An improved stiffness assessment of cracked steel beam and thick tube with a transverse crack

Abstract

For rotating machines maintenance and industrial inspection, cracking phenomena are unpredictable events, since the occurrence of a crack and its growth are unavoidable during the system time-life.

Current models for cracks behaviour in shafts or tubes were developed thanks to beams theory and finite element method. Even though the beam models are simplified, they provide acceptable results for most cases encountered in practice. The modelling of cracked beams and tubes is based on the addition of a local flexibility due to the crack to the intact structure. However, when the crack depth exceeds the beam’s radius, the models results diverge from experimental observations; in general, the models give a higher global rigidity and do not perform a complete assessment of the local flexibility of the tube. This is true in the particular case of thick tubes. As a result, the lifetime is underestimated.

Therefore, an investigation overview of the models behaviour origins demonstrates that the rigidity’s rise was unrealistic. The investigation shows that the generic models underestimate the effect of the section’s geometry change in stiffness’s computation, during the crack depth growing process.

The present paper proposes a new approach to improve these models’ accuracy. Therefore, some modification to the models has been performed in order to correct the beam’s model and to complete the assessment of the local flexibility calculations of the tube. Those modifications have been tested within the two beam models. The assessment of the updated models has shown good agreement for the stiffness even when the crack depth exceeded the beam’s radius.

Keywords

Crack modelling beam thick tube experimentation verification transverse crack local and global flexibility correction

Introduction

Cracked shaft modelling is performed commonly using the beam model or even more complicated ones (such as for example 2D/3D models). The 2D/3D models use methods like FEM, XFEM, etc., which allow finer discretization and allow resolving many problems like the breathing effect of the transverse crack, while the beam model is more simple and only few versions do consider the effects cited above.

Experience feedbacks showed that the cracks are local problems and their effect along the axis of the shaft is often smaller compared to the shaft’s size. Those cracks appear at geometrical discontinuities, like diameter change, blades, welding, etc. Therefore, the representation of a cracked cross section by a discrete node by some authors is justified (El-Arem and Maitournam, 2007). In addition, when higher accuracy is not required, or for only two or three vibration modes approximation, the structure can be presented by a simple equivalent discrete system (Ewins, 1984).

Indeed, even though very simple, the beam models provide acceptable results for most localized cracks encountered in practical cases. However, some issues remain open regarding cracked cross section modelling (Dimarogonas, 1996; Dimarogonas and Paipettis, 1983). In particular, we noticed a deviation between experimental and theoretical rigidity and the tube’s local flexibility. We noticed that these methods do not perform local flexibility calculations at the final cracking stage. Consequently, the deviation from experimental observations becomes significant especially for the thick tube. As a result, the life time is underestimated.

Therefore, the purpose of this paper is the demonstration that the global rigidity, calculated using these simpler models, is overestimated due to the fact that local flexibility calculations are not performed at the final cracking stage for beams and thick tubes. After that, a correction model is proposed in order to get results that are closer to the experimental data.

The results showed a problem about initial intact rigidity valuation for beam; in fact, the modification of the section’s geometry, introduced by the crack growth at cross section, was not considered in calculation. When a simple correction is introduced into geometrical properties, results became closer to experimental results. For the tube, results have proved that when the crack traversed almost all the section, except the thickness, the calculation of the local flexibility diverged, but when the combined model is used, results became closer to experimental results.

In this paper, we will present some usual models of cracked beam and tube models, critics, corrections, combined model of tube, and the experimental and numerical tests, results and discussions.

Investigation of usual model of cracked beam

In the first step, simple geometrical methods to resolve cracked beam problems are used. It comes out that, even if the fracture mechanics theory was not considered, acceptable solutions were obtained.

In this framework, several models from the literature are considered, starting with models based only on the geometrical modifications to those based on nodal element and fracture mechanics.

Geometrical models

The Christides and Barr (Sinha et al., 2002) method is based on the introduction of the crack as a change of the inertia moment of prismatic beam. This method gives a good approximation of the phenomenon but the other changes are roughly estimated.

Green and Casey (2005) presented a diagram and showed that there is no crack opening and that, as illustrated in Figure 1, only the geometrical changes control the bending and govern the stiffness reduction.

It is clear that the suppression of all parts of the beam, which do not contribute to the bearing effort to the crack presence inside the beam, can be acceptable, but the approximation remains poor towards the other modifications.

