# Shrinkage-Induced Response of Composite Steel-Concrete Slabs: A State-of-the-Art Review
## 4 Time-Dependent Modelling of Composite slabs  
### Numerical Model  
The time-dependent behavior of composite slabs has been commonly represented by means of full shear interaction theory [7; 8; 60; 61] that assumed relative movement to occur between the slab and the profiled steel sheeting. Based on the age-adjusted effective modulus method (AEMM), reference [62] adopted a constant uniform shrinkage profile through the thickness of the composite slab. A closed-form solution to calculate the deflections and stresses in one-way composite steel-concrete slabs over time was proposed in [63; 64]. The formulation was based on the principle of virtual work and the model can account for different shrinkage profiles, including non-uniform shrinkage distributions such as those that can occur in composite slabs. Other methods of analysis to capture the composite time-dependent response accounting for shrinkage gradients were presented in [17; 55; 59; 65; 66]. In [17], the influence of non-uniform shrinkage gradients on the deflection response was included in the analysis by assuming the composite slab cross-section to be discretized into several layers. In this study, four different shrinkage profiles were considered and these consisted of a uniform shrinkage profile, a triangular shrinkage profile, a bilinear shrinkage profile and a parabolic shrinkage profile. Full shear interaction was also assumed between the concrete and profiled steel sheeting. The long-term behavior of post-tensioned composite slabs based on the Euler-Bernoulli beam theory was presented in [14]. Long-term analyses were performed by means of the step-by-step procedure in which the time domain is discretized in different time steps and is widely used in the long-term predictions of concrete structures as, for example, outlined in [8]. Measured shrinkage gradients that occurred through the thickness of the slab were considered to calculate the long-term deflection of the composite slabs. Comparisons were made between experimental and calculated deflections. With the same beam assumptions, a non-linear solution strategy was considered in [18; 19] based on the formulations described in [67]. Based on their experimental results, full interaction between concrete and steel sheeting were considered in the analysis. Linear shrinkage gradients were assumed to occur through the thickness of the composite slab to account for the non-uniform shrinkage strains. Figure 6: Mid-span deflections measured for the continuous specimens (LVDT-L = mid-span deflection of left span; LVDT-R = mid-span deflection of right span) [53]. The non-uniform shrinkage effects on the time-dependent behavior of composite and post-tensioned slabs were presented in [68]. A hygro-thermo-chemical-mechanical model has been proposed to calculate the more refined non-uniform shrinkage profiles from the temperature and relative humidity distribution through the thickness of the composite slab [69]. The time-dependent response of the composite slab was then obtained from the cross-sectional analysis introduced in [8]. In this study, measured long-term deflections of composite slabs presented in [54] were used to validate the numerical predictions. Comparisons with measured deflections are shown in Figure 7. For this purpose, three different shrinkage profiles were considered and included (Figure 7): (i) a uniform shrinkage profile typically used in reinforced concrete solid slabs; (ii) a linearly varying shrinkage gradient that was specified in accordance with the specifications of the Australian composite guidelines [12; 13] (and suitable for routine design)--this case is referred to as'simplified approach' in Figure 7; and (iii) a shrinkage gradient that is determined from the hygro-thermo-chemical-mechanical model based on the inverse analysis procedure described in [69].