Progressive failure analysis of thin-walled composite columns subjected to uniaxial compression

Accepted Manuscript Progressive failure analysis of thin-walled composite columns subjected to uniaxial compression Gliszczynski Adrian, Kubiak Tomasz PII: DOI: Reference:

S0263-8223(16)32109-2 http://dx.doi.org/10.1016/j.compstruct.2016.10.029 COST 7855

To appear in:

Composite Structures

Received Date: Accepted Date:

10 October 2016 11 October 2016

Please cite this article as: Adrian, G., Tomasz, K., Progressive failure analysis of thin-walled composite columns subjected to uniaxial compression, Composite Structures (2016), doi: http://dx.doi.org/10.1016/j.compstruct. 2016.10.029

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PROGRESSIVE FAILURE ANALYSIS OF THIN-WALLED COMPOSITE COLUMNS SUBJECTED TO UNIAXIAL COMPRESSION Gliszczynski Adrian, Kubiak Tomasz Department of Strength of Materials Lodz University of Technology, 90-924 Lodz, Poland [email protected], [email protected] Abstract: The paper deals with estimating the load capacity of thin-walled composite columns with C-shaped cross-section subjected to uniform compression. The discussed columns were made of eight-layer GFRP laminate. Three symmetric arrangements of layers were taken into consideration: [0/-45/45/90]S, [90/0/90/0]S, [45/-45/45/-45]S. The experimental research was conducted with the use of the ultimate testing machine and the DIC system Aramis®. Additionally, the numerical analyses were performed employing the Ansys® software. The numerical calculations were conducted with the implementation of the progressive failure algorithm, based on the material property degradation method and implementation of the Hashn's criterion as the damage initiation criterion. In all analyzed cases high consistency of numerical and experimental results was achieved and the failure mechanism included the initiation of the fiber failure in the corner of the columns and its propagation in the direction of the web and the flange of the columns.

Keywords: FEM; GFRP; progressive failure analysis; material property degradation method; uniaxial compression. 1. Introduction The application of advanced composite materials in responsible construction types based on the improvement of analytical and computational tools, which are able to predict the thermomechanical behavior of composites under general loading conditions and geometries. Due to the lack of precise analytical models, the design process must be based on the expensive mechanical tests [1] and empirical knockdown factors. The prediction of ultimate strength remains the main challenge in the simulation of the mechanical response of composite materials [2]. The analysis of size effects on the strength of composites is especially important and relevant [3–7] because the reliable analytical or numerical approaches must reflect the decrease of the load-capacity with the increasing dimensions of structure [8]. Many different types of damage can be observed in composite materials, including rupture and kinking of the fibers, cracking and crushing of the matrix or fiber-matrix de-bonding. However, the implementation of all of this information in the case of practical applications proves to be a difficult and time-consuming task. In the literature many papers devoted to the definition of the impact of these defects on the behavior of composite structures with respect to time and nature of the load, impact damage occurring to reduce stiffness and durability can be found. In a continuum damage mechanics (CDM) the failure modes are represented and modeled by the degradation of the material stiffness on the meso-scale (lamina level) [9]. The work of Ribeiro et al. [10] presents the development of a damage model and its application to simulate a progressive failure of flat composite laminates made using a filament winding process. In international literature the papers which deal with the progressive failure analysis (PFA) of composites structure on the basis of the theory of micro-mechanics of failure (MMF) [11] and also by means of physically based phenomenological models [12] can

