Materials and Structures DOI 10.1617/s11527-008-9422-6
ORIGINAL ARTICLE
Geometrical and mechanical aspects of fabric bonding and pullout in cement composites C. Soranakom Æ B. Mobasher
Received: 22 February 2008 / Accepted: 21 August 2008 RILEM 2008
Abstract Fabric reinforced cement composites are a new class of cementitious materials with enhanced tensile strength and ductility. The reinforcing mechanisms of 2-D fabric structures in cement matrix are studied using a fabric pullout model based on nonlinear finite difference method. Three main aspects of the composite are evaluated: nonlinear bond slip characteristic at interface; slack in longitudinal warp yarns, and mechanical anchorage provided by cross yarn junctions. Parametric studies of these key parameters indicate that an increase in the interfacial bond strength directly increases the pullout strength. Grid structures offering mechanical anchorage at cross yarn junctions can substantially increase the pullout resistance. Presence of slack in the yarn geometry causes an apparently weaker and more compliant pullout response. The model was calibrated using a variety of test data on the experimental pullout response of AR-Glass specimens, manufactured by different techniques to investigate the relative force contribution from bond at interface and from cross yarn junctions of alkaline resistant glass fabric reinforced cement composites.
C. Soranakom B. Mobasher (&) Department of Civil & Environmental Engineering, Arizona State University, Tempe, AZ, USA e-mail:
[email protected] C. Soranakom e-mail:
[email protected]
Keywords Fiber pullout Textile reinforced concrete Fabric reinforced cement composites Bond properties Debonding Pullout
1 Introduction There is a growing interest in the use of fabrics as the main reinforcement in cement based composites. This new class of materials exhibits superior tensile strength and ductility in comparison to other discrete fiber reinforced composites such as glass fiber reinforced concrete (GFRC) and engineering cementitious composite (ECC) [1]. The enhanced behavior of fabric reinforced cement composites is primarily governed by interfacial bond characteristics between fabric and matrix. It has been well known that fiber-matrix interface plays an important role in controlling the mechanical performance of cementitious composites. Bond mechanisms of straight, hooked, and crimped fibers have been well studied by experimental and analytical techniques [2–6]. Due to their 2- and 3-D nature, fabrics show a significant improvement over 1-D straight or deformed fibers in the development of mechanical bond. This is partly due to the anchorage of the longitudinal (warp) yarns to the transverse (fill) yarns. The scanning electron microscopic (SEM) pictures of different types of fabrics structure shown in Fig. 1a–c are used to explain the mechanical and geometrical interlock effects observed in these composites. Figure 1a presents the mechanical anchorage
Materials and Structures
Fig. 1 Scanning electron microscope (SEM) pictures of different fabric reinforced cement composites; (a) nonlinear geometry of longitudinal yarns and cross yarn junction; (b) tortuous crack propagation; (c) densification
resistance due to curvature in the geometry of longitudinal yarns. The interlocking effect results in additional resistance offered by the transverse yarns at the cross yarn junctions of the woven fabric. Figure 1b demonstrates the interaction of interfacial debonding with a propagating matrix crack that result in crack deflection and induces tortuosity in the path of crack propagation. This mechanism leads to crack branching, bifurcation, and multiple cracking. These mechanisms increase the fracture energy and lead to enhanced strength and ductility. In addition to the geometrical effect, the effectiveness of fabric embedment in the matrix is also important. Processing methods such as pultrusion technique with a matrix modified using flyash are quite effective means of increasing matrix infiltration in the interstitial spaces of individual fibers, in between parallel yarns, and furthermore in the vicinity of junction point (Fig. 1c). The paste infiltration is affected by the rheology of the mixture and the dense structure can significantly reduce the potential of debonding at the interface zone. Ultimately, a better bond increases the ability of cross yarn junctions to carry the load, leading to the improved pullout resistance. The pullout load versus slip response of various fabric structures (woven, warp knitted and bonded) were characterized earlier [7]. Results show resemblance to the pullout response of a straight fiber [8– 10], however, the bond strength parameters are quite high due to the geometrical anchorage mechanisms offered by the transverse yarns. Figure 2 compares pullout responses of a single yarn when embedded as a straight yarn and as a fabric. Note that the response with a fabric configuration is significantly higher. This is attributed to the interlock mechanism of transverse yarns which help redistribute the applied forces to the supporting matrix. Pullout models derived for a straight fiber have often been used to
