The bending resistance of the element is obtained using a developed numerical procedure that enables the relationship between sectional curvatures and bending moment to be calculated, based on the equilibrium equations, using the hypothesis of Navier-Bernoulli (Equations 4.4, with contribution of prestressing strands). The developed algorithm enables arbitrary material laws, σ(ε), and variable sectional dimensions, b(y), to be considered (Figure 7.18). Prestressing is introduced as the initial deformation of the prestressing strand. When the localisation of deformations in a discrete crack occurs, crack opening and deformations are related using Equation 4.74, which is in accordance with [SETRA, AFGC 2002] and [JSCE 2006], as discussed in § 4.3.5.5.
ξ
σ(ε)
χ ε
hts b(y)
Figure 7.18: Equilibrium of a cross section Prestressing concept
In ordinary RC structures, prestressing is usually designed in order to restrict or eliminate cracking due to dead load (by limiting tension stress in the concrete section) or to fulfil ULS requirements with initially predefined ordinary reinforcement ratio [Menn 1990]. Following the same concept, in UHPFRC structures prestressing can be of much higher efficiency, due to higher admissible compressive stress in comparison to ordinary concrete, and a low creep coefficient in the case of thermally treated UHPFRC elements (0.2 instead of 0.5 to 0.8) [Toutlemonde et al. 2005]. However, in the present study, for the first design iteration, the quantity of prestressing will be imposed by the geometry of the slab, that is by the position of ribs. The prestressing force is designed to maximally exploit the strength of the strands, not exceeding allowed compressive stress in the concrete.
According to technical documentation on Freyssinet prestressing strands and Article 4.1.5.2 of [SIA 2003b], stress in the prestressing steel is limited to σp,max ≤ 0.75 fpk during prestressing, and to σp ≤ 0.7 fpk immediately after prestressing. Maximum compressive stress in the concrete, σc, must not be higher than 0.6 fck. At time t , prestressing force is calculated by taking into account prestressing losses due to concrete shrinkage (εcs), creep (εcc) and relaxation of prestressing steel. The losses are estimated using εcs(t)= 10μm/m, for thermally treated elements, as explained in § 3.4.1, and
el c cc(t) ϕ(t,t0) ε ,
ε = ⋅ (7.12)
with creep coefficient ϕ(t, t0) = 0.2 for thermally treated elements (§7.3.2.1) and
c c
p p el
c A E
A
⋅
≈ ,0⋅
,
ε σ . (7.13)
According to article 3.3.2.7 [SIA 2003b], prestressing losses due to strands relaxation, for σp,0 = 0.7 fpk, are estimated as Δσpr / σp,0 ≈ 7 %. Thus, total prestressing losses are
Δσp / σp,0 = (εcs Ep+εcc Ep+Δσpr ) / σp,0 . (7.14) For a minimal cross-section area, with closely spaced ribs and small rib height (beff=0.4 m and
h=0.1 m), Δσp / σp,0=7.4% with one strand per rib and Δσp / σp,0= 7.64 % with two strands per rib.
For distant high ribs (beff =1 m and h = 1 m), Δσp / σp,0 = 7.26 % with one strand per rib and Δσp / σp,0= 7.33% with two strands per rib. In case when no thermal curing is provided, the value for εcs(t)= 500μm/m and ϕ(t, t0) = 0.6 result in an increase of losses, leading to Δσp / σp,0 =15.6 % for the smallest section. For further analysis, the prestressing losses are considered to be 10% of the initial prestressing force.
