CHAPTER 10 - SPECIAL PRESCRIPTIONS AND JUSTIFICATIONS
IV. P LATE DESIGN
The design of plates (particularly bridge slabs and webs) brings up a number of recurring questions: which stress values to consider? How to combine them? How to combine bending, shear stress, diffusion steels? Is there an overall performance model for plates, in reinforced concrete, subject to bending and shear forces?
The paragraphs detail Eurocode 2’s answers to these questions, together with useful additional information on those questions not dealt with in Eurocode 2.
IV.1.Stress calculations IV.1.1.Analysis method
For bridges, calculation of stresses in the plates is generally done by linear elastic analysis. Eurocode 2 part 1-1 permits other methods (elastic analysis with fixed moment redistribution, plastic analysis, non-linear analysis).
These methods are essentially used in buildings, where the risks from cracking are low. In civil engineering structures the use of plastic analysis is not recommended except in special cases. Analyses with fixed redistribution are possible subject to effective crack control at SLS.
IV.1.2.Calculation value used for design
Eurocode does not specify the moment value to use for plate design, particularly where concentrated loads are applied: is it the bending moment resulting from the analysis, or could a mean value be used? In compliance with past French practice, an average moment value might be used on a width equal to twice the thickness of the slab (average taken transversely in the direction of the stress considered).
Special prescriptions and justifications Chapter 10 - Special prescriptions and justifications
In practice, for thicknesses and loads currently used in civil engineering structures, there is very little difference between the two calculations. The bending moments of the abacuses in the bridge slabs published by Sétra have taken into consideration the point bending moments.
IV.1.3.Stress combinations
Model LM1 of Eurocode 1 part 2 leads to a combination of overall and local effects. For example, in an upper slab of a box girder bridge or in a composite bridge slab, at an intermediate support, an overall tension due to the negative moment created by the UDL loads distributed along the bay, and local bending of the slab created by a TS tandem situated near the section, may exist simultaneously.
There are two possible solutions:
• Deal in an exact manner with the load concomitants (particularly when it is not possible to dissociate overall and local effects, for example for slab bridges), or
• Take account of concomitances in a simplified way, from extreme stresses.
Eurocode 2 has no information regarding the combination of overall and local stresses. On the other hand, annex E of Eurocode 3 part 2 gives a combination factor ψ, a function of the span length. The combination rules proposed in this annex may be used to treat the examples cited above.
Fig./Tab.1.1): Combination factor depending on span length [EC3-2 Fig.E.2]
IV.2.General problematic of design of concrete plates
The figure below, from annex LL of Eurocode 2 part 2, represents the different stresses that might act on a slab element:
Fig./Tab.1.1): Different stresses that might act on a slab element
• 3 plate components nEdx , nEdy , nEdxy = nEdyx
• 3 slab components mEdx , mEdy , mEdxy = mEdyx
• 2 transverse shear forces vEdx , vEdy
In the case of a bridge slab, there are typically:
• a longitudinal stress nEdx , created by the global bending in the longitudinal direction,
• a stress nEdy, created by a possible transverse prestress,
• bending moments mEdx and mEdy created by TS loads applied on the slab,
• an in-plane shear flux nEdxy, created by the global torsion,
• transverse shear vEdx and vEdy, created by the TS loads applied on the slab.
In the case of a web of a box bridge, there are typically:
• a longitudinal stress nEdx , created by the longitudinal prestress,
• a bending moment mEdy created by transverse bending (effect of TS loads applied on the slab),
• an in-plane shear flux nEdxy, due to the beam shear/torsion combination.
The other components are generally small.
Eurocode 2 allows dimensioning of the necessary reinforcement for each of these stresses taken individually, according to the rules presented in the preceding chapters of this guide.
There is thus the question of the reinforcement combination obtained. Eurocode 2 deals with this problem in several ways:
• fixed combination rules for some special cases (particularly combination of in-plane bending and shear),
Special prescriptions and justifications Chapter 10 - Special prescriptions and justifications
• a general method of justification of a plate under a unity of concomitant stresses.
These combination rules are all formulated at ULS.
IV.3.Fixed combination rules for reinforcement
IV.3.1.Combination of shear stress steels/torsional stress/prestressing dispersion steels
The combination of shear stress and torsional stress steels is covered in chapter 6.Eurocode 2 gives no rules concerning the combination rules of prestressing dispersion steels. Previous practices might for example be used (see the Sétra design guide on prestressed concrete bridges built using balanced cantilever method):
Applications: slab subjected to a combination of bending/transverse shear stress.
