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Finite element modelling of z-pinned composite T-joints
Composites Science and Technology 73 (2012) 48–56
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Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech
Finite element modelling of z-pinned composite T-joints F. Bianchi a, T.M. Koh b, X. Zhang a, I.K. Partridge c, A.P. Mouritz b,⇑ a
School of Engineering, Cranfield University, Cranfield, Bedford, England MK43 0AL, United Kingdom School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, GPO Box 2476, Melbourne 3001, Australia c School of Applied Sciences, Cranfield University, Cranfield, Bedford, England MK43 0AL, United Kingdom b
a r t i c l e
i n f o
Article history: Received 14 January 2012 Received in revised form 30 August 2012 Accepted 11 September 2012 Available online 24 September 2012 Keywords: A. Polymer matrix composites (PMCs) C. Finite element analysis (FEA) C. Multiscale modelling Z-Pinned joints
a b s t r a c t This paper presents a finite element model (FEM) to analyse the structural deformation, strength properties and delamination fracture behaviour of composite T-joints reinforced with z-pins. The FE modelling involves multi-level analysis of a pinned joint using unit cell and macro-scale structural models. Unit cell model is used to calculate the crack bridging traction loads generated by the elastic and interfacial friction (pull-out) stresses of a single pin within a representative unit cell of the joint. Macro-scale analysis involves modelling delamination crack growth in the pinned joint using a cohesive zone model, which is based on the traction load analysis of a single pin. The FE model was validated using experimental results for a pinned carbon fibre/epoxy T-joint subjected to tensile (stiffener pull-off) loading. The model accurately calculated the crack initiation load, ultimate load, and fracture mode of the pinned joint. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Many thin-walled composite structures used in aircraft (e.g. wing panels, fuselage sections) and other applications are reinforced with internal stiffeners to increase the buckling load. The most common design is the T-shaped stiffened joint, which consists of a thin laminate skin reinforced with a T-section laminate stiffener, which is joined to the skin by co-curing or adhesive bonding. A long-standing problem with T-joints is delamination cracking between the skin and stiffener caused by out-of-plane (through-thickness) loads, impact, or environmental deterioration of the bonded region due to moisture ingress. Recent research has revealed that the delamination resistance of composite joints can be improved by through-thickness reinforcement of the bonded region with z-pins, and this increases the strength and toughness properties [1–8]. Experimental testing has shown that the ultimate load, damage tolerance and strain energy absorption of T-joints are increased greatly by pinning, particularly at relatively high pin contents. For example, Koh et al. [5] recently reported that the ultimate load of T-joints was doubled when reinforced at the pin content of 4% by volume. The joint properties are improved by the pins generating bridging traction loads across delamination cracks when damage propagates along the interface region between the skin and stiffener flange. Numerous studies
⇑ Corresponding author. E-mail address: [email protected] (A.P. Mouritz). 0266-3538/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compscitech.2012.09.008
have developed finite element or analytical models to calculate the delamination fracture toughness of flat composite panels reinforced with pins (e.g. [10–13]). Such a model is essential for the optimised design of damage resistant joints reinforced with pins. However models for analysing the properties of pinned joints (which are geometrically more complex) are limited [7,14]. Toral Vazquez et al. [7] developed an FE model of a pinned joint using plane strain elements to model the laminates and non-linear springs for the pins that connect the skin and T-stiffener. Allegri and Zhang [14] also developed an FE model for pinned joints with the laminates modelled with shell elements and the pins using one-dimensional non-linear elements, whose constitutive law was governed by the pin bridging traction law that was derived from an analytical model. Bridging law parameters were estimated by calibrating the FE model with the experimentally measured load–displacement response. The objective of this study is the development of a finite element (FE) modelling approach to analyse the structural properties and delamination failure of pinned T-joints under tensile (stiffener pull-off) loading. The model is based on two levels of analysis: unit-cell model of a single pin in the joint to calculate the bridging traction force and a structure-level model of the T-joint to calculate the damage initiation load, ultimate failure load and fracture behaviour. The modelling approach is validated by experimental observation and results of a pinned T-joint under tensile loading. The practical outcome of this research is a validated FE modelling approach that can be used in the design of pinned T-joints with enhanced strength and delamination toughness properties.
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Fig. 1. Geometry and loading of the pinned T-joint.
2. Finite element model of pinned T-joints 2.1. FE modelling approach A FE model was developed to analyse the structural properties and failure of a z-pinned T-shaped composite joint subjected to a tensile (pull-off) load applied to the stiffener, as shown schematically in Fig. 1. The analysis was based on a multi-level modelling approach performed at the unit cell and structural levels, as represented in Fig. 2. The joint was reinforced along the skin–stiffener flange region only, which is where delamination failure occurs [5], and outside of this region the joint was not pinned. The unit cell model consists of a single pin embedded in the orthogonal direction within the laminate material used in the joint. The unit cell model was used to calculate the bridging traction loads generated by a single pin under modes I and II delamination crack growth conditions within the joint laminate. The macro-scale model applied the traction loads calculated using the unit cell model to analyse the pin bridging mechanics at the structural level of the joint. Cohesive zone modelling (CMZ) was used to calculate the initiation and growth of delamination cracks in the joint. 2.2. Unit cell model The pin traction loads generated during delamination crack growth along the skin–flange region to the joint were calculated using a unit-cell model developed by Bianchi and Zhang [12]. It is for a single pin under two stress conditions: tensile stress normal to the crack plane (induced by mode I interlaminar stress) and shear stress along the crack plane (induced by mode II interlaminar stress). Under tensile loading of the T-joint, the pins along the skin–stiffener flange connection were predominantly loaded by a mode I stress during delamination crack extension. The bridging traction laws for a single pin under pure mode I loading have been
determined by Dai et al. [15] and Zhang et al. [16], and assume that under increasing crack opening displacement the traction load increases by elastic deformation and then the load decreases due to debonding and pull-out of the pin, as represented by the two-stage traction force-crack displacement curve given in Fig. 2. The strain energy due to pin debonding is usually negligible compared to the energy due to pin pull-out [9], and therefore the traction load during the pin pull-out phase was assumed to be completely due to friction. While the analysis presented here assumed the pin failed by pull-out, the bridging traction law used in the unit cell model can be modified to analyse when the pin fails by tensile rupture. Pin pull-out was assumed to occur when the applied axial force was higher than the friction load generated at the pin/laminate interface whereas pin rupture occurs when the maximum axial stress exceeds the pin material strength. While the loading of pins in the T-joint was predominantly mode I, a mixed mode I/II stress condition exists at the delamination crack tip due to bending of the skin–stiffener flange section under tensile loading. When a single pin is loaded in mode II the unit cell model analysed its shear deformation as a rod supported by an elastic foundation (i.e. laminate), as shown schematically in Fig. 3. The stiffness of the foundation was determined by the elasticity of the laminate. An elevated compressive stress was generated due to shear deformation of the pin into the laminate near the delamination crack plane, which is known as the pin snubbing effect [13], and this increased the friction stress opposing pin pullout. When the shear stress acting on the pin reached its shear strength, the unit cell model assumes the pin splits into several ligaments and thereby loses its bending rigidity. From this point, it was assumed that the pin can undergo large-scale sliding displacement under the mode II loading; however it is only capable of carrying the axial stress. The laminate around the pin, due to the large shear sliding displacement of the damaged pin, was assumed to behave as a perfectly plastic foundation.
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Fig. 2. Schematic of the multi-level FE modelling approach [11]: (a) unit-cell model for measuring or calculating ‘‘pin force vs. displacement’’ relation, which is then deduced to a ‘‘traction–stress vs. separation–displacement’’ law; (b) structure-level model with the ‘‘traction–separation’’ law implemented at pin locations.
Fig. 3. Schematic of unit cell model of a single pin under mode II interlaminar loading [12].