Other authors use a variable spring model at the cracked zone. Loutridis et al. (2005) introduced a spring as a punctual node. El Arem and Maitournam (2007) proposed intact beam elements linked by a nodal element (zero length) to model the crack, where the nodal element has a variable and nonlinear stiffness. Rosales et al. (2004) presented a similar model and Ouahabi et al. (2006) used an interpolation function to model stiffness of a beam with crack breathing. Sinou and Lees (2005) proposed a harmonic approach to model the damaged stiffness.

It is obvious that considering only geometrical changes cannot govern the behaviour of a cracked beam. In fact, as already known, the presence of a crack modifies drastically the mechanical properties of the structure. The crack presence tends to create a stress concentration at the crack tip and adds an additional local flexibility.

It comes out that in order to improve the assessment of mechanical properties and to be as closer as possible to the reality; the introduction of fracture mechanics theory became necessary.

The first attempts to introduce the fracture mechanics were done by Irwin, Bueckner, and Westmann who have linked local flexibility to the stress intensity factor (El Arem, 2006). Later on, Dimarogonas and his team continued to improve the initial model (Dimarogonas, 1996; Dimarogonas and Paipettis, 1983; Sinha, 2002).

Dimarogonas model

Dimarogonas (1996), Dimarogonas and Paipettis (1983), and Sinha (2002) computed an additional local flexibility based on the fracture mechanics theory and then added it to the global flexibility of the intact beam (undamaged beam). The obtained flexibility showed a great dependence for small and large crack, as it will be verified below.

Model of Papadopoulos

Following the same approach, Papadopoulos (2008) proposed the following form to calculate the global flexibility

[C_{g}] = [G_{1}] [C_{cr}] [G_{2}] + [C_{i}]

(1)

where C_cr is the local flexibility due to crack effect, C_i is the flexibility of intact beam, and G₁, G₂ a diagonal matrix (for the simply supported beam) expressed as follows

[G_{1}] = diag [111 l / 2 l / 2 1], [G_{2}] = diag [111 l / 4 l / 4 1]

(2)

and [C_{i}] = [\frac{l}{AE} \frac{l}{4 AG} \frac{l}{4 AG} \frac{l 3}{48 EI} \frac{l 3}{48 EI} \frac{l}{{GI}_{p}}]

(3)

The global stiffness is expressed then in the form

[K_{g}] = [C_{g}] - 1

(4)

For the last three decades, the theory of strain energy release rate, combined to linear fracture mechanics and the rotor dynamics (Papadopoulos, 2008; Papadopoulos and Dimarogonas, 1987), is used to model the behaviour at the cracked zone. This method allows to calculate the resulting compliance to the cracked cross section. However, the question remains: can those models describe correctly the behaviour of the cracked beam?

Critical review of usual models

The review focuses on the methods that consider the addition of local flexibility.

The Dimarogonas model offers the advantage of being easily integrated in a numerical algorithm; however, the accuracy of this model is not well established.

According to EL Arem, several authors (Varé et al., Abraham et al., and Dimarogonas) have highlighted convergence issues when the crack depth exceeds the radius of the rotor section (El Arem, 2009; El-Arem and Maitournam, 2007).

Dimarogonas considered that many issues remain open, especially regarding the closing cracks in rotating shafts (Dimarogonas, 1996; Dimarogonas and Paipettis, 1983). The Dimarogonas model does not consider the geometrical change of the section after the damage.

In addition, some critical remarks have risen around this methodology and approach (Ouali and Dougdag, 2008).

As described previously, the usual model is based on both materials strength and fracture mechanics theories. In the first domain, the beam section is supposed unchanged. In the second domain, the crack effect is considered as a local change (Figure 2).

According to the definition of the stress intensity factor K, the stress singularity at the crack tip makes energy of dissipation nonlinear when the crack lips are opened. So, this indicates that an additional part of energy is released in addition to the elastic strain energy. As a consequence, under loading, the beam deformation is followed by an additional displacement. However, this approach does not reveal that the section at the crack tip is modified after cracking. Nevertheless, a numerical model with finite element can reveal these aspects.

Numerical demonstration

A simple simulation by finite element method of the crack growth effect shows two aspects (Figure 3): the stress strengthening at crack tip and the shifting of the neutral axis.

The shifting proves that the mechanical properties have changed after the crack growth; i.e. the second moment of area became modified.

