Modeling Local Pressure on Inclined Rovings in Textile-Reinforced Concrete

(ProQuest: ... denotes formulae omitted.)

INTRODUCTION

Technical textiles such as alkali-resistant (AR) glass, carbon, or aramid have become an attractive alternative to steel reinforcement because they enable, due to their corrosionresistant properties, the production of rather thin-walled structural elements. This leads to entirely new fields of application of reinforced concrete. There are many types of textiles available differing in shape and basic materials. The standard type is bidirectional fabrics consisting of long fiber strands, known as rovings. It is possible to impregnate the rovings, for example, with epoxy, to increase the bond between the individual fibers (filaments). In this paper, only nonimpregnated rovings are considered. As an example, Fig. 1 shows the AR glass fabric MAG-07-03, made of 2400 tex (1.34 × 10^sup -4^ lb/in. [2.4 g/m]) rovings in both warp (0 degrees) and weft (90 degrees) directions.

When designing structural elements made of textilereinforced concrete (TRC), it is preferable to align the rovings with the expected direction of the maximum principal (tensile) stresses. If this is the case, the crack-bridging behavior of the rovings is optimal. This cannot always be realized, however, either due to practical reasons or because the direction of the principal stresses leading to the first cracks is not known a priori; for example, if there are different loading cases. Therefore, it is important for the design of TRC to investigate and model the load-bearing behavior of fabrics with a slope relative to the tensile direction. Several results regarding this issue have already been published,1-4 all of them mainly focusing on the maximum strength of the fabrics at different slopes.

For the development of a model describing the behavior of TRC with inclined textile orientation, not only the strength of the composite but also the crack-bridging stresses at intermediate strain states are of importance. In the case of short, randomly distributed fibers, such models are available.5,6 These are, however, not directly applicable to TRC due to the continuity of the filaments and their interaction.

The authors are of the opinion that there are three effects relating to the textile slope that have to be considered: the partial alignment of the filaments with the direction of the load, the local lateral pressure on the rovings, and the damage of the filaments.

RESEARCH SIGNIFICANCE

Currently, especially within the scope of two collaborative research centers in Germany, great efforts are being made to understand the basic mechanisms leading to the macroscopically observed material behavior of TRC. To date, there is no model available comprising all of the effects that need to be considered in a general model for TRC. Due to the complexity of the material's structure, it is necessary to set up models on different resolution scales, that is, microlevels, mesolevels, and macrolevels. Whereas on the microlevel the interactions of single filaments and particles of the matrix are considered, on the mesolevel, the average behavior of filament groups (core and sleeve fiber, respectively) is modeled. This paper deals with the effect of the lateral pressure at the crack edge using a rather simple twoparameter model. It is derived for a single filament but can be used analogously for a group of filaments as a part of a more complex mesolevel model.

The model can easily be incorporated in a finite element model by explicitly discretizing the textile as one or more separate layers, depending on the resolution scale. In this paper, a method of reducing a particular source of mesh size dependency that is related to the increase in bond stiffness is proposed.

EFFECT OF TEXTILE SLOPE

Partial alignment of filaments with load direction

Figure 2(a) shows a textile reinforcement that is inclined relative to the direction of the load. The Axis A at the crack and the Axis B between two cracks are axes of symmetry recurring periodically at the average crack spacing s^sub rm^. The orientation of the outer filaments is nearly parallel to the direction of the crack-bridging force F. This is due to the more-or-less fixed position of the outer filaments that are embedded in the matrix. The inner filaments do not align as much as the outer filaments because they can change their position within the cross section of the roving, that is, the voids within the roving permit the core fibers to remain partially out of alignment relative to the bridging force. There is a smooth transition of the degree of alignment from the inner to the outer filaments. An idealization of the described effect is depicted in Fig. 2(b). The idealization distinguishes between sleeve and core of the roving, each having a constant orientation (φ^sub s^ and φ^sub c^, respectively) at the crack.

