Experimental and Numerical Assessment of the Service Behaviour of an Innovative Long-Span Precast Roof Element

A time-step procedure was implemented for the correct estimation of the deformative behaviour of pre-stressed elements in visco-elastic range. The procedure is based on the time-step evaluation of the visco-elastic phenomena linked to the member longitudinal shortening and its subsequent deflection profile. The geometrical non-linearity of the problem displays due to the following considerations:

Furthermore, the viscous axial shortening of pre-stressed elements is particularly relevant, since they are subjected to strong axial loads applied in an early stage of concrete hardening due to the daily schedule typical of the industrial production.

With reference to the behaviour in storage of pre-stressed elements, the elastic contribution always has a monotonic decreasing tendency, due to the combination of the following phenomena:while the creep contribution, if the elastic component is always positive, is also always positive with a monotonic increasing trend. The combination of those terms determines the overall deformation behaviour of the element, which could also be subjected to trend reversal of the deflection in the range of downwards values, difficultly predictable without an accurate evaluation of the element shortening evolution in time.

The employed methodology solves the problem by means of semi-analytical techniques through a discretisation in the time domain, under the hypotheses of plane sectional strain profile and homogeneous section. In particular, the longitudinal behaviour of the element and its deflection profile are considered as uncoupled, and the formulation is corrected with linear terms in order to take into account in a simplified way the real coupling. As a consequence of such an hypothesis, it is possible to formulate the longitudinal shortening first, and subsequently compute the deflection history as a function of the shortening tendency.

The deflection profile of a member is calculated in accordance with the well-known visco-elastic model having the integral form of Eq. (1) (see Branson 1977; Migliacci and Mola 1985, Mola 1997; Ghali et al. 2011; Mola and Pellegrini 2012):where the visco-elastic deflection v at time t is expressed as the sum of a contribution of the elastic deformation computed at time t to which the contribution of creep, depending on the first derivative of the creep coefficient φ(t,t ₀ ), has to be summed. A similar equation defines the visco-elastic longitudinal strain equation for pre-stressed members with constant cross-section [Eq. (2)], depending on the elastic [Eq. (3)], creep [Eq. (4)] and shrinkage strains: where the mean stress in the pre-stressing reinforcement can be defined as per Eq. (5), taking into account the relaxation, thermal and shortening losses:where an anchorage factor φ _anch is simply defined in average terms for the distribution of debonding sleeves and anchorage of the cables [Eq. (6)]:and φ _χ is a weighted factor for curvature loss taking into account the combination of parabolic-shaped deformation profile due to pre-stressing and fourth order polynomial function deformation profile due to distributed loads (self-weight), to which the maximum bending moments M’ e M*, respectively, belong [Eq. (7)]. The φ _χ factor is always lower than unity.

The non-linearity of Eq. (2) is clear, due to the presence of the unknown ε _long in the integral belonging to the creep term. The solution is not achievable in closed form, and the equation needs to be solved with the aid of numerical techniques.

It is though possible to identify an approximated solution with independent variables estimating the loss for shortening of the member, through a fictitious stiffening of the axial deformability.

By eliminating the unknown from the formulation of σ _p [Eq. (8)]:and by making its contribution explicit in the equation of the elastic deformation ε _el , the latter can be written as follows [Eq. (9)]:By operating the proper simplifications, it is then possible to define the elastic deformation term ε _el as per Eq. (10):defining the equivalent axial stiffness EA _eq as follows [Eq. (11)]:In this way, the integral equation is fully explicit and can be solved without the need of iteration.

The deformation of the case study element during its storage phase was monitored with low precision instruments (rolling tape for shortening and analogue deflection gauge for mid-span deflection), typical of the standard control of production, having as objective a global check. Figure 6 reports the measures.