Fig. 2 Pullout load slip response of Polypropylene embedded as a straight yarn compared to a woven fabric
characterize and obtain equivalent bond properties of fabric reinforced cement. Sueki et al. [11] modified Naaman’s pullout model [4] to analyze the pull-out test results and quantify the equivalent bond properties of various fabric types, mixtures, processing methods and embedded lengths. Zastrau et al. [8] proposed a triple linear stress-slip relation for modeling of textile reinforced concrete. Banholzer [12] proposed an N-piecewise linear bond law that can describe nonlinear bond-slip characteristics of various fiber-matrix interface encountered in modeling. In addition to analytical approaches, an alternative numerical framework such as finite element method using discrete bond model [13] has also been employed in characterization of bond between fabric and mortar. Fabric reinforced cement composites have three main characteristics differentiating them from
Materials and Structures
unidirectional monofilament reinforced composites. The first is the inhomogeneous nature of the bond between the filaments within the outer perimeter of a yarn and surrounding matrix. The relative proportion of the bonded outer fibers and the unbonded inner fibers within a yarn affects the overall load transfer mechanisms. Mix designs that introduce flyash and/or silica fume to densify the microstructure result in higher bond strength between yarn and matrix [14]. Pultrusion technique applies normal and shear stress to impregnate fabrics during manufacturing process and reduces porosity in matrix, leading to better bonding at interface. The second factor is the slack in longitudinal yarns which is due to the initial curvature of the yarns and commonly observed in hand-lay up manufacturing when there is no pre-stretching of the fabric during casting. The uncrimping of individual filaments as they are loaded results in a low initial stiffness. The third aspect is the mechanical anchorage provided at the intersection of longitudinal and transverse yarns, which may have various forms of cross-yarn junctions such as woven, bonded, and knitted. Each type may have different load transfer mechanism from longitudinal yarn to the junction and then to the surrounding matrix. The restraining force offered at these junctions can be simplified to a nonlinear spring model.
Fig. 3 Schematic drawing of fabric pullout mechanism; (a) interface bond slip model; (b) longitudinal yarn model; (c) spring model simulating anchorage strength at cross yarn junction; (d) fabric pullout specimen
represents the additional stiffness g provided at cross yarn junction. This feature is only active at the junction points, representing the geometrical feature of the fabric and depending on the weave density. With similarity of using multi-linear-segments to describe material models and using secant modulus to control material constitutive behaviors (Fig. 3a–c), all three material models can be implemented as piecewise linear functions in the finite difference model.
2 Free form nonlinear material models
3 Finite difference fabric pullout model
In order to develop a flexible and unified material law, the interfacial bond constitutive law, the tensile stress strain response of the longitudinal yarn, and the restraint at the cross junction of longitudinal and transverse yarns are described as piecewise linear functions and shown in Fig. 3. Figure 3a represents nonlinear bond slip model of the pre and post peak response of the shear stress s as a function of slip s. The secant modulus k enforces the material constitutive behavior of local shear stress and slip at each node of the finite difference model. Figure 3b presents a typical longitudinal yarn stress strain model that may begin with initial slack. This simulates the delay in load-deformation response of the yarn. Similar to the bond slip model, secant modulus Ey is used to ensure stress strain relationship of the longitudinal yarn in a finite difference model. Finally, a nonlinear spring model as shown in Fig. 3c
Due to the complicated nature of the nonlinear fibermatrix interface and competing failure mechanisms, analytical solutions may be tedious or difficult to obtain. Finite difference method presents an alternative numerical framework that treats mechanical aspects into individual equations and assembles them in order to represent a numerical pullout model. Nodal equilibrium equations can be obtained from free body diagram of typical nodes. Material constitutive behaviors are ensured by a material stiffness updating algorithm. 3.1 Finite difference equilibrium equation In formulating the nodal equilibrium equations, a variable slip s is defined as the relative difference between the elongation of the longitudinal yarn and cement matrix.