This combination rule was developed in the chapter relative to the justifications of the shear stress. It is the well-known 'shift rule', where the calculation of longitudinal reinforcement is based on an offset moment of length al in the unfavorable direction:
• al = d if the slab requires no shear stress reinforcement
• al = z × cotθ / 2 if not, θ being the inclination of the shear stress struts.
IV.3.3.Combination bending / in-plane shear
These rules are given in Eurocode 2 for the connection of a slab to a concrete web. It is however quite acceptable to apply them in a more general way to webs and slabs.
Applications:
• connection of a slab to a concrete web or a steel web
• web subjected to a shear stress/transverse bending combination.
3.3.0.a)Rule for reinforcement
The corresponding combination rules are given in [EC2-1-1 6.2.4]. The combination rule may be summarized as follows: A = Max { Acis; ½ Acis + Aflex }, where
Acis is the reinforcing steel section required to balance the maximum shear/torsion and prestressing dispersion stresses [chapter 10–IV.3.1]
Aflex is the reinforcing steel section necessary to balance the maximum bending moment
This formula is written to illustrate the case of the figure (6.7) of Eurocode 2 part 1-1 (connection of a slab to a concrete web), where a single bending moment is considered to bend the upper fiber.
In the more general case where there are two bending moments of opposite signs, the distribution between the two layers of reinforcement must respect the following rules:
Asup + Ainf ≥ Max { Acis; ½ Acis + Aflex,sup; ½ Acis + Aflex,inf } where
Aflex,sup is the upper layer reinforcement section necessary to balance the corresponding bending moment Aflex,inf is the lower layer reinforcement section necessary to balance the corresponding bending moment
By proposing this combination rule, Eurocode considers there to be no simultaneous instance of maximum bending and maximum shear. However, Eurocode ignores the contribution of compressed concrete to shear resistance, which is on the safe side.
Eurocode does not specify how to distribute the steels between the two layers. The following distribution may, for example, be adopted:
Asup ≥ Acis / 4 + Aflex,sup
Ainf ≥ Acis / 4 + Aflex,inf
Asup + Ainf ≥ Acis
Other distribution is always possible, particularly for the evaluation of existing structures.
Example 1:
Aflex,sup = 20cm² Aflex,inf = 0cm² Acis = 12cm²
The total area of reinforcement should exceed max (12; 20+12/2) = 26.0 cm², to divide between the two layers. By adopting the recommended distribution, 23cm² will be placed on the upper layer and 3 cm² on the lower layer.
Example 2:
Aflex,sup = 5cm² Aflex,inf = 5cm² Acis = 36cm²
The total area of reinforcement should exceed max (36; 5+36/2) = 36 cm², to divide between the two layers while respecting a minimum of 14cm² on the upper layer and 14cm² on the lower layer. There will thus be 18cm² per layer.
3.3.0.b)Case of exemption from combination
When shear is low, that is when vEd < 0,4fctd [EC2-1-1 6.2.4(6)], only the steels necessary relative to bending need to be put in place.
3.3.0.c)Combination rules relative to compression in concrete
Eurocode 2 part 2 imposes in [EC2-2 6.2.4(105)] a combination rule for compression in concrete. According to this rule, when the concrete is raised to its plastic resistance to balance bending, it cannot also contribute to the resistance of the shear stress struts. It is advisable to verify the compression in the struts based on a reduced height of concrete, namely hf,red , where:
hf,réd = slab height – compressed height used in ULS bending
IV.4.General method of plate verification
Annexes F and LL of Eurocode 2 part 1-1 and Eurocode 2 part 2 present a method that may be used to verify in a general way the resistance at ULS s of a plate subjected to a given combination of concomitant stresses.
Special prescriptions and justifications Chapter 10 - Special prescriptions and justifications
The method is directly applicable to a plate with an orthogonal reinforcement, and it may be used with certain adaptations in the case of a plate with skew reinforcement.
The general principle of this verification consists in splitting the plate into three layers:
• 2 outer layers that balance the bending stresses and in-plane shear,
• 1 intermediate layer that balances the transverse shear
The plate stresses are broken down into the three layers. The 2 outer layers thus work as a membrane, which means they are subjected only to in-plane stresses, with no bending moments.
The articulation of the various parts of the Eurocode 2 is as follows:
• Annex F Eurocode 2 part 1-1 (modified by Eurocode 2 part 2, and completed by clause 6.109) shows the justification of a concrete membrane
• Annex LL of Eurocode 2 part 2 describes the way the stresses are broken down into the three layers, and the way the plate is justified
• Annex MM of Eurocode 2 part 2 is a simplified application of annex LL for webs
IV.4.1.General verification principle
The proposed method is iterative. It is much simpler to explain and to apply for a verification, but it may also be used in dimensioning. The approach to verification is described below.