be found. New proposals of PFA are focused on the energy descriptions [13], including also the inter-ply failure modes e.g. delaminations between plies [14]. The PFA is currently most widely used by researchers to study the mechanisms of single- or double-shear bearing [15-18] of composite materials and to investigate the low velocity impacts in composite plates [19] or panels [20-21], also under various temperature conditions [22] or with the assumptions of preload [23]. Due to the search of the optimum discretization parameters, the study of the behavior of composite structures within specified ranges of dimensions and in order to create practical procedures for proper handling of the PFA algorithm, the published numerical results are currently more often complemented by sections devoted to the influence of the applied finite element size and the scale effect [13, 2425]. Nowadays, the PFA is successfully implemented to the automotive industry, characterized by the highest restrictions relating to the repeatability and reliability of the numerical results [26], and to the design process of the responsible structures characterized by the specific working conditions e.g. fiber winding hydrogen storage vessel [27-28], offloading hoses [29]. In addition, the algorithm can reliably reflect the experimental results of the analysis of bonded composite repairs [30]. In the literature examples of the implementation of PFA to the design process of the responsible composite structures [26-29] can be found. However, in the author's opinion the number of validated complex systems is still insufficient and requires further work. This claim is also confirmed by subsequent editions of World Wide Failure Exercise (WWFE) [31-33]. The WWFE editions were organized in order to assess the practical usefulness of the failure criteria and the methods of their implementation in numerical subroutines and to validate the predicted results with experimental research in many characteristic schemes of the load. 2. Progressive failure analysis (PFA) The damage may be interpreted as the occurrence of the micro-voids or micro-cracks in the material. A more precise is the definition of the damage material in terms of volume density of micro-cracks or loss of effective cross-sectional area caused by micro-cracks. This approach was proposed by Kachanov [34] by assuming the scalar damage parameter d, which takes values from the range <0;1>, and the value of the damage parameter equals to unity means the destruction of the sample. This interpretation of the damage assumes that the load is transferred only by undamaged part of the cross sectional area, from which it follows the definition of the effective stress (1): 1 σ= σ (1) 1− d where: σ - apparent (nominal, Cauchy) stresses. Due to heterogeneity of the composite materials, the application of the fracture mechanics is more complex compared to its application to isotropic materials. The anisotropy of the material causes that the orientation of the cracks and their growth depends not only on the load, geometry and boundary conditions but also on the morphology of the material. In the fiber reinforced laminate a different mechanisms of fracture can be observed, the type and

size of which depend not only on the material properties but also on the sequence of layers. These factors result in the need to apply not one but several damage parameters d, responsible for the destruction of its individual elements in the failure analysis of composites: d F - fiber damage variable, dM - matrix damage variable, dS - shear damage variable. Similarly, the effective stress tensor in composite materials can be defined. The relation between the nominal stress and the effective stress can be presented in the following form: σ = Mσ (2) where: M - the damage operator. Therefore, the damage effect tensor in Voigt notation is given as (3):  1  0 0   (1 − d F )   σ11  σ11     1    (3) 0  σ 22  σ22  =  0 (1 − d M ) σ    σ   12   12   1  0 0   (1 − d S )   It can be noticed that when d F = 0 and dM = dS =1, equation (3) represents the degradation scheme of the ply discount method (PDA). The progressive failure analysis (PFA) requires a declaration of the material model, the damage initiation criterion and the damage evolution law. 2.1 The Damage Initiation Criterion

Strength-based failure criteria are commonly used to predict failure events in composite structures. Many of continuum-based failure criteria have been derived to relate to material strength limits to the experimental identification of the first damage (first ply failure - FPF). In the Par´ıs work [35] a detailed analysis of the nature of the formulation of most strengthbased criteria can be found. Hashin [36-37] is credited with establishing the need for failure criteria that are based on failure mechanisms. In his 1973 proposal [36], Hashin used his experimental observations of failure of tensile specimens to propose two different failure criteria, one related to fiber failure and the other related to matrix failure. The criteria assume a quadratic interaction between the tractions acting on the plane of failure. In 1980 [37], he introduced fiber and matrix failure criteria that distinguish between tension and compression failure. Given the difficulty in obtaining the plane of fracture for the matrix compression mode, Hashin used a quadratic interaction between stress invariants. Although the Hashin criteria were developed for unidirectional laminates, they have also been applied successfully to progressive failure analyses of laminates. The two-dimensional versions of the failure criteria proposed by Hashin in 1973 and 1980, in a space of the effective stresses, are summarized in Table 1.