Materials and Structures
s¼
Zxiþ1
ey em dx
ð1Þ
xi
where ey and em are yarn and matrix strain, respectively, and dx is a finite length between two consecutive nodes (i and i ? 1) in the longitudinal x-direction. For fabric reinforced cement composites with a relatively low fiber volume fraction (\4%), the axial stiffness of the yarn AyEy is significantly lower than the axial stiffness of the matrix AmEm; thus, the effect of matrix strain can be ignored and the slip is simplified to: s¼
Zxiþ1 ey dx
and
s0 ¼
ds ¼ ey dx
ð2Þ
condition, implying that the yarn strain or derivative of slip vanishes. At the right end, node n, the nodal slip is prescribed incrementally, simulating displacement control. As the loading progresses, the part of the longitudinal yarn that slips out of the matrix has no frictional bond resistance; thus, yarn elongation is the only term in that section. The extruding part can be easily implemented by checking the amount of slip versus the embedded length of each node. If the slip is greater than the embedded length, zero bond stress is applied to that node. For n number of nodes used in the finite difference model, only three typical nodes labeled ‘‘A’’, ‘‘B’’ and ‘‘C’’ are necessary to derive nodal equilibrium equations; other nodes are replicates of these nodes.
xi
Figure 4a shows a finite difference model of a fabric pullout specimen in Fig. 3d. The embedded length L is discretized into ‘‘n’’ nodes with equal spacing of h. The bond stress is assumed constant over the small spacing h for each node within each linear domain. Two types of nodes are used; nodes along the length of the yarn which receive contribution from the interface bond forces, and nodes at each junction point which contain an additional spring due to the transverse yarns. Two boundary conditions are imposed. At the left end, node 1, force in longitudinal yarn is imposed to be zero, simulating stress free
3.1.1 Boundary condition at the left end Figure 4b shows a free body diagram of a force boundary condition at the typical node ‘‘A’’. The force in longitudinal yarn at distance ?h/2 from the first node 1, is defined as: F1?1/2 and must be balanced with the bond resistance force over the first node 1, B1. F1þ12 B1 ¼ 0
ð3Þ
F1?1/2 is calculated by using the yarn area Ay, average secant modulus at node 1 and 2 and the yarn strain at node 1, which is approximated by forward difference method: Ey;1 þ Ey;2 ðs2 s1 Þ F1þ12 ¼ Ay ð4Þ h 2 Bond stress at node 1 is determined by the secant modulus k1 and slip s1. The resistance bond force B1 is then obtained by integrating the bond stress on the surface area of the yarn defined by the perimeter w over a half spacing h/2. 1 B1 ¼ wk1 s1 h 2
Fig. 4 Finite difference fabric pullout model; (a) discretized fabric pullout model under displacement control; (b–d) free body diagram of three typical nodes ‘‘A’’, ‘‘B’’ and ‘‘C’’
ð5Þ
By substituting Eqs. 4 and 5 into Eq. 3 and rearranging terms, the equilibrium of force for the typical node ‘‘A’’ can be expressed as: Ey;1 þ Ey;2 Ey;1 þ Ey;2 1 2 Ay þ wk1 h s1 þ Ay s2 2 2 2 ¼0
ð6Þ
Materials and Structures
3.1.2 Interior nodes Figure 4c and d show two typical nodes ‘‘B’’ and ‘‘C’’ which describe two possible types of interior node i; those connected and those not connected to the cross yarn junctions. The force equilibrium for these two types of nodes can be represented by the following equations: Fiþ12 Fi12 Bi Gi ¼ 0
ð7Þ
Fiþ12 Fi12 Bi ¼ 0
ð8Þ
The forces at the half right (?h/2) and half left –h/ 2) of the node i are calculated by: Ey;i þ Ey;iþ1 ðsiþ1 si Þ ð9Þ Fiþ12 ¼ Ay h 2 Ey;i1 þ Ey;i ðsi si1 Þ Fi12 ¼ Ay ð10Þ h 2 The bond force at node i is calculated from the distributed bond stress over a spacing of h: Bi ¼ wki si h
ð11Þ
The spring force at node i can be expressed as a function of slip si and the secant modulus of the material model gi. G i ¼ gi s i
ð12Þ