The traction loads generated by a single pin under modes I and II loading were determined experimentally by Koh et al. [17] using pin pull-out and by Cartié et al. [18] using pin shear tests, respectively. Using the loads, the unit cell model can then be solved to estimate the bridging traction laws for pins with different diameters or embedded lengths within the laminate. The traction loads for a pin when delamination cracks occurred on different planes to the laminate was also analysed by assuming different pin embedded lengths, and therefore the pin traction load was simply a function of the location of the delamination crack plane. The mode II failure stress of the pin was measured to be 1040 MPa by interlaminar shear testing of a single pin [18]. The measured traction load value was high because of the shear fracture mode of the pin, which deformed by firstly rotating towards the interlaminar shear loading direction (i.e. parallel with the delamination crack plane) and then fracturing in transverse shear. The high shear fracture stress of the carbon fibres within the pin is therefore responsible for the high mode II traction stress of the pin. The foundation modulus (KII) of the laminate was assumed to be 5 1013 N/m3, which was measured from the mode II pin traction test performed by Cartié et al. [18]. The test by Cartié and colleagues was performed on a different type carbon–epoxy laminate with a different ply stacking sequence to the material used in the joints, although the pins were the same. The KII value of 5 1013 N/m3 in the FE analysis is taken to be acceptable because the foundation stiffness is determined mainly by the resin-rich
pocket surrounding the pin, which was similar for the laminates tested by Cartié and colleagues and used in the joints. Also, a sensitivity study (not reported in this paper) showed that the mode-II bridging parameters (including KII) have small influence on delamination propagation because the pins fails predominantly in mode I. A mixed-mode failure criterion was used to calculate the fracture toughness for a pin [2]:
Gpin I Gpin IC
þ
Gpin II Gpin IIC
ð1Þ
where Gpin and Gpin I II are the instantaneous traction strain energy values of the pin under crack opening and crack sliding displacements for mode I and II loading, respectively. These values were determined by the area under the traction load–displacement curves at pin a specific displacement value. Gpin IC and GIIC are the total traction strain energy upon pull-out failure of the pin under mode I and II pin loading, respectively. Gpin IC and GIIC were determined by the entire area under the mode I and II traction load–displacement curves, which were measured experimentally [17,18]. Two traction load–displacement laws were used for the laminate and pin. Each law was defined by three parameters: K, T0 and GC for the laminate and K pin , T pin and Gpin for pin bridging; C 0 where K, T and G are the elastic stiffness, peak load and stored strain energy properties (see Fig. 2). K, T0 and GC were based on
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the delamination toughness properties for the unpinned laminate and K pin , T pin and Gpin were calculated using the bridging traction C 0 curve obtained from the unit cell model. The pin traction law specified in the FE model is defined by the interfacial traction properties between the pin and laminate. In reality, the traction law is determined by the pull-out of the pin in the through-thickness direction of the laminate. However, Bianchi and Zhang [11] have shown that the derived interfacial traction law is equivalent to the pin pull-out process in terms of strain energy release rate, and therefore defining the traction laws based on interfacial effects between the pin and laminate is numerically accurate. Further details can be found in [11,12]. Another factor considered in the unit cell model was the compressive stress field that occurs at the pin/laminate interface due to residual stress generated from the elevated temperature curing of the laminate. This compressive stress increases the friction stress generated along the pin/laminate interface during the pullout phase of the pin failure process, and consequently increases the bridging traction load. As summarised in [11,12] the pin bridging law is a function of the materials and geometry, particularly the pin embedded length. Therefore, the pin bridging effect varies depending on which plane delamination is taking place. The model parameters were obtained from the single-pin test [14,15], although the parameter can also be determined using an analytical or numerical model of the single-pin configuration. Once a calculated pin bridging law is validated by test result, the model can be used to calculate bridging laws of other pin configurations, which is then implemented in a structural model at locations where delamination is likely to occur. The bridging traction-separation laws employed are constructed by the following procedure: (1) obtain the single-pin force vs. displacement curve (Fig. 2 insert top left) and divide the force by pin cross-sectional area to obtain a traction stress vs. displacement curve (Fig. 2 insert lower right); (2) find the initial stiffness (K pin ) that is the slope of the straight line tangent to the curve in the origin; (3) cohesive strength (T pin 0 ) equals to the maximum traction stress exerted by the pin; (4) cohesive energy (Gpin C ) equals to the integration area of the stress–displacement curve. 2.3. Macro-scale FE model: unpinned joint A finite element model of the unpinned joint was created using quadratic plane strain elements with reduced integration, as shown in Fig. 4. The model was developed to calibrate the cohesive element properties (stiffness, K and cohesive strength, To) along the delamination crack planes and to validate the cohesive zone model
Table 1 Properties of the laminate, cohesive elements and z-pin in the FE model of the T-joint. Property Laminate (unidirectional carbon fibre/epoxy) In-plane Young’s modulus (E1) Transverse Young’s moduli (E2, E3) Shear moduli (G1, G2, G3) Poisson’s ratio (c12, c13) Poisson’s ratio (c23)
Value a
Cohesive element properties of laminate Mode I traction stiffness of laminate (KI) Mode II traction stiffness of laminate (KII) Mode I failure load of laminate (TI0) Mode II failure load of laminate (TII0) Mode I fracture toughness of laminate (GIC) Mode II fracture of laminate (GIIC)
120 GPa 7.5 GPa 3 GPa 0.32 0.32 5 1013 N/m3 5 1013 N/m3 30 MPa 70 MPa 600 J/m2 1200 J/m2
Cohesive element properties of pin Mode I traction stiffness of pin for a1-type crack (K pin I ) Mode Mode Mode Mode Mode Mode Mode Mode Mode Mode Mode
II traction stiffness of pin for a1-type crack (K pin II ) I traction stiffness of pin for a2-type crack (K pin I ) I traction stiffness of pin for a2-type crack (K pin II ) I traction load of pin for a1-type crack (T pin I0 ) II traction load of pin for a1-type crack (T pin II0 ) I traction load of pin for a2-type crack (T pin I0 ) II traction load of pin for a2-type crack (T pin II0 ) I fracture toughness of pin for a1-type crack (Gpin IC ) II fracture toughness of pin for a1-type crack (Gpin IIC ) I fracture toughness of pin for a2-type crack (Gpin IC ) II fracture toughness of pin for a2-type crack (Gpin IIC )
1 108 N/m3 4.2 1010 N/m3 7.5 1010 N/m3 3.1 1010 N/m3 450 MPa 1040 MPa 255 MPa 580 MPa 400 kJ/m2 210 kJ/m2 100 kJ/m2 170 kJ/m2
a VTM 264 (T700 fibres, HS200 resin by Advanced Composites Group). The laminate properties were measured in-house at RMIT University.
used for analysing crack growth. Mesh sensitivity and element sensitivity analysis was performed, and the mesh shown in Fig. 4 was found to give the best compromise between numerical accuracy and computation time. The orthotropic elastic properties of the cross-ply [0/90]s carbon fibre/epoxy laminate used for the skin, flange and stiffener of the joint are given in Table 1. While z-pinning can reduce slightly (typically under 10%) the in-plane properties of laminates [19,20], it was assumed in the FE model that the pins did not alter the laminate properties. The triangular D-fillet region at the stiffener base was filled with unidirectional laminate with the fibre direction perpendicular to the modelling plane. The triangular region at the taper run-out at the flange ends was assumed to have the mechanical properties of epoxy resin (E = 3 GPa, m = 0.4).
Fig. 4. (a) Unpinned joint model. (b) Location of discrete delamination cracks.
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Fig. 5. Unit-strip FE model of the pinned joint.
Fig. 6. Comparison of the calculated and measured applied force–displacement curves for the unpinned joint. Diagrams on the right-side indicate the onset of vertical cracking along the centre-line of the stiffener at point 1 and horizontal cracking along the skin–flange interface at point 2.