Constituent

Table 1. Hashin criteria for plane stress. Tension Compression

2

2

Matrix

σ  σ  f M =  22  +  12   T2   S12 

Fiber

σ  σ  f F =  11  +  12   T1   S12 

2

2

1973 [36]:

σ  σ  f M =  22  +  12   C2   S12 

1980 [37]:

2 2 2 σ  σ22   σ12   C2    22 fM =  + + − 1      2S S 2S  C2  12   12   12 

2

2

fF = −

σ11 C1

The Hashin's criterion, implemented to analyze the progressive failure analysis, is applied to the identification of the damage initiation, and in his description it takes into account the following four damage modes: fiber tension (rupture), fiber compression (kinking), matrix tension (cracking), and matrix compression (crushing). In accordance with the quantities given in the Table 1, the symbol σij refers to elements of the effective stress tensor, T1 and C1 are respectively the tensile strength and the compressive strength in the direction of the fibers, T2, and C2 refer respectively to the tensile strength and compressive strength in the direction perpendicular to the fibers and S12 is the shear strength in the plane orthotropy. Several researchers have proposed modifications to Hashin’s criteria to improve their predictive capabilities. The Chang-Chang criterion [38-39], after the adoption of certain assumptions reduces directly to the Hashin's criterion. Works in which the elements responsible for the shear stress are associated with additional coefficients (usually denoted as α), multiplying or reducing their influence on the laminate failure are also known. In this work these coefficients have been assumed as equal to unity. In accordance with the WWFE conclusions [31-33] most criteria were unable to capture some of the trends in the failure envelopes of the experimental results. Nevertheless, due to the physical character of the Hashin's criterion, which separate the effects of the fiber failure and the matrix destruction, and the increasing applicability of this criterion in normative issues, the authors decided to apply the Hashin's criterion to the progressive failure analysis. 2.2 Degradation of stiffness of laminate and the Damage Evolution Law

The initiation of the damage of laminate implies a reduction of the stiffness of the damaged parts of layers. To determine the degraded element of the stiffness matrix of the composite layer, the model proposed by Matzenmillera [40] has been adapted. Using the relation (2) and the quantitative evaluation of the degradation of the Poisson's ratio (Laws et al. [41]), the damaged compliance matrix [S] can be presented in the following form (4):   1 ν − 21 0   E2  (1 − d F ) E1    ν12 1 S=  − 0 (4)  E1 (1 − d M ) E 2     1 0 0   (1 − dS ) G12  

The corresponding damaged elasticity matrix [D] can be written as (5): (1 − d F ) E1 (1 − d F )(1 − d M )ν 21 E1 0   1  D= (1 − d F )(1 − d M )ν12 E1 (1 − d M ) E 2 0  A  0 0 A(1 − dS ) G12 

(5)

where: A = 1 − ν12 ν21 (1 − d F )(1 − d M ) Ply elastic properties (E1, E2, G12, ν12) and ply strengths (T1, T2, C1, C2, S12) can be measured using test standards defined by ASTM [42-45]. The determined material properties are presented in Table 2. Table 2. Mechanical characteristics of the GFRP composite [46-47]. E1 E2 G12 ν12 T1 T2 S12 C1 C2 [GPa] [GPa] [GPa] [-] [MPa] [MPa] [MPa] [MPa] [MPa] 38.5 8.1 2.0 0.27 792 39 108 679 71 The matrix and the fiber damage parameters (d F, dM) can have a different values in tension and in compression, and therefore these values are described by additional subscripts: T tension, C - compression. The damage variables (6) for calculating the damaged elasticity matrix are determined as follows: for σˆ 1 ≥ 0 d d F =  FT d FC for σˆ 1 < 0 for σˆ 2 ≥ 0 d d M =  MT d MC for σˆ 2 < 0 d S = 1 − (1 − d FT )(1 − d FC )(1 − d MT )(1 − d MC )

(6)

In the applied degradation model, the shear damage variable d S is not independent and can be expressed as a function of the remaining damage variables (6). The PFA was performed with the use of material property degradation method (MPDG), which is an “instant stiffness reduction” method and the material stiffness is instantly reduced based on the damage variables. Damage can progress through the model into other elements in the mesh with the increasing load, but the damage within a particular element is modeled as a step function: either damaged or undamaged. The reduction of the stiffness of the material is controlled by the damage variables (dF, dM, dS) which take the values from the range <0;1>, wherein 0 means the undamaged state and 1 refers to a full damaged state of the corresponding type of the failure. The description of the damage evolution is based on the fracture energy dissipated per unit area. This approach is a generalization of the concept of evolution proposed by Camanho et al. [48] for modeling the delamination in the fiber-reinforced composites. The additional information regarding to the attached degradation model and the procedures of selecting the damage variables can be found in the works of Camanho et al. [48-49] and Barbero et al. [50-51].