Therefore, cracking causes not only a local modification but also the change of mechanical properties. For those reasons, the geometrical effect should be considered in the global flexibility calculation. To highlight that, a simple method, which uses a correction factor, is investigated in “Application to a simple case” section.

Investigations about cracked tube

Several works were devoted to the cracked tube.

Gang and Xing (1989) mentioned that in engineering, there are many cylindrical bars and tubes with transverse surface cracks for which the solution of stress intensity factors is very difficult even by means of the finite element method. Therefore, there are no results about them. Gang and Xing (1989) use the method of the energy released rate and the model of thin annuli of cracked cylindrical tube.

He et al.(2009) settled that the stress intensity factor in a cracked pipe is calculated by considering a combination of series of thin annuli to obtain the theoretical solution for the local flexibility. Other authors, such as Liu et al.(2003), Yoon and Son (2004, 2005), used the same approach.

Regarding thick tubes, in addition to previous remarks about the beam model, the thin annuli method does not show any comparisons (similarities or differences) with the method of local flexibility of cylindrical beam.

In general, when the transverse crack crosses the thick tube thickness (partial cut), the local flexibility should be closer than the beam’s one, if the weight is not considered. When the crack crosses the hole (throughout the cut); the flexibility of the tube will be different. However, after crossing the hole (final stage), the flexibility of the tube should be the same for both because they have the same crack tip geometry and the same section, which remain intact. Those models do not report any discontinuities at these borders, as well. On the other hand, the thin annuli models are limited to crack depths smaller than (R_in + R).

Analytical model development

In order to improve the usual model, an analytical development is needed to introduce corrections.

The cracked beam model (Dimarogonas and Paipettis, 1983) used for an open crack is based on the elasticity theory (Lee, 1993; Timoshenko, 1968) with consideration of an additional local flexibility (Dimarogonas, 1976; Dimarogonas and Paipettis, 1983). This flexibility is obtained by the strain calculation according to Castigliano’s theorem and by generalization of the Paris’ equation (5) and with using the strain energies by the stress intensity factors (Dougdag et al.,1992; Owen and Fawkes, 1983; Sabot, 1993).

Before starting, an analytical development, the “breathing” problem for the case of rotating machine should be reviewed.

Some authors (Gash and Henry) modelled the nonlinear mechanism due to the “breathing” by applying the De Laval principle rotor. Mayes and Davies (1976) improved this technique by the introduction of the Paris’ principal energy rule. Grabowski (1979) showed that nonlinearity due to the closing and opening of the crack does not influence the equation of motion. Varé and Andrieu (2005) and Andrieux and Varé (2002) proposed a new method, which consists to start from the 3D calculations and taking into account the unilateral contact between the crack lips, a nonlinear behaviour relation between bending and displacements, compatible with the beams theory in order to allow the introduction into a “wire model”. Audebert and Voinis have validated this approach experimentally. Nevertheless, Bachschmid et al.(2004) showed that applying load torque to the sections that can crack presented a significant component of shearing, requires the integration of the shear loads into the model.

Analytical model of Dimarogonas

The study of a beam (prismatic or round bar) subjected to bending, shearing, and tensile loads in the presence of a transverse crack depth (a) gives additional displacements (u_i) along the (P_i) loading axis as presented in Figures 4 and 5 (Dimarogonas and Paipettis, 1983).

In this part, a brief and illustrative presentation of the model is given. We notice here that the model is slightly modified by considering the torsional force.

The prismatic beam model

The prismatic beam is presented in the global coordinate system as follows:

The strain energy density general function is defined by

J (a) = \frac{1}{E'} [(\sum_{m = 1}^{6} K_{I . m}) 2 + (\sum_{m = 1}^{6} K_{II . m}) 2 + κ (\sum_{m = 1}^{6} K_{III . m}) 2]

(5)

with: E’ = E (plane stress, m = 1, 4, 5), E′ = E/(1 – ν²) (plane strain, m = 1, 4, 5),

E’ = G (shear, m = 2, 3, 6) and κ = 1 + ν

Flexibility influence coefficient is expressed as

c_{ij} = \frac{1}{E' b} \int_{0}^{a} [\frac{\partial 2}{\partial P_{i} \partial P_{j}} \sum_{n}^{III} \int_{0}^{b} (e_{n} \sum_{m}^{6} K_{nm}) 2 d y] \cdot d a,

(6)

where n = I, II, and III are the three cracking modes.

e_n = κ for n = III and e_n = 1 for n = I and n = II.