According to experimental observation of the failure of T-joints under tensile loading [5], discrete delamination cracks grow along the interface between the skin and flange (a1 – horizontal path), centre-line of the stiffener (a1 – vertical path), and centre of the web (a2), as illustrated in Fig. 4b. These crack paths were modelled by placing a layer of cohesive elements between the plies. Cohesive elements were used to model the delamination cracks along these paths, with the cohesive interface element size being one-fifth of the adjacent ply element size in order to achieve numerical stability [21]. Chen [23] also reports that cohesive elements need to be much finer than surrounding elements for modelling multidirectional cracks in braided composite T-joint. Chen gives details on obtaining cohesive parameter values and selecting mix-mode failure criterion. In this present paper, cohesive element property values were selected based on the delamination toughness properties of the carbon/epoxy laminate used in the joint (GIC = 0.6 and GIIC = 1.2 kJ/m2). The parameters and their values used for the cohesive fracture analysis are provided in Table 1. The FE simulations were run under displacement-controlled loading by applying a monotonically increasing tensile
displacement as the boundary condition on the nodes at the upper extremity of the stiffener. In order to avoid hourglass deformation of cohesive elements on the symmetry plane, the y-direction displacement of each cohesive element node was constrained to be the same as its corresponding node on the other face of the element (i.e. no shear strain due to symmetry). The ends of the skin were clamped to prevent vertical displacement and rotation of the joint. However, it was found that assuming perfect axial displacement constraint (ideal clamping) or assuming the clamp does not react in the axial direction (sliding clamps) resulted in the joint being either too stiff or too compliant compared to the experimental results (which are presented later). The clamp was therefore assumed to constrain the axial displacement until a maximum axial force. From this point the joint was assumed to slide axially with a constant friction force. This boundary condition was modelled using a non-linear spring element with an initial stiffness of 1 108 N/m and maximum axial force of 9 kN. The initial stiffness and maximum axial force used as the boundary condition was calibrated against measured values determined by experimental testing of the unpinned T-joint.
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2.4. Macro-scale FE model: z-pinned joint A FE model was created for a pinned joint that had the same geometry as the unpinned joint. A unit-strip model was constructed with the pins spaced at regular intervals (1.75 mm) along the skin–flange region, as shown in Fig. 5. Because a unit-strip model was used, it was possible to model the pins as one-half of a single row of pins. This is based on FE modelling results in [11,12]. Due to the pinned joint having a higher ultimate load limit than the unpinned joint, the central interlaminar planes of the stiffener are important for delamination damage progression after crack growth between the skin and stiffener flange. For this reason an extra layer of cohesive elements was added between the plies close to the centre-line of the stiffener (illustrated as crack a2 in 4b). Each sub-laminate was modelled using one layer of continuum shell elements. Periodical boundary conditions were applied to the longitudinal planes that delimit the strip. These boundary conditions consist in a symmetry boundary condition and a constrain to the nodes belonging to the opposite one to remain within the same plane. The pins were modelled using cohesive elements for the pin traction load–displacement law (TSL) determined using the unit cell model. For mode II loading near the delamination crack tip, the non-linear shear deformation of the pins was neglected and a simplified bi-linear TSL was used [12]. Cohesive element properties used for the FE analysis are provided in Table 1. 3. Experimental testing of T-joints T-joint specimens were made of unidirectional T700 carbon/ epoxy prepreg tape (VTM 264 supplied by Advanced Composites Group). The material properties of the carbon/epoxy laminate are provided in Table 1, and it was assumed they were not affected by the pins. The joint geometry and dimensions are shown in Fig. 1, and further information is provided by Koh et al. [5,8]. The plies to the skin, stiffener and flange were stacked in a cross-ply [90/0/90/0/90]S pattern. The thickness of each ply was approximately 0.125 mm. The D-fillet region at the stiffener base was filled with the unidirectional prepreg tape. Before curing, the skin–stiffener flange section to the T-joint was pinned with pultruded rods of unidirectional T300 carbon fibre/bismaleimide (Albany Engineered Composites Pty. Ltd.). The pins were inserted in the orthogonal (through-thickness) direction of the flange section using an ultrasonic hand-held device operated at the frequency of 20 kHz. The pin tip was chamfered to aid insertion through the laminate while the trailing end of the pin was blunt. The chamfered end of the pin was embedded in the flange and the blunt end was in the skin. The entire length of the flange connection was reinforced with pins, whereas the regions outside of this connection were not pinned (as indicated in Fig. 1). Koh et al. [5] give a full description of the z-pinning process used to
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reinforce the T-joint. The volume content and diameter of the pins was 2% and 0.28 mm, respectively. The pins were arranged in a square grid pattern aligned along the length and across the width of the flange section to the joint. The spacing between the pins both along and across the joint was 1.75 mm. Control T-joint specimens (without pins) were also made with the same geometry as the pinned specimens. Both the unpinned and pinned joint specimens were cured in an autoclave at 120 °C and 620 kPa for 1 h. The joints were bonded by co-curing without the use of adhesive. The average volume content of carbon fibres in the skin, flange and stiffener of the joints was about 60%. A stiffener pull-off test was performed on the T-joint specimens to validate the FE model. A tensile load was applied to the stiffener using a 50 kN Instron machine at a constant displacement rate of 1 mm/min until final failure. The ends of the skin were clamped to a rigid support plate, although some slippage occurred during testing which was accounted for by the FE model. At least five specimens of the unpinned and pinned joints were tested under identical conditions to validate the FE model. 4. Results and discussion 4.1. Unpinned T-joint Fig. 6 shows the applied force-vertical displacement curves for the unpinned joint that were calculated using the FE model and measured using the stiffener pull-off test. The measured result was first reported by Koh et al. [5]. There was some variability in the measured curves based on repetitions of the stiffener pull-off test performed on five samples of the unpinned joint. The standard deviation to the ultimate load was about 15% of the average value; the average value for the ultimate load was 1656 N and the standard deviation was 252 N based on repeat tests performed by Koh et al. on five joint specimens [5]. Fig. 6 shows there is relatively good agreement between the calculated and measured curves: the FE model accurately predicted the non-linear rise in the stiffness and the peak load of the unpinned joint, after which it was predicted that the load capacity would drop abruptly due to splitting cracking along the centre-line of the stiffener (at point 1 in Fig. 6) followed by delamination cracking along the skin–flange interface (at point 2). These two fracture modes were observed in the unpinned test specimens, as shown in Fig. 7. Only a slight difference in stiffness between the model and experiment can be noticed before the load drop. This is probably due to the clamp boundary condition in the model, which overestimates the clamping axial force, thus predicting stiffer response at large deformations. Fig. 8 shows the normal (y-direction) stress contour map for the unpinned joint calculated using the FE model. The numerical analysis showed that crack initiation occurred when the normal tensile stress exceeded the peel strength of the cohesive elements in the FE model. It was determined that the splitting crack along the
applied load
stiffener crack
skin/stiffener flange crack Fig. 7. Fractured unpinned joint showing splitting cracking along the centre-line of the stiffener and delamination cracking along the skin–flange interface.