Fig. 1. Typical progressive failure analysis methodology. Typical methodology for a progressive failure analysis is illustrated in Figure 1. At each load step, a nonlinear analysis is performed to account for the geometrically nonlinear response of the structure. Using this nonlinear solution, the local lamina stresses are determined and checked against a failure criteria to determine whether any failures have occurred for this load increment. If no failure is detected, the applied load is increased and the analysis continues. When failure in the lamina occurs, a change in the stiffness tensor is calculated based on the material degradation model. This adjustment accounts for the nonlinearity associated with material damage. Static equilibrium needs to be re-established by repeating the geometrically nonlinear analysis at the current load step, using the new material properties. This process is repeated until no additional lamina failures are detected. The load step is then incremented until final failure of the structure is detected. In most models final structural failure is identified at the load level at which static equilibrium can no longer be reached. Because it generally converges quite rapidly, the most popular iterative scheme for the solution of the nonlinear FE equations is the Newton–Raphson procedure. 3. Object of the analysis

The analyzed C-shaped columns were subjected to uniform compression (uniform shortening of the profile). The columns taken into consideration (see: Fig. 2) were 250 mm long, with the following cross-section dimensions: width of the flange: 40 mm, width of the web: 80 mm and wall thickness: approximately 2 mm (see Fig. 2). The thin-walled composite profiles consisted of eight-layer GFRP laminate and were made of unidirectional pre-preg band (denoted as SE70/EGL/300g/400mm/35%/PoPa). The considered profiles were produced by autoclaving technique, which allows to manufacture highly-efficient composite structures

[52]. A diagram of the performed compression tests, describing the support and the load conditions and characteristic geometrical dimensions is presented in Figure 2a-2b. Figure 2c presents the column on the test stand.

a)

b)

c)

Fig. 2. The diagram of the support and the load (a) with characteristic dimensions (b) and the test stand (c) of columns subjected to compression. The uniform compression was performed with the use of a plate and a table. The boundary conditions of the column's end were assumed as fixation (restraint of rotation in respect to the neutral axis of cross section). The plate was mounted to the upper jaw and has only one degree of freedom (vertical movement). The boundary conditions of the column's end were assumed as fixation (restraint of rotation in respect to the neutral axis of cross section). It was assumed that the examined profiles had walls consisting of eight layers with three symmetric ply arrangements denoted as follows: C1 - [0/-45/45/90]S, C3 - [90/0/90/0]S, C5 - [45/-45/45/45]S. It was assumed that all layups under consideration had their general axis of orthotropy parallel to the wall edges (0 0 – fibers along the longitudinal direction or 90 0 – fibers in the transverse direction) or inclined to the wall edges by 450. Three sets of data were collected: from the displacement of the upper jaw of the ultimate testing machine (further denoted as Cx_EXP_ZWICK_UTM and "x" refers to analyzed ply system) and in the case of numerical calculations and experimental research with the use of digital image correlation system Aramis®, the curves were determined from the corner of the C-shaped column (respectively: Cx_FEM and Cx-EXP_ARAMIS). Aramis® system was correlated with the ultimate testing machine with the result that the forces for the successive loading points were the same but the recorded displacements were different. 4. FEM model with the assumed boundary conditions

Material model, geometrical model, the method of discretisation, boundary conditions and the way of loads implementation are the integral elements of the numerical model. Numerical calculations were conducted in the environment of Ansys® software based on the finite element method. The nonlinear analysis of stability was performed using the NewtonRaphson algorithm. In the analyzed cases the small initial geometrical imperfection corresponding to buckling mode with amplitude equal to 1/10 of thickness of the profile walls was assumed. The geometry of the analyzed profile with the assumed boundary condition is shown in Figure 3.

Fig. 3. Discrete model of compressed column with the assumed boundary conditions. It was assumed that the column is subjected to compression. The load was modeled in such a manner as it was carried on the testing machine i.e. the loaded edges displace uniformly along the column (uniform shortening of the column). It was assumed that the end edges of the walls of the column are simply supported at both ends. On the nodes lying on the end edges of the column's walls, the zero values of displacement in the direction perpendicular to the surface of the column's walls were assumed (respectively: ux = 0 or uz = 0, cf. Fig. 3). For all the nodes on the supported end of the column, the possibility of the displacement in the longitudinal direction (uy = 0) was taken away and on the loaded end of the column. The load was implemented in the form of a concentrated compressive force F together with the constant value of the displacement along the axis of the column for all the nodes of loaded edges of column (uy = const., cf. Fig. 3). The size of the finite element was set at 2 mm. This approach determines 20 elements along the width of the flanges and along the width of the web, and 125 elements along the length of the analyzed column. A four-node multi-layered shell element with six degrees of freedom at each node was taken for discretisation. According to the ANSYS® documentation [53] this finite element may be used for layered applications for modeling of laminated composite shells or sandwich construction. The accuracy in modeling of composite shells is governed by the first order shear deformation theory (Mindlin-Reissner shell theory). The created numerical model was used in the linear

