Eliminating Shrinkage Effect from Moment Curvature and Tension Stiffening Relationships of Reinforced Concrete Members

The techniques are on the basis of the following approaches and assumptions: (1) smeared crack approach; (2) linear strain distribution within the depth of the section implying perfect bond between concrete and reinforcement; (3) uniform shrinkage within the section; (4) shrinkage-induced stresses do not exceed tensile strength of concrete $f_{ct}(t)$; and (5) all concrete fibers in the tension zone follow a uniform stress-strain tension-stiffening law.

Shrinkage of an isolated plain concrete member would merely shorten it without causing any stresses [see Figs. 1(, a and c)]. Reinforcement embedded in a concrete member provides restraint to shrinkage deformation, leading to compressive stresses in reinforcement and tensile stresses in concrete [see Figs. 1(, b and d)]. If the reinforcement is asymmetrically placed on a section, shrinkage causes a nonuniform stress-strain distribution within the height of the section. The maximal tensile stresses appear on the extreme concrete fiber, close to the larger concentration of reinforcement [see Figs. 1(f, g)].

Because shrinkage is a long-term effect, creep always relieves shrinkage-induced stresses. Concrete is an aging material, so its strength and modulus of elasticity increase with time, as does the shrinkage strain. Analysis is also complicated by interdependence between a stress history and a creep strain. The Trost-Bazant method, called the age-adjusted modulus method (Bazant 1972), gives a simple procedure for computing a strain under a varying stress. The method introduces the aging coefficient $\chi (t,t_0)$, which assesses that the stress $\Delta {\sigma }_c(t,t_0)$ gradually applied from time $t_0$ to $t$ will cause smaller strain than the stress $\Delta {\sigma }_c(t,t_0)$ instantly applied at time $t_0$ and kept constant until time $t$. For shrinkage analysis of a noncracked member assuming zero initial stress at time $t_0$, the strain in concrete can be expressed as follows (Gilbert 1988):in which $E_{ea}(t,t_0)$ = age-adjusted effective modulus of concrete; $E_c(t_0)$ = modulus of elasticity of concrete at time $t_0$; $\phi (t,t_0)$ = creep factor; and ${\epsilon }_{cs}(t,t_0)$ = mean free shrinkage strain of concrete assumed negative.

Following references (Kaklauskas et al. 2009; Gribniak 2009), shrinkage is modeled by means of fictitious axial force $N_{cs}(t,t_0)$. For a noncracked member, it is expressed asin which $A_c$ = area of concrete net section [see Fig. 2(a)]. Fictitious shrinkage force $N_{cs}(t,t_0)$ is applied on the centroid of a concrete section [see Fig. 2(b)]. As shown in Fig. 2(c), the fictitious force in a nonsymmetrical section acts with an eccentricity exerting bending upon the member as follows:in which $y_{c,c}$ and $y_{c,rc}$ = centroid coordinates of plain and transformed reinforced sections, respectively. Curvature ${\kappa }_{cs}(t,t_0)$ and strain ${\epsilon }_{i,cs}(t,t_0)$ at any fiber $i$ [see Fig. 2(d)] attributable shrinkage with creep taken into account can be calculated by the following formulas:in which $A_{rc}(t,t_0)$, $I_{rc}(t,t_0)$ = area and the second moment of area of the reinforced section, respectively.