Local lateral pressure on roving

The equilibrium between the textile forces bridging the crack and the textile forces in the initial orientation lead to lateral forces U as shown in Fig. 3. One can distinguish between U^sub s^ and U^sub c^ acting on the sleeve and core, respectively (Fig. 2(b)). The values of the lateral forces depend on the crack-bridging forces and on the difference between the angles α (initial textile orientation) and φ (orientation of the textile at the crack)

U^sub s^ = F^sub s^ [the square root of]2(1 - cos(α - φ^sub s^)) (1)

U^sub s^ = F^sub c^ [the square root of]2(1 - cos(α - φ^sub c^)) (2)

where F^sub s^ and F^sub c^ are the forces in the sleeve and core, respectively. The lateral forces U^sub s^ and U^sub c^ are equivalent to lateral stresses acting on the roving along a certain length k, herein called the redirecting length, in the close range of the crack edge as shown in Fig. 2(a). Most likely, these lateral stresses lead to an increased bond performance and, hence, to increased normal stresses within the filaments at the same global strain.

Damage of filaments

As a result of the lateral forces and of the deflection of the roving at the crack, the filaments fail earlier than they would without textile inclination. Therefore, as a rule, the maximum load carried by the composite decreases with increasing textile slope.3 The quantity of damage differs significantly among the available textile materials. Aramid, for instance, is much more resistant to the lateral loading than AR glass. A comprehensive model for TRC requires the consideration of the damage of the fibers, probably as a function of the crack opening and the textile inclination. The analytical model presented next describes solely the behavior of the undamaged filaments-that is, it is meant to be used as a part of more complex models.

ANALYTICAL INVESTIGATION

In the following text, a model is proposed taking into account the increased bond performance of a filament along the redirecting length k.

Assumptions

The numerical value of the lateral stresses cannot be determined because the exact geometry of each filament within the embedded roving is unknown. To determine the lateral stress, it would be necessary to exactly determine the force, the deflection, and the contact area of each filament at every position along the roving. Therefore, the following simplifying assumptions are made:

1. The lateral stress σ^sub u^ is proportional to the lateral force U; and

2. The lateral stress σ^sub u^ is constant along the redirecting length k.

The assumptions lead to the following relation between the normal stress σ^sub 0^ of a filament at the crack and the lateral stress σ^sub u^

... (3)

where ξ is a factor of dimensions 1/mm (1/in.) taking into account the unknown contact area per unit length of the filament. The increased bond capacity τ* within the redirecting length (refer to Fig. 4) can then be formulated as a function of the lateral stress

... (4)

If the cross section of the filament Af and the factor ξ are assumed to be constant, they can be joined together with the parameter (mm^sup 2^/N [in.^sup 2^/lb]) and the factor to the new parameter β (mm^sup 3^/N [in.^sup 3^/lb])

... (5)

Moreover, if the notional lateral pressure p^sub [perpendicular]^ (N/mm^sup 3^ [lb/in.^sup 3^]) is introduced in Eq. (4) as

... (6)

then the increased bond capacity τ*, which depends on the normal stress of the filament at the crack, can be written as

... (7)

Figure 5 shows the scaling-law for the bond stress τ, which is determined by the parameter β. It should be noted that a constant bond stress-slip relation is assumed herein. Strictly speaking, this is only valid if the filament has debonded over the entire length l.

Derivation of differential equation

The equilibrium at the differential d^sub x^ shown in Fig. 6 leads to the basic differential equation for a filament embedded in the concrete matrix7

... (8)

For common reinforcement ratios of TRC (for example, two layers of MAG-07-03 and a wall thickness of 1 cm (0.39 in.), the stiffness of the concrete matrix is much larger than the stiffness of the textile. Thus, with the exception of very highly reinforced members, the second term in the brackets of Eq. (8) can be neglected. Because a constant bond stress τ independent of the slip between filament and matrix is assumed, the differential equation can be simplified to

... (9)

The differential equation within the range of the redirecting length k is the same but, due to the better bond performance, requires an increased constant C* on the right-hand side.