and the non-linear analysis of stability with implementation of the progressive failure algorithm (data were denoted as Cx_FEM_PFA), and without the use of additional algorithm which means that the failure loads were determined with the implementation of the failure criterion applied to non-degraded structure (Cx_FEM_STANDARD). 5. Analysis of the results

The variety of composite materials, the lack of standardized test procedures and experimental techniques and the multiplicity of destruction mechanisms implies significant differences in the obtained results, as well as in the degree of correlation between the experimental results and the values determined from the numerical simulation. In view of the mentioned factors, the authors have decided to appoint failure values assuming two algorithms of the solution: the progressive failure analysis (PFA) and a standard geometrically nonlinear analysis with the implementation of the Hashin's criterion. In the case of the second algorithm the failure modes are calculated based on nominal stress state (STANDARD). The experimental validation included the comparison of the curves of compressive force in a function of the shortening of the analyzed columns and the comparison of the loadcarrying capacity of the structures. In the case of progressive failure analysis the load carrying-capacity PMAX was the maximum compressive force transmitted through the column determined from the curve of the compressive force in a function of the shortening of the profile. In the case of the analysis denoted as a STANDARD the failure load PMAX, leading to the global collapse of the analyzed column, has been determined in accordance with Hashin's criterion with the assumption that the column fails if in one layer of the finite element the mechanism of the fiber destruction was initiated. In practice, the global failure of the composite structure is not determined by the destruction of the first layer. In most cases, laminates can still safely carry the load, even several times higher, compared to the value identical to the destruction of the first layer (first ply failure). In the light of the adopted conditions, this assumption can be understood as the upper estimation of the load-carrying capacity of the channel section column but the investigated structures were loaded to the global failure point and the attention of the authors was focused mainly on a comparison of the load-carrying capacity. In the first stage of the analysis of the obtained results, the progressive failure analysis results for quasi-isotropic arrangement of layers (C1) are shown. In order to determine the influence of the damage variables on the load capacity of the analyzed profile and the relative differences of the experimental and numerical results, a number of numerical analysis were performed. In the conducted analyses, the same values of the damage variables for the stretched and the compressed fibers (d FT = d FC) and for the stretched and the compressed matrix (dMT = d MC) were assumed. The numerical calculations, with the implementation of the progressive failure algorithm, were carried out in the range of the damage variables changes from 0.1 to 1every 0.1 for each damage variable (dF, d M). The damage variable equals to unity should be understood as a complete reduction of the corresponding elements in the matrices (4) and (5) (cf. Section 2.2).

25-27

27-29

29-31

31-33

33-35

35-37

37

PMAX [kN]

35 33 31 29

0.1 0.3 0.5

27 25 0.1

0.7 0.2

0.3

0.4

0.5

0.9

0.6

0.7

0.8

0.9

1

Fig. 4. Influence of the damage parameters on the load capacity - PMAX (C1). The comparison of the values of load capacity PMAX of the analyzed C-shaped profile with quasi-isotropic arrangement of layers denoted as C1 is shown in Fig. 4. Analyzing the obtained results, it can easily be seen that, depending on the assumed values of the damage variables, the maximal compressive force varies in the range from slightly more than 35.3 kN to about 25.7 kN. The relative difference of numerical results in relation to the failure loads, determined from the experimental research, lies in the range from 30.6% to approximately 4.4%. 20-30

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Fiber damage variable dFT=dFC [-]

40

30-40 0.1

1

C1 [0/-45/45/90]s C3 [90/0/90/0]s C5 [45/-45/45/-45]s

35 30 Relative difference [%]

10-20

Matrix damage variable dMT=dMC [-]

Rel. difference [%]: 0-10

25 20 15 10 5 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Damage variable dFT=dFC=dMT=dMC [-] a) b) Fig. 5. Influence of the damage parameters on the relative differences of numerical and experimental results for the analyzed ply systems: C1 (a), C1, C3 and C5 (b).