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applied load
splitting crack
(a) applied load
delamination
(b) Fig. 8. FE stress contour map of the normal stress in the y-direction of the unpinned joint (the coordinate system follows laminate orientation).
centre-line of the stiffener was due to a pure mode I stress, which at the crack tip was about 30 MPa. The delamination crack at the skin–flange interface propagated under a mixed mode I/II stress condition, and therefore the peel stress was calculated to be slightly lower at about 18 MPa. 4.2. Z-pinned T-joint Fig. 9 compares the calculated and measured applied load–displacement curves for the pinned joint, and again there is excellent agreement. The measured curves are taken from Koh et al. [5], who determined the average and standard deviation in the ultimate load of the pinned joint to be 2777 N and 683 N, respectively, based on five test measurements. The stiffness of the pinned joint was calculated using the FE model to be the same as the unpinned joint, and this was confirmed by experimental testing. Pins do not significantly alter the in-plane tensile modulus of carbon/epoxy laminate [19,20,22], and therefore the stiffness of the joint was not changed by pinning. The pinned joint experienced an initial load drop (at d = 4 mm), and the FE model determined that it was caused by the initiation from the D-fillet region of a splitting crack
Fig. 10. Pinned joint showing (a) splitting cracking along the stiffener centre-line at the initial load drop and (b) delamination cracking along the skin–flange interface at the second (ultimate) load drop.
along the centre-line of the stiffener. This was immediately followed with the initiation of a delamination crack along the skin–flange interface. Again, this was confirmed by experimental testing with both stiffener splitting and skin–flange delamination cracking spreading from the D-fillet region of the pinned joint specimen following the initial load drop (Fig. 10). The FE model predicted that the pinned joint does not fail catastrophically at the initial load drop point (unlike the unpinned joint) due to bridging traction loads generated by the pins along the delamination crack between the skin and flange. The pin traction loads caused a recovery in strength and consequently the pinned joint was able to withstand further loading up to the ultimate load limit of about 3800 N, which was over twice as high as the unpinned joint. The FE model determined that the second (and much larger load drop at d = 10.7 mm) was caused by the formation of a second crack within the web region (a2), and this was also observed experimentally. Between the initial and second load drop points, the FE model predicted that an increasing number of pins generated traction loads as the delamination crack grew in length along the skin– flange interface, and this was the cause for the progressive increase
Fig. 9. (a) FEA and measured applied force vs. displacement curves for the pinned joint. (b) FEA of the crack length vs. applied displacement and number of active pin rows along the skin–stiffener flange delamination crack.
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Fig. 11. Pin bridging stresses immediately after the first load drop, applied displacement d = 6 mm. (a) Normal stress in z-direction (through-thickness). (b) Transverse shear stress. The stresses represent the bridging traction force per pin unit area (unit: MPa).
Fig. 12. Pin bridging stresses at ultimate load, applied displacement d = 10.67 mm. (a) Normal stress in z-direction (through-thickness). (b) Transverse shear stress. The stresses represent the bridging traction force per pin unit area (unit: MPa).
in joint strength between the two points, as shown in Fig. 9b. The spacing between the pin rows along the skin–stiffener interface was 1.75 mm, and it was calculated that seven rows of pins were required to create a fully established bridging traction zone along the delamination crack. The FE analysis revealed that once the delamination crack length exceeded the pin bridging traction zone length of about 12.25 mm, the last row of pins in the crack wake failed by pull-out from the skin as a new row of pins began to generate traction loads near the crack tip. Again, this was confirmed by experimental testing with the pin bridging zone along the skin– flange interface being 10–15 mm long. The FE analysis assumed that all the pins were pulled from the flange and retained in the skin, which was found for the majority of pins in the experimental tests. The pins were preferentially pulled from the flange because of their chamfered tip which reduced the bonded region with the laminate. If, however, the pins were pulled out from the skin and retained in the flange then the a2 crack is effectively an unpinned interface with low toughness, and this could affect the ultimate load of the pinned joint. FE analysis was performed to calculate the pin traction loads along the skin–flange delamination crack at two points: (a) when the pins first formed a fully-developed bridging zone which occurred at d = 6 mm (Fig. 11) and (b) at the maximum load point which occurred at d = 10.7 mm (Fig. 12). The FE model computed both the normal tensile and transverse shear traction stresses for each row of pins along the delamination crack. (The stresses
represent the traction loads per pin unit area). The analysis showed that the traction stresses were non-uniformly distributed between the pin rows along the crack. Both the tensile and shear traction stresses increased with distance behind the crack tip due to increasing crack opening/sliding displacement and reached a maximum at about one-half along the crack length. The traction stresses then decreased towards the rear of the bridging zone due to the reduced friction stress generated by the pins when they were nearly pulled out from the laminate. Fig. 11 also shows that the normal traction stresses generated along the upper (a2) crack were lower than those generated along the skin–flange interface crack (a1). This was due to the smaller opening displacement in the a2 crack and the shorter embedded pin length which induced a lower shear load along the pin/laminate interface. However, the transverse shear stresses were higher along the a2 crack due to the higher shear sliding displacement compared to the a1 crack. This analysis revealed the complexity of the bridging traction loads generated in pinned joints which increased the strength and toughness properties.
5. Conclusions A validated FE model has been developed which can analyse the structural properties and fracture behaviour of pinned T-shaped composite joints when subjected to tensile (stiffener pull-off) loading up to the maximum load point. The model is demonstrated
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capable of predicting the rising portion of the force–displacement curve, which corresponds to damage initiation and propagation up to the ultimate load of the joint. The FE model analyses the crack bridging traction loads of a single pin under modes I and II interlaminar stresses at the unit cell level, and this analysis is then used to calculate the strength and delamination fracture properties of pinned joints at the structural level. The FE model provides important insights into the strengthening and toughening of pinned joints, such as failure initiation being determined by mode I splitting cracking along the vertical stiffener and the ultimate strength being controlled by the modes I/II pin traction loads, which are unevenly distributed along the bridging zone between the skin and flange. Splitting of the vertical stiffener can be avoided by z-pinning, although it reduces the ultimate failure load of the Tjoint [5]. The FE model also revealed that both the traction loads generated within the web region (a2-type crack) and along the skin–flange interface (a1 crack) are important in the strengthening and toughening of pinned joints. The FE model computed that the traction loads are mainly mode II shear stresses in the a2 crack and mode I tensile stresses in the a1 cracks. Despite the complexities of the internal stress distribution, pin traction laws and crack growth behaviour for the pinned joint, the FE model accurately calculated the stiffness, strength and fracture modes when compared to experimental test results. This validation study demonstrates that the FE model can be used in the design of pinned T-joints when subjected to tensile loading. References [1] Cartié DDR, Dell’ Anno G, Poulin E, Partridge IK. 3D reinforcement of stiffenerto-skin T-joints by z-pinning and tufting. Eng Fract Mech 2006;73:2532–40. [2] Rugg KL, Cox BN, Massabò R. Mixed mode delamination of polymer composite laminates reinforced through the thickness by z-fibres. Composites 2002;33:177–90. [3] Chang P, Mouritz AP, Cox BN. Tensile properties and failure mechanisms of zpinned composite lap joints. Compos Sci Technol 2006;66:2163–76. [4] Byrd LW, Birman V. Effectiveness of z-pins in preventing delamination of cocured composite joints on the example of a double cantilever test. Composites 2006;37B:365–78.
[5] Koh TM, Feih S, Mouritz AP. Experimental determination of the structural properties and strengthening mechanisms of z-pinned composite T-joints. Compos Struct 2011;93:2222–30. [6] Park YB, Lee BH, Kweon JH, Choi JH, Choi IH. The strength of composite bonded T-joints transversely reinforced by carbon pins. Compos Struct 2012;94:625–34. [7] Toral Vazquez J, Castanie B, Barrau JJ, Swiergiel N. Multi-level analysis of lowcost z-pinned composite joints. Part 2: joint behaviour. Composites 2011;42A:2082–92. [8] Koh TM, Isa MD, Chang P, Mouritz AP. Improving the structural properties and damage tolerance of composite joints using z-pins. J Compos Mater. http:// dx.doi.org/10.1177/0021998312437233. [9] Cartié DDR, Troulis M, Partridge IK. Delamination of z-pinned carbon fibre reinforced laminates. Compos Sci Technol 2006;66:855–61. [10] Yan W, Liu HY, Mai YW. Numerical study of the mode I delamination toughness of z-pinned laminates. Compos Sci Technol 2003;63:1481–93. [11] Bianchi F, Zhang X. A cohesive zone model for predicting delamination suppression in z-pinned laminates. Compos Sci Technol 2011;71(16):1898–907. [12] Bianchi F, Zhang X. Predicting mode-II delamination suppression in z-pinned laminates. Compos Sci Technol 2012;72(8):924–32. [13] Cox BN. Snubbing effects in the pullout of a fibrous rod from a laminate. Mech Adv Mater Struct 2005;12(2):85–98. [14] Allegri G, Zhang X. On the delamination suppression in structural joints by zfibre pinning. Composites 2007;38A:1107–15. [15] Dai SC, Yan W, Liu HY, Mai YW. Experimental study on z-pin bridging law by pullout test. Compos Sci Technol 2004;64:2451–7. [16] Zhang A, Liu HY, Mouritz AP, Mai YW. Experimental study and computer simulation on degradation of z-pin reinforcement under cyclic fatigue. Composites 2008;39A:406–14. [17] Koh TM, Feih S, Mouritz AP. Strengthening mechanics of thin and thick composite T-joints reinforced with z-pins. Composites 2012;43A:1308–17. [18] Cartié DDR, Cox BN, Fleck NA. Mechanisms of crack bridging by composite and metallic rods. Compos A 2004;35(11):1325–36. [19] Mouritz AP. Review of z-pinned composite laminates. Composites 2007;38A:2383–97. [20] Mouritz AP, Cox BN. A mechanistic interpretation of the comparative in-plane mechanical properties of 3D woven, stitched, and pinned composites. Composites 2010;41A:709–28. [21] Diehl T. On using a penalty-based cohesive-zone finite element approach, part II: inelastic peeling of an epoxy-bonded aluminium strip. Int J Adhes Adhes 2008;28(4–5):256–65. [22] Chang P, Mouritz AP, Cox BN. Properties and failure mechanisms of z-pinned laminates in monotonic and cyclic tension. Composites 2006;37A:1501–13. [23] Chen J. Simulation of multi-directional crack growth in braided composite Tpiece specimens using cohesive models. Fatigue Fract Eng Mater Struct 2011;34:123–30.