Assume that a shrunk member at time $t$ is exposed to external short-term bending. If the member is not cracked, strains and stresses attributable to shrinkage and external bending moment $M$ can be calculated on a basis of superposition. After cracking, superposition is not valid and the stress-strain state can be assessed from nonlinear analysis on the basis of the layer section model [see Fig. 2(e)]. Cracking and nonlinear material properties are taken into account by means of secant deformation modulus. For the purposes of numerical analysis, it has been assumed that shrinkage effect is a short-term action, that is, the fictitious force is applied instantly as follows:in which $A_{c,\sec }(t)$ = transformed concrete section; $n$ = total number of layers; $b_i$ and $t_i$ = width and thickness of the $i$-th layer; ${\alpha }_{i,\sec }(t)$ = modular ratio that assesses reduction in stiffness of the $i$-th layer after cracking; and $k_i$ = material number assumed to be 1 for concrete and 0 for reinforcement layers. In Eq. (5), the effective shrinkage strain ${\epsilon }_{cs}^{*}(t)$ has been introduced. As the creep effect, responsible for stress relief, is neglected, the reduced shrinkage strain ${\epsilon }_{cs}^{*}(t)$ is used to obtain adequate stresses. It can be defined from the condition of equality of the steel strains calculated from the short-term and the long-term analyses of a noncracked member. For a symmetrical section, ${\epsilon }_{cs}^{*}(t)$ can be obtained explicitly (Kaklauskas et al. 2009; Gribniak 2009); for asymmetrical sections, it is recommended to be numerically defined from the following condition:in which ${\epsilon }_{s,cs}^{\prime }(t,t_0)$ and ${\epsilon }_{s,cs}^{\prime }(t)$ = strains in the most strained reinforcement, including and excluding creep effect, respectively.

Curvature $\kappa (t)$ and strain ${\epsilon }_i(t)$ at any fiber $i$ [see Fig. 2(e)] are calculated by the following formulas:in which $M_{cs}^{*}(t)$ = moment induced by shrinkage; and $A_{rc,\sec }(t)$, $I_{rc,\sec }(t)$ = area and the second moment of area of the transformed reinforced section, respectively. The latter cross-sectional characteristics are calculated by the following formulas:in which $y_{c,i}$ = coordinate of the centroid of the $i$-th layer. Coordinates of the centroids $y_{c,rc}(t)$ and $y_{c,c}(t)$ are calculated as follows:

After cracking, fictitious force $N_{cs}^{*}(t)$ and the coordinates of the centroids $y_{c,rc}(t)$ and $y_{c,c}(t)$ decrease with increasing loads primarily because of reduction in the secant deformation modulus of concrete in tension.

The direct analysis is performed iteratively through the following steps:

Although the analysis neglects creep, adequate stresses in concrete and reinforcement are obtained. The curvature component attributable to creep is as follows:in which $I_{rc}(t,t_0)$ is calculated as for a noncracked section. Then a total curvature is as follows:

In contrast to the direct analysis, which results in prediction of structural response by using a specified constitutive model, the inverse analysis aims at determining parameters of the model on the basis of the response of the structure. The present investigation deals with the behavior of RC flexural members and aims at developing a technique for selecting parameters of the tension-stiffening model. The selected parameters will allow modeling the same moment-curvature response as it was measured at the tests. This section sketches a solution of the inverse problem, discussing major aspects only, and presents a step-by-step application example of the proposed inverse technique.

The first writer in coauthorship (Kaklauskas and Ghaboussi 2001) has formulated the principles of the inverse technique for deriving tension-stiffening relationships by using the test data of RC flexural members. For a given experimental moment-curvature/strain curve, a stress-strain tension-stiffening relationship was computed from the equilibrium equations of axial forces and bending moments. The layer section model was used for computation of the internal forces. Analysis was performed with incrementally increasing bending moment. The two equilibrium equations were solved for each loading stage, yielding a solution for the coordinate of neutral axis and the concrete stress in the extreme tension fiber. Because the extreme fiber had the largest strain, other tension fibers of concrete had smaller strains falling within the portion of the stress-strain diagram, which had already been determined. The tension-stiffening relationship was progressively derived assuming portions obtained from the previous increments.

The present research uses the inverse technique modified by the second writer (Gribniak 2009). Modifications were aimed at improving computational efficiency and reducing the influence of scatter of the test data points of moment-curvature diagrams on the analysis results. The analysis is on the basis of the direct technique and the previously mentioned concept of a progressive calculation of the tension-stiffening relationship for the extreme tension fiber of concrete. The assumption of uniform stress-strain tension-stiffening law for different layers allows reduction of the dimension of the solution to a single nonlinear equation. For a given load increment, an initial value of the secant deformation modulus of a tension-stiffening relationship is assumed and a curvature is calculated. If the calculated and experimental curvatures differ, methods of iterative analysis are used to obtain the solution for the secant deformation modulus.