Design of bending resistance: case study
Two critical sections are considered: a section at the support (longitudinal bridge girder), designed for the maximal negative moment, and the mid-span section, designed for the maximal positive moment. As an example, positive and negative resistant bending moments of a UHPFRC T-shaped cross section with dimensions h=0.4, br=0.6, brw=0.07, hts=0.05 (values similar to those of the ribbed deck slab presented in [Toutlemonde et al. 2005]) and without prestressing force are shown in Figure 7.19. It is interesting to notice that, due to material mechanical properties (tensile ductility in the first place), higher bending resistance is attained by the elements with a larger sectional surface subjected to tensile stresses, which is contrary to the concept of RC beams. The points corresponding to the end of the elastic phase, the beginning of local tensile softening, and the maximal bending moment are noted as A, B and C in Figure 7.19. It can be seen that in the beginning of the development of the pseudo-plastic phase in tension, significant increase in load-bearing capacity is achieved with a very slight decrease of the initial elastic stiffness, which is an important property at service states.
0 0.005 0.01
Χ m1 0
50 100
MkNm
M
M
br
brw
hst
σ σ
fct f ,ct εu
σ A B C
h
Figure 7.19: Simulated bending resistance of T-shaped cross section made of UHPFRC with neither ordinary nor prestressing reinforcement: positive bending moment (thick line), negative bending moment (tin line)
Prestressing strands are placed as shown in Figure 7.20 for sections subjected to negative and positive bending moment respectively. Concrete cover of 25 mm is assumed as sufficient and minimal rib width is fixed at 70 mm.
B, C
A
B, C A
a) b)
σ
pstrand
UHPFRC
σ
pstrand UHPFRC
Figure 7.20: Position of prestressing strands in ribbed deck slab and distribution of prestressing stress, for the section subjected to: a) negative bending moment; b) positive bending moment
The contribution of the prestressing force to bending strength of a T-shaped beam is shown in Figure 7.21, using simulated moment-curvature relationship for the same beam geometry as in Figure 7.19. Figure 7.21 a) shows the response of the beam prestressed with one T15S strand, at σp=0.7 fpkand with prestressing losses of 10 % (black line), in comparison to the response obtained assuming an elastic-perfectly brittle behaviour for concrete in tension (dashed line), without prestressing strand (thin grey line) and with one prestressing strand T15S with no initial prestressing force (thick grey line). The relative contributions of strands and concrete in tension to ultimate moment, as well as the contribution of prestressing force to elements stiffness, clearly result from the plot. Figure b) shows the contribution of prestressing with variable number or disposition of strands:
a section with no strand (thin grey line), with one strand in the lower part of the rib (black line) or with two strands with different positions, one at the lower part of the rib and one in the thin slab (thin black line, nearly coinciding with the response of the cross-section with one strand), or two strands in the lower part of the rib (thick grey line).
a) b)
0 0.02 0.04
Χ m1 0
50 100 150
MkNm
M
M
0 0.01 0.02
Χ m1 0
100 200
MkNm
M
M M
M
Figure 7.21: Simulated bending resistances of UHPFRC sections: contribution of prestressing strands, prestressing force and concrete tensile force
It must be noted that the maximal concrete compressive stresses remains very low, also in the case when two prestressing strands are used, and bending strength is governed by the capacity of prestressing strands and concrete in tension. This suggests that strands of higher yielding strength could be beneficial in these sections.
It is assumed that the design value of the resistant moment, MRd, corresponds to the beginning of yielding of prestressing reinforcement. Figure 7.22 shows a comparison between the nominal moment-curvature relationship (grey line) and the design moment-curvature relationship (black line) obtained with K=1.25, γb=1.3 and γp =1.15. The maximal nominal moment (MR) and the design value of the resistant moment MRd are also indicated in the same plot. It is important to note that, in both cases, yielding of prestressing steel is attained before the tensile softening in concrete occurs, due to the pre-compression of concrete. As a consequence, the assumed definition of MRd is not influenced by the stress-crack opening relationship, neither for concrete nor for prestressing steel.
σP=0.7fpk
fct=0 for ε >εel
σP=0 Ap=0
This do not always apply, when strands of higher strengths are used or in case of sections with a different geometry.