Contents lists available at SciVerse ScienceDirect
Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech
Finite element modelling of z-pinned composite T-joints F. Bianchi a, T.M. Koh b, X. Zhang a, I.K. Partridge c, A.P. Mouritz b,⇑ a
School of Engineering, Cranfield University, Cranfield, Bedford, England MK43 0AL, United Kingdom School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, GPO Box 2476, Melbourne 3001, Australia c School of Applied Sciences, Cranfield University, Cranfield, Bedford, England MK43 0AL, United Kingdom b
a r t i c l e
i n f o
Article history: Received 14 January 2012 Received in revised form 30 August 2012 Accepted 11 September 2012 Available online 24 September 2012 Keywords: A. Polymer matrix composites (PMCs) C. Finite element analysis (FEA) C. Multiscale modelling Z-Pinned joints
a b s t r a c t This paper presents a finite element model (FEM) to analyse the structural deformation, strength properties and delamination fracture behaviour of composite T-joints reinforced with z-pins. The FE modelling involves multi-level analysis of a pinned joint using unit cell and macro-scale structural models. Unit cell model is used to calculate the crack bridging traction loads generated by the elastic and interfacial friction (pull-out) stresses of a single pin within a representative unit cell of the joint. Macro-scale analysis involves modelling delamination crack growth in the pinned joint using a cohesive zone model, which is based on the traction load analysis of a single pin. The FE model was validated using experimental results for a pinned carbon fibre/epoxy T-joint subjected to tensile (stiffener pull-off) loading. The model accurately calculated the crack initiation load, ultimate load, and fracture mode of the pinned joint. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Many thin-walled composite structures used in aircraft (e.g. wing panels, fuselage sections) and other applications are reinforced with internal stiffeners to increase the buckling load. The most common design is the T-shaped stiffened joint, which consists of a thin laminate skin reinforced with a T-section laminate stiffener, which is joined to the skin by co-curing or adhesive bonding. A long-standing problem with T-joints is delamination cracking between the skin and stiffener caused by out-of-plane (through-thickness) loads, impact, or environmental deterioration of the bonded region due to moisture ingress. Recent research has revealed that the delamination resistance of composite joints can be improved by through-thickness reinforcement of the bonded region with z-pins, and this increases the strength and toughness properties [1–8]. Experimental testing has shown that the ultimate load, damage tolerance and strain energy absorption of T-joints are increased greatly by pinning, particularly at relatively high pin contents. For example, Koh et al. [5] recently reported that the ultimate load of T-joints was doubled when reinforced at the pin content of 4% by volume. The joint properties are improved by the pins generating bridging traction loads across delamination cracks when damage propagates along the interface region between the skin and stiffener flange. Numerous studies
⇑ Corresponding author. E-mail address: [email protected] (A.P. Mouritz). 0266-3538/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compscitech.2012.09.008
have developed finite element or analytical models to calculate the delamination fracture toughness of flat composite panels reinforced with pins (e.g. [10–13]). Such a model is essential for the optimised design of damage resistant joints reinforced with pins. However models for analysing the properties of pinned joints (which are geometrically more complex) are limited [7,14]. Toral Vazquez et al. [7] developed an FE model of a pinned joint using plane strain elements to model the laminates and non-linear springs for the pins that connect the skin and T-stiffener. Allegri and Zhang [14] also developed an FE model for pinned joints with the laminates modelled with shell elements and the pins using one-dimensional non-linear elements, whose constitutive law was governed by the pin bridging traction law that was derived from an analytical model. Bridging law parameters were estimated by calibrating the FE model with the experimentally measured load–displacement response. The objective of this study is the development of a finite element (FE) modelling approach to analyse the structural properties and delamination failure of pinned T-joints under tensile (stiffener pull-off) loading. The model is based on two levels of analysis: unit-cell model of a single pin in the joint to calculate the bridging traction force and a structure-level model of the T-joint to calculate the damage initiation load, ultimate failure load and fracture behaviour. The modelling approach is validated by experimental observation and results of a pinned T-joint under tensile loading. The practical outcome of this research is a validated FE modelling approach that can be used in the design of pinned T-joints with enhanced strength and delamination toughness properties.
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Fig. 1. Geometry and loading of the pinned T-joint.
2. Finite element model of pinned T-joints 2.1. FE modelling approach A FE model was developed to analyse the structural properties and failure of a z-pinned T-shaped composite joint subjected to a tensile (pull-off) load applied to the stiffener, as shown schematically in Fig. 1. The analysis was based on a multi-level modelling approach performed at the unit cell and structural levels, as represented in Fig. 2. The joint was reinforced along the skin–stiffener flange region only, which is where delamination failure occurs [5], and outside of this region the joint was not pinned. The unit cell model consists of a single pin embedded in the orthogonal direction within the laminate material used in the joint. The unit cell model was used to calculate the bridging traction loads generated by a single pin under modes I and II delamination crack growth conditions within the joint laminate. The macro-scale model applied the traction loads calculated using the unit cell model to analyse the pin bridging mechanics at the structural level of the joint. Cohesive zone modelling (CMZ) was used to calculate the initiation and growth of delamination cracks in the joint. 2.2. Unit cell model The pin traction loads generated during delamination crack growth along the skin–flange region to the joint were calculated using a unit-cell model developed by Bianchi and Zhang [12]. It is for a single pin under two stress conditions: tensile stress normal to the crack plane (induced by mode I interlaminar stress) and shear stress along the crack plane (induced by mode II interlaminar stress). Under tensile loading of the T-joint, the pins along the skin–stiffener flange connection were predominantly loaded by a mode I stress during delamination crack extension. The bridging traction laws for a single pin under pure mode I loading have been
determined by Dai et al. [15] and Zhang et al. [16], and assume that under increasing crack opening displacement the traction load increases by elastic deformation and then the load decreases due to debonding and pull-out of the pin, as represented by the two-stage traction force-crack displacement curve given in Fig. 2. The strain energy due to pin debonding is usually negligible compared to the energy due to pin pull-out [9], and therefore the traction load during the pin pull-out phase was assumed to be completely due to friction. While the analysis presented here assumed the pin failed by pull-out, the bridging traction law used in the unit cell model can be modified to analyse when the pin fails by tensile rupture. Pin pull-out was assumed to occur when the applied axial force was higher than the friction load generated at the pin/laminate interface whereas pin rupture occurs when the maximum axial stress exceeds the pin material strength. While the loading of pins in the T-joint was predominantly mode I, a mixed mode I/II stress condition exists at the delamination crack tip due to bending of the skin–stiffener flange section under tensile loading. When a single pin is loaded in mode II the unit cell model analysed its shear deformation as a rod supported by an elastic foundation (i.e. laminate), as shown schematically in Fig. 3. The stiffness of the foundation was determined by the elasticity of the laminate. An elevated compressive stress was generated due to shear deformation of the pin into the laminate near the delamination crack plane, which is known as the pin snubbing effect [13], and this increased the friction stress opposing pin pullout. When the shear stress acting on the pin reached its shear strength, the unit cell model assumes the pin splits into several ligaments and thereby loses its bending rigidity. From this point, it was assumed that the pin can undergo large-scale sliding displacement under the mode II loading; however it is only capable of carrying the axial stress. The laminate around the pin, due to the large shear sliding displacement of the damaged pin, was assumed to behave as a perfectly plastic foundation.
