Models Based on LEFM

All the LEFM models are rooted in the model first proposed by Carpinteri (1981, 1984; also 1986, Sec. 6.2). Figs. 10.4.2a-c show the basic superposition in Carpinteri’s approach: the reinforced beam with a crack of length a subjected to bending (Fig. 10.4.2a) is approximated by a beam subjected to the bending

Models Based on LEFM
Models Based on LEFM

Figure 10.4.1 Influence of steel ratio on tile load-displacement curves: (a) data from Bosco, Carpinteri and Debernardi (1990b); (b) data from Hcdcdal and Kroon (1991); (c) data fromPlanas, Ruiz and Elices (1995) for microconcrete, (d) General trend of the load-displacement curve according to Ruiz, Planas and Elices (see the text for details), (e) Influence of bond on the response (after Planas, Ruiz and Elices 1995). (d) Experimental double peaks for a relatively thick cover (data from Ruiz and Planas 1995).

Подпись: Figure 10.4.2 Carpinteri’s LEFM approximation.

moment and to the steel force applied remotely from the crack plane (Fig. 10.4.2b). Next, the steel action is decomposed in a standard way into a bending moment and a centric force (Fig. 10.4.2c).

Подпись: ft N1 к I - ~w^DkM{a) ~ ^VDk„(a) , Подпись: -2- , M1 = M — F. Подпись: С, (Ю.4.1)

With this decomposition, we can easily write the stress intensity factor as:

where км{оі) and ka{a) are the shape function for pure bending and for a uniform remote tension, respectively, and cs is the steel cover (see Fig. 10.4.2). An approximate, but accurate, expression for км (a) is given, for example, by (3.1.1) for S/D — eo; an expression for ka{() can be found in most stress intensity factor manuals (e. g., Tada, Paris and Irwin 1985).

Carpinteri also calculated the additional rotation caused by to the crack 0C according to the method described in Section 5.5.2 with P = M, 0C = U, and Pj = F, with the result

°c = ШБі‘иМм(‘а"> ~ WbDVMa^ ’ M’ = M ‘ FsD Yl " 7) (Ю.4.2)

Models Based on LEFM Models Based on LEFM Подпись: (10.4.3)

where 7 = cs/D is the relative cover thickness and

Подпись: ONS/D BcE'sfD Klc Подпись: r— -F ("з -67 + Np км V км) VMM , кдУмм - к МУ Ма км км Подпись: (10.4.4) (10.4.5)

Carpinteri assumed that the steel behavior was elastic-pcrfectly plastic, and that the crack was closed (Qc = 0) while the steel remained elastic. Therefore, the crack growth takes place only when the steel yields and, simultaneously, Ki = K]c. With these conditions, it is easy to obtain the parametric equations of the moment-rotation curves (with parameter a). Indeed, setting Fs = pbDfy (where p is the steel ratio and fy the steel yield stress) and K[ = K;c in the foregoing equations, the solutions can be written as

Подпись: Np = p Подпись: (10.4.6)
Models Based on LEFM

where, as always, ітдг = 6M/bD2, and Лгр is Carpinteri’s brittleness number for reinforced beams in bending defined as

Here we have introduced the length (p to emphasize the similitude of this brittleness number to those based on Irwin’s or Hillerborg’s characteristic size: the only change is to replace the tensile strength of concrete by the tensile strength of the reinforcement fy.

One of the limitations of this model is that, due to the simplifications involved in its derivation, the crack cannot grow while the steel remains elastic and does not slip (in reality, it must slip). This limitation was removed, using very different methods, by Baluch, Azad and Ashmawi (1992) and by Bosco and Carpinteri (1992).

Figure 10.4.3 Model of Baluch, Azad and Ashmawi (1992): (a) stress-strain curve for concrete; (b) strain distribution; (c) stress distribution; (d) comparison of experimental and theoretical curves of load vs. crack length for lightly reinforced beams with a notch.