The above-described numerical procedure was applied to the roof element under investigation. In order to make its solution comparable with the simplified formulation of EN 1992-1-1:2005 (2005) (Eurocode 2), all data related to time-dependent properties of materials was taken from that standard. The numerical curves related to the evolution in time of axial shortening and mid-span deflection are reported in Fig. 6 in comparison with the experimental measurements. The axial behaviour is well predicted by the numerical algorithm, and the deflection shows a tendency to stabilise on a plateau, in accordance with the experimental data, but on a slightly over-estimated value. It can be observed that the difference between the numerical and experimental plateau value of the mid-span camber, of about 14 mm, is limited to only 1/3000 of the span.

EN 1992-1-1:2005 (2005) part 1-1 provides a closed formulation for the calculation of elastic and time-dependant losses in accordance with the multiplier method, reported in Eq. (12):The prediction of the evolution in time of the deformation calculated in accordance with this simplified criterion leads to a curve that is sensibly different from that obtained with the proposed formulation, with a relevant under-estimation of the mean pre-stressing losses, to which a larger camber of the element in time is associated. Figure 7 shows the comparison of the two loss curves plotted with a logarithmic function of the time. At the day of execution of the tests, 14 days from casting, the mean pre-stressing loss estimated through the time-step proposed formulation is equal to 15.0%, while the simplified formulation leads to 11.7% of loss.

The roof element monitored in storage was later subjected to load testing at the Antonio Basso company production plant in Treviso, Italy. Two loading tests were carried out in order to verify the mechanical behaviour of the element under combined flexure and torsion in elastic field and under pure flexure in pre- and post-cracking fields. Figure 8 shows sketches of the loading apparatus and of the measurement setup. Pictures of the experimental setup are shown in Figs. 9 and 10 shows particular views of the instrumentation. 30 hydraulic jacks were used in parallel to apply a quasi-uniformly distributed load to the element. They have been positioned along ten sections distanced by 3.70 m and connected at each section at both wing edges and at the centre. Repartition steel beams were placed at the edges to better distribute the load. The jacks were connected with bundles of rebars placed underneath the element, used as counter-acting weight, having an approximated global mass equal to 60 tons. One vertical displacement transducer was positioned in correspondence of each of the jacks positioned at the quarters of the span, for a total of nine vertical LVDTs, with three additional horizontal transducers placed at the top of the element at the quarters of the span with the aim to measure the relative opening of the wings. The vertical displacements are subtracted to the small deformation of the element in correspondence of the supports, measured by two vertical LVDTs with short stroke. The load tests were carried out by 4EMME (Altinier 2015).

The first test carried out was the one of combined flexure and torsion, where the load was applied to one row of jacks placed in correspondence of one wing only. In such a way, the vertical load was applied with an eccentricity equal to half of the width of the element with respect to its centre of the mass. The test is aimed at simulating the positioning of the structural completing shells on top of the roof element (Fig. 2), which is typically performed one span per time, studying the stability of the element under this temporary load condition.

Under an equivalent distributed load of 1.66 kN/m, a maximum torsional rotation of 0.23 rad was measured at mid-span, together with a wing distancing limited to half a millimetre in the same section. This shows the efficiency of the full collaboration of the cross-section. A computer model using 8-node hexahedra brick elastic elements (32 elements were used to describe the cross-section, each 250 mm long, for a total of 5248 brick elements) was developed with Straus7 (G+D Computing 2010) for the simulation of the test. The end shaping of the element provides a torsional clamp at the element ends, which in real constructions is also enhanced by distanced dowels, not used in the load test. The deformation profile of the mid-span cross-section of the beam torsionally clamped at the supports is shown in Fig. 11. Using an elastic modulus equal to 37 GPa, which was obtained from the elastic modulus growth curve in time provided by EC2 for a concrete in between classes C55/67 and C60/75 at 14 days, the numerical simulation provides vertical displacements equal to 12-17-21 mm in the monitored points. This displacement profile yields to a torsional rotation angle of 0.21 rad, which is close to the measured one.