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Fig. 2. Schematic of the multi-level FE modelling approach [11]: (a) unit-cell model for measuring or calculating ‘‘pin force vs. displacement’’ relation, which is then deduced to a ‘‘traction–stress vs. separation–displacement’’ law; (b) structure-level model with the ‘‘traction–separation’’ law implemented at pin locations.
Fig. 3. Schematic of unit cell model of a single pin under mode II interlaminar loading [12].
The traction loads generated by a single pin under modes I and II loading were determined experimentally by Koh et al. [17] using pin pull-out and by Cartié et al. [18] using pin shear tests, respectively. Using the loads, the unit cell model can then be solved to estimate the bridging traction laws for pins with different diameters or embedded lengths within the laminate. The traction loads for a pin when delamination cracks occurred on different planes to the laminate was also analysed by assuming different pin embedded lengths, and therefore the pin traction load was simply a function of the location of the delamination crack plane. The mode II failure stress of the pin was measured to be 1040 MPa by interlaminar shear testing of a single pin [18]. The measured traction load value was high because of the shear fracture mode of the pin, which deformed by firstly rotating towards the interlaminar shear loading direction (i.e. parallel with the delamination crack plane) and then fracturing in transverse shear. The high shear fracture stress of the carbon fibres within the pin is therefore responsible for the high mode II traction stress of the pin. The foundation modulus (KII) of the laminate was assumed to be 5 1013 N/m3, which was measured from the mode II pin traction test performed by Cartié et al. [18]. The test by Cartié and colleagues was performed on a different type carbon–epoxy laminate with a different ply stacking sequence to the material used in the joints, although the pins were the same. The KII value of 5 1013 N/m3 in the FE analysis is taken to be acceptable because the foundation stiffness is determined mainly by the resin-rich
pocket surrounding the pin, which was similar for the laminates tested by Cartié and colleagues and used in the joints. Also, a sensitivity study (not reported in this paper) showed that the mode-II bridging parameters (including KII) have small influence on delamination propagation because the pins fails predominantly in mode I. A mixed-mode failure criterion was used to calculate the fracture toughness for a pin [2]:
Gpin I Gpin IC
þ
Gpin II Gpin IIC
ð1Þ
where Gpin and Gpin I II are the instantaneous traction strain energy values of the pin under crack opening and crack sliding displacements for mode I and II loading, respectively. These values were determined by the area under the traction load–displacement curves at pin a specific displacement value. Gpin IC and GIIC are the total traction strain energy upon pull-out failure of the pin under mode I and II pin loading, respectively. Gpin IC and GIIC were determined by the entire area under the mode I and II traction load–displacement curves, which were measured experimentally [17,18]. Two traction load–displacement laws were used for the laminate and pin. Each law was defined by three parameters: K, T0 and GC for the laminate and K pin , T pin and Gpin for pin bridging; C 0 where K, T and G are the elastic stiffness, peak load and stored strain energy properties (see Fig. 2). K, T0 and GC were based on
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the delamination toughness properties for the unpinned laminate and K pin , T pin and Gpin were calculated using the bridging traction C 0 curve obtained from the unit cell model. The pin traction law specified in the FE model is defined by the interfacial traction properties between the pin and laminate. In reality, the traction law is determined by the pull-out of the pin in the through-thickness direction of the laminate. However, Bianchi and Zhang [11] have shown that the derived interfacial traction law is equivalent to the pin pull-out process in terms of strain energy release rate, and therefore defining the traction laws based on interfacial effects between the pin and laminate is numerically accurate. Further details can be found in [11,12]. Another factor considered in the unit cell model was the compressive stress field that occurs at the pin/laminate interface due to residual stress generated from the elevated temperature curing of the laminate. This compressive stress increases the friction stress generated along the pin/laminate interface during the pullout phase of the pin failure process, and consequently increases the bridging traction load. As summarised in [11,12] the pin bridging law is a function of the materials and geometry, particularly the pin embedded length. Therefore, the pin bridging effect varies depending on which plane delamination is taking place. The model parameters were obtained from the single-pin test [14,15], although the parameter can also be determined using an analytical or numerical model of the single-pin configuration. Once a calculated pin bridging law is validated by test result, the model can be used to calculate bridging laws of other pin configurations, which is then implemented in a structural model at locations where delamination is likely to occur. The bridging traction-separation laws employed are constructed by the following procedure: (1) obtain the single-pin force vs. displacement curve (Fig. 2 insert top left) and divide the force by pin cross-sectional area to obtain a traction stress vs. displacement curve (Fig. 2 insert lower right); (2) find the initial stiffness (K pin ) that is the slope of the straight line tangent to the curve in the origin; (3) cohesive strength (T pin 0 ) equals to the maximum traction stress exerted by the pin; (4) cohesive energy (Gpin C ) equals to the integration area of the stress–displacement curve. 2.3. Macro-scale FE model: unpinned joint A finite element model of the unpinned joint was created using quadratic plane strain elements with reduced integration, as shown in Fig. 4. The model was developed to calibrate the cohesive element properties (stiffness, K and cohesive strength, To) along the delamination crack planes and to validate the cohesive zone model
Table 1 Properties of the laminate, cohesive elements and z-pin in the FE model of the T-joint. Property Laminate (unidirectional carbon fibre/epoxy) In-plane Young’s modulus (E1) Transverse Young’s moduli (E2, E3) Shear moduli (G1, G2, G3) Poisson’s ratio (c12, c13) Poisson’s ratio (c23)
Value a
Cohesive element properties of laminate Mode I traction stiffness of laminate (KI) Mode II traction stiffness of laminate (KII) Mode I failure load of laminate (TI0) Mode II failure load of laminate (TII0) Mode I fracture toughness of laminate (GIC) Mode II fracture of laminate (GIIC)
120 GPa 7.5 GPa 3 GPa 0.32 0.32 5 1013 N/m3 5 1013 N/m3 30 MPa 70 MPa 600 J/m2 1200 J/m2
Cohesive element properties of pin Mode I traction stiffness of pin for a1-type crack (K pin I ) Mode Mode Mode Mode Mode Mode Mode Mode Mode Mode Mode
II traction stiffness of pin for a1-type crack (K pin II ) I traction stiffness of pin for a2-type crack (K pin I ) I traction stiffness of pin for a2-type crack (K pin II ) I traction load of pin for a1-type crack (T pin I0 ) II traction load of pin for a1-type crack (T pin II0 ) I traction load of pin for a2-type crack (T pin I0 ) II traction load of pin for a2-type crack (T pin II0 ) I fracture toughness of pin for a1-type crack (Gpin IC ) II fracture toughness of pin for a1-type crack (Gpin IIC ) I fracture toughness of pin for a2-type crack (Gpin IC ) II fracture toughness of pin for a2-type crack (Gpin IIC )
1 108 N/m3 4.2 1010 N/m3 7.5 1010 N/m3 3.1 1010 N/m3 450 MPa 1040 MPa 255 MPa 580 MPa 400 kJ/m2 210 kJ/m2 100 kJ/m2 170 kJ/m2
a VTM 264 (T700 fibres, HS200 resin by Advanced Composites Group). The laminate properties were measured in-house at RMIT University.
used for analysing crack growth. Mesh sensitivity and element sensitivity analysis was performed, and the mesh shown in Fig. 4 was found to give the best compromise between numerical accuracy and computation time. The orthotropic elastic properties of the cross-ply [0/90]s carbon fibre/epoxy laminate used for the skin, flange and stiffener of the joint are given in Table 1. While z-pinning can reduce slightly (typically under 10%) the in-plane properties of laminates [19,20], it was assumed in the FE model that the pins did not alter the laminate properties. The triangular D-fillet region at the stiffener base was filled with unidirectional laminate with the fibre direction perpendicular to the modelling plane. The triangular region at the taper run-out at the flange ends was assumed to have the mechanical properties of epoxy resin (E = 3 GPa, m = 0.4).