0 0.01 0.02
Χ m1 0
50 100 150
MkNm
Figure 7.22: Design value of resistant bending moment (MRd, black line), in comparison to nominal resistant moment (MR, grey line)
A variation of the slab thickness in the interval 40-60 mm (§ 7.3.4.1) changes insignificantly the value of MRd(for h=100 mm, ΔMRd= ± 6 % with respect to MRd for hts=50 mm, while for h=800 mm there is no change in MRd). Influence of variation of rib spacing br on MRd expressed per T-girder is also not significant (Figure 7.23 a)), however, when MRd is represented per unit slab with (Figure 7.23 b)) the influence of rib spacing on slab bearing capacity is obvious. Two cases of design values of resistant bending moments are presented in Figure 7.23: for rib height h = 200 mm (grey line) and h = 400 mm (black line), with one prestressing strand per rib. As expected, the most relevant geometric parameter for MRd is the depth of the rib, h. Differently from ordinary concrete, this is not only due to the increased level arm, but also to the increased contribution of the tensile force sustained by concrete and to the fact that a sufficient compressive strength is provided.
a) b)
0 0.5 1
brm
0 50 100
MRdkNm
h = 400mm h = 200mm
0 0.5 1
brm
0 250 500
MRdkNmm
h = 400mm h = 200mm
Figure 7.23: Influence of rib spacing on design value of the positive resistant bending moment:
a) resistance per one T-girder; b) resistance per unit slab width.
Similar considerations apply to negative moment bending resistance, for which the results will be shown in the context of the following design step
Comparison of design and resistant moment values
Design values of bending moments due to dead and traffic load are obtained using the previously described 3D FEM model (§7.3.3). The number of ribs, nb(Figure 7.24) is varied whereas the slab width is kept constant, bd=12m. The considered geometries consisted of 14 to 28 ribs, with the rib height 0.2 to 0.4 m, resulting in equivalent slab thickness, hequiv in the range of 0.07-0.1m for 0.2 m ribs and 0.1 to 0.15 m for 0.4 m high ribs. The length of the cantilever parts of the slabs varies in the
MR
MRd
range of 2.3 to 3.5 m, as a function of rib spacing. For the same rib spacing, two different positions of the supports (main bridge girders, defined as a function of rib na1 position in Figure 7.24) are considered. Examples of disposition of traffic wheel loads are presented in Figure 7.24 a): L1, causing maximal negative moment on the cantilever part and L2 causing maximal positive moment for the mid-span section; the rest of the traffic load is disposed according the model presented in Figure 7.11. An illustration of simulated response of the slab under wheel loads is shown in Figure 7.24 b).
a) b)
L1
L2
Figure 7.24: Numerical slab model: a) parametric definition of slab geometry and example of disposition of concentrated traffic loads L1 (grey squares) for maximal negative moment, L2 (voided squares) for maximal positive moment; b) view of a beam response – deformations obtained using 3D finite element model of ribbed deck slab The results of the analysis for the slab with 0.4 m high ribs are presented in Figures 7.25 and 7.26. In Figure 7.25 design moment values (dots) issued from the calculation are compared with previously calculated design values of resistant moments (lines), for the mid-span section (Figure a)), and for the section at the support, (Figure b)). It can be noted that slabs with ribs height of 0.4 m and ribs spacing up to 0.7 m can be designed to sustain and transfer loads to the main girders. Presented design resistance moments are obtained with one strand T15S, thus, the small difference that exist between the design values can be overtaken by a slightly change in geometry or by application of prestressing strands of higher yielding strength. Thus further optimisations are possible, but the principal trends are well illustrated by the presented results. Slabs with ribs of smaller heights, on the contrary, are less appropriate to meet ULS design requirements for the given support disposition; for different support arrangements however, slabs with less deep ribs can represent attractive solutions.
a) b)
0 0.5 1
brm
0 250 500
MRd,MdkNmm
0 0.5 1
brm
0 250 500
MRd,MdkNmm
Figure 7.25: Resistant design moments (line) and design moments (dots) per unit slab with: a) in the mid-span; b) at the support