Fig. 3 presents a flow chart of the inverse technique. On the basis of geometrical parameters of the cross section, the layer section is composed. Stress-strain material laws for steel and compressive concrete are assumed. Computations are performed iteratively for an incrementally increasing bending moment. At each moment increment $i$, an initial value of the secant deformation modulus of the tension-stiffening relationship under derivation is assumed equal to zero ($E_{i,0}=0$). Curvature ${\kappa }_{th,i}$ is calculated by the direct analysis. If the agreement between the calculated and the experimental curvature ${\kappa }_{\mathrm{obs},i}$ is not within the assumed tolerance $\Delta (|{\delta }_{i,k}|=|({\kappa }_{th,i}/{\kappa }_{\mathrm{obs},i})-1|>\Delta )$, that is, condition 1 is not fulfilled (see Fig. 3), the analysis is repeated by using the hybrid Newton-Raphson and Bisection procedure (Gribniak 2009) until Condition 2 is satisfied. At each iteration $k$, a secant deformation modulus $E_{i,k}$ is calculated. If the solution is found, that is, Condition 1 is satisfied, the obtained value of $E_{i.k}$ is fixed and used for next load increments. If the limit iteration number is exceeded ($k>N=30$), the calculated $E_{i,30}$ is rejected, meaning that the secant deformation modulus $E_i$ is not defined at the moment increment $i$, and the analysis moves to the next load step. The calculation is terminated when the ultimate loading step is reached (Condition 3). The analysis results in the derived tension-stiffening relationship. Application of this relationship in the direct analysis would give the original moment-curvature diagram.

As described, the inverse analysis is performed incrementally by using the constitutive law obtained at previous loading stages. Consequently, the error made at a given moment increment will have influence on the shape of the remaining part of the constitutive law under the construction. Although in absolute terms similar at all loading stages, the errors of curvature measurements at early stages have a much greater relative effect on the resulting tension-stiffening curve. Therefore, particular care should be taken in the early stages of analysis associated with small curvatures. Because before cracking concrete both in compression and tension essentially behaves elastically, a limitation on curvature has been introduced; the curvature calculated by using the elastic material properties should exceed the experimental one. If this limitation is not satisfied, the test curvature is replaced by the calculated one.

To illustrate the proposed technique, a detailed step-by-step numerical analysis has been performed for beam S-1 (see Table 1) tested by Gribniak (2009). The beam had a relatively small amount of tensile reinforcement ($p=0.4\%$); therefore, its deformation behavior was highly affected by the tension-stiffening effect. The beam was tested under a four-point bending scheme with small load steps, approximately 80 increments in total. Such testing allows obtaining moment-curvature diagrams suitable for the solution of inverse problems.

The constitutive laws assumed in the analysis are shown in Fig. 4. Both measured and idealized stress-strain relationships for steel are shown in Fig. 4(a). The compressive behavior of concrete was modeled by Eurocode 2 [see Fig. 4(b)]. The test moment-curvature diagram is shown in Fig. 4(c). The constitutive stress-strain relationship under construction is shown in Fig. 4(d). In this example, the elastic curvature limitation was imposed. Because of this, no credible conclusions can be drawn regarding the cracking point corresponding to the start of the softening behavior in the tension-stiffening relationship [see Figs. 4(c, d)].

Consider load increment 20 marked in the moment-curvature diagram. The constitutive stress-strain relationship and the corresponding secant modulus derived at load stages 1–19 are presented in Fig. 4(d). At load increment 20, the convergence was reached after eight iterations. As seen, the initial value of secant deformation modulus was assumed equal to zero. Intermediate solution points along with the initial point ($E_{20,0}=0$) and the portion of stress-strain diagram derived for load increment 20 are shown in Fig. 4(d).

Because of the discrete cracking phenomenon and stochastic nature of the test data, experimental moment-curvature relationships have some oscillations. The tension-stiffening relationships obtained from the inverse analysis also have oscillations that may become dramatic because of the accumulative nature of the proposed procedure. To reduce the influence of extreme measurement points on the results, a smoothing procedure on the basis of the Monte Carlo simulation and the modified running-average method has been developed (Gribniak 2009). A MATLAB code of the proposed inverse procedure is given in the latter reference.