Fig. 4. (a) Unpinned joint model. (b) Location of discrete delamination cracks.
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Fig. 5. Unit-strip FE model of the pinned joint.
Fig. 6. Comparison of the calculated and measured applied force–displacement curves for the unpinned joint. Diagrams on the right-side indicate the onset of vertical cracking along the centre-line of the stiffener at point 1 and horizontal cracking along the skin–flange interface at point 2.
According to experimental observation of the failure of T-joints under tensile loading [5], discrete delamination cracks grow along the interface between the skin and flange (a1 – horizontal path), centre-line of the stiffener (a1 – vertical path), and centre of the web (a2), as illustrated in Fig. 4b. These crack paths were modelled by placing a layer of cohesive elements between the plies. Cohesive elements were used to model the delamination cracks along these paths, with the cohesive interface element size being one-fifth of the adjacent ply element size in order to achieve numerical stability [21]. Chen [23] also reports that cohesive elements need to be much finer than surrounding elements for modelling multidirectional cracks in braided composite T-joint. Chen gives details on obtaining cohesive parameter values and selecting mix-mode failure criterion. In this present paper, cohesive element property values were selected based on the delamination toughness properties of the carbon/epoxy laminate used in the joint (GIC = 0.6 and GIIC = 1.2 kJ/m2). The parameters and their values used for the cohesive fracture analysis are provided in Table 1. The FE simulations were run under displacement-controlled loading by applying a monotonically increasing tensile
displacement as the boundary condition on the nodes at the upper extremity of the stiffener. In order to avoid hourglass deformation of cohesive elements on the symmetry plane, the y-direction displacement of each cohesive element node was constrained to be the same as its corresponding node on the other face of the element (i.e. no shear strain due to symmetry). The ends of the skin were clamped to prevent vertical displacement and rotation of the joint. However, it was found that assuming perfect axial displacement constraint (ideal clamping) or assuming the clamp does not react in the axial direction (sliding clamps) resulted in the joint being either too stiff or too compliant compared to the experimental results (which are presented later). The clamp was therefore assumed to constrain the axial displacement until a maximum axial force. From this point the joint was assumed to slide axially with a constant friction force. This boundary condition was modelled using a non-linear spring element with an initial stiffness of 1 108 N/m and maximum axial force of 9 kN. The initial stiffness and maximum axial force used as the boundary condition was calibrated against measured values determined by experimental testing of the unpinned T-joint.
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2.4. Macro-scale FE model: z-pinned joint A FE model was created for a pinned joint that had the same geometry as the unpinned joint. A unit-strip model was constructed with the pins spaced at regular intervals (1.75 mm) along the skin–flange region, as shown in Fig. 5. Because a unit-strip model was used, it was possible to model the pins as one-half of a single row of pins. This is based on FE modelling results in [11,12]. Due to the pinned joint having a higher ultimate load limit than the unpinned joint, the central interlaminar planes of the stiffener are important for delamination damage progression after crack growth between the skin and stiffener flange. For this reason an extra layer of cohesive elements was added between the plies close to the centre-line of the stiffener (illustrated as crack a2 in 4b). Each sub-laminate was modelled using one layer of continuum shell elements. Periodical boundary conditions were applied to the longitudinal planes that delimit the strip. These boundary conditions consist in a symmetry boundary condition and a constrain to the nodes belonging to the opposite one to remain within the same plane. The pins were modelled using cohesive elements for the pin traction load–displacement law (TSL) determined using the unit cell model. For mode II loading near the delamination crack tip, the non-linear shear deformation of the pins was neglected and a simplified bi-linear TSL was used [12]. Cohesive element properties used for the FE analysis are provided in Table 1. 3. Experimental testing of T-joints T-joint specimens were made of unidirectional T700 carbon/ epoxy prepreg tape (VTM 264 supplied by Advanced Composites Group). The material properties of the carbon/epoxy laminate are provided in Table 1, and it was assumed they were not affected by the pins. The joint geometry and dimensions are shown in Fig. 1, and further information is provided by Koh et al. [5,8]. The plies to the skin, stiffener and flange were stacked in a cross-ply [90/0/90/0/90]S pattern. The thickness of each ply was approximately 0.125 mm. The D-fillet region at the stiffener base was filled with the unidirectional prepreg tape. Before curing, the skin–stiffener flange section to the T-joint was pinned with pultruded rods of unidirectional T300 carbon fibre/bismaleimide (Albany Engineered Composites Pty. Ltd.). The pins were inserted in the orthogonal (through-thickness) direction of the flange section using an ultrasonic hand-held device operated at the frequency of 20 kHz. The pin tip was chamfered to aid insertion through the laminate while the trailing end of the pin was blunt. The chamfered end of the pin was embedded in the flange and the blunt end was in the skin. The entire length of the flange connection was reinforced with pins, whereas the regions outside of this connection were not pinned (as indicated in Fig. 1). Koh et al. [5] give a full description of the z-pinning process used to
53
reinforce the T-joint. The volume content and diameter of the pins was 2% and 0.28 mm, respectively. The pins were arranged in a square grid pattern aligned along the length and across the width of the flange section to the joint. The spacing between the pins both along and across the joint was 1.75 mm. Control T-joint specimens (without pins) were also made with the same geometry as the pinned specimens. Both the unpinned and pinned joint specimens were cured in an autoclave at 120 °C and 620 kPa for 1 h. The joints were bonded by co-curing without the use of adhesive. The average volume content of carbon fibres in the skin, flange and stiffener of the joints was about 60%. A stiffener pull-off test was performed on the T-joint specimens to validate the FE model. A tensile load was applied to the stiffener using a 50 kN Instron machine at a constant displacement rate of 1 mm/min until final failure. The ends of the skin were clamped to a rigid support plate, although some slippage occurred during testing which was accounted for by the FE model. At least five specimens of the unpinned and pinned joints were tested under identical conditions to validate the FE model. 4. Results and discussion 4.1. Unpinned T-joint Fig. 6 shows the applied force-vertical displacement curves for the unpinned joint that were calculated using the FE model and measured using the stiffener pull-off test. The measured result was first reported by Koh et al. [5]. There was some variability in the measured curves based on repetitions of the stiffener pull-off test performed on five samples of the unpinned joint. The standard deviation to the ultimate load was about 15% of the average value; the average value for the ultimate load was 1656 N and the standard deviation was 252 N based on repeat tests performed by Koh et al. on five joint specimens [5]. Fig. 6 shows there is relatively good agreement between the calculated and measured curves: the FE model accurately predicted the non-linear rise in the stiffness and the peak load of the unpinned joint, after which it was predicted that the load capacity would drop abruptly due to splitting cracking along the centre-line of the stiffener (at point 1 in Fig. 6) followed by delamination cracking along the skin–flange interface (at point 2). These two fracture modes were observed in the unpinned test specimens, as shown in Fig. 7. Only a slight difference in stiffness between the model and experiment can be noticed before the load drop. This is probably due to the clamp boundary condition in the model, which overestimates the clamping axial force, thus predicting stiffer response at large deformations. Fig. 8 shows the normal (y-direction) stress contour map for the unpinned joint calculated using the FE model. The numerical analysis showed that crack initiation occurred when the normal tensile stress exceeded the peel strength of the cohesive elements in the FE model. It was determined that the splitting crack along the
applied load
stiffener crack
skin/stiffener flange crack Fig. 7. Fractured unpinned joint showing splitting cracking along the centre-line of the stiffener and delamination cracking along the skin–flange interface.