The first step is to know if the plate is cracked or not. For this, the stresses are calculated on the whole thickness of the plate, on the upper face, in the center and on the bottom face of the plate, assuming it not to be cracked. Equation (LL101) then gives a criterion that determines if the plate is cracked or not. This criterion is expressed according to the principal stresses, which generalize in 3 dimensions the criteria of cracking by shear/tension.
If the plate is completely uncracked, it is sufficient to simply verify that the maximum principal compressive stress is less than fcd.
The plate will now be assumed to be cracked by application of the preceding criterion. If the thickness of the layers is known, the verification principle is as follows:
• The plate stresses are broken down into membranous stresses
• It is verified that each layer may balance the stresses to which it is subjected, by applying the rules of annex F for the external layers, and the rules of annex LL for the intermediate layer.
If there is a combination of layer thicknesses such that verification of each of them is satisfied, the plate is capable of resisting the applied stresses.
The justification models of the various layers are shown below:
• Intermediate layer (annex LL)
• External layers (membrane model of annex F)
The order of this presentation stems from the fact that the stresses in the external layers depend upon the choices made for justification of the intermediate layer.
IV.4.2.Verification of intermediate layer
The intermediate layer is a plate subjected to two transverse shear stresses vEdx and vEdy. These two shear stresses combine vectorially into one principal shear stress vEd0 in direction ϕ0 [EC2-2 Anx.LL.(109), Expr.(LL.121) and (LL122)].
The justification is then brought back to that of a slab subjected to a shear stress vEd0. The applicable rules are those of clause 6.2 of Eurocode 2, modified by the national annex for slabs.
Firstly the necessity of shear stress reinforcement is evaluated, by applying the formula (6.2a). This formula brings in the rate of longitudinal reinforcement in the slab direction. Here the longitudinal reinforcement make an angle ϕ0 with the direction of the principal shear stress, and it is possible to demonstrate that the rate of reinforcement to use is given by the formula (LL.123):
ρl = ρlx×cos²ϕ0 + ρly×sin²ϕ0 = (Aslx/h) ×cos²ϕ0 + (Asly/h) ×sin²ϕ0
where Aslx is the area of reinforcement per linear meter in direction x, Asly is the area of reinforcement per linear meter in direction y.
If the formula (6.2a) is verified, it is not necessary to use shear stress reinforcement.
If not, the truss model developed in 6.2.3 to dimension the shear stress steels is applied and the direction of the struts θ is chosen.
In the formula (6.2a), the longitudinal reinforcement in the outer layers should be taken into account.
There results from this an additional longitudinal stress vEd0×cotθ in the plate that will be equally distributed on the two outer layers. This stress is oriented toward the direction ϕ0. Its projection according to directions x and y gives the equations (LL.124 to LL.127). For example, for the additional stress in direction x:
nEdxc = (vEd0×cotθ)×cos²ϕ0 = (vEd0×cotθ)× (vEdx/vEd0)² = vEdx²/vEd0×cotθ (LL.126)
IV.4.3.Membrane model
The external layers are verified using the membrane model developed in annex F, completed by clause [EC2-2 6.8.7 Expr.(6.109)].
The application of this annex poses no particular difficulties. The designer is referred to the text. The justification principle consists of superposing two models independently:
• The membranous tensions are taken up by the reinforcement
• The shear stresses are balanced by a system of struts and ties; the compression in the struts is verified by taking account of the general compression using an interaction criterion (6.109), and the tensions developed by the ties are added to the stresses in the reinforcement.
The figure below shows the interaction criterion (6.109) for a bi-compressed membrane, and compares it with the simpler criterion in annex F of Eurocode 2 part 1-1 (σ < fcd).
Special prescriptions and justifications Chapter 10 - Special prescriptions and justifications
Fig./Tab.1.1): Domain of resistance for a bi-compressed membrane
The application of this model shows that even a slab tensioned in the two directions can balance the shear stresses. In other words, there is no interaction between the resistance to shear at the level of the slab and the traction.
The model proposed allows in theory the choice of a completely free orientation of the struts, which would assume a perfectly plastic concrete. In practice, the struts orientation is conditioned by the behaviour of the membrane at SLS. A large rotation may lead unacceptable cracking. For this, annex F has a limitation given at the end of clause [EC2-2 Anx.F.1(104)].
IV.4.4.Breakdown of stresses on membranes
In the previous paragraphs it was seen how to justify the different layers. It remains to describe the way the plate stresses are broken down into membranous stresses. The formulae are given by equations (LL137) to (LL148).
The general principle consists in writing the equilibrium in axial force and in bending moment between