The flexure test has been carried out by applying the same oil pressure to all hydraulic jacks. The graph of Fig. 12 shows the experimental equivalent distributed load versus mean mid-span deflection calculated as the average of the three measurements taken at wings and centre. Two load cycles were performed.

After an elastic branch in which all sections of the element were fully compressed, cracking was attained at about 10 N/mm, after which the experimental curve shows a softening tendency due to the diffusion of cracking from mid-span towards the supports.

A picture of the mid-span area of the element with the flexural crack pattern post-marked with the aid of the computer is shown in Fig. 13a. The cracks were all of modest opening and difficult to be observed.

The test was interrupted at the attainment of an equivalent distributed load of 14.0 N/mm (the SLS load for this element was calculated to be 10.0 N/mm), due to the uplift of the counter-acting strand bundles. Cracks at mid-span reached a height of about 600 mm from the bottom (Fig. 13a). The maximum wing opening was measured equal to 13 mm at mid-span, corresponding to practically negligible shallowing. The element, once the load was removed, showed negligible residual deflection and completely closed the cracks developed during the test, which suggests that the strands were not plasticised.

A non-linear semi-analytical mechanical model was implemented with the aim to compare the experimental results. The model is based on the evaluation of the non-linear moment vs curvature diagram of the current cross-section in pre-stressed concrete and on the later solution of the equation of the inelastic curve. A Sargin model (Sargin 1971) was used to model concrete in compression considering the above deduced base compressive strength (Fig. 14a), an elastic–plastic model with parabolic plastic branch was used for grade B450C mild steel (Fig. 14b) and an elastic–plastic relationship with linear hardening was used for pre-stressing steel having f _p(0.1)k equal to 1860 MPa (Fig. 14c). The strength values were not divided by safety coefficients, in order to catch the experimental behaviour of the element.

Figure 15a shows the non-linear moment vs curvature diagrams of the cross-section corresponding to mean pre-stressing losses equal to 0, 15 and 30% obtained through the solution of the translational and rotational equilibria of the section neglecting the tensile strength of concrete. It can be observed that, as well known by the designers, the pre-stressing losses do not influence neither the resistance of the element nor its ductility, due to the plastic properties of steel. To be noted that the analytical model developed does not take into account possible preliminary failures due to local instability. On the contrary, a relevant influence of the losses is observed with respect to the cracked moment, diminishing with higher losses, which is a fundamental parameter for a good behaviour of the manufact in service. The expected concrete stress distribution along the depth of the section is reported in Fig. 13b either considering or neglecting the tensile strength of concrete. A better correspondence with the height attained by the cracks observed on the element is obtained considering the tensile strength of concrete. It can be observed that the expected crack moment M _crack of the section with mean losses of 15%, as estimated in the previous paragraph after 14 days of ageing, is equal to about 4000 kNm. The uniformly distributed load corresponding to first cracking at mid-span is equal to q _crack  = 8M _crack /L ²  = 19.5 N/mm. In the experimental curve obtained with the flexure test an inflection point corresponding to cracking initiation is observed for a load of about 10 N/mm (Fig. 12). By subtracting to the total load the structural self-weight, equal to 10.0 N/mm, which is already acting before the test, the crack load is estimated with good precision. This confirms the estimation of about 15% of losses, providing a validation of the results of the time-step numerical procedure illustrated above.

The numerical prediction matches with good precision the experimental curve corresponding to the second cycle, slightly underestimating the cracked stiffness of the first cycle which is larger due to the contribution of the tensile strength of concrete, not anymore playing a role in the second cycle. The obtained non-linear moment vs curvature diagram is used to get the deflection profile of the element by solving the inelastic curve formula v’’ = θ(M) governed by a 2nd order differential equation. Figure 15b shows the numerical deflection profile corresponding to the application of 24 N/mm, equal to the sum of the structural weight load and the maximum equivalent distributed load given by the jacks, depurated by the pre-camber. The experimental points included in the graph show that the estimation is sound.