is first observed that the agreement between the two theoriesis better for this problem, although far from ideal. However, it isnoted that results have opposite trends in terms of uncertaintypropagation: the propagation of uncertainty in random beam

height is larger for the Euler–Bernoulli response than for the Tim-oshenko displacements. Hence, for this example, Ew � Tw.

To understand this result, the first term of Eq. (1) can be writtenin the following form:

d2

dx2 EI � d/dx

� �¼ f : ð25Þ

When this equation is solved for /, and the result is used in Eq.(2) to find the transverse displacement w, one notes that the solu-tion is proportional to h�2. For the Euler–Bernoulli beam, this dis-placement is proportional to h�3. This explains the differences inbeam height uncertainty propagation for the two beam models,and why the propagation is larger for the Euler beam.

Comparing Figs. 7 and 3, it is observed that the agreement be-tween the two theories is better, for this example, in comparisonto the random elasticity modulus. Comparing Figs. 8 and 4, it is ob-served that the variance is smaller for the random beam heightexample.

7. Effect on reliability and risk analysis

From the results presented in Section 6, it is clear that differ-ences in uncertainty propagation will affect any reliability or riskanalysis based on the Euler or Timoshenko beam theories. This isconfirmed in this section, and quantified for the example problemsconsidered in the study.

Fig. 6. Envelope of samples for transverse (left) and angular (right) beam displacements.

Fig. 7. Mean value of transverse (left) and angular (right) beam displacements.

Fig. 8. Variance of transverse (left) and angular (right) beam displacements.

24 A.T. Beck, C.R.A. da Silva Jr. / Structural Safety 33 (2011) 19–25

Table 1 shows failure probability results obtained for the twobeam theories, and for an admissible mid-spam displacement ofwADM ¼ � l

200 (Eq. (18)). These results were obtained via simpleMonte Carlo simulation. It is clear that the results are completelydependent on the beam theory used in the analysis.

A qualitative assessment of failure probability results can bedrawn from Figs. 2, 5, 6 and 9. In Fig. 2, it can be observed that,

for the random elasticity modulus example, Ew \ B ¼ Ø. This im-plies that, for the Euler beam model, the probability of event B iszero, that is, the probability of failure is zero. On the other hand,for the Timoshenko beam theory, there is some probability associ-ated to this event. This probability can be drawn from Fig. 5, and isgiven in Table 1. For the case of random beam height, it can be ob-served in Fig. 6 that Ew \ Tw \ B–Ø. Hence, the failure probabilities

Fig. 9. Cumulative distribution functions of transverse beam displacements.

Table 1Effect of beam theory on failure probability results.

Problem P̂f kl ¼ wADMlwðL=2Þ

Euler–Bernoulli Timoshenko Euler Timoshenko

Random E 0.0000 0.2310 2.78 2.33Random h 0.1007 0.0208 2.70 2.70

A.T. Beck, C.R.A. da Silva Jr. / Structural Safety 33 (2011) 19–25 25

are nonzero for both beam models. These failure probabilities canbe drawn from Fig. 9, and are given in Table 1.

Apart from the minor (3%) difference between the deterministicEuler and Timoshenko solutions of this problem, the safety coeffi-cient for the deterministic problem is given by:

k ¼ wADM

w l2

� � ¼ 0:005w l

2

� � ¼ 2:78: ð26Þ

This coefficient is the same for both Euler and Timoshenkobeam formulations: hence, it clearly does not take into accountthe differences in uncertainty propagation and in failure probabil-ities. The central safety coefficients, which are given in Table 1, arealso not sufficient to provide uniform reliability for this problem.

8. Conclusions

In this paper, it was shown that two beam theories, whichseemed perfectly equivalent when compared in terms of determin-istic response, behave radically different in terms of uncertaintypropagation. Hence, the very notion that the theories are equiva-lent is limited to the realm of determinacy, and is unfounded whenuncertainty propagation is considered.

Two very simple examples were presented to illustrate the is-sue, involving the Timoshenko and Euler–Bernoulli beam theories.A mid-length beam was considered, and it was shown that deter-ministic displacement responses obtained by the two theoriesagreed within 97%. However, uncertainty in the elasticity moduluspropagates much largely for the Timoshenko beam, in comparisonto the Euler beam. When uncertainty in beam height is considered,propagation to the displacement response is larger for the Euler

beam than for the Timoshenko beam. Hence, although the Timo-shenko and Euler–Bernoulli beam theories appear to be equivalentfor the mid-length beam considered, the propagation of uncer-tainty to the beams displacement response is radically different.As a consequence, any reliability or risk analysis becomes com-pletely dependent on the theory employed.

There are no pitfalls in the Timoshenko or Euler–Bernoulli beamtheories presented herein. What the title of the manuscript sug-gests is that there are pitfalls in using pure deterministic judgmentwhen comparing these formulations, in order to choose one ofthem for a reliability or risk analysis.

Acknowledgements

Sponsorship of this research project by the São Paulo StateFoundation for Research – FAPESP (Grant No. 2008/10366-4) andby the National Council for Research and Development – CNPq(Grant No. 305120/2006-9) is greatly acknowledged.

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