... (10)

where

... (11)

Initially, the notional lateral pressure p^sub [perpendicular]^ is not known because it depends on the normal stress at the crack σ^sub 0^. Hence, the parameter C* is also unknown.

The solution of the differential Eq. (9) and (10) must satisfy the boundary conditions Eq. (12a) through (12d). Thereby the slip within the redirecting length is described by the function s^sub 1^(x) and in the remaining length by the function s^sub 2^(x).

The slip at the crack edge is set to s^sub 0^

s^sub 1^(x = 0) = s^sub 0^ (12a)

The slip is continuous at the position k

s^sub 1^(x = k) = s^sub 2^(x = k) (12b)

The strain is continuous at the position k

... (12c)

Position l is at an axis of symmetry as shown in Fig. 4

s^sub 2^(x = l) = 0 (12d)

The solution of the two differential equations provides the functions s^sub 1^(x) and s^sub 2^(x) for the relative displacement between matrix and filament, whereby C* is still unknown.

Within the redirecting length (0 ≤ x ≤ k)

... (13)

Outside the redirecting length (k ≤ x ≤ l)

... (14)

The filament stress σ at any position x can be calculated by Eq. (15)

... (15)

Using Eq. (15), the normal stress σ^sub 0^ leading to the slip s^sub 0^ at the crack can be determined as

... (16)

where C* is a function of σ^sub 0^. If the value of C* from Eq. (11) is substituted in Eq. (16) and then solved for σ^sub 0^, then

... (17)

Now, C* can also be calculated using Eq. (11) and substituted in Eq. (13) and (14) to determine the slip.

Numerical example

In the following numerical example, the derived equations are evaluated for an AR glass filament. The material parameters are listed in Table 1. Due to the imperfect embedment of the filaments, the perimeter of the filament p^sub f^ has been reduced by 80%. The bond stress τ is taken according to Brameshuber and Banholzer.8 Figure 7 shows the distribution of the relative displacement between filament and matrix along half the inclined crack spacing l = s^sub rm^/(2cos(α - φ)). The calculations are performed for two values of β: 0.0 and 0.004 mm^sup 3^/N (1.09 × 10^sup -6^ in.^sup 3^/lbf). For both values of the parameter β, the boundary conditions are fulfilled, that is, the slip at the crack s^sub 0^ is 0.025 mm (0.001 in.) and in the middle between two cracks s^sub 0^ is zero. A larger value of β leads to smaller relative displacements along the axis of the roving due to the increased bond performance within the redirecting length.

The influence of β on the distribution of filament stresses σ is clearly shown in Fig. 8. The stress σ^sub 0^ at the crack increases with increasing value of β. In this example, a value of 0.004 mm^sup 3^/N (1.09 × 10^sup -6^ in.^sup 3^/lbf) for β results in an increase of approximately 22% in the stress compared with the basic calculation using β = 0.

Effect of redirecting length k

In the model, the resulting filament stress is controlled by the redirecting length k and the parameter β. Both parameters are actually unknown. If the model was to be incorporated in a finite element code where the cracks in the concrete are represented by localized strains in one row of elements (crack band),9 then it would be convenient to scale the bondstress relation locally in these elements only. The calculational redirecting length k' is than equal to half the element length (refer to Fig. 9). In general, k' is different from k, which causes a mesh dependency.

To obtain results for the filament stress σ^sub 0^ that is independent of the element length, it is necessary to adjust the parameter β of the scaling law. This may be considered as an analogy to a mesh-adjusted stress-strain relationship in a local model for concrete where the crack opening is smeared over the element length.10 In other words, the aim is to get a constant value for σ^sub 0^ for different arbitrary values of k' if the parameters E^sub f^, s^sub 0^, C, α, φ, and l are given. For this purpose, Eq. (17) is solved for β as a function of k.