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F. Bianchi et al. / Composites Science and Technology 73 (2012) 48–56
applied load
splitting crack
(a) applied load
delamination
(b) Fig. 8. FE stress contour map of the normal stress in the y-direction of the unpinned joint (the coordinate system follows laminate orientation).
centre-line of the stiffener was due to a pure mode I stress, which at the crack tip was about 30 MPa. The delamination crack at the skin–flange interface propagated under a mixed mode I/II stress condition, and therefore the peel stress was calculated to be slightly lower at about 18 MPa. 4.2. Z-pinned T-joint Fig. 9 compares the calculated and measured applied load–displacement curves for the pinned joint, and again there is excellent agreement. The measured curves are taken from Koh et al. [5], who determined the average and standard deviation in the ultimate load of the pinned joint to be 2777 N and 683 N, respectively, based on five test measurements. The stiffness of the pinned joint was calculated using the FE model to be the same as the unpinned joint, and this was confirmed by experimental testing. Pins do not significantly alter the in-plane tensile modulus of carbon/epoxy laminate [19,20,22], and therefore the stiffness of the joint was not changed by pinning. The pinned joint experienced an initial load drop (at d = 4 mm), and the FE model determined that it was caused by the initiation from the D-fillet region of a splitting crack
Fig. 10. Pinned joint showing (a) splitting cracking along the stiffener centre-line at the initial load drop and (b) delamination cracking along the skin–flange interface at the second (ultimate) load drop.
along the centre-line of the stiffener. This was immediately followed with the initiation of a delamination crack along the skin–flange interface. Again, this was confirmed by experimental testing with both stiffener splitting and skin–flange delamination cracking spreading from the D-fillet region of the pinned joint specimen following the initial load drop (Fig. 10). The FE model predicted that the pinned joint does not fail catastrophically at the initial load drop point (unlike the unpinned joint) due to bridging traction loads generated by the pins along the delamination crack between the skin and flange. The pin traction loads caused a recovery in strength and consequently the pinned joint was able to withstand further loading up to the ultimate load limit of about 3800 N, which was over twice as high as the unpinned joint. The FE model determined that the second (and much larger load drop at d = 10.7 mm) was caused by the formation of a second crack within the web region (a2), and this was also observed experimentally. Between the initial and second load drop points, the FE model predicted that an increasing number of pins generated traction loads as the delamination crack grew in length along the skin– flange interface, and this was the cause for the progressive increase
Fig. 9. (a) FEA and measured applied force vs. displacement curves for the pinned joint. (b) FEA of the crack length vs. applied displacement and number of active pin rows along the skin–stiffener flange delamination crack.
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Fig. 11. Pin bridging stresses immediately after the first load drop, applied displacement d = 6 mm. (a) Normal stress in z-direction (through-thickness). (b) Transverse shear stress. The stresses represent the bridging traction force per pin unit area (unit: MPa).
Fig. 12. Pin bridging stresses at ultimate load, applied displacement d = 10.67 mm. (a) Normal stress in z-direction (through-thickness). (b) Transverse shear stress. The stresses represent the bridging traction force per pin unit area (unit: MPa).
in joint strength between the two points, as shown in Fig. 9b. The spacing between the pin rows along the skin–stiffener interface was 1.75 mm, and it was calculated that seven rows of pins were required to create a fully established bridging traction zone along the delamination crack. The FE analysis revealed that once the delamination crack length exceeded the pin bridging traction zone length of about 12.25 mm, the last row of pins in the crack wake failed by pull-out from the skin as a new row of pins began to generate traction loads near the crack tip. Again, this was confirmed by experimental testing with the pin bridging zone along the skin– flange interface being 10–15 mm long. The FE analysis assumed that all the pins were pulled from the flange and retained in the skin, which was found for the majority of pins in the experimental tests. The pins were preferentially pulled from the flange because of their chamfered tip which reduced the bonded region with the laminate. If, however, the pins were pulled out from the skin and retained in the flange then the a2 crack is effectively an unpinned interface with low toughness, and this could affect the ultimate load of the pinned joint. FE analysis was performed to calculate the pin traction loads along the skin–flange delamination crack at two points: (a) when the pins first formed a fully-developed bridging zone which occurred at d = 6 mm (Fig. 11) and (b) at the maximum load point which occurred at d = 10.7 mm (Fig. 12). The FE model computed both the normal tensile and transverse shear traction stresses for each row of pins along the delamination crack. (The stresses
represent the traction loads per pin unit area). The analysis showed that the traction stresses were non-uniformly distributed between the pin rows along the crack. Both the tensile and shear traction stresses increased with distance behind the crack tip due to increasing crack opening/sliding displacement and reached a maximum at about one-half along the crack length. The traction stresses then decreased towards the rear of the bridging zone due to the reduced friction stress generated by the pins when they were nearly pulled out from the laminate. Fig. 11 also shows that the normal traction stresses generated along the upper (a2) crack were lower than those generated along the skin–flange interface crack (a1). This was due to the smaller opening displacement in the a2 crack and the shorter embedded pin length which induced a lower shear load along the pin/laminate interface. However, the transverse shear stresses were higher along the a2 crack due to the higher shear sliding displacement compared to the a1 crack. This analysis revealed the complexity of the bridging traction loads generated in pinned joints which increased the strength and toughness properties.
5. Conclusions A validated FE model has been developed which can analyse the structural properties and fracture behaviour of pinned T-shaped composite joints when subjected to tensile (stiffener pull-off) loading up to the maximum load point. The model is demonstrated
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capable of predicting the rising portion of the force–displacement curve, which corresponds to damage initiation and propagation up to the ultimate load of the joint. The FE model analyses the crack bridging traction loads of a single pin under modes I and II interlaminar stresses at the unit cell level, and this analysis is then used to calculate the strength and delamination fracture properties of pinned joints at the structural level. The FE model provides important insights into the strengthening and toughening of pinned joints, such as failure initiation being determined by mode I splitting cracking along the vertical stiffener and the ultimate strength being controlled by the modes I/II pin traction loads, which are unevenly distributed along the bridging zone between the skin and flange. Splitting of the vertical stiffener can be avoided by z-pinning, although it reduces the ultimate failure load of the Tjoint [5]. The FE model also revealed that both the traction loads generated within the web region (a2-type crack) and along the skin–flange interface (a1 crack) are important in the strengthening and toughening of pinned joints. The FE model computed that the traction loads are mainly mode II shear stresses in the a2 crack and mode I tensile stresses in the a1 cracks. Despite the complexities of the internal stress distribution, pin traction laws and crack growth behaviour for the pinned joint, the FE model accurately calculated the stiffness, strength and fracture modes when compared to experimental test results. This validation study demonstrates that the FE model can be used in the design of pinned T-joints when subjected to tensile loading. References [1] Cartié DDR, Dell’ Anno G, Poulin E, Partridge IK. 3D reinforcement of stiffenerto-skin T-joints by z-pinning and tufting. Eng Fract Mech 2006;73:2532–40. [2] Rugg KL, Cox BN, Massabò R. Mixed mode delamination of polymer composite laminates reinforced through the thickness by z-fibres. Composites 2002;33:177–90. [3] Chang P, Mouritz AP, Cox BN. Tensile properties and failure mechanisms of zpinned composite lap joints. Compos Sci Technol 2006;66:2163–76. [4] Byrd LW, Birman V. Effectiveness of z-pins in preventing delamination of cocured composite joints on the example of a double cantilever test. Composites 2006;37B:365–78.
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