A thorough analysis of the relative differences of experimental and numerical results proves that only the implementation of the high values of the damage parameters provides a relatively close agreement with experimental studies (see. Fig. 5a). This conclusion is also confirmed by the results of analyses carried out for the two remaining arrangements of layers denoted as C3 and C5 . According to the results included in Figure 5b, the acceptable differences of experimental and numerical results were achieved for values of the damage variables greater than or equal to 0.9. For all analyzed cases, the courses of the curves in Figure 5b have very similar character. In the range of low values of the damage variables, the most significant relative differences are observed for the cross-ply laminate (C5 ). Nevertheless, in a range of high values of the damage variables all curves have an almost the same course and the relative difference of numerical and experimental failure loads falls below 10%. The conducted identification of the numerical model allowed to establish that the highest level of correlation of experimental and numerical results is achieved with the declaration of all damage parameters equal to unity. These values successfully reflect the results of experimental studies and, in the authors' opinion, can be recommended for other composite structures subjected to compression, analyzed with the implementation of a progressive failure algorithm. Simultaneously, the authors want to pay attention to the fact that it is also necessary to perform similar considerations for other schemes of load and a validation of their results by experiment. The authors also want to emphasize that the progressive failure analysis can be carried out with the implementation of the continuum damage mechanics method (CDM), in which the problem of the selection of the appropriate damage variables comes down to the declaration of the four additional mechanical properties energies dissipated per unit area from: tensile fiber damage, compressive fiber damage, tensile matrix, damage, compressive matrix damage. These parameters are determined by means of additional tests and complement the material characteristics of the additional variables describing the mechanism of failure. However, they have a physical character and do not have to be empirically selected. CDM based modeling of damage and fracture mechanisms provides a continuum change of the damage variables values with increasing level of load.

30 #1

#7 #8

25

#2

#3

#4 #5

#6

P [kN]

20 15 10

C1_EXP_ARAMIS C1_EXP._ZWICK_UTM C1_FEM_PFA C1_FEM_STANDARD Damage initation

5 0 0.0

0.5

1.0 1.5 2.0 2.5 3.0 Shortening of the column [mm] Fig. 6. The compressive force PMAX in a function of shortening of the column (C1).

3.5

In the remaining part of the article the authors' attention will be focused on the detailed analysis of the behavior of the analyzed cases with the implementation of all the damage variables equal to unity. Analyzing the curves of compressive force in a function of the shortening of the channel section profile with quasi-isotropic ply system it can be noticed that the experimental and numerical curves have a very similar course (see Fig.6). In the initial state of the load (below 15 kN), when none of the layers is degraded yet, we can observe the same buckling mode in experimental research and numerical analysis (cf. Fig.6). The black dots included on the figures (Fig.6-8) correspond to the damage initiation in successive layers (their number were denoted with "#" on the figures, see Fig.6-8) of the analyzed profiles and the increasing depth of the yellow color means that the damages initiation and their propagation includes the increasing number of layers of the laminate.

30 #8 #4 #5 25

P [kN]

20 15 10 5 #1 #7 0 0.0

0.5

#2

#3 #6

C3_EXP_ARAMIS C3_EXP._ZWICK_UTM C3_FEM_PFA C3_FEM_STANDARD Damage initation

1.0 1.5 2.0 2.5 Shortening of the column [mm] Fig. 7. The compressive force PMAX in a function of shortening of the column (C3).

3.0

In the case of cross-ply laminate (C3) we can notice small differences in the courses of the curves determined from numerical and experimental results with the use of Aramis® system, and slightly higher in comparison to the curve determined from the universal testing machine. This curve's shift results from the clearance elimination during the initial loading of the column. Close to the loaded edges and the corners of the channel section profile it can be also observed a significant local inflection area. This effect has been observed both in experimental research and numeric analysis.

20 #3

#7

16

P [kN]

12

8

4 #1 0 0.0

1.0

#6 #8

#2 #4 #5

C5_EXP_ARAMIS C5_EXP._ZWICK_UTM C5_FEM_PFA C5_FEM_STANDARD Damage initation

2.0 3.0 4.0 Shortening of the column [mm] Fig. 8. The compressive force PMAX in a function of shortening of the column (C5).