... (18)

Now, an adjustment factor ν can be calculated for the parameter β, which has to be applied if the redirecting length k is changed by a factor η

... (19)

As Eq. (19) reveals, the adjustment factor ν is independent of the material parameters C and Ef. It is also independent of the angles α and φ and of the slip at the crack s^sub 0^. Only the redirecting length k' and half the crack spacing l have an influence on the adjustment factor ν. If l is relatively large compared with the redirecting length, then ν results in

... (20)

That is, for a large crack spacing, the parameter β does not need to be adjusted. The result for the filament stress at the crack is independent of k'. If the crack spacing is relatively small, there is a size effect of k' and therefore β needs to be changed. Figure 10 compares the adjustment factors ν for different lengths l. Thereby the redirecting length k was assumed to be 1.0 mm (0.039 in.), hence

... (21)

The effectiveness of the adjustment of the parameter β was validated exemplarily, choosing the parameter β = 0.002 mm^sup 3^/N (5.43 × 10-7 in.^sup 3^/lbf). Half the crack spacing l was set to 5 mm (0.197 in.), and a slip s^sub 0^ of 0.025 mm (0.001 in.) was applied. Figure 11 shows the results for different calculative redirecting lengths k' and the corresponding parameters β that were adjusted according to Eq. (19). All calculations result in the same filament stress σ^sub 0^ of 630 MPa (91.4 ksi).

SUMMARY AND CONCLUSIONS

An analytical model that takes into account the increased bond capacity due to lateral pressure on the roving at the crack edge has been developed. This is one of the principle effects that must be considered when modeling TRC with inclined textile orientation along with consideration of roving alignment with respect to the load direction and textile damage at the cracks.

Two important parameters were introduced: 1) the redirecting length, along which the roving changes its direction and is subjected to lateral pressure; and 2) the parameter β describing the increase in bond performance depending on the lateral pressure. The effect of these parameters was demonstrated by numerical examples.

The model can easily be incorporated into a finite element code. If, for instance, the crack-band model is used for the concrete matrix, then the strain localizes in one row of elements.7 In the case of an inclined textile orientation, the bond in those cracked elements may be automatically increased. The calculative redirecting length k' is then equal to half the element length. Mesh-independent results for the stresses of the textile at the crack can be obtained applying an adjustment of the parameter β according to Eq. (19).

ACKNOWLEDGMENTS

The authors gratefully acknowledge the German Research Foundation for the financial support of this project, which is part of the collaborative research center SFB 532, Textile Reinforced Concrete, Development of a New Technology. This paper was written during a research visit of A. Sherif to the University of Aachen financed by the Alexander von Humboldt Foundation. The support of the Alexander von Humboldt Foundation is deeply appreciated.

NOTATION

A^sup c^ = cross section of concrete matrix

A^sup f^ = cross section of filament

E^sup c^ = Young's modulus of concrete matrix

E^sup f^ = Young's modulus of filament

F^sup 0^ = normal force in filament at crack

k = redirecting length

k' = calculative redirecting length in FE model

p^sup f^ = perimeter of one filament

p^sup [perpendicular]^= notional lateral pressure, input value for scaling law

s = slip between filament and matrix

s^sub 0^ = slip between filament and matrix at crack

s^sup rm^ = average crack spacing

U = lateral force acting on filament along redirecting length

U^sup c^ = lateral force acting on core at crack

U^sup s^ = lateral force acting on sleeve at crack

α = slope of textile relative to load direction

β, β = parameter influencing scaling law

η = multiplier of k

φ = slope of textile in crack relative to load direction

ν = adjustment factor for β if k is multiplied by η

σ = normal stress of filament

σ^sub 0^ = normal stress of filament at crack

σ^sub u^ = lateral stress acting on filament along redirecting length

τ = bond stress

τ^sup max^ = maximum bond stress

τ^sup max^* = scaled maximum bond stress

ξ = parameter taking into account unknown contact area between filament and matrix