K_nm : stress intensity factor for the modes (n) under the loadings P_m for m = 1, 2,.., 6

The stress intensity factor is expressed by

K_{nm} = σ_{m} \sqrt{π a} F_{l} (a / h)

(7)

where l depends to n(5) as follows : l = 1, 2 for mode I and l = II, III for modes II, III

We have zero values of K(1,4,5,6) for the following cases:

K_{nm} = 0 for (n, m) = {(I, 2), (I, 3), (I, 6), (II, 1), (II, 2), (II, 4), (II, 5), (II, 6), (III, 1), (III, 3), (III, 4), (III, 5)}

where stresses are defined by

σ_{1} = \frac{P_{1}}{A} = \frac{P_{1}}{bh} σ_{2} = \frac{P_{2}}{A} = \frac{P_{2}}{bh} σ_{3} = \frac{P_{3}}{A} = \frac{P_{3}}{bh} σ_{4} = \frac{P_{4}}{I_{4} / ξ} = \frac{12 P_{4}}{b 3 h} ξ σ_{5} = \frac{P_{5}}{I_{5} / η} = \frac{12 P_{5}}{h 3 b} η σ_{6} = \frac{P_{6}}{γ hb 2}

(8)

where η = h/2 and

γ = \frac{1}{3 + 1.8 \frac{b}{h}}

. γ is the correction factor, which depends on the ratio (b/h) (Timoshenko, 1968), a is the crack depth and F_l(a/h) is the shape function factor dependent on the cracking modes and the load type (Dimarogonas and Paipettis, 1983; Dougdag, 2001).

Equation (5) contains the added torque stress (equation 8) applied to prismatic section (Timoshenko, 1968; Bourgain et al.,1988).

For this case, the flexibility coefficients result primarily from the strain energy that is dependent on the stress intensity factors. The crack width (b) covers all the beam width.

The strain energy density related to the stress intensity factors becomes

J = \int_{0}^{+ b} [\frac{1}{E'} [K_{I 1} + K_{I 4} + K_{I 5}] 2 + \frac{1}{G} K_{II 3}^{2} + \frac{κ}{G} [K_{III 2} + K_{III 6}] 2] d ξ

(9)

The substitution of K_mn by their expressions in the last equations and integrating them in the range width of [0, b], the equation of flexibilities coefficients becomes

c_{ij} = \frac{Φ_{kl}}{E' b p h q} (1)

(10)

with : Φ_{kl} = \int_{0}^{a} π (\frac{a}{h}) F_{k} (a / h) F_{l} (a / h) d (a / h)

(11)

where p and q are defined between 1 and 3.

While, the flexibility matrix [C] is symmetric (Dimarogonas and Paipettis, 1983; Sabot, 1993), the global flexibility at crack location is defined by the addition of the local flexibility to the intact one

[C_{g}] = [C_{cr}] + [C_{i}] = > [K_{g}] - 1 = [K_{cr}] - 1 + [K_{i}] - 1

(12)

Round bar models

The study of a round bar segment (AB) with the following properties: a diameter (D) and a transverse crack depth (a). This segment of round bar is subjected to forces as shown in Figure 5.

The calculation of the local flexibility is done following the previous approach with some differences; indeed, in addition to the section difference, there is also a difference of crack tip form, which makes one more weak than the other.

The passage from prismatic to cylindrical geometry requires a change of variables to determine the effective parameters, as sketched in Figure 6.

The change of variables is

a = a (ξ) = \sqrt{R 2 - ξ 2} - (R - a), h = h (ξ) = 2 \sqrt{R 2 - ξ 2} and b = \sqrt{R 2 - (R - a) 2}

(13)

The local flexibility is then expressed in a non-dimensional and local coordinate system (ζ, ξ, η) as follows

c_{ij} = \frac{\partial 2}{\partial P_{i} \partial P_{j}} \int_{0}^{α} \int_{- b}^{+ b} J (R, a / h, P_{i}, P_{j}, F_{mn} (a / h)) d a d ξ

(14)

After transformation, the flexibility becomes

c_{ij} = \frac{b p h q Δ Θ Φ_{kl}}{E' π 2 R 8}

(15)

with : Φ_{kl} = \int_{0}^{α} π (\frac{α}{h}) F_{k} (α / h) F_{l} (α / h) d (α / h)

(16)

Θ = \int_{- b}^{b} \sqrt{R 2 - ξ 2} d ξ = - R 2 \int_{ϕ_{0}}^{ϕ_{1}} \sin 2 ϕ . d ϕ = \frac{R 2}{2} [(\cos (2 ϕ_{1}) - \cos (2 ϕ_{0})) + 2 (ϕ_{0} - ϕ_{1})]

(17)

Δ = \int_{- b}^{b} ξ \sqrt{R 2 - ξ 2} d ξ = - R 3 \int_{ϕ_{0}}^{ϕ_{1}} \cos ϕ \sin 2 ϕ . d ϕ

(18)

ξ = R . cos ϕ, ϕ_{0} = a cos (b / R) and ϕ_{1} = π - ϕ .

(19)

(i, j) have a relation with (k, l) written in equation (10) and p, q, and s are defined between the values 1 and 6.

Thick tube model

For the thick tube model, it is proposed to combine the round bar and the thin annuli’s models (He et al.,2009). The motivation of this choice is to take advantage from each of the methods. The first method was able to model more efficiently the small crack depth and the second method more efficiently the large crack depth (depth exceeding the radius, see Figure 7(c)).

The thin model method subdivides the tube on (n) thin annuli (He et al., 2009) and the crack growth passes through three situations as illustrated in Figure 7:

when the crack depth is smaller than the tube wall thickness

t = \frac{a}{n} and R_{i} = (R - a) + t (i - \frac{1}{2})

(20)

when the crack depth is bigger than the tube wall thickness

t = \frac{(R - R_{in})}{n} and R_{i} = R + t (i - \frac{1}{2})

(21)

In those stages, we applied equation (14) for the bending case of the (i^th) thin annuli located between (–ϕ_o) and (+ϕ_o) with a thickness of (t/n). The flexibility becomes (He et al., 2009)

c_{ξ} = \frac{4 \sqrt{2} (1 - υ 2)}{E π 2 t} \sum_{i = 1}^{n} (\frac{(\frac{I_{i}}{I}) 2}{R_{i}^{2} ɛ_{i}} \int_{0}^{ϕ_{i}} F (ϕ) 2 d ϕ)

(22)

where ϕ_{i} = \arccos (\frac{R - a}{R_{i}}) I = I = \frac{π}{4} (R 4 - R_{in}^{4}), I_{i} = \frac{π}{4} ((R_{in} + \frac{t}{2}) 4 - (R_{in} + \frac{t}{2}) 4)

(23)

ɛ_{i} = \frac{\frac{t}{R_{i}}}{\sqrt{12 (1 - υ 2)}} and F (ϕ) = \sin (ϕ) [1 + \frac{1}{2} \frac{ϕ - \cot (ϕ) (1 - θ . \cot (ϕ))}{2 . \cot (ϕ) + \sqrt{2} \cot (\frac{(π - ϕ)}{\sqrt{2}})}]

(24)

when the crack depth exceeds the hole (a ≥ R + R_i_n)

The previous equation (15) of round bar (“Analytical model of Dimarogonas” section) is applied.

The beam model corrections

Principle

The deduction of the correction factors is extracted from the comparison of the modified and intact beam cross sections. The correction factor will be introduced into the initial intact flexibility.

[C_{g}] = [C_{cr}] + [C_{i}^{m}]

(25)

C_{i}^{m} = fact . C_{i}, fact isthefactor .

(26)

Correction factors of quadratic moment

Some reductions are done in order to simplify the correction factor computation of a cylindrical section, as illustrated in Figure 8.

The modified moment I_az could be approximated by the subtraction of the second moment of area of the rectangular surface, as illustrated with the dashed line in Figure 8

I_{az} = I_{z} - α \frac{ba 3}{12} if a < = R and I_{az} = I_{z} - α \frac{bR 3}{12} and ifa >

(27)

with

with : α = \frac{3 π}{16} and I_{z} = \frac{π R 4}{4} .

The correction factor could have the following approximated form

{fact}_{z} \approx 1 + (\frac{I_{z} - I_{az}}{I_{az}}) β

(28)

with β express the power (= 2 or 3)

Application to a simple case

A simple example of a shaft under pure bending loads, according to the plane stress case, is studied in order to simplify the demonstration.

The local flexibility for the shaft calculation gives (Dimarogonas and Paipettis, 1983)

c_{ξ} = \frac{(1 - ν 2)}{3 E' π 2 R 3} \int_{0}^{a} \int_{- b}^{b} 32 [1 - (ξ / R) 2] π (\frac{α}{h}) F_{k} (α / h) F_{l} (α / R) d (α / R)

(29)

The calculation of equation (29) gives local flexibilities as a function of the crack depth, where the numerical double integral calculation of the term Φ₂₂ (equation (16)) has used the Simpson method (Sabot, 1993).

The geometric correction factor used is defined by

I_{ξ}^{m} = \frac{1}{{fact}_{z}} . I_{ξ}

(30)

This factor will be introduced in flexibility calculation as follows

C_{ξ}^{m} = {fact}_{z} . C_{ξ}

(31)

For the simple case of the De Laval’s rotor with a shaft of a length (l) and an initial stiffness k_i (without crack), the global shaft stiffness K_g becomes (Dimarogonas and Paipettis, 1983)

K_{g} = \frac{k_{i}}{(1 + \frac{c_{i}^{m} k_{i} l 2}{8})}

(32)

For the validation of this theoretical model, an experimental test is used.

Experimental model

Testing device

The experimental device shown in Figure 9 is designed to support beams with different sections and according to several modes of maintaining. It is constituted of the two vertical strong supports connected by three robust beams. Vertical supports are fixed at the concrete slabs.

Beam models description

The beam models are fixed on the bench with screws. The steel bars used for these experimental tests have a length of 800 mm and various sections (Figure 10). Material properties are the Young modulus E = 2.10¹¹N/m², the density ρ = 7800 kg/m³, and the Poisson’s ratio is ν = 0.33.

Figure 10 represents the numerical model, the analytical test beam, and all the beam sections that were used. The figure shows all positions of applied forces, displacement measurement, and crack. The figure shows also that the numerical model respects the same crack sizes and position of the test beam as shown in Figure 9.

Results

Flexibilities and stiffness variation

The experimental tests started with different cantilever steel beams (prismatic, round, and tube). During the tests, static forces were applied at various positions and the displacements were measured (Figure 10). In order to make sure that the bar behaviour remains elastic when the thickness becomes thin, lower forces were applied. The measurements have been repeated several times to commensurate with the uncertainties.

The obtained results for round bar are presented by a series of stiffness element K_ij as illustrated in Figure 11, where (i,j) indices represent respectively positions (I = 2 to 8) for forces and (j = 8) for displacement positions (Figure 10). The stiffness K₂₈ is the greater one and K₁₈ cannot be measured.

Figure 11 shows that the effect of the crack growth on the flexibility is increasingly important at final stage of cracking and at the position near the fixation point. In our example, only the flexibility C₈₈ is used. In additive, the slight stiffness increase occasioned at small crack size (at beginning), what can be explained by measurement errors of displacement.

Validation of the theoretical model

Initial intact flexibility correction

In order to highlight the correction effect, equations (28), (29), (31), and (32) are used and compared to the geometrical model of Christides and Barr (Sinha et al., 2002).

Figure 12 shows that the Christides and Barr curve can be obtained by using an estimated factor f_C&B represented by the black curve.

This curve gives a strongly conservative approach of the flexibility change; the effect of the crack growth is visible from the first stages. However, the use of a correction factor (equation (31)), with power 2 or 3, seems to induce a very limited damage of the initial (intact) flexibility at first stages and becomes stronger at the final cracking stage; this approach is in agreement with experimental results (see next section).

Comparison of the global flexibility curves

A finite elements model (Figure 10) is used to support the operation of verification (as a tool of reinforcement). The obtained global stiffness in experimental tests is compared to the numerical one. The results are presented in Figure 13.

Figure 13 shows that the numerical curve is, in general, closer to the experimental data in the range of (a/R = 0 to 1.6) the crack depth, which corresponds to 80% of the diameter. Above that, slight discrepancies are observed. This result confirms that the numerical model is in good agreement with the experimental data in this range.

To highlight the geometric correction effect, a comparison is done between the initial (without correction), the corrected, and experimental global flexibilities. The results are presented in Figure 14.

Figure 14 illustrates the curves variation of global flexibilities and stiffness for the initial, corrected, and experimental. One can notice deviations of the curves when the crack depth exceeds the radius. The deviation is more important for the initial flexibilities that grow and remain constant until fracture. Similar observations were reported by other authors (El Arem, 2009; El-Arem and Maitournam, 2007). On the opposite, the deviation between the corrected and the experimental curves becomes smaller and the two curves are closer at the end of stage.

In conclusion, the benefit of the addition of a correction is confirmed by the experimental observation and the present model behaves better than usual models.

Flexibility of the thick tube

Local flexibility of the thick tube

For the two models of cracked round bar and thin annuli, the results are given in Figure 15 for different ratios (R_in/R). It is shown, as expected, that all the curves of the thin annuli method, from various ratios (R_in/R), tend to join the cylindrical beam curve, when the crack depth reaches the value of (R_in + R). Afterwards, it diverges (dashed lines) because this method is not able to calculate large crack depth. In addition, it is noted that the thin annuli method is not appropriate for the case of round bar (R_in/R = 0), as shown by the gap with the curve of round bar, especially for the small crack depth.

On the other hand, the thin annuli method gives curves with minor and punctual discontinuities (small impulse which is observed when the crack depth reaches the borders (a = t and a = R_in + R)); those discontinuities are due to the geometrical change and/or to the change of variables in calculation.

It should be mentioned that local flexibility results of both round bar and tube were successfully compared to literature cases (Dimarogonas and Paipettis, 1983; He et al., 2009). He et al. (2009) have checked successfully the method of the thin annuli experimentally for a small crack size and the Dimarogonas method is widely used.

Therefore, we consider combining the two methods and propose a new alternative method to correct for local flexibility computation.

One should note that the correction becomes strongly recommended when the crack depth exceeds (R_in + R), especially for great wall thicknesses of tube, for example t/R > 50%.

Application to fundamental frequencies measurement

Considering experimental observations, some dynamic tests were conducted until maximal crack depth before rupture (Dougdag, 2014). In those tests, the two methods were used. The first one considers measurement of fundamental frequencies by spectral analysis at each stage of cracking under random excitation. The second method (theoretical–experimental) considers using the rigidity obtained by experience in static mode to evaluate the fundamental natural frequencies using the Rayleigh’s method approximation (Harris and Piersol, 2002). As in previous section, the results are presented for the cases with and without correction and compared to experimental data, for round bar, thin annuli, and combined model.

Figure 16(a) shows a good agreement at the beginning of cracking for all curves. However, later a slight gap settled and became reducing continuously until rupture. The theoretical–experimental results are in good agreement with the corrected curves, while the experimental curve is in better agreement with the curve without correction.

Those experimental results show that round bar is stronger than what is illustrated by stiffness damage in Figure 14; the frequencies measurement errors by spectral analysis can be the cause of this situation.

Figure 16(b) shows that the experimental and the theoretical–experimental curves for the frequency is more sensitive than the round bar, while all the analytical curves remain less affected compared to experimental data. However, at the final cracking stage, the curves are more or less converging.

If the geometric correction is added to the combined model, the curve is in better agreement with experimental data than to the other analytical curves. This correction is strongly sensitive to the wall thickness. When all the curves tend to the zero value, the thin annuli do not follow the same trend because it is limited by the condition of (a < R_in + R).

It comes out that in general, the results for the combined model are more interesting than for round bar in viewpoint of spectral analysis and those results prove that the calculations of the tube rigidity should be more carefully considered at the final stage.

Conclusions

The review of current models for cracked beam and tube showed disagreement with experimental observations in the sense that the effect of the section’s geometry change in stiffness’s computation, during the crack depth growing process is underestimated.

Therefore, a new model is proposed in order to improve this evaluation and some corrections have been introduced in the beam model based on a combination of the two generic models. As a result, the stiffness assessment became better; in fact, the results for the obtained stiffness were in good agreement with experimental data, when the crack depth exceeded the beam’s radius for both models studied. In addition, good similarities are achieved between curves of local flexibility of developed model and those found in literature (Dimarogonas and Paipettis, 1983; He et al., 2009) giving a solid theoretical base to this analytical development.

In conclusion, these findings enable us to consider that the modified model is capable to improve slightly the stiffness assessment of both the beam and the tube for all the stages of cracking. In particular, the improvement is more apparent when the crack depth exceeds the beam’s radius for the thick tube. Furthermore, this model brings a solution when others diverge.

Consequently, this new model will help for a better assessment of the tube lifetime.

Footnotes

Acknowledgement

Thanks to Mr. Tewfik Hamidouche for his contribution.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

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