5.0

In the case of angle-ply laminate (C5 ) as in the case of quasi isotropic arrangement of layers (C1 ), we can observe the same buckling modes and very similar courses of experimental and numerical curves. It can also be noticed that the range in which the damage initiation comprises from one to eight layers is lower than in the other two cases analyzed. This indicates the fact that even significant damage of the matrix does not have a major impact on the global behavior of the analyzed structure. If as a reference point the relative difference of global failure loads, determined from the PFA and STANDARD analysis for quasi-isotropic ply system (C1) will be assumed at the level of 2%, it turns out that the significant differences in the courses of the curves are observed above 20.8 kN, which represents about 85% of the load capacity determined from PFA. Using analogous assessment criterion for the remaining ply systems, it can be noticed that the considerable differences of the curves courses can be observed only above 23.79 kN for the C3 and above 17.74 kN for the C5, which in both cases is almost 100% of the failure loads obtained from PFA. Nevertheless, in the range of the initiation of the damages induced by the fiber failure, the courses of the curves differ significantly in terms of quality. In the case of PFA algorithm, there appears the characteristic inflection of the curve, whose the highest point corresponds to the maximum compressive force transmitted through the cross-section. In the case of geometrically nonlinear analysis without the implementation of progressive failure algorithm, the increase of the compressive force corresponds to the increase of the shortening of the profile. The qualitative assessment of the obtained results shows that, despite the application of the progressive failure algorithm, the global failure for the all analyzed cases is characterized by a mild decrease of the compressive force whereas the experimental results indicate an immediate loss of the load

capacity, as it is in the case of the basic strength tests, conducted in order to determine the material characteristics. In the remaining part of the article the authors' attention will be focused on the analysis of the failure loads of the C-shaped profile and (on) the character of the initiated damages. The comparison of the failure loads was included in the Table 3. Table 3. The values of the load-carrying capacities. Failure load [kN] C1 [0,-45,45,90]S C3 [90,0,90,0]S C5 [45,-45,45,-45]S FEM_STANDARD 25.9 26.5 19.5 FEM_PFA 25.7 23.8 17.8 EXPERIMENT_MV. 24.6 24.8 17.6 Analyzing the data included in the Table 3, it is easy to notice that in all analyzed arrangement of layers the values of global failure loads, determined with the implementation of the PFA algorithm, were in closer correlation with the experimental results than the results obtained with the use of STANDARD procedure. The relative differences of the global failure loads were within in the range from 1.1% for the C1 ply system to 4.3% in the case of arrangement of layers denoted as C5. In the standard procedure of the determination of failure loads, the corresponding differences were more than 5.0% in the case of quasi-isotropic arrangement of layers (C1) and more than 9.7% in the case of the cross-ply laminate (C5). It turns out that the acceptance of the arbitrary assumption about the loss of the capacity of the compressed plate system after the fibers failure in any element of the analyzed model is in close correlation with the results obtained from the PFA as well as when compared to the experimental studies. The numerical analysis with the use of PFA algorithm are more timeconsuming in terms of the calculation time than the standard geometrically nonlinear analysis, in which the area and the character of the damage is determined on the basis of the nondegraded state of stress and strain. Nevertheless, they are characterized by a much shorter time of the results compilation and the load capacity determination. In addition, the curves determined from the PFA represent the global behavior of the analyzed structures and they can be directly compared with experimental results.

a)

b)

c)