By substituting Eqs. 9–12 in Eqs. 7 and 8, two equilibrium equations for interior nodes with and without spring are obtained: Ey;i1 þ Ey;i si1 Ay 2 Ey;i1 þ 2Ey;i þ Ey;iþ1 2 þ wki h þ gi h si Ay 2 Ey;i þ Ey;iþ1 siþ1 ¼ 0 þ Ay ð13Þ 2 Ey;i1 þ Ey;i si1 Ay 2 Ey;i1 þ 2Ey;i þ Ey;iþ1 þ wki h2 si Ay 2 Ey;i þ Ey;iþ1 siþ1 ¼ 0 þ Ay ð14Þ 2 3.1.3 Boundary condition at the right end In simulation of displacement control, the slip s* at the right end is incrementally imposed for each
load step until it reaches a prescribed value. Since this slip is known in advanced, no equilibrium equation for the end node n is required; but, it will be implemented in the equilibrium equation of node n – 1. This can be accomplished by replacing the node index i – 1, i, i ? 1 with n – 2, n – 1, n and the slip sn with s* in Eqs. 13 or 14. The next step is to rearrange the unknown terms to the left and move the known terms to the right as the driving force. This yields two equations for the node n – 1 that is connected and not connected to spring:
Ey;n2 þ Ey;n1 Ay sn2 2 Ey;n2 þ 2Ey;n1 þ Ey;n Ay 2 Ey;n1 þ Ey;n 2 s þ wkn1 h þ gn1 h sn1 ¼ Ay 2 ð15Þ
Ey;n2 þ Ey;n1 sn2 Ay 2 Ey;n2 þ 2Ey;n1 þ Ey;n þ wkn1 h2 sn1 Ay 2 Ey;n1 þ Ey;n s ¼ Ay ð16Þ 2
3.2 Matrix assemblage The 1-D model is obtained by discritizing an embedded length L into n nodes and applying appropriate nodal equilibrium equations expressed in Eqs. 6, 13–16 for each type of node from node 1 to n – 1 and the assemblage of the nodal equations in the following form: ½M ðn1Þ;ðn1Þ fSgðn1Þ ¼ fPgðn1Þ
ð17Þ
where [M] is a coefficient matrix containing coefficient terms. {S} = {s1, s2, s3,…, sn–1} is an unknown nodal slip vector and {P} = {0,0,0,…–Ay(Ey,n–1 ? Ey,n)s*/2} is the known driving force vector. The coefficient matrix [M] is a tridiagonal matrix containing a bandwidth of three non zero terms; thus, this system of equation can be solved efficiently by Thomas’s algorithm [15].
Materials and Structures
3.3 Material stiffness updating and convergence The algorithm for updating material stiffness and checking for convergence are presented as a flow chart in Fig. 5. For each imposed slip, the system of equations is solved for slip values at all nodal points. According to the Eq. 2, the yarn strain at each node can be computed using forward, central, and backward finite difference for node 1, 2 to n – 1 and n, respectively. Then the slip and yarn strain are used to calculate the distribution of bond stress, yarn tensile stress, and spring forces according to their material models. Next, the secant stiffness k, Ey and g are calculated by dividing the current forces with deformations. In order to meet the convergence criterion, all updated secant stiffness values must be sufficiently close to the previous iteration. To calculate the relative change of secant stiffness between the two iterations, the steepest positive slopes of material models were used as a reference secant modulus kref, Eyref and gref. If all the relative changes are less than a specified tolerance level at all nodes, the constitutive laws are satisfied for the whole domain. The next slip
Fig. 5 An algorithm for updating material stiffness and checking for convergence of local force deformation at each node for each imposed slip increment
increment can be imposed and the process continues until the total slip reaches the specified final value. If the secant modulus at some nodes does not converge, the next secant modulus k(j?1), E(j?1) and g(j?1) will y be estimated from the previous values k(j–1), E(j?1) y (j) and g(j–1) and the current value k(j), E(j) y and g by the weighting factors nk, nEy and ng, which vary between 0 and 1. Small weighting factor gradually change the next secant modulus, leading to stable process but slower convergence while the larger values lead to faster convergence but less stable. For highly nonlinear problems, the rapid changes of modulus between two iterations may result in no convergence at all. The default values for weighting factor between 0.2 and 0.3 were found to be efficient for most moderate to severe nonlinear problems.