To study the shrinkage effect on short-term deformations and tension stiffening, tests on three RC beams have been performed. The specimens were of rectangular section, with the nominal length 3,280 mm (span 3,000 mm), depth 300 mm, and width 280 mm. The beams were tested under a four-point bending scheme with the concentrated forces dividing the span into three equal parts. Main geometrical and material characteristics of the beams S1-3, S2-3, and S3-2-3 are presented in Table 1. In the tension zone, the beams were reinforced with three deformed bars of 18, 14, and 10 mm in diameter, respectively. Stirrups in the shear span and top reinforcement were also provided.

Concrete mix proportion was assumed to be uniform for all the beams. Ordinary Portland cement and crushed aggregate (16 mm maximum nominal size) were used. Water/cement and aggregate/cement ratios by weight were measured as 0.42 and 2.97, respectively. Before the short-term tests of the beams, measurements on the concrete shrinkage and creep were performed. Free shrinkage measurements were performed on the fragments of the test beams ($280\times 300\times 350\mathrm{mm}$).

Experimental moment-curvature diagrams needed for deriving tension-stiffening relationships were obtained in two ways: from deflections and from concrete surface strains, both recorded in the pure bending zone. Concrete surface strains were measured throughout the length of the pure bending zone on a 200-mm gauge length by using mechanical gauges. Four continuous gauge lines were located at different depths, with two extreme lines placed along the top and bottom reinforcement. Measured strains were averaged along each gauge line. The tests were performed with small increments (2 kN) and paused for short periods (approximately 2 min) to take readings of gauges and to measure crack development. In total, it took from 40 to 60 load increments. Deflections were automatically recorded at 1-kN load increments. Good agreement was achieved for the moment-curvature diagrams obtained from deflection and strain measurements. The present analysis is on the basis of the data obtained from average strains.

The test moment-curvature diagrams of the beams are shown in Fig. 7. Because of small load increments, the diagrams did not possess a clear horizontal part corresponding to the start of formation of major cracks. Moment-curvature diagrams after eliminating the shrinkage effect by the proposed technique are also presented in Fig. 7 by a gray line. Quite a striking difference between the transformed and the original curvature diagrams (at least not acceptable for constitutive modeling purposes) can be seen for the beams with a greater reinforcement ratio.

The derived tension-stiffening relationships without eliminating the shrinkage effect are shown in Fig. 8(a). Maximal stresses attributable to the effect causing additional tension in the bottom concrete surface and consequent reduction of the cracking resistance were well below the tensile strength. It should be pointed out that the tension-stiffening diagrams derived for the beams with a greater reinforcement ratio (the beams S1-3 and S2-3) contained portions of negative stresses. The descending branches of the tension-stiffening relationships characterized by the limit strain (corresponding to zero stress) were quite different for the beams. Because the amount of tensile reinforcement was the only different parameter between one beam and another, it is expected to be responsible for the difference. Previous research has shown the limit strain dependence on the reinforcement ratio as follows (Kaklauskas and Ghaboussi 2001):

Normalized limit strains, calculated by Eq. (12), were 12.4, 18.7, and 25.1 for beams S1-3, S2-3, and S3-2-3, respectively, and well agreed with the ones obtained from present tests [see Fig. 8(a)].

The derived tension-stiffening relationships after eliminating the shrinkage effect are shown in Fig. 8(b). On the basis of this figure, two observations can be made. First, maximal stresses have approached the tensile strength. Second, the tension-stiffening relationships obtained for different beams have approached one another and the portions of negative stresses practically disappeared.

The preceding analysis shows that the shrinkage occurring before the short-term loading has quite a substantial effect on deformations of RC members and the tension-stiffening relationships derived from the tests. Therefore, the shrinkage effect should be disregarded neither in constitutive modeling nor in the design problems requiring high accuracy. Future tests aimed at the constitutive modeling should either eliminate shrinkage or perform shrinkage and associated creep recordings for the subsequent assessment of this effect.