Fig. 9. Maps of the damage status for the ply system denoted as: C1 (a), C3 (b), C5 (c). For all analyzed cases 32 maps of the damage status (8 layers x 4 damage modes) were prepared and they were put all together on one another, taking the fiber damage effects as the most important (see Fig. 9). It should be understood that these damage maps were at the top of the layer stack. Maps were made for the step when the numerical model achieves the maximum compressive load and they were compared to the experimental forms of failure. It turns out that in the analyzed C-shaped columns subjected to uniform compression the largest areas of the damage were induced by the destruction of the matrix caused by tensile stresses. This results from the very low tensile strength of the matrix, which is more than twice less than the compressive strength of the matrix. Nevertheless, a reduction of the compressive force transmitted through the column is caused by the appearance of the damage involving the fibers. In all analyzed cases it was noticed that the failure mechanism will include the initiation of the fiber destruction in the corner of the column and its propagation in the direction of the web and the flange of the column. The local character of the fiber failure and its small area, located in the close proximity to the supported and loaded edges (cf. Fig. 9), can induce a mechanism formation and a change of boundary conditions. Therefore, the shape and the number of the buckling half-waves observed at the moment of the loss of the capacity is different in comparison to the buckling modes observed in the initial stages of loading (cf. Fig. 6-8). Taking the courses of the curves of compressive loads in the function of the shortening of the columns as a additional assessment point (Fig. 6-8), it can be concluded that the large area of the matrix failure do not have significant impact on the courses of these curves and in practice the larger differences can be noticed only after the initiation of the first damage, involving degradation of the fibers (Fig. 9). 6. Conclusions

Within the present study, experimental and numerical investigations of the C-shaped profile subjected to uniform compression were conducted. The analyzed columns were made of GFRP laminate and simply supported on both ends. Three arrangements of layers were analyzed: quasi-isotropic laminate (C1 ), cross-ply laminate (C3 ) and angle-ply laminate (C5 ). The experimental research was conducted with the use of the ultimate testing machine and the digital image correlation system Aramis® and the numerical analyses have been performed in the Ansys® program, based on the finite element method. The numerical calculations were conducted with the implementation of the progressive failure algorithm and with the use of a standard geometrically nonlinear analysis with the application of the Hashin's criterion. The progressive failure analyses were performed with the use of material property degradation method and implementation the Hashn's criterion as the damage initiation criterion and in the case of the second algorithm the failure modes were calculated based on nominal stress state. Based on the performed experimental and numerical studies it has been concluded that:



















In all analyzed cases of arrangements of layers very good consistence of numerical and experimental results was achieved, both in terms of the failure loads and determined equilibrium paths. It is further noted that the use of additional DIC system Aramis® can allow to identify or confirm the phenomenon of clearance elimination during the loading, influencing on the curve courses collected by the testing machines (cf. Fig. 7). A significant damage of the matrix does not have a major impact on the global behavior of the analyzed structure (cf. Fig.6-9) and the loss of the load capacity is affected mainly by the initiation of the damage, involving degradation of the fibers. The load-carrying capacity, determined with the implementation of the PFA algorithm, was in closer correlation with the experimental results than the failure loads obtained from STANDARD analysis. The relative differences of the global failure loads were within the range from 1.1% (C1) to 4.3% (C5) in the case of the PFA algorithm and from 5.0 % (C1) to 9.7 % (C5 ) with the application of the procedure described as a STANDARD. In all analyzed cases, the failure mechanism included the initiation of the fiber destruction in the corner of the columns and its propagation in the direction of the web and the flange of the column. In terms of quality, the courses of the equilibrium paths, determined from the PFA and a STANDARD procedure, differ significantly in the range of the initiation of the damages induced by the fiber failure. The acceptable relative differences of experimental and numerical results (below 10%) were achieved for values of the damage variables greater than or equal to 0.9 but the identification of the numerical model allowed to establish that the highest level of correlation of experimental and numerical results is achieved with the declaration of all damage parameters equal to unity.

In the authors' opinion the damage variables assumed as equal to unity successfully reflect the results of experimental studies and can be recommended for other composite structures subjected to compression, analyzed with the implementation of a progressive failure algorithm. Simultaneously, the authors wish to pay attention to the fact that it is also necessary to perform a similar considerations for other schemes of load and a validation of their results by experiment. The authors also want to emphasize that the progressive failure analysis can be carried out with the implementation of the continuum damage mechanics method (CDM), in which the problem of the selection of the appropriate damage variables comes down to the declaration of the four additional mechanical properties - energies dissipated per unit area from: tensile fiber damage, compressive fiber damage, tensile matrix, damage, compressive matrix damage. These parameters are determined by means of additional tests and complement the material characteristics of the additional variables describing the mechanism of failure. However, they have physical character and do not have to be empirically selected. CDM based modeling of damage and fracture mechanisms provides a continuum change of the damage variables values with increasing level of load. Acknowledgements

The studies have been conducted as a part of the research project financed by the National Centre for Science - decision number UMO 2015/17/B/ST8/00033. References

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