4 Parametric studies Parametric studies are conducted to address three main characteristics of fabric reinforced cement composites: bond slip relationship, slack in a yarn, and mechanical anchorage provided at the cross yarn junctions. To demonstrate these effects, benchmark mechanical and geometrical parameters are assumed as follows: the Young’s modulus of the yarn of 5000 MPa, the longitudinal yarn diameter of 0.5 mm, an embedded length of 12 mm. For the case study of the cross yarn junctions, a weave density of one yarn per 6 mm were assumed; thus, two springs are placed symmetrically at nodal locations 3 and 9 mm from the left end. Figure 6a–c shows three material models and Table 1 shows different combinations of material models used in parametric studies. The first row shows a base numerical model, which consists of a base bond model (smax_base = 2 MPa) and a yarn stress strain model without slack. Subsequent models below show other combinations, in which the base model is modified; the bond strength may be increased; the yarn model may be added with different degree of slack; or the longitudinal yarn is connected to springs at different degree of restraining. In brief notations, ‘‘base’’ refers to a basic material model needed to be included in a base numerical model while ‘‘opt’’ refers to optional parameter or material model that can be added to the base numerical model. The magnitude of base/opt models can be amplified by a scaling factor.
Materials and Structures
Fig. 6 Material constitutive laws; (a) bond stress slip model; (b) longitudinal yarn stress strain model; (c) spring force slip model
Table 1 Numerical models for parametric study and the results of simulations
Input material models
Predicted responses Yarn model eslack Spring model (mm/mm) SFmax (N)
Slip at max. load (mm)
Max. load (N)
–
0.35
35.5
–
–
0.56
44.5
–
–
0.55
53.1
Model name
Bond model smax (MPa)
Base model
2.0
–
1.25 9 smax_base 2.5 1.50 9 smax_base 3.0 ?1.0 9 eslack_opt 2.0
?0.050
–
0.75
34.1
?1.5 9 eslack_opt 2.0
?0.075
–
1.00
32.2
?2.0 9 eslack_opt 2.0
?0.100
–
1.25
30.0
?1 9 SFmax_opt 2.0
–
?5.0
0.65
43.9
?2 9 SFmax_opt 2.0
–
?10.0
0.65
52.1
?3 9 SFmax_opt 2.0
–
?15.0
0.70
59.6
4.1 Effect of slack and cross yarn junction to the pullout responses The first set of parametric study compares the pullout responses of the base numerical model with its two modified models: including slack (?1 9 eslack_opt) and including springs (?1 9 SFmax_opt). Figure 7 shows the distribution of slip, bond stress and yarn force along the length of the fabric. Figure 7a and b present these responses at two stages: the slip at the right end reaches 0.5 and 1.0 mm. These two values correspond to the two controlling points, peak and post peak specified in the base bond model (smax_base) (Fig. 6a). At the bottom of Fig. 7a, the slips at the right ends of the three models are all equal to 0.5 mm as prescribed in the displacement control. When compared to the base model, it can be seen that the effect of slack in longitudinal yarn significantly decreases
the slip distribution while the spring slightly decreases. The above subplot shows the corresponding bond stress. At the right end, bond stresses on the extruding parts of distance 0.5 mm are zero. The next parts to the left which slips vary between 0.1 and 0.5 mm have bond stresses vary in narrow range of 1.9 and 2.0 MPa as specified in the base bond model. The effect of slack considerably decreases the bond stress in the left half as the slip is very low while the effect of spring is negligible. The top subplot shows the yarn force distribution. It is noted that the effect of slack decreases the force distribution especially in the left half due to lower bond stress. The effect of spring is observed at two nodal spring locations 3 and 6 mm, which add more resistant force in addition to the bond stress, resulting in higher pullout force at the right end. At the bottom of Fig. 7b, the slip at the right end reaches 1.0 mm in the post peak of base bond slip
Materials and Structures Fig. 7 Distribution of slip, bond stress and yarn force along the embedded length; (a) and (b) imposed slip at the right end reaches 0.5 mm and 1.0 mm
model. The trend of slip distribution remains the same such that a variation in the slack yields the least slip while the spring yields results much closer to those of the base model. In the middle plot, it reveals that the base model and the modified model including springs are very close to one another but significantly lower than the modified model including slack. This due to the slip distributions of the first two are in the post peak region (slip [ 0.5 mm) while the slip of last model vary in pre and post peak (0.3 mm \ slip \ 1 mm). The top subplot shows that the base model has the lowest yarn force distribution due to low bond stress while the modified model with springs has higher force due to additional
restraint forces from the springs. The modified model with slack yields the highest force; however, this number is still lower than the other two models measured at slip of 0.5 mm (Fig. 7a). 4.2 The effect of bond strength, degree of slack and transverse yarn strength The second set of parametric study is shown in Fig. 8, which demonstrates the effect of bond strength, degree of slack in longitudinal yarn and junction strength to the pullout load–slip response. Material models at three levels of scaling factors previously shown in Fig. 6a–c are used to create numerical
Fig. 8 Parametric studies of material models to pullout load slip response; (a) the effect of bond strength; (b) the effect of slack; (c) the effect of spring strength
Materials and Structures
models for this study and the details are presented in Table 1. It can be seen from Fig. 8a that the increase in bond strength from the base level by 25% and 50% results in proportional increase of peak load (35.5/ 35.5 = 1.00, 44.5/35.5 = 1.25 and 53.1/35.5 = 1.50). Furthermore, it is noted that the shapes of the pullout slip responses resemble to the shape of input bond slip models. Figure 8b shows the effect of slack in longitudinal yarn in lowering the apparent stiffness and ultimate capacity as compared to the base model. Figure 8c confirms the study in previous section that the presence of cross yarn junctions increases the pullout resistant. By increasing the strength of springs, it can be seen that the ultimate load increases but at a decreasing rate (35.5/35.5 = 1.00, 43.9/ 35.5 = 1.24 and 52.1/35.5 = 1.47 and 59.6/35.5 = 1.68). This is due to the fact that bond and spring contribute to the pullout response independently according to slip level in the material models.
5 Comparison with other fiber pullout models To further verify the process, the finite difference fabric pullout model was compared to an analytical fiber pullout model using N-piecewise linear bond law proposed by Banholzer [12]. Experimental pull out results of a steel rebar from a concrete cylinder was used. The description of the bond model and other parameters for simulation are shown in Fig. 9a; the embedded length, L = 40 mm; rebar diameter, d = 16 mm; Young’s modulus of steel, Es = 210 GPa; Young’s modulus of concrete matrix, Em = 35 GPa, and its cross section area Am = 10,000 mm2. Fig. 9 Comparison of pullout responses of steel rebar in normal strength concrete between Npiecewise linear bond law model and the proposed model; (a) bond stress slip models for both models; (b) the predicted pullout response
In the case of steel rod, slack and spring were not used in the finite difference model. It should be noted that the analytical model considers the concrete strain while the proposed numerical model ignores this parameter in the simulation. Figure 9b shows that the predicted response of the analytical model using input bond parameters obtained from material calibration tests agrees very well with the experimental results. The agreement of finite difference model (FDM) with the analytical model and experimental results is quite well up to the peak, however, the FDM approach is slightly less stiff in the post peak response. This is due to the difference in assumptions between the two methods. In derivation of analytical solution, the protruding fiber from the matrix is still assumed to have bond stress as described by the bond slip relation while the finite difference model assumes this part with zero bond stress. A decrease in bond stress distribution yields smaller post peak forces which become more evident when the protruding section becomes longer. Note that in the present case, the axial stiffness ratio between matrix and fabric (AmEm/AfEf = 350/42.2) is relatively high at 8.3, therefore the simplified assumption in the proposed model that neglects the matrix strain still yields a reliable prediction of the pullout response.
6 Case study of AR-Glass fabric reinforced cement The fabric pullout was conducted using the test setup shown in Fig. 10a by simultaneous pulling of 8 yarns (four pairs) out of a cement paste block. The
Materials and Structures
during casting) and vacuum (negative pressure deairing prior to casting) were used. For all processing methods, the fabrics were pre-stretched to eliminate slack in the longitudinal direction. The details of sample preparations are provided in the earlier work [7]. Labels GC, GP and GV represent using control mix in cast, pultrusion, and vacuum processes while GF represents using flyash mix in cast process.
Fig. 10 Fabric pullout test; (a) test setup; (b) geometry of ARglass fabric
specimen was 8.1 mm thick, 25.4 mm wide and 12.7 mm long, which is the same as embedded length. Figure 10b shows the geometry of the AR glass fabric that contains five pairs of yarn per inch in each direction. Each yarn consists of 400 glass filaments, 13 lm in diameter each, with a Young’s modulus of 78,600 MPa. The filaments were impregnated with a polymeric sizing and bundled to form an equivalent diameter of 0.27 mm in longitudinal direction. Two individual yarns with a relatively flat shape and width of 1.778 mm are used in the transverse direction. The experimentally obtained averaged modulus of the fabric in longitudinal direction was 58,605 MPa, which was less than theoretical modulus of individual glass filaments making up the yarn; this value was used in the simulation. Four sets of AR-glass samples were prepared from two types of cement pastes: control (100% Portland cement) and flyash mix (replacing cement with 40% by volume of class F flyash) [11]. Three types of manufacturing methods: cast (no shear stress during casting), pultrusion (applied shear stress Fig. 11 Simulation of fabric pullout using equivalent single fiber pullout model; (a) best fit equivalent bond slip parameters; (b) simulations of the experimental results
6.1 Numerical simulation of fabric pullout response Four sets of samples (GC, GP, GV and GF) have previously been analyzed [11] using a single fiber pullout model to obtain equivalent bond slip parameters as shown in Fig. 11a and Table 2. In this case, the resistance provided by transverse yarn was treated as smeared bond stress addition to the bond stress at matrix-longitudinal yarn interface, the obtained bond values from this calibration were higher than that would obtain from straight yarn embedded in cement matrix. The predictions of pullout responses using the calibrated parameters are shown in Fig. 11b. It was observed that due to shear lag, the apparent stiffness of a yarn inside matrix was much lower than the stiffness obtained from testing of plain fabric; thus, the efficiency factor g, which accounts for the inefficiency of individual filament to contribute to the axial stiffness of a yarn (gAyEy), were introduced in the material calibration process. The parameter g obtained for sample GC, GP, GV and FG were 5.0%, 5.7%, 9.5% and 4.0%. Reanalysis of the same data using the proposed fabric pullout model allows for better characterization
Materials and Structures Table 2 Equivalent bond slip parameters for a smooth fiber pullout model
Glass cast (GC) s (mm)
s (MPa)
Glass vacuum (GV) s (mm)
s (mm)
s (MPa)
Glass flyash (GF) s (mm)
s (MPa)
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.56
3.41
0.48
3.96
0.46
2.96
0.75
5.58
0.56 0.89
1.24 1.24
0.48 0.65
2.07 2.07
0.46 0.71
1.14 1.14
0.75 1.71
1.93 1.93
0.89
1.59
12.7
1.59
0.65 12.7
of the response and computation of the properties of the cross yarn junctions. All samples had the same embedded length of 12.7 mm with a 5.08 mm spacing between two lines of transverse yarns, each intersects 8 longitudinal yarns, creating 8 junctions. In material calibration process, it was assumed that two springs, each has equivalent strength of 8 junctions, were placed symmetrically, in which the location of the first and second were at 3.81 and 8.89 mm, respectively. To obtain material parameters for the spring model, a plain AR fabric was tested by pulling a longitudinal yarn until the junction failure. The average failure load of a junction point was 9.0 N. Since cement paste at the junctions provides high fixity against movement, it was assumed that spring force-slip response begins immediately after fiber is loaded and after reaching the peak, load drops to 10% of the ultimate capacity in the post peak region. The values of slips at peak and at post peak region of the response were determined in the calibration process. Figure 12a and b show the bond and spring models obtained from the calibration processes. The bond
Fig. 12 Calibrated material parameters for AR-glass fabric reinforced cement; (a) best fit bond slip model; (b) best fit spring model
s (MPa)
Glass pultrusion (GP)
2.62 2.62
0.71 12.7
1.24 1.24
1.71 12.7
1.52 1.52
parameters are also given in Table 3. Figures 13a–d show the simulations of pullout response due to bond as well as the combined bond and spring at junctions, which indicate that the relative contribution of the transverse yarns (springs) in carrying the applied load is quite significant. It can be seen from Fig. 13a–d that the response has two peaks in which the first one is higher than the second one. This simulation represents the sequential failure of the junction as the slip propagates through the embedded length. At initial loading, the slip in longitudinal yarn is relatively small between 0.0 and 0.66 mm (Fig. 12b) and two junctions are active; thus, the first peak of the response is contributed by spring and bond models. Two junctions are expected to fail one after another as they have a little post peak capacity. Therefore, the second peak is contributed solely be the bond model. The bond parameters obtained using a straight yarn model as shown in Table 2 cannot be directly compared to the present results in Table 3 since in the first case an equivalent bond parameter which smears
Materials and Structures Table 3 Bond-slip parameters for a fabric pullout model
Glass cast (GC) s (mm)
s (MPa)
Glass vacuum (GV) s (mm)
s (MPa)
Glass pultrusion (GP) s (mm)
s (MPa)
Glass flyash (GF) s (mm)
s (MPa)
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.15
0.95
0.20
1.70
0.35
0.90
0.40
2.80
2.39 5.90
1.86 0.35
2.15 4.75
3.00 0.85
2.25 2.90
1.40 0.60
1.30 4.00
2.80 1.00
12.7
0.1
12.7
0.10
5.00 12.7
0.30 0.05
6.00 12.7
0.35 0.10
Fig. 13 Simulation of pullout response of ARglass fabric; (a) cast sample, (b) pultrution sample; (c) vacuum process sample; (d) mix with fly ash
the junction effects over the entire length of fiber is used while the second model differentiates the contribution of the two mechanisms. It is interesting to note that both models yield similar trends; the bond of mixtures containing fly ash, or those subjected to vacuum processing are higher than the cast and pultrusion techniques. Comparison the simulations of pullout responses between Figs. 11b and 13a–d, indicates that the proposed fabric pullout model can capture the response better than the fiber pullout model using equivalent bond parameters.
7 Conclusions A fabric pullout model based on a nonlinear finite difference method is proposed for characterization of interface properties between fabric and matrix. Nonlinear characteristics of interface, longitudinal yarns and the mechanical anchorage offered by transverse yarns are described. Parametric studies show that the increasing bond strength proportionately increases the pullout strength while the slack in longitudinal yarns causes an apparently weaker and more flexible
Materials and Structures
pullout response. With the presence of a bonded fill yarn to serve as mechanical anchor to the longitudinal yarn, pullout strength was substantially increased. In the verification process, the proposed fabric pullout model was compared to the piecewise linear bond law model in the prediction of pulling a steel rebar out of concrete matrix. They both agree with the experimental result; however, the proposed model predicts slightly weaker post peak response because it assumes the part of yarn that comes out of the matrix has no shear resistance while the other model assumes it has according to the bond law model. Finally, the proposed model was used to analyze the test results of AR-glass fabric reinforced cement to quantitatively identify the contribution between bond and cross yarn junctions. It clearly shows that the first phase of pullout resistance was contributed by bond and transverse springs while the second phase which was after the sequential failure of the springs was due to bond only. Acknowledgements The authors acknowledge the US National Science Foundation, program 0324669-03 Program managers Drs. P. Balaguru and K. Chong for supporting this project.
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