The model of Baluch, Azad and Ashmawi keeps Carpinteri’s solution after steel yielding, but relaxes the assumption that the crack remains closed while the steel is elastic. In the elastic regime for steel, the model retains the stress intensity equation (10.4.1) and the condition Kj — К и, which can be rewritten as

Kic – км (a) – ^/Z> [(3 — 6”) kM(a) + (10.4.7)

which provides one equation with two unknowns, namely, M and Fs (a is given in this context). To determine Fs, Baluch et al. introduced a classical analysis based on a stress-strain formulation with the following assumptions: (1) the stress-strain curve is as depicted in Fig. 10.4.3a, parabolic in compression and linear in tension down to the failure stress fr which is taken to coincide with the modulus of rupture rather than with the tensile strength; (2) the softening in tension is linear, as depicled by the dolled line in Fig. 10.4.3a, but the softening slope depends on the geometry as indicated later; (3) the strain distribution is linear (Fig. 10.4.3b); (4) the fracture process zone is represented by a linear distribution of stress which is zero at the crack tip as shown in Fig. 10.4.3c. Note that the essential difference with respect to other formulations is that here the softening curve for concrete in tension is not related to the strain in a predefined way; rather, the form of the spatial distribution of stress is postulated. With these hypotheses, and given the stress-strain curve of the steel, it is possible to determine a relationship between Fs and M. For elastic behavior of the steel, the strain distribution is obtained from Fig. 10.4.3b:


Подпись: crack length (mm)
Models Based on LEFM

Fs_______ г

-4$ F – • x

where As is the steel cross section and Д, its elastic modulus; x is the depth of the neutral axis. From this, the stress distribution can be determined as sketched in Fig. 10.4.3c. Then, the equilibrium of forces provides an equation with the two unknowns Fs and x, and the equilibrium of moments a further equation with the three unknowns Fs, x, and M. Complementing these two equations with (10.4.7), we get a system that determines the three aforementioned unknowns. This system must be solved numerically. Baluch, Azad and Ashmawi (1992) use two iteration loops; given a, they assume a value for Fs and solve iteratively for x from the condition of equilibrium of forces (inner iteration loop); then they compute M from the equilibrium of moments and from (10.4.7); if the two values coincide, this is the solution for the given crack depth; if not, they start over with a new value of F„ (outer iteration loop).

Baluch, Azad and Ashmawi (1992) checked their model by comparing the load-crack length curves for two lightly reinforced notched beams tested in three-point bending (the determination of load-displacement

Models Based on LEFM

Figure 10.4.4 LEFM approximation of Bosco and Carpintcri (1992).

or moment-rotation curves were not included as part of the formulation). Fig. 10.4.3d shows the exper­imental results and the theoretical predictions (the dashed curves correspond to Carpinteri’s model; note that the prediction of both models coincide after the yielding of steel). Note also that, much like what we indicated for the load-displacement curves in Fig 10.4.1c, the theory predicts a sharp change of slope upon steel yielding, while the experiments show a rounded transition.

Bosco and Carpinteri (1992) adopted an approach radically different front the one just discussed. They modified the initial Carpinlcri’s model by letting the force of the reinforcement act on the crack faces rater than remotely from the crack plane, as shown in Figs. 10.4.4a-b. I lowevcr, the slip of reinforcement which must occur near the crack faces was neglected (even though elasticity indicates infinite stress at the point of intersection of the steel bar with the crack face). With this, the expression for the stress intensity factor reads

I<i =—’/D kM(a) ~ ^s/D kF(a,’y) , a = Jj ’ 7 " ^ (10.4.9)

where km (a) is the same as in Eq. (10.4.1), and kp(a,’f) is the shape factor for a pair of forces acting on the faces of the crack; a closed form expression for this shape factor can be found in Tada, Paris and Irwin (1985).

Now it is no longer necessary to assume that the crack is closed everywhere while the steel is elastic; it is enough to assume that the crack is closed at the point where the reinforcement crosses it. Allowance for bond slip could also be made; however, Bosco and Carpinteri did not consider this possibility. The method in Section 5.5.2 is used to determine expressions for the rotation and the crack opening at the reinforcement level with Pi == M, u = 0C and Pi = —Fs, u2 = ws, where w„ is the crack opening at the reinforcement level. The resulting expressions are as follows:

6M, , F3 , ,

вс ~ E’bD2VMM^a’ E’bDVMF^a’^


M t t F> t л Ws = "E, bDVMF^a’’y) E, bvFF(rin)


where vmm(a) is given by the first of (10.4.3), while vmf{oi, 7) and vpjr(a, 7) are given by

ra ra

vmf(oi, 7) = 12 / км{а’,і)кр{ос’,7) da’ , vff(o/, 7) = 2 / k2F(a 7) da’ (10.4.12) J 7 7 7

The integration is carried out over the crack portion in excess of cover thickness. This is so because for shorter cracks the stress intensity factor caused by the point loads is zero, i. e., kF(a, 7) = 0 for a < 7.

Assuming that the functions vmmP’mf and vff havebeendetermined, Eqs. (10.4.9)-(10.4.11)com – pletely solve the problem of crack propagation. Two cases can arise:

Case 1: The steel is still in elastic state. In this situation, we set ws — 0 in (10.4.11), solve for Fs from the resulting equation and substitute it in (10.4.9), simultaneously setting I(j — Ktc then wc solve

Models Based on LEFM

the resulting equation for M. Finally, the rotation follows from (10.4.9). The final results are:

where the arguments of the shape functions have been dropped for brevity, and as is the stress in the steel. Note that the right hand sides of these equations are independent of the brittleness number Np defined in (10.4.6). If the value of a3 resulting from the foregoing equations exceeds the steel yield stress fy, then we move to the next case.

Models Based on LEFM Models Based on LEFM Подпись: for w, > 0 Подпись: (10.4.14)

Case 2: The steel yields. In this case, we set Fs — bDfy in (10.4.9)—(10.4.11) and solve for M (or cr, v) and 6C (it might be useful to also check that ws > 0; otherwise, we are in case 1). The resulting equations then are

The plot of the ctn(0c) curves in terms of the nondimensional variables X = 6cE’/~D/Kjc and Y = oprs/D/Kic consists of two parts as sketched in Fig. 10.4.5a. The arc MNP is a part of the fixed curve LMNPQ which is given by (10.4.13); this curve depends only on the relative steel cover 7, but is independent of the beam size, of the amount and quality of steel, and of the properties of concrete; it is a pure geometrical property. The arc PT corresponds to the solution for yielded steel and its shape is concave with a horizontal asymptote which corresponds to fully broken concrete. This branch PT depends only on the brittleness number Np as sketched in the figure (and also on the relative cover thickness which is constant for geometrically similar beams).

In the foregoing equations, the rotation includes only the additional rotation caused by the crack at

remote cross sections. When dealing with the load-displacement curves, one can cither subtract the clastic displacement from the total displacement to isolate the rotation due to the crack, or conversely, one can add the elastic displacement (analytically computed) to the concentrated rotation determined by the foregoing theory. This last approach was used in a recent work by Massabo (1994) to analyze the experimental results of Bosco, Carpinteri and Debernardi (1990a, b).

Massabo (1994) determined the values of vmm, vmf> and Vpp for each л and 7 by numerically performing the integrations in (10.4.3) and (10.4.12). The integration for vpp deserves further comments because as written in (10.4.12), its value is infinite. This is so because, for simplicity, the load was assumed to be applied at a single point, which always gives a logarithmic singularity at the load-point. In reality, the action of the reinforcement is distributed over a certain area which is of the order of the diameter of the bars. This problem can be easily handled by assuming, for example, that the force is uniformly distributed and using the general formulas to determine the crack opening profile as given in Section 5.2. But this requires a double integration which greatly complicates the solution of the problem. Therefore, Massabo (1994) proposed to take this effect into account by performing the integral over an interval that does not include the load-point, as follows:

Подпись:vpp(a, j) — 2 f k2F(a’,7) da’

J 7 + e

Here, for a single layer of steel bars, t is a small value proportional (but not identical) to the ratio Db/D, with Db = diameter of the bars.

Fig. 10.4.5b shows the kind of agreement with the experiments attained with the model of Bosco and Carpinteri. Note that the postpeak behavior is reasonably well predicted, but the model predicts a very large initial strength (for short cracks). This is a general limitation for the LEFM-based models because the stress intensity factor is always zero for uncracked specimens, implying that, in strict LEFM, a crack can never initiate in an unnotched specimen. Therefore, all these models must be interpreted as approximately describing the evolution of the fracture after the crack has formed; the crack initiation itself can be described only by recourse to nonlinear fracture mechanics, as in the models to be described next.