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The dynamic stiffness matrix (DSM) of axially loaded multi-cracked frames
Accepted Manuscript Title: The Dynamic Stiffness Matrix (DSM) of axially loaded multi-cracked frames Authors: Salvatore Caddemi, Ivo Calio, Francesco Cannizzaro PII: DOI: Reference:
S0093-6413(17)30010-1 http://dx.doi.org/doi:10.1016/j.mechrescom.2017.06.012 MRC 3180
To appear in: Received date: Revised date: Accepted date:
5-1-2017 25-5-2017 17-6-2017
Please cite this article as: Caddemi, Salvatore, Calio, Ivo, Cannizzaro, Francesco, The Dynamic Stiffness Matrix (DSM) of axially loaded multi-cracked frames.Mechanics Research Communications http://dx.doi.org/10.1016/j.mechrescom.2017.06.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Publication Office:
Elsevier UK
Mechanics Research Communications. Year Editor-in-Chief: A. Rosato New Jersey Institute of Technology, Newark, New Jersey, USA [email protected]
The Dynamic Stiffness Matrix (DSM) of axially loaded multi-cracked frames Salvatore CADDEMI1*, Ivo CALIO’1, Francesco CANNIZZARO1 1
Dipartimento Ingegneria Civile e Architettura, University of Catania, Via Santa Sofia 64, Catania, Italy
*Corresponding author [email protected] Tel.:+39-095-738-2266; fax: +39-095-738-2297
Highlights
Study of the influence of the axial load on free vibrations of damaged frames.
The Dynamic Stiffness Matrix (DSM) approach is adopted.
A closed form expression of the DSM for frames with an arbitrary number of cracks is presented.
The presented DSM allows the evaluation of the natural frequencies of damaged frames.
The results can be achieved with the desired accuracy with the Wittrick&Williams algorithm.
Abstract The presence of axial load, due to mechanical loading or to temperature effects, represents a strong peculiarity of frame applications both in direct and inverse problems involving the presence of cracks. The lack of explicit formulations of the Dynamic Stiffness Matrix (DSM) for cracked beam elements accounting for the influence of axial loads inspired and motivated the present work and the calculations herein reported. In this paper, the transcendental, frequency dependent, exact explicit expression of the DSM of an Euler-Bernoulli beam-column in presence of an arbitrary number of open cracks is presented. The advantage of the derived expression consists in a fourth order DSM whose dimension stays consistent independently as the number of cracks along the beam increases. Assembling DSMs of single beam-column elements, according to any required frame layout, leads to a number of degrees of freedom of the overall damaged frame structure exactly the same as those of the equivalent undamaged structure. Among other recent achievements on the vibration of damaged frames, the presented formulation provides a further original contribution, in the context of the dynamic stiffness method, which, in conjunction with the Wittrick & Williams algorithm, allows the exact evaluation of the frequencies and vibration modes of damaged frame structures in presence of axially loaded elements. Keywords: Cracks, dynamic instability, dynamic stiffness matrix, damaged beam-columns, damaged frame, Wittrick-Williams algorithm.
0093-6413© 2015 The Authors. Published by Elsevier Ltd.
Publication Office:
Mechanics Research Communications. Year
Elsevier UK
Editor-in-Chief: A. Rosato New Jersey Institute of Technology, Newark, New Jersey, USA [email protected]
1. Introduction The study of vibrations of continuous systems may be conducted by means of the formulation of the Dynamic Stiffness Matrix (DSM) of the structure that is capable of retaining the infinite number of degrees of freedom of the problem and extracting exact results. In fact, the DSM is a frequency dependent matrix, obtained by the solution of the differential equations of the continuous problem, that leads to a transcendental eigenproblem. Many important contributions on the field of the dynamic stiffness method applied to beam-like and frame structures are reported in the literature in the context of inhomogeneous and discontinuous beams [1-11] where peculiarities and advantages of the dynamic stiffness method have been clearly provided. In the latter papers Banerjee and co-workers highlighted how the dynamic stiffness method, unlike the Finite Element Method (FEM), is related to exact frequency dependent shape functions which contain both the mass and stiffness properties. Furthermore, unlike the FEM the results obtained from the DSM are independent of the number of elements used in the analysis. Despite its uncompromising accuracy, the literature on the DSM is not as broad or diverse as the FEM and it is unknown to many scientists focused on FEM analyses. Very recently, scientists involved in the analysis of vibration of continuous systems have been encouraged by Banerjee [12] to provide explicit formulations of the DSM for those cases yet uncovered. The same appeal has been further confirmed in a successive broader context in [13]. The authors have been involved in the past in the application of the DSM for beams and frames in the presence of multiple cracks. In fact, the exact stiffness matrix for the stability and the dynamic of frame structures affected by the presence of multiple cracks have been proposed in [14] and [15], respectively. The latter two papers are deeply based on the explicit formulation of the solution of the governing equations of columns [16] and beams [17] in presence of cracks without recourse to finite-element discretization. The DSM previously formulated by the authors for the case of multi-cracked beams provided a fruitful basis for the exact evaluation of frequencies and modes of cracked frames. However, in all the previous studies, despite the efficacious formulation, the latter stiffness matrix suffers the limitation of neglecting the presence of any axial load acting on the beam elements. Recently also Labib et al. [18] faced the direct problem of calculating the natural frequencies of cracked beams and frames by using a DSM that was obtained in a recursive manner by applying partial Gaussian elimination. Furthermore, Labib et al. in [19] applied the latter procedure to solve the inverse problem regarding the identification of a single crack in a frame structure by using natural frequency degradations also in presence of contaminated measurements and applying efficaciously interval arithmetic. Furthermore, more recently, Failla [20] analysed the frequency response of beams and plane frames with an arbitrary number of Kelvin-Voigt visco-elastic dampers by evaluating the exact DSM of the structure taking advantage on the use of generalised functions. Once the DSM for the structure under study is conveniently formulated, the Wittrick and Williams (W&W) algorithm [2123] is generally adopted to evaluate the exact eigenvalues. The W&W algorithm represents a robust and reliable alternative to finding the roots of the DSM determinant by means of cumbersome zero search procedures. In particular the algorithm starts from an initial choice of the frequency and subsequently evaluates how many frequency values are smaller than the chosen value. Successive iterations aim at finding the frequency value such that no frequencies between the actual and the previous one are found. Among the procedures to choose successive trial frequency values to converge on any required frequency of the transcendental eigenvalue problem, the bisection method is the most popular, however other techniques for faster convergence methods have been devised [24,25]. Another reliable procedure for eigenvalue evaluation is based on the definition of the “member stiffness determinant” defined as stiffness matrix determinant of an element discretised by infinite finite elements with the end sections clamped [26]. By doing so the frequency dependent global stiffness matrix is not affected by poles. Once the desired frequencies of the continuous system are evaluated, the W&W algorithm requires a matrix inversion to compute the correspondent eigenvectors. A recent alternative to the original W&W algorithm based on a congruent transformation of the DSM is introduced whose last element is denoted as last energy norm [27]. The trial frequency that makes the latter norm vanish is an eigenfrequency of the system and the last column of the congruent stiffness matrix tends to the corresponding vibration mode. By doing so, the matrix inversion generally required by the W&W algorithm to infer the vibration mode can be avoided. Applications of the W&W algorithm to cracked frame structures and beams can be found in [14,15,28].
0093-6413© 2015 The Authors. Published by Elsevier Ltd.
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This work provides a novelty in the field of the dynamic stiffness method that consists in the evaluation of the explicit closed form formulation of the DSM of axially loaded frames with multiple cracks. No formulation of the dynamic stiffness matrix of multiple cracked axially loaded frames is available in the current literature. The explicit formulation of DSM for axially loaded cracked frames herein presented improves significantly the standard procedure commonly applied in the literature for the construction of the DSM. In fact, the standard approach available in the literature relies on the adoption of the DSM for the continuous beam with no cracks for each segment between two successive cracks. The latter matrices are then assembled by accounting for the continuity conditions at the cracked cross-sections. The latter subdivision is inevitably affected by a degree of freedom growth of the assembled structure with respect to the undamaged one dependent on the number of cracks. A simple application of the standard approach with augmented degrees of freedom is shown in [29] where a single crack in a portal frame is considered with the aim to investigate the crack detection inverse problem. Hence the improvement implied by the proposed approach with respect to the standard procedure is that the latter requires an increasing dimension of the DSM with the number of cracks while the proposed expression incorporates an arbitrary number of cracks by leaving unaltered the matrix dimension as that of frame composed of undamaged elements. Furthermore, even though previous formulations of DSM for damaged beam elements available in the literature including the presence of cracks are considered such as [15], the effect of axial loads cannot be included hence the analysis will lead to inaccurate results. The authors dealt with axially loaded beam elements affected by the presence of multiple cracks according to a non standard approach based on the use of generalised functions in [30,31]. However, in the latter papers, no DSM for the analysis of more complex structures, such as damaged frames, was therein formulated. Hence, the novel expression of the DSM to account for the presence of axial load in damaged frames represents a significant non trivial advancement with respect to both the standard approach and the generalised function approach. The distributional approach has been also used by the present authors to investigate on the importance of the effect of axial loads with regard to tensile/compressive flutter and divergence instabilities on discontinuous beams particularly in [31-33] while experimental evidence of the latter phenomena, somehow not quite well understood in the past, can be found in [34,35]. The exact closed-form expression proposed in this work can be used also as a benchmark or conveniently adopted for extensive parametric analysis. However, the main advantage consists in the use of four degrees of freedom at the end crosssections of a multi-damaged frame element in analogy to the standard undamaged element. Moreover, the proposed approach allows modelling of arbitrary distributions of cracks in frames without any variation of the global matrix dimension. The evaluation of the exact natural frequencies of damaged frames is then conducted by customizing the W&W algorithm to include the effect of the axial load in multi-cracked frames. The correspondent eigenvectors of the overall system are also straightforwardly obtained by explicit expressions without any additional discretisation or matrix inversions. A numerical application is reported to show how the theoretical results can handle cases with an increasing number of cracks both in terms of natural frequencies and vibration mode shapes. In this study cracks are considered always open and the behaviour of the cracked frame is linear. However during vibrations, particularly in case of the presence of axial compressive load, nonlinearities due to the closing/opening phenomenon (switching cracks) may arise. An example of vibration analysis of a simple beam with switching cracks can be found in [36] where the non linear analysis was conducted as a sequence of multiple linear regions according to the number of cracks in the open state. The assumption of open cracks in vibration of cracked frames under axial load, and the presented explicit DSM expression, considered in this work are fundamental achievements to address the more complex non linear problem of switching cracks in frames. In fact, as the number of active open cracks varies during vibrations the DSM changes and could be evaluated with the proposed expression by updating the number of active open cracks. With respect to the results herein presented the influence of the crack closure and contact is expected to consist in that the fundamental frequencies are collocated between those corresponding to the always-open and to the always-closed (undamaged) crack condition. 2. The exact explicit solution of the multi-cracked beam-column The presence of cracks introduces a local flexibility in beams whose quantification allows a simple and effective representation of the vibrating behaviour of cracked beams and frames [37]. The analysis and evaluation of the crack-induced local flexibility has been conducted by various authors [38-40] and related to the stress intensity factor, describing the intensity of the stress field about the tip of the crack. Based on the previous studies, the evaluation of the stress intensity factor was determined by experimental methods based on measuring the local flexibility (or, alternatively, the local bending stiffness) [4143]. On the other hand, according to a macroscopic approach, widely accepted in the literature, the effect on the local flexibility introduced by a crack in a beam element can be modelled by an equivalent rotational spring with stiffness K eq connecting the two adjacent segments of the beam, as for example discussed in [44,45]. A mechanical justification of the macroscopic model of rotational elastic spring commonly used to describe the presence of an open crack in a beam under bending deformation has been provided in [46]. Within the latter approach, different models have been presented in the literature proposing expressions of the spring stiffness K eq equivalent to the crack for a large number of cases, concerning different geometry of the cross-section and different crack shapes. As a matter of example, when a lateral crack of uniform depth d is present in a rectangular cross-section of width b and height h, the following expression for the stiffness K eq Eo I o / [hC ( )] can be adopted where d / h is defined
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as the ratio between the crack depth d and the cross-section height h, Eo and Io represent respectively the Young’s modulus and the moment of inertia, and C(β) is a dimensionless local flexibility which can take different forms according to the chosen damage model as summarized in Appendix D of [17]. The flexibility C ( ) of the equivalent spring represents the macroscopic effect of the local flexibility introduced by a crack and is hence related to the stress intensity factor. In this work cracks are modelled as concentrated flexural stiffness reductions of the beam element by means of Dirac’s delta generalised functions, which proved to be equivalent to the macroscopic rotational spring approach [17]. Assuming that the amplitude of the deformation is such as to maintain the crack always open, the assumed model offers the great advantage to be linear. The results of the linear model represent an indispensable basis to study more advanced non linear problems implying the closure of cracks. Following this approach, the presence of a crack provides a slope discontinuity at the location of the crack that can be modelled through generalised functions. Namely, the presence of an arbitrary number of cracks can be modelled by considering Dirac’s deltas, centred at the crack positions, superimposed to a uniform flexural stiffness as follows [47]: n E ( x) I ( x) Eo I o 1 ˆi ( x xi ) i 1
(1)
where x is the spatial abscissa spanning from 0 to the length L of the beam. Furthermore, in Eq.(1), the n singularities, given by Dirac’s deltas ( x xi ) centred at abscissae xi , i 1, , n , represent n cracks and the parameters ˆi , i 1, , n , multiplying the Dirac’s deltas, are related to the rotational stiffness of the equivalent internal springs, which simulates the cracks, as specified in the referenced papers [16,17]. Accounting for Eq.(1), the free vibrations of a uniform beam-column in presence of multiple cracks, with a uniformly distributed mass m, subjected to a constant axial compressive force N, are governed by the following normalised differential equation: n NL2 mL4 1 ( ) u ( , t ) u ( , t ) u ( , t ) 0 i i Eo I o Eo I o i 1
(2)
where ξ = x/L is the normalised spatial abscissa, t is the time variable, i ˆi / L are the normalised crack parameters and
u( , t ) is the transversal displacement. The apex indicates derivatives with respect to ξ, while the superimposed dot indicates the time derivative. By adopting a standard separation of variables, Eq.(2) can be formulated in terms of spatial vibration mode ( ) as: n NL2 mL4 2 0 1 i ( i ) Eo I o i 1 Eo I o
(3)
where is the cyclic frequency. By introducing the axial load parameter 2 NL2 / Eo I o and the frequency parameter 4 2 mL4 / Eo I o variable, Eq.(3) may be written in the non-homogeneous dimensionless form as follows:
IV 2 4 B( )
(4)
where the function B( ) collects all the terms with the Dirac’s deltas and their derivatives as follows: n
n
i 1
i 1
B i IV ( i ) 2 i ( i ) n
i ( i )
(5)
i 1
The general explicit solution of Eq.(4) has been derived by the authors in [30], however it is here written under a different form more suitable for the explicit derivation of the DSM as presented in the next section. In fact, the frequency/axial load dependent vibration mode representing the solution of Eq.(4) is written as follows: 4
; , Ck f k ; ,
(6)
k 1
where , are frequency/axial load dependent parameter defined as , 2 / 2 [ 4 / 4 4 ]1/ 2
1/ 2
.
In Eq.(6) Ck , k 1, , 4, are integration constants to be easily evaluated by the relevant boundary conditions, while the generalised functions f k ( ; , ), k 1, , 4, are given by:
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f k ; ,
1 2 2
5 n
i 1
i ki
( , ) Si ; , U ( i )
(7)
ak ; , where the crack intensity parameters i , i 1..., n , dependent on i as fully detailed in [30], but also related to the standard dimensionless crack local compliance C(β) as follows i C (i )h / L , have been introduced. All the terms appearing in Eq.(7) are fully defined as follows: ki
i 1 1 3 sin i j 3 sinh i j bki (8) 2 j kj j 1 2
Si ; , sin i sinh i a1 ; sin ; a2 ; cos ; a3 ; sinh ; a4 ; cosh ;
(9)
b1i 2 sin i ; b2i 2 cos i ; b3i 2 sinh i ; b4i 2 cosh i .
It is worth to notice that each function f k ( ; , ), k 1,
, 4, expressed by Eq.(7), can also be written as the sum of its
generalised part gˆ k ( ; , ) plus the trigonometric or hyperbolic terms ak ; , as follows:
f k ; , gˆ k ; , ak ; ,
(10)
where the generalised part, in view of Eq.(7), is defined as: gˆ k ; ,
1 2 2
n
, S ; , U ( ) i 1
i ki
i
i
(11)
while the terms ak ; , , defined in Eq.(9), are relative to the solution of the undamaged uniform beam-column, that is obtained as particular case when all the crack intensities approach zero, i 0; i 1..., n . It is worth noting that the solution expressed by Eq.(6) is valid for the overall beam-column and for any number and positions of cracks. Furthermore it preserves the same analytical structure of the undamaged beam, being a function of four integration constants only irrespective of the number and the positions of the cracked cross sections. This form of the exact solution allows the derivation, in closed form, of the exact DSM of a multi-cracked beam-column, as reported in the following section. 3. The exact DSM of the beam-column The DSM of a beam-column affected by the presence of multiple cracks can be evaluated by imposing the boundary conditions on the displacements, rotations, shear forces and bending moments at the end cross sections evaluated on account of the closed form solution in Eq.(6). By denoting the displacement and rotation at the end cross-sections 0, 1 as uo , o , u1 , 1 the relevant boundary conditions are written as follows: uo (0) Lo (0) u1 (1) L1 (1) 0 1 0 0 g1 s g2 c g3 S d c d s d 2 3 C 1
1 C1 0 C2 g 4 C C3 d 4 S C4
(12)
In Eq.(12) the following quantities have been defined: s sin , c cos , S sinh , C cosh , gk gˆ k 1; ,
dk gˆ k 1; , , k 1,
v Wc
, 4 . Eq.(12) can also be written as follows: (13)
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where the vectors and matrices can be easily inferred by comparing Eq.(13) with Eq.(12). By denoting the shear forces and bending moments at the end cross-sections 0, 1 as To , M o , T1 , M1 the relevant boundary conditions are written as follows: To L M o Eo I o T1 L L2 M 1
(0) (0) (1) (1) 2 3 0 3 2 0 C1 2 2 0 0 C Eo I o 3 3 3 3 2 2 [ c q1 [ C q3 [ s q2 [ S q4 C3 L 2 (d1 c)] 2 (d 2 s )] 2 (d3 C )] 2 (d 4 S )] C4 2 p2 2c p3 2 S p4 2C p1 s
where the following quantities have been introduced: pk gˆ k 1; , , qk gˆ k1, , , k 1,
(14)
,4.
Eq.(14) can also be written as follows:
F Qc
(15)
where the vectors and matrices can be easily inferred by comparing Eq.(15) with Eq.(14). Evaluation of the vector of integration constants c from Eq.(13) and substitution in Eq.(15) leads to the following expression:
F Q W1 v K v
(16)
where K denotes the DSM of the beam column with multiple cracks while W 1 can be obtained by explicit inversion of W appearing in Eq.(13). The product Q W1 , appearing in Eq.(16), can be explicitly evaluated providing the sought expression of the DSM of the beam with multiple cracks as follows:
Eo I o 1 K QW 2 L
*
*
*
ˆ
ˆ
(17)
ˆ
with the following explicit expressions of each component:
D
2
2 d 4 g 2 d 2 g 4 d 4 c d 2C g 4 C s g 2 c S
2c+ 2C g 2 g 4 s S g1 g 3 D + d1 d 3 g 2 g 4 c C g 3 S + g1 s d 4 d 2
2 2 1 cC 2 sS
D
D
2
2
2 d 4 d 2 s S
2
2 g 4 -g 2 c C
2
g1 s d 3 C g 3 S d1 c D 2 2 d 3 d1 C c * D
2
2 D
*
g
1
g 3 S s
1 q3 q1 2C 2 c d 2 d 4 s S D
q2 q4 3 s 3 S d1 d 3 c C
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1 q3 q1 2 2 d 3 d1 2C 2c D g 2 g 4 c C q4 q2 2 2 d 4 d 2 S s g1 g 3 S s
ˆ
ˆ
1 p1 p3 s S g 4 g 2 C c D
p2 p4 2 c 2C g1 g 3 S s
(18)
All the matrix elements in Eq.(18) possess a common denominator given by:
D 2 1 cC 2 2 sS d 4 d 2 g1 g3 d1 d3 g 2 g 4 d1 d3 g 2 g 4 c C (19) d 4 d 2 s S g1 g3 s S
Eqs.(18) and (19) provide explicit expressions of DSM elements so as no additional computations has to be performed to account for cracks in presence of axial load. It can be easily verified that the previous elements reduce to those regarding the undamaged beam-column when no cracks are considered. Once the DSM of a damaged beam column is provided explicitly as in Eqs.(17)-(19), the DSM, denoted as Kg, of an entire frame structure composed of multi-cracked elements can be built by assembling the matrices of the damaged beams according to the structural scheme. Given the DSM Kg of the assembled structure, the adoption of the W&W algorithm [21-23,48,49] for the evaluation of the natural frequencies is suggested in order to determine the exact values, rather than application of the finite element method affected by approximation. The details of the W&W algorithm, particularized to the case of frame structures composed of multi-cracked beams, are summarized in the following section. Furthermore, the exact and complete global mode shapes of the assembled structures can be easily evaluated in accordance to the following steps: i) For each frequency, from the global structure relation K g vg 0 , the nodal values of the corresponding structural global mode shape are obtained by the vector vg. ii) From the global vector vg the sub-vector v for each beam can be extracted and the vector of the integration constants c C1 C2
C3
C4 are obtained from Eq.(13) as c W1 v . T
iii) Finally, besides the nodal values of the corresponding global mode shape of the structure collected in the vector vg, the mode shapes of each damaged beam are obtained by substituting the constants C1, C2, C3, C4, in Eq.(10). It is not trivial to state that the latter described steps, based on the explicit solution of the continuous beam elements in Eq.(10), avoid any discretisation of the structural system to evaluate the complete mode shapes. In particular, formulation of step ii), as above detailed, implies the advantage of avoiding any additional matrix inversion to evaluate each beam element mode shape. 4. The application of the Wittrick and Williams (W&W) algorithm Assembling the novel explicit expression of the DSM for single beam-columns in the formulation of the global DSM Kg of a damaged frame structure in presence of axial load implies to maintain the same global degrees of freedom as the undamaged frame. The latter advantage may be exploited to formulate a convenient expression of the characteristic equation of the fundamental frequencies embedding the presence of cracks. However, the search of the zeros of the latter equation does not allow to distinguish multiple frequencies or the possible presence of poles, and becomes cumbersome for high frequencies. Hence, the well known W&W algorithm [21-23] for the extraction of the fundamental frequencies is here preferred in view of its capability to provide the fundamental frequencies with the desired accuracy and to detect both multiple frequencies and poles. In this section, first, the basic strategy of the W&W algorithm is briefly recalled for the benefit of the interested reader, then, an original customization to the case of damaged frames under study is introduced in sub-section 4.1. In his original formulation the W&W algorithm was meant for calculation of buckling loads of elastic structures however the potentiality of the algorithm towards evaluation of natural frequency was promptly highlighted [21-23] In particular the W&W algorithm starts from an initial choice of the frequency, say , called trial frequency value, and evaluates the number J of frequency values smaller than as follows:
J ( ) J k ( ) J 0 ( )
(20)
where J k ( ) is the number of negative eigenvalues of Kg at the trial frequency , and J 0 ( ) is the number of frequencies smaller than of all the beam-column elements, with clamped-clamped boundary conditions, of the frame written as follows:
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J 0 ( ) r beams J br ( ) 1 N
8
(21)
where Nbeams is the number of beam-column elements in the structure, while J br ( ) is the number of frequencies lower than * for the rth beam-column element with clamped-clamped boundary conditions. Once J for the trial frequency is evaluated by means of Eq.( 20) and (21), by successive iterations the algorithm aims at finding the fundamental frequencies of the structure as those transition values of implying a change of J. The change of the number J provides the multiplicity of the fundamental frequency. As for the evaluation of J 0 given by Eq.(21) it can be stated that, for undamaged Euler-Bernoulli beam-columns in the frame,
J br is given by the following closed form expression: ( ) 1 2 J br ( ) int 2 sign( ) sign 2
(22)
where , , given in Eq.(18), are evaluated for the undamaged beam by neglecting the terms dependent on the cracks, hence given as follows:
2
2 D
sC cS ,
2
2 D
S s
(23)
On the other hand, there is no analogous expression for J br regarding beam-columns with multiple cracks. To this aim the substructuring strategy is proposed in the next section to customize the standard W&W algorithm usually requiring that each beam element is not affected by any crack. 4.1 Beam-column sub-structuring The evaluation of the terms J br for a trial frequency regarding beam-columns with multiple cracks, under clampedclamped boundary conditions, is not trivial. To this purpose the latter single-beam elements are suggested to be treated as simple sub-structures obtained by assemblage of undamaged sub-beams connected by internal hinges equivalent to the cracks. A simple example of such a sub-structuring approach is given in Fig.1, where the global degrees of freedom ( u1 , 2 , 3 ) of the overall frame indicating the size of the DSM are distinct from the local degrees of freedom (indicated with numbers) adopted for the sub-structuring. By doing so, a local W&W algorithm for the evaluation of J br of those beam elements affected by cracks can be performed. The local W&W algorithm requires for a generic damaged beam-column, under clamped boundary conditions, the construction of the 3nx3n DSM of the sub-structure with n cracks. The latter matrix for a generic beam-column with n cracks, characterized by a recurrent analytical scheme, has been easily implemented in a computer code, however, it is not here reported for brevity reasons. Differently from standard applications, the adoption of the above mentioned sub-structuring approach allows to leave unaltered the global structural degrees of freedom and the dimension of the global DSM corresponds to that of the undamaged structure. 5. Numerical Applications A numerical application to a frame structure in presence of multiple cracks, based on the construction of the DSM according to section 3, to show the effect of the axial load on the natural frequencies is presented in this section. As discussed in section 4, the adoption of the DSM with reduced degrees of freedom in conjunction with the application of the W&W algorithm results in a convenient and computational efficient procedure since no additional degree of freedom is required by the crack occurrences. Based on the explicit closed-form expression for the DSM proposed in Eqs.(17)-(19) a parametric analysis to assess the effect of cracks and the axial load on the dynamics of frame structures is conducted. In particular, a portal frame with cracks occurring along the two compressed columns, depicted in Fig.1, is analysed. The two columns and the beam of the portal frame have the same cross-section and span length. The DSM of the structure is constructed with three degrees of freedom, regardless of the number of cracks, by disregarding the axial deformation of the elements. In Fig.2 the effect of the axial load on the first natural frequency of the frame under study is reported for n 1, 2,5,10 cracks, evenly distributed along the two columns. In particular, the first frequency parameter 4 is plotted in Fig.2 against the axial load parameter 2 , for increasing values of the crack intensity parameter (equal for all the cracks), showing the frequency drop as the axial load increases. Furthermore, Fig.2 shows that occurrence of cracks causes both frequency and critical load decrement (attained at zero frequency) particularly at high level of damage.
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Moreover, the critical load parameter 2 is plotted against the crack intensity parameter in Fig.3 for an increasing number of cracks along the two compressed columns . From Fig.3 the critical load reduction due to the crack occurrences is evident and particularly significant as the number of cracks increases even for low crack extent. The latter case can be considered as representative of the early stage of diffused damage development very common in many engineering problems. The instability mode shapes of the structure affected by n 1, 2,10 cracks in the columns with intensity 2.5 , obtained as detailed in section 3, from i) to iii), are depicted in Fig.4. In particular, in Fig.4, the transition from a flexing behavior ( n 1 ) to a straight configuration ( n 10 ) of the beam together with a hinged shaped ( n 1 ) to flexing deformation ( n 10 ) of the columns is represented. The latter behaviour of the two columns, already encountered for the case of single columns [16], is also related to the crack extent. In fact, for high crack intensities and few cracks, the structural elements deform as beams with elastic internal hinges and the instability mode shape is clearly characterised by pronounced rotation discontinuities. In particular in Fig.4b ( n 2 ) the instability mode shows a deformation of the columns with a quasi-rigid configuration between successive cracks. However, if the crack intensity decreases the latter effect tends to disappear. On the other hand, by increasing the number of cracks, as in Fig. 4c (n 10) , the columns follow the deformation proper of elements with diffused elasticity while the instability mode shape is characterised by quasi-rigid horizontal beam since it is not affected by any damage. 6. Conclusions Natural frequency evaluation, together with the relevant mode shapes, of frame structures in presence of multiple cracks in presence of axially loaded elements, is still an open problem in the current literature. This paper shows that a distributional approach to model cracks allows an explicit formulation of the Dynamic Stiffness Matrix (DSM) of multi-cracked vibrating beams where the effect of the axial load is included. The inclusion of the effect of the axial load in the formulation of the DSM for damaged beams is the main novelty of this work. This result is of great importance and allows deriving to derive the global DSM of frame structures with multiple cracks along its members. The derivation of the DSM in presence of axial load is not trivial and provides an important tool to evaluate the frequency shift and the loss of load bearing capacity of axially loaded frames affected by multiple cracks. The formulation proposed for the DSM for the damaged frame implies the adoption of the same degrees of freedom required by the standard approach for undamaged frame structures regardless of the number and position of the cracks. The possibility of exploiting the same degrees of freedom of the undamaged frame, besides a computational advantage, implies a great facility for the implementation effort. On the explicit formulation of the global DSM, the W&W algorithm for the evaluation of the exact frequencies has been ad hoc implemented and the steps to evaluate the complete global structural mode shapes have been also featured. The application reported in this work regards a portal frame in presence of different numbers and distribution of cracks. The presented results aim at showing how the influence of axial load and the cracks on the damaged members of the structure can be easily assessed by means of the proposed expression of the DSM. The presented study is based on the assumptions that cracks remain always open and, furthermore, the crack depth remains constant during vibration. Non linear behaviour of cracked frames due to either closing/opening phenomenon or to evolution phenomenon through the cross section of the crack requires ad hoc studies that can be started from the explicit solution proposed in this study for the linear case. In the first case, examples of switching cracks, however limited to simple beams, have been discussed in section 1. While, with regard to evolution phenomena, a novel procedure dealing with development of plastic deformation in beams has been proposed in [50]. The latter procedure, starting from the closed form solution of the linear case, is based on adaptive linear solutions to analyse the non linear case. The same strategy could be applied in case the crack evolution phenomenon, rather than, diffusion of plastic deformations is to be considered. 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[9] J.R.Banerjee, H.Su, C.Jayatunga, A dynamic stiffness element for free vibration analysis of composite beams and its application to aircraft wings, Computers and Structures 86(6) (2008) 573-579. [10] A.Y.T.Leung, W.E.Zhou, Dynamic stiffness analysis of non-uniform Timoshenko beams, Journal of Sound and Vibration 181 (1995) 447-456. [11] J.R.Banerjee, The Dynamic Stiffness Method: Theory, Practice and Promise, Computational Technology Reviews 11 (2015) 31-57. [12] J.R.Banerjee, The Dynamic Stiffness Method: A review of the state of the art., 10th International Symposium on Vibrations of Continuous Systems ISVCS10, Stanley Hotel, Estes Park, Colorado, USA, July 26 –31 (2015). [13] J.R.Banerjee, W.D.Gunawardana, Development of a dynamic stiffness element for free vibration analysis of a moving functionally graded beam, in J.Kruis, Y.Tsompanakis, B.H.V.Topping, (Eds.), Proceedings of the Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing CC2015, Civil-Comp Press, Stirlingshire, Scotland, 2015, Paper 106. [14] S.Caddemi, I.Caliò, The exact stability stiffness matrix for the analysis of multi-cracked frame structures, Computers and Structures 125 (2013) 137-144. [15] S.Caddemi, I.Caliò, The exact explicit dynamic stiffness matrix of multi-cracked Euler-Bernoulli beam and applications to damaged frame structures, Journal of Sound and Vibration 332(12) (2013) 3049-3063. [16] S.Caddemi, I.Caliò, Exact solution of the multi-cracked Euler-Bernoulli column, International Journal of Solids and Structures 45(16) (2008) 1332-1351. [17] S.Caddemi, I.Caliò, Exact closed-form solution for the vibration modes of the Euler-Bernoulli beam with multiple open cracks, Journal of Sound and Vibration 327 (2009) 473-489. [18] A.Labib, D.Kennedy, C.Featherston, Free vibration analysis of beams and frames with multiple cracks for damage detection. Journal of Sound and Vibration 333(20) (2014) 4991-5003. [19] A.Labib, D.Kennedy, C.A.Featherston, Crack localisation in frames using natural frequency degradations, Computers and Structures 157 (2015) 51-59. [20] G.Failla, An exact generalised function approach to frequency response analysis of beams and plane frames with the inclusion of viscoelastic damping, Journal of Sound and Vibration 360 (2016) 171-202. [21] F.W.Williams, W.H.Wittrick, An automatic computational procedure for calculating natural frequencies of skeletal structures, International Journal of Mechanical Science 12(9) (1970) 781–791. [22] W.H.Wittrick, F.W.Williams, A general algorithm for computing natural frequencies of elastic structures, Quarterly Journal of Mechanics and Applied Mathematics 24 (1971) 263–284. [23] F.W.Williams, W.H.Wittrick, Efficient calculation of natural frequencies of certain marine structures, International Journal of Mechanical Science 15 (1973) 833–843. [24] F.W.Williams, D.Kennedy, Reliable use of determinants to solve non-linear structural eigenvalue problems efficiently, International Journal for Numerical Methods in Engineering 26 (1988) 1825–1841. [25] F.W.Williams, D.Kennedy, More efficient use of determinants to solve transcendental structural eigenvalue problems reliably, Computers and Structures 41 (1991) 973–979. [26] F.W.Williams, D.Kennedy, M.S.Djoudi, The member stiffness determinant and its uses for the transcendental eigenproblems of structural engineering and other disciplines, Proceedings of the Royal Society A 459 (2003) 1-19. [27] Q.Zhaohui, D.Kennedy, F.W.Williams, An accurate method for transcendental eigenproblems with a new criterion for eigenfrequencies, International Journal of Solids and Structures 41 (2004) 3225-3242. [28] W.P.Howson, J.R.Banerjee, F.W.Williams, Concise equations and program for exact eigensolutions of plane frames including member shear, Advances in Engineering Software 5 (1983) 137–141. [29] A.Greco, A.Pau, Damage identification in Euler frames, Computers and Structures 92–93 (2012) 328–336. [30] S.Caddemi, I.Caliò, The influence of the axial force on the vibration of the Euler-Bernoulli beam with an arbitrary number of cracks, Archive of Applied Mechanics 82(6) (2012) 827-839. [31] S.Caddemi, I.Caliò, F.Cannizzaro, Flutter and divergence instability of the multi-cracked cantilever beam-column, Journal of Sound and Vibration 333(6) (2014) 1718-1733. [32] S.Caddemi, I.Caliò, F.Cannizzaro, Advances in dynamic instability: can a beam-column undergo tensile flutter?, Journal of Vibration and Control 23(8) (2017) 1309-1320. [33] S.Caddemi, I.Caliò, F.Cannizzaro, Tensile and compressive buckling of columns with shear deformation singularities, Meccanica 50(3) (2014) 707-720. [34] D.Bigoni, G.Noselli, Experimental evidence of flutter and divergence instabilities induced by dry friction, Journal of the Mechanics of Physics and Solids 59(10) (2010) 2208–2226. [35] D.Zaccaria, D.Bigoni, G.Noselli, D.Misseroni, Structures buckling under tensile dead load, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467(2130) (2011) 1686–1700. [36] S.Caddemi, I.Caliò, M.Marletta, The non-linear dynamic response of the Euler–Bernoulli beam with an arbitrary number of switching cracks, International Journal of Non-Linear Mechanics 45 (2010) 714–726. [37] R.D.Adams, P.Cawley, C.J.Pye, B.J.Stone, A vibration technique for nondestructively assessing the integrity of structures, Journal of Mechanical Engineering Science 20(2) (1978) 93–100. [38] G.R. Irwin, Analysis of stresses and strains near the end of a crack traversing a plate, Journal of Applied Mechanics 24 (1957) 361–364. [39] H.F. Bueckner, The propagation of cracks and the energy of elastic deformation, Transaction of ASME 80 (1958) 1225–1229. [40] R.A. Westmann, W.H. Yang, Stress analysis of cracked rectangular beams, Journal of Applied Mechanics 32 (1967) 693–701. [41] H. Liebowitz, H. Vanderveldt, D.W. Harris, Carrying capacity of notched columns, International Journal of Solids and Structures 3 (1967) 489–500. [42] H. Liebowitz, W.D. Claus, Failure of notched columns, Engineering Fracture Mechanics 1 (1968) 379–383. [43] H. Okamura, H.W. Liu, C.S. Chu, H. Liebowitz, A cracked columns under compression, Engineering Fracture Mechanics 1 (1969) 547–564. [44] P.Gudmnundson, The dynamic behaviour of slender structures with cross sectional cracks, Journal of the Mechanics and Physics of Solids 31 (1983) 329– 345. [45] J.K.Sinha, M.I.Friswell, S.Edwards, Simplified models for the location of cracks in beam structures using measured vibration data, Journal of Sound and Vibration 251(1) (2002) 13–38. [46] S.Caddemi, A.Morassi, Multi-cracked Euler–Bernoulli beams: Mathematical modeling and exact solutions, International Journal of Solids and Structures 50 (2015) 944-956. [47] B.Biondi, S.Caddemi, Closed form solutions of Euler-Bernoulli beams with singularities, International Journal of Solids and Structures 42 (2005) 30273044. [48] F.W.Williams, W.H.Wittrick, Exact buckling and frequency calculations surveyed, Journal of Structural Engineering (ASCE) 109(1) (1983) 169–87 [49] F.W.Williams, D.Kennedy, Historic, recent and ongoing applications of the Wittrick–Williams algorithm, Computational Technology Reviews 2 (2010) 223–246. [50] B.Pantò, D.Rapicavoli, S.Caddemi, I.Caliò, A Smart Displacement Based (SDB) beam element with distributed plasticity, Applied Mathematical Modelling 44 (2017) 336–356.
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Fig. 1 A portal frame with global degrees of freedom and cracked beam-column sub-structures.
n 1
n2
0, 0.25, 0.5,1, 2
0, 0.25, 0.5,1, 2
n 10
n5
0, 0.25, 0.5,1, 2
0, 0.25, 0.5,1, 2
Fig. 2 First frequency parameter vs axial load parameter for different numbers n of evenly distributed cracks in the columns
undamaged
n 1
n2 n 10
n5
n 20
2 Fig. 3 The critical load parameter for different numbers n of evenly distributed cracks in the columns versus the crack intensity parameter
n 1
(a)
n2
(b)
n 10
(c)
Fig. 4 Buckling shapes of a damaged portal frame in presence of evenly distributed cracks in the columns
11
S0093-6413(17)30010-1 http://dx.doi.org/doi:10.1016/j.mechrescom.2017.06.012 MRC 3180
To appear in: Received date: Revised date: Accepted date:
5-1-2017 25-5-2017 17-6-2017
Please cite this article as: Caddemi, Salvatore, Calio, Ivo, Cannizzaro, Francesco, The Dynamic Stiffness Matrix (DSM) of axially loaded multi-cracked frames.Mechanics Research Communications http://dx.doi.org/10.1016/j.mechrescom.2017.06.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Publication Office:
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Mechanics Research Communications. Year Editor-in-Chief: A. Rosato New Jersey Institute of Technology, Newark, New Jersey, USA [email protected]
The Dynamic Stiffness Matrix (DSM) of axially loaded multi-cracked frames Salvatore CADDEMI1*, Ivo CALIO’1, Francesco CANNIZZARO1 1
Dipartimento Ingegneria Civile e Architettura, University of Catania, Via Santa Sofia 64, Catania, Italy
*Corresponding author [email protected] Tel.:+39-095-738-2266; fax: +39-095-738-2297
Highlights
Study of the influence of the axial load on free vibrations of damaged frames.
The Dynamic Stiffness Matrix (DSM) approach is adopted.
A closed form expression of the DSM for frames with an arbitrary number of cracks is presented.
The presented DSM allows the evaluation of the natural frequencies of damaged frames.
The results can be achieved with the desired accuracy with the Wittrick&Williams algorithm.
Abstract The presence of axial load, due to mechanical loading or to temperature effects, represents a strong peculiarity of frame applications both in direct and inverse problems involving the presence of cracks. The lack of explicit formulations of the Dynamic Stiffness Matrix (DSM) for cracked beam elements accounting for the influence of axial loads inspired and motivated the present work and the calculations herein reported. In this paper, the transcendental, frequency dependent, exact explicit expression of the DSM of an Euler-Bernoulli beam-column in presence of an arbitrary number of open cracks is presented. The advantage of the derived expression consists in a fourth order DSM whose dimension stays consistent independently as the number of cracks along the beam increases. Assembling DSMs of single beam-column elements, according to any required frame layout, leads to a number of degrees of freedom of the overall damaged frame structure exactly the same as those of the equivalent undamaged structure. Among other recent achievements on the vibration of damaged frames, the presented formulation provides a further original contribution, in the context of the dynamic stiffness method, which, in conjunction with the Wittrick & Williams algorithm, allows the exact evaluation of the frequencies and vibration modes of damaged frame structures in presence of axially loaded elements. Keywords: Cracks, dynamic instability, dynamic stiffness matrix, damaged beam-columns, damaged frame, Wittrick-Williams algorithm.
0093-6413© 2015 The Authors. Published by Elsevier Ltd.
Publication Office:
Mechanics Research Communications. Year
Elsevier UK
Editor-in-Chief: A. Rosato New Jersey Institute of Technology, Newark, New Jersey, USA [email protected]
1. Introduction The study of vibrations of continuous systems may be conducted by means of the formulation of the Dynamic Stiffness Matrix (DSM) of the structure that is capable of retaining the infinite number of degrees of freedom of the problem and extracting exact results. In fact, the DSM is a frequency dependent matrix, obtained by the solution of the differential equations of the continuous problem, that leads to a transcendental eigenproblem. Many important contributions on the field of the dynamic stiffness method applied to beam-like and frame structures are reported in the literature in the context of inhomogeneous and discontinuous beams [1-11] where peculiarities and advantages of the dynamic stiffness method have been clearly provided. In the latter papers Banerjee and co-workers highlighted how the dynamic stiffness method, unlike the Finite Element Method (FEM), is related to exact frequency dependent shape functions which contain both the mass and stiffness properties. Furthermore, unlike the FEM the results obtained from the DSM are independent of the number of elements used in the analysis. Despite its uncompromising accuracy, the literature on the DSM is not as broad or diverse as the FEM and it is unknown to many scientists focused on FEM analyses. Very recently, scientists involved in the analysis of vibration of continuous systems have been encouraged by Banerjee [12] to provide explicit formulations of the DSM for those cases yet uncovered. The same appeal has been further confirmed in a successive broader context in [13]. The authors have been involved in the past in the application of the DSM for beams and frames in the presence of multiple cracks. In fact, the exact stiffness matrix for the stability and the dynamic of frame structures affected by the presence of multiple cracks have been proposed in [14] and [15], respectively. The latter two papers are deeply based on the explicit formulation of the solution of the governing equations of columns [16] and beams [17] in presence of cracks without recourse to finite-element discretization. The DSM previously formulated by the authors for the case of multi-cracked beams provided a fruitful basis for the exact evaluation of frequencies and modes of cracked frames. However, in all the previous studies, despite the efficacious formulation, the latter stiffness matrix suffers the limitation of neglecting the presence of any axial load acting on the beam elements. Recently also Labib et al. [18] faced the direct problem of calculating the natural frequencies of cracked beams and frames by using a DSM that was obtained in a recursive manner by applying partial Gaussian elimination. Furthermore, Labib et al. in [19] applied the latter procedure to solve the inverse problem regarding the identification of a single crack in a frame structure by using natural frequency degradations also in presence of contaminated measurements and applying efficaciously interval arithmetic. Furthermore, more recently, Failla [20] analysed the frequency response of beams and plane frames with an arbitrary number of Kelvin-Voigt visco-elastic dampers by evaluating the exact DSM of the structure taking advantage on the use of generalised functions. Once the DSM for the structure under study is conveniently formulated, the Wittrick and Williams (W&W) algorithm [2123] is generally adopted to evaluate the exact eigenvalues. The W&W algorithm represents a robust and reliable alternative to finding the roots of the DSM determinant by means of cumbersome zero search procedures. In particular the algorithm starts from an initial choice of the frequency and subsequently evaluates how many frequency values are smaller than the chosen value. Successive iterations aim at finding the frequency value such that no frequencies between the actual and the previous one are found. Among the procedures to choose successive trial frequency values to converge on any required frequency of the transcendental eigenvalue problem, the bisection method is the most popular, however other techniques for faster convergence methods have been devised [24,25]. Another reliable procedure for eigenvalue evaluation is based on the definition of the “member stiffness determinant” defined as stiffness matrix determinant of an element discretised by infinite finite elements with the end sections clamped [26]. By doing so the frequency dependent global stiffness matrix is not affected by poles. Once the desired frequencies of the continuous system are evaluated, the W&W algorithm requires a matrix inversion to compute the correspondent eigenvectors. A recent alternative to the original W&W algorithm based on a congruent transformation of the DSM is introduced whose last element is denoted as last energy norm [27]. The trial frequency that makes the latter norm vanish is an eigenfrequency of the system and the last column of the congruent stiffness matrix tends to the corresponding vibration mode. By doing so, the matrix inversion generally required by the W&W algorithm to infer the vibration mode can be avoided. Applications of the W&W algorithm to cracked frame structures and beams can be found in [14,15,28].
0093-6413© 2015 The Authors. Published by Elsevier Ltd.
Author name / Mechanics Research Communications 00 (2015) 000–000
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This work provides a novelty in the field of the dynamic stiffness method that consists in the evaluation of the explicit closed form formulation of the DSM of axially loaded frames with multiple cracks. No formulation of the dynamic stiffness matrix of multiple cracked axially loaded frames is available in the current literature. The explicit formulation of DSM for axially loaded cracked frames herein presented improves significantly the standard procedure commonly applied in the literature for the construction of the DSM. In fact, the standard approach available in the literature relies on the adoption of the DSM for the continuous beam with no cracks for each segment between two successive cracks. The latter matrices are then assembled by accounting for the continuity conditions at the cracked cross-sections. The latter subdivision is inevitably affected by a degree of freedom growth of the assembled structure with respect to the undamaged one dependent on the number of cracks. A simple application of the standard approach with augmented degrees of freedom is shown in [29] where a single crack in a portal frame is considered with the aim to investigate the crack detection inverse problem. Hence the improvement implied by the proposed approach with respect to the standard procedure is that the latter requires an increasing dimension of the DSM with the number of cracks while the proposed expression incorporates an arbitrary number of cracks by leaving unaltered the matrix dimension as that of frame composed of undamaged elements. Furthermore, even though previous formulations of DSM for damaged beam elements available in the literature including the presence of cracks are considered such as [15], the effect of axial loads cannot be included hence the analysis will lead to inaccurate results. The authors dealt with axially loaded beam elements affected by the presence of multiple cracks according to a non standard approach based on the use of generalised functions in [30,31]. However, in the latter papers, no DSM for the analysis of more complex structures, such as damaged frames, was therein formulated. Hence, the novel expression of the DSM to account for the presence of axial load in damaged frames represents a significant non trivial advancement with respect to both the standard approach and the generalised function approach. The distributional approach has been also used by the present authors to investigate on the importance of the effect of axial loads with regard to tensile/compressive flutter and divergence instabilities on discontinuous beams particularly in [31-33] while experimental evidence of the latter phenomena, somehow not quite well understood in the past, can be found in [34,35]. The exact closed-form expression proposed in this work can be used also as a benchmark or conveniently adopted for extensive parametric analysis. However, the main advantage consists in the use of four degrees of freedom at the end crosssections of a multi-damaged frame element in analogy to the standard undamaged element. Moreover, the proposed approach allows modelling of arbitrary distributions of cracks in frames without any variation of the global matrix dimension. The evaluation of the exact natural frequencies of damaged frames is then conducted by customizing the W&W algorithm to include the effect of the axial load in multi-cracked frames. The correspondent eigenvectors of the overall system are also straightforwardly obtained by explicit expressions without any additional discretisation or matrix inversions. A numerical application is reported to show how the theoretical results can handle cases with an increasing number of cracks both in terms of natural frequencies and vibration mode shapes. In this study cracks are considered always open and the behaviour of the cracked frame is linear. However during vibrations, particularly in case of the presence of axial compressive load, nonlinearities due to the closing/opening phenomenon (switching cracks) may arise. An example of vibration analysis of a simple beam with switching cracks can be found in [36] where the non linear analysis was conducted as a sequence of multiple linear regions according to the number of cracks in the open state. The assumption of open cracks in vibration of cracked frames under axial load, and the presented explicit DSM expression, considered in this work are fundamental achievements to address the more complex non linear problem of switching cracks in frames. In fact, as the number of active open cracks varies during vibrations the DSM changes and could be evaluated with the proposed expression by updating the number of active open cracks. With respect to the results herein presented the influence of the crack closure and contact is expected to consist in that the fundamental frequencies are collocated between those corresponding to the always-open and to the always-closed (undamaged) crack condition. 2. The exact explicit solution of the multi-cracked beam-column The presence of cracks introduces a local flexibility in beams whose quantification allows a simple and effective representation of the vibrating behaviour of cracked beams and frames [37]. The analysis and evaluation of the crack-induced local flexibility has been conducted by various authors [38-40] and related to the stress intensity factor, describing the intensity of the stress field about the tip of the crack. Based on the previous studies, the evaluation of the stress intensity factor was determined by experimental methods based on measuring the local flexibility (or, alternatively, the local bending stiffness) [4143]. On the other hand, according to a macroscopic approach, widely accepted in the literature, the effect on the local flexibility introduced by a crack in a beam element can be modelled by an equivalent rotational spring with stiffness K eq connecting the two adjacent segments of the beam, as for example discussed in [44,45]. A mechanical justification of the macroscopic model of rotational elastic spring commonly used to describe the presence of an open crack in a beam under bending deformation has been provided in [46]. Within the latter approach, different models have been presented in the literature proposing expressions of the spring stiffness K eq equivalent to the crack for a large number of cases, concerning different geometry of the cross-section and different crack shapes. As a matter of example, when a lateral crack of uniform depth d is present in a rectangular cross-section of width b and height h, the following expression for the stiffness K eq Eo I o / [hC ( )] can be adopted where d / h is defined
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as the ratio between the crack depth d and the cross-section height h, Eo and Io represent respectively the Young’s modulus and the moment of inertia, and C(β) is a dimensionless local flexibility which can take different forms according to the chosen damage model as summarized in Appendix D of [17]. The flexibility C ( ) of the equivalent spring represents the macroscopic effect of the local flexibility introduced by a crack and is hence related to the stress intensity factor. In this work cracks are modelled as concentrated flexural stiffness reductions of the beam element by means of Dirac’s delta generalised functions, which proved to be equivalent to the macroscopic rotational spring approach [17]. Assuming that the amplitude of the deformation is such as to maintain the crack always open, the assumed model offers the great advantage to be linear. The results of the linear model represent an indispensable basis to study more advanced non linear problems implying the closure of cracks. Following this approach, the presence of a crack provides a slope discontinuity at the location of the crack that can be modelled through generalised functions. Namely, the presence of an arbitrary number of cracks can be modelled by considering Dirac’s deltas, centred at the crack positions, superimposed to a uniform flexural stiffness as follows [47]: n E ( x) I ( x) Eo I o 1 ˆi ( x xi ) i 1
(1)
where x is the spatial abscissa spanning from 0 to the length L of the beam. Furthermore, in Eq.(1), the n singularities, given by Dirac’s deltas ( x xi ) centred at abscissae xi , i 1, , n , represent n cracks and the parameters ˆi , i 1, , n , multiplying the Dirac’s deltas, are related to the rotational stiffness of the equivalent internal springs, which simulates the cracks, as specified in the referenced papers [16,17]. Accounting for Eq.(1), the free vibrations of a uniform beam-column in presence of multiple cracks, with a uniformly distributed mass m, subjected to a constant axial compressive force N, are governed by the following normalised differential equation: n NL2 mL4 1 ( ) u ( , t ) u ( , t ) u ( , t ) 0 i i Eo I o Eo I o i 1
(2)
where ξ = x/L is the normalised spatial abscissa, t is the time variable, i ˆi / L are the normalised crack parameters and
u( , t ) is the transversal displacement. The apex indicates derivatives with respect to ξ, while the superimposed dot indicates the time derivative. By adopting a standard separation of variables, Eq.(2) can be formulated in terms of spatial vibration mode ( ) as: n NL2 mL4 2 0 1 i ( i ) Eo I o i 1 Eo I o
(3)
where is the cyclic frequency. By introducing the axial load parameter 2 NL2 / Eo I o and the frequency parameter 4 2 mL4 / Eo I o variable, Eq.(3) may be written in the non-homogeneous dimensionless form as follows:
IV 2 4 B( )
(4)
where the function B( ) collects all the terms with the Dirac’s deltas and their derivatives as follows: n
n
i 1
i 1
B i IV ( i ) 2 i ( i ) n
i ( i )
(5)
i 1
The general explicit solution of Eq.(4) has been derived by the authors in [30], however it is here written under a different form more suitable for the explicit derivation of the DSM as presented in the next section. In fact, the frequency/axial load dependent vibration mode representing the solution of Eq.(4) is written as follows: 4
; , Ck f k ; ,
(6)
k 1
where , are frequency/axial load dependent parameter defined as , 2 / 2 [ 4 / 4 4 ]1/ 2
1/ 2
.
In Eq.(6) Ck , k 1, , 4, are integration constants to be easily evaluated by the relevant boundary conditions, while the generalised functions f k ( ; , ), k 1, , 4, are given by:
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f k ; ,
1 2 2
5 n
i 1
i ki
( , ) Si ; , U ( i )
(7)
ak ; , where the crack intensity parameters i , i 1..., n , dependent on i as fully detailed in [30], but also related to the standard dimensionless crack local compliance C(β) as follows i C (i )h / L , have been introduced. All the terms appearing in Eq.(7) are fully defined as follows: ki
i 1 1 3 sin i j 3 sinh i j bki (8) 2 j kj j 1 2
Si ; , sin i sinh i a1 ; sin ; a2 ; cos ; a3 ; sinh ; a4 ; cosh ;
(9)
b1i 2 sin i ; b2i 2 cos i ; b3i 2 sinh i ; b4i 2 cosh i .
It is worth to notice that each function f k ( ; , ), k 1,
, 4, expressed by Eq.(7), can also be written as the sum of its
generalised part gˆ k ( ; , ) plus the trigonometric or hyperbolic terms ak ; , as follows:
f k ; , gˆ k ; , ak ; ,
(10)
where the generalised part, in view of Eq.(7), is defined as: gˆ k ; ,
1 2 2
n
, S ; , U ( ) i 1
i ki
i
i
(11)
while the terms ak ; , , defined in Eq.(9), are relative to the solution of the undamaged uniform beam-column, that is obtained as particular case when all the crack intensities approach zero, i 0; i 1..., n . It is worth noting that the solution expressed by Eq.(6) is valid for the overall beam-column and for any number and positions of cracks. Furthermore it preserves the same analytical structure of the undamaged beam, being a function of four integration constants only irrespective of the number and the positions of the cracked cross sections. This form of the exact solution allows the derivation, in closed form, of the exact DSM of a multi-cracked beam-column, as reported in the following section. 3. The exact DSM of the beam-column The DSM of a beam-column affected by the presence of multiple cracks can be evaluated by imposing the boundary conditions on the displacements, rotations, shear forces and bending moments at the end cross sections evaluated on account of the closed form solution in Eq.(6). By denoting the displacement and rotation at the end cross-sections 0, 1 as uo , o , u1 , 1 the relevant boundary conditions are written as follows: uo (0) Lo (0) u1 (1) L1 (1) 0 1 0 0 g1 s g2 c g3 S d c d s d 2 3 C 1
1 C1 0 C2 g 4 C C3 d 4 S C4
(12)
In Eq.(12) the following quantities have been defined: s sin , c cos , S sinh , C cosh , gk gˆ k 1; ,
dk gˆ k 1; , , k 1,
v Wc
, 4 . Eq.(12) can also be written as follows: (13)
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where the vectors and matrices can be easily inferred by comparing Eq.(13) with Eq.(12). By denoting the shear forces and bending moments at the end cross-sections 0, 1 as To , M o , T1 , M1 the relevant boundary conditions are written as follows: To L M o Eo I o T1 L L2 M 1
(0) (0) (1) (1) 2 3 0 3 2 0 C1 2 2 0 0 C Eo I o 3 3 3 3 2 2 [ c q1 [ C q3 [ s q2 [ S q4 C3 L 2 (d1 c)] 2 (d 2 s )] 2 (d3 C )] 2 (d 4 S )] C4 2 p2 2c p3 2 S p4 2C p1 s
where the following quantities have been introduced: pk gˆ k 1; , , qk gˆ k1, , , k 1,
(14)
,4.
Eq.(14) can also be written as follows:
F Qc
(15)
where the vectors and matrices can be easily inferred by comparing Eq.(15) with Eq.(14). Evaluation of the vector of integration constants c from Eq.(13) and substitution in Eq.(15) leads to the following expression:
F Q W1 v K v
(16)
where K denotes the DSM of the beam column with multiple cracks while W 1 can be obtained by explicit inversion of W appearing in Eq.(13). The product Q W1 , appearing in Eq.(16), can be explicitly evaluated providing the sought expression of the DSM of the beam with multiple cracks as follows:
Eo I o 1 K QW 2 L
*
*
*
ˆ
ˆ
(17)
ˆ
with the following explicit expressions of each component:
D
2
2 d 4 g 2 d 2 g 4 d 4 c d 2C g 4 C s g 2 c S
2c+ 2C g 2 g 4 s S g1 g 3 D + d1 d 3 g 2 g 4 c C g 3 S + g1 s d 4 d 2
2 2 1 cC 2 sS
D
D
2
2
2 d 4 d 2 s S
2
2 g 4 -g 2 c C
2
g1 s d 3 C g 3 S d1 c D 2 2 d 3 d1 C c * D
2
2 D
*
g
1
g 3 S s
1 q3 q1 2C 2 c d 2 d 4 s S D
q2 q4 3 s 3 S d1 d 3 c C
Author name / Mechanics Research Communications 00 (2015) 000–000
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1 q3 q1 2 2 d 3 d1 2C 2c D g 2 g 4 c C q4 q2 2 2 d 4 d 2 S s g1 g 3 S s
ˆ
ˆ
1 p1 p3 s S g 4 g 2 C c D
p2 p4 2 c 2C g1 g 3 S s
(18)
All the matrix elements in Eq.(18) possess a common denominator given by:
D 2 1 cC 2 2 sS d 4 d 2 g1 g3 d1 d3 g 2 g 4 d1 d3 g 2 g 4 c C (19) d 4 d 2 s S g1 g3 s S
Eqs.(18) and (19) provide explicit expressions of DSM elements so as no additional computations has to be performed to account for cracks in presence of axial load. It can be easily verified that the previous elements reduce to those regarding the undamaged beam-column when no cracks are considered. Once the DSM of a damaged beam column is provided explicitly as in Eqs.(17)-(19), the DSM, denoted as Kg, of an entire frame structure composed of multi-cracked elements can be built by assembling the matrices of the damaged beams according to the structural scheme. Given the DSM Kg of the assembled structure, the adoption of the W&W algorithm [21-23,48,49] for the evaluation of the natural frequencies is suggested in order to determine the exact values, rather than application of the finite element method affected by approximation. The details of the W&W algorithm, particularized to the case of frame structures composed of multi-cracked beams, are summarized in the following section. Furthermore, the exact and complete global mode shapes of the assembled structures can be easily evaluated in accordance to the following steps: i) For each frequency, from the global structure relation K g vg 0 , the nodal values of the corresponding structural global mode shape are obtained by the vector vg. ii) From the global vector vg the sub-vector v for each beam can be extracted and the vector of the integration constants c C1 C2
C3
C4 are obtained from Eq.(13) as c W1 v . T
iii) Finally, besides the nodal values of the corresponding global mode shape of the structure collected in the vector vg, the mode shapes of each damaged beam are obtained by substituting the constants C1, C2, C3, C4, in Eq.(10). It is not trivial to state that the latter described steps, based on the explicit solution of the continuous beam elements in Eq.(10), avoid any discretisation of the structural system to evaluate the complete mode shapes. In particular, formulation of step ii), as above detailed, implies the advantage of avoiding any additional matrix inversion to evaluate each beam element mode shape. 4. The application of the Wittrick and Williams (W&W) algorithm Assembling the novel explicit expression of the DSM for single beam-columns in the formulation of the global DSM Kg of a damaged frame structure in presence of axial load implies to maintain the same global degrees of freedom as the undamaged frame. The latter advantage may be exploited to formulate a convenient expression of the characteristic equation of the fundamental frequencies embedding the presence of cracks. However, the search of the zeros of the latter equation does not allow to distinguish multiple frequencies or the possible presence of poles, and becomes cumbersome for high frequencies. Hence, the well known W&W algorithm [21-23] for the extraction of the fundamental frequencies is here preferred in view of its capability to provide the fundamental frequencies with the desired accuracy and to detect both multiple frequencies and poles. In this section, first, the basic strategy of the W&W algorithm is briefly recalled for the benefit of the interested reader, then, an original customization to the case of damaged frames under study is introduced in sub-section 4.1. In his original formulation the W&W algorithm was meant for calculation of buckling loads of elastic structures however the potentiality of the algorithm towards evaluation of natural frequency was promptly highlighted [21-23] In particular the W&W algorithm starts from an initial choice of the frequency, say , called trial frequency value, and evaluates the number J of frequency values smaller than as follows:
J ( ) J k ( ) J 0 ( )
(20)
where J k ( ) is the number of negative eigenvalues of Kg at the trial frequency , and J 0 ( ) is the number of frequencies smaller than of all the beam-column elements, with clamped-clamped boundary conditions, of the frame written as follows:
Author name / Mechanics Research Communications 00 (2015) 000–000
J 0 ( ) r beams J br ( ) 1 N
8
(21)
where Nbeams is the number of beam-column elements in the structure, while J br ( ) is the number of frequencies lower than * for the rth beam-column element with clamped-clamped boundary conditions. Once J for the trial frequency is evaluated by means of Eq.( 20) and (21), by successive iterations the algorithm aims at finding the fundamental frequencies of the structure as those transition values of implying a change of J. The change of the number J provides the multiplicity of the fundamental frequency. As for the evaluation of J 0 given by Eq.(21) it can be stated that, for undamaged Euler-Bernoulli beam-columns in the frame,
J br is given by the following closed form expression: ( ) 1 2 J br ( ) int 2 sign( ) sign 2
(22)
where , , given in Eq.(18), are evaluated for the undamaged beam by neglecting the terms dependent on the cracks, hence given as follows:
2
2 D
sC cS ,
2
2 D
S s
(23)
On the other hand, there is no analogous expression for J br regarding beam-columns with multiple cracks. To this aim the substructuring strategy is proposed in the next section to customize the standard W&W algorithm usually requiring that each beam element is not affected by any crack. 4.1 Beam-column sub-structuring The evaluation of the terms J br for a trial frequency regarding beam-columns with multiple cracks, under clampedclamped boundary conditions, is not trivial. To this purpose the latter single-beam elements are suggested to be treated as simple sub-structures obtained by assemblage of undamaged sub-beams connected by internal hinges equivalent to the cracks. A simple example of such a sub-structuring approach is given in Fig.1, where the global degrees of freedom ( u1 , 2 , 3 ) of the overall frame indicating the size of the DSM are distinct from the local degrees of freedom (indicated with numbers) adopted for the sub-structuring. By doing so, a local W&W algorithm for the evaluation of J br of those beam elements affected by cracks can be performed. The local W&W algorithm requires for a generic damaged beam-column, under clamped boundary conditions, the construction of the 3nx3n DSM of the sub-structure with n cracks. The latter matrix for a generic beam-column with n cracks, characterized by a recurrent analytical scheme, has been easily implemented in a computer code, however, it is not here reported for brevity reasons. Differently from standard applications, the adoption of the above mentioned sub-structuring approach allows to leave unaltered the global structural degrees of freedom and the dimension of the global DSM corresponds to that of the undamaged structure. 5. Numerical Applications A numerical application to a frame structure in presence of multiple cracks, based on the construction of the DSM according to section 3, to show the effect of the axial load on the natural frequencies is presented in this section. As discussed in section 4, the adoption of the DSM with reduced degrees of freedom in conjunction with the application of the W&W algorithm results in a convenient and computational efficient procedure since no additional degree of freedom is required by the crack occurrences. Based on the explicit closed-form expression for the DSM proposed in Eqs.(17)-(19) a parametric analysis to assess the effect of cracks and the axial load on the dynamics of frame structures is conducted. In particular, a portal frame with cracks occurring along the two compressed columns, depicted in Fig.1, is analysed. The two columns and the beam of the portal frame have the same cross-section and span length. The DSM of the structure is constructed with three degrees of freedom, regardless of the number of cracks, by disregarding the axial deformation of the elements. In Fig.2 the effect of the axial load on the first natural frequency of the frame under study is reported for n 1, 2,5,10 cracks, evenly distributed along the two columns. In particular, the first frequency parameter 4 is plotted in Fig.2 against the axial load parameter 2 , for increasing values of the crack intensity parameter (equal for all the cracks), showing the frequency drop as the axial load increases. Furthermore, Fig.2 shows that occurrence of cracks causes both frequency and critical load decrement (attained at zero frequency) particularly at high level of damage.
Author name / Mechanics Research Communications 00 (2015) 000–000
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Moreover, the critical load parameter 2 is plotted against the crack intensity parameter in Fig.3 for an increasing number of cracks along the two compressed columns . From Fig.3 the critical load reduction due to the crack occurrences is evident and particularly significant as the number of cracks increases even for low crack extent. The latter case can be considered as representative of the early stage of diffused damage development very common in many engineering problems. The instability mode shapes of the structure affected by n 1, 2,10 cracks in the columns with intensity 2.5 , obtained as detailed in section 3, from i) to iii), are depicted in Fig.4. In particular, in Fig.4, the transition from a flexing behavior ( n 1 ) to a straight configuration ( n 10 ) of the beam together with a hinged shaped ( n 1 ) to flexing deformation ( n 10 ) of the columns is represented. The latter behaviour of the two columns, already encountered for the case of single columns [16], is also related to the crack extent. In fact, for high crack intensities and few cracks, the structural elements deform as beams with elastic internal hinges and the instability mode shape is clearly characterised by pronounced rotation discontinuities. In particular in Fig.4b ( n 2 ) the instability mode shows a deformation of the columns with a quasi-rigid configuration between successive cracks. However, if the crack intensity decreases the latter effect tends to disappear. On the other hand, by increasing the number of cracks, as in Fig. 4c (n 10) , the columns follow the deformation proper of elements with diffused elasticity while the instability mode shape is characterised by quasi-rigid horizontal beam since it is not affected by any damage. 6. Conclusions Natural frequency evaluation, together with the relevant mode shapes, of frame structures in presence of multiple cracks in presence of axially loaded elements, is still an open problem in the current literature. This paper shows that a distributional approach to model cracks allows an explicit formulation of the Dynamic Stiffness Matrix (DSM) of multi-cracked vibrating beams where the effect of the axial load is included. The inclusion of the effect of the axial load in the formulation of the DSM for damaged beams is the main novelty of this work. This result is of great importance and allows deriving to derive the global DSM of frame structures with multiple cracks along its members. The derivation of the DSM in presence of axial load is not trivial and provides an important tool to evaluate the frequency shift and the loss of load bearing capacity of axially loaded frames affected by multiple cracks. The formulation proposed for the DSM for the damaged frame implies the adoption of the same degrees of freedom required by the standard approach for undamaged frame structures regardless of the number and position of the cracks. The possibility of exploiting the same degrees of freedom of the undamaged frame, besides a computational advantage, implies a great facility for the implementation effort. On the explicit formulation of the global DSM, the W&W algorithm for the evaluation of the exact frequencies has been ad hoc implemented and the steps to evaluate the complete global structural mode shapes have been also featured. The application reported in this work regards a portal frame in presence of different numbers and distribution of cracks. The presented results aim at showing how the influence of axial load and the cracks on the damaged members of the structure can be easily assessed by means of the proposed expression of the DSM. The presented study is based on the assumptions that cracks remain always open and, furthermore, the crack depth remains constant during vibration. Non linear behaviour of cracked frames due to either closing/opening phenomenon or to evolution phenomenon through the cross section of the crack requires ad hoc studies that can be started from the explicit solution proposed in this study for the linear case. In the first case, examples of switching cracks, however limited to simple beams, have been discussed in section 1. While, with regard to evolution phenomena, a novel procedure dealing with development of plastic deformation in beams has been proposed in [50]. The latter procedure, starting from the closed form solution of the linear case, is based on adaptive linear solutions to analyse the non linear case. The same strategy could be applied in case the crack evolution phenomenon, rather than, diffusion of plastic deformations is to be considered. References [1] J.R.Banerjee, Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness method, Computers and Structures 69 (1998) 197208. [2] A.Pagani, E.Carrera, M.Boscolo, J.R.Banerjee, Refined dynamic stiffness elements applied to free vibration analysis of generally laminated composite beams with arbitrary boundary conditions, Composite Structures 110(1) (2014) 305-316. [3] J.R.Banerjee, Free vibration of centrifugally stiffened uniform and tapered beams using the dynamic stiffness method, Journal of Sound and Vibration 233(5) (2000) 857-875. [4] J.R.Banerjee, F.W.Williams, Exact Bernoulli-Euler dynamic stiffness matrix for a range of tapered beams, International Journal for Numerical Methods in Engineering 21(12) (1985) 2289-2302. [5] J.R.Banerjee, Free vibration of sandwich beams using the dynamic stiffness method, Computers and Structures 81(18-19) (2003) 1915-1922. 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[9] J.R.Banerjee, H.Su, C.Jayatunga, A dynamic stiffness element for free vibration analysis of composite beams and its application to aircraft wings, Computers and Structures 86(6) (2008) 573-579. [10] A.Y.T.Leung, W.E.Zhou, Dynamic stiffness analysis of non-uniform Timoshenko beams, Journal of Sound and Vibration 181 (1995) 447-456. [11] J.R.Banerjee, The Dynamic Stiffness Method: Theory, Practice and Promise, Computational Technology Reviews 11 (2015) 31-57. [12] J.R.Banerjee, The Dynamic Stiffness Method: A review of the state of the art., 10th International Symposium on Vibrations of Continuous Systems ISVCS10, Stanley Hotel, Estes Park, Colorado, USA, July 26 –31 (2015). 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[37] R.D.Adams, P.Cawley, C.J.Pye, B.J.Stone, A vibration technique for nondestructively assessing the integrity of structures, Journal of Mechanical Engineering Science 20(2) (1978) 93–100. [38] G.R. Irwin, Analysis of stresses and strains near the end of a crack traversing a plate, Journal of Applied Mechanics 24 (1957) 361–364. [39] H.F. Bueckner, The propagation of cracks and the energy of elastic deformation, Transaction of ASME 80 (1958) 1225–1229. [40] R.A. Westmann, W.H. Yang, Stress analysis of cracked rectangular beams, Journal of Applied Mechanics 32 (1967) 693–701. [41] H. Liebowitz, H. Vanderveldt, D.W. Harris, Carrying capacity of notched columns, International Journal of Solids and Structures 3 (1967) 489–500. [42] H. Liebowitz, W.D. Claus, Failure of notched columns, Engineering Fracture Mechanics 1 (1968) 379–383. [43] H. Okamura, H.W. Liu, C.S. Chu, H. Liebowitz, A cracked columns under compression, Engineering Fracture Mechanics 1 (1969) 547–564. [44] P.Gudmnundson, The dynamic behaviour of slender structures with cross sectional cracks, Journal of the Mechanics and Physics of Solids 31 (1983) 329– 345. [45] J.K.Sinha, M.I.Friswell, S.Edwards, Simplified models for the location of cracks in beam structures using measured vibration data, Journal of Sound and Vibration 251(1) (2002) 13–38. [46] S.Caddemi, A.Morassi, Multi-cracked Euler–Bernoulli beams: Mathematical modeling and exact solutions, International Journal of Solids and Structures 50 (2015) 944-956. [47] B.Biondi, S.Caddemi, Closed form solutions of Euler-Bernoulli beams with singularities, International Journal of Solids and Structures 42 (2005) 30273044. [48] F.W.Williams, W.H.Wittrick, Exact buckling and frequency calculations surveyed, Journal of Structural Engineering (ASCE) 109(1) (1983) 169–87 [49] F.W.Williams, D.Kennedy, Historic, recent and ongoing applications of the Wittrick–Williams algorithm, Computational Technology Reviews 2 (2010) 223–246. 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Author name / Mechanics Research Communications 00 (2015) 000–000
Fig. 1 A portal frame with global degrees of freedom and cracked beam-column sub-structures.
n 1
n2
0, 0.25, 0.5,1, 2
0, 0.25, 0.5,1, 2
n 10
n5
0, 0.25, 0.5,1, 2
0, 0.25, 0.5,1, 2
Fig. 2 First frequency parameter vs axial load parameter for different numbers n of evenly distributed cracks in the columns
undamaged
n 1
n2 n 10
n5
n 20
2 Fig. 3 The critical load parameter for different numbers n of evenly distributed cracks in the columns versus the crack intensity parameter
n 1
(a)
n2
(b)
n 10
(c)
Fig. 4 Buckling shapes of a damaged portal frame in presence of evenly distributed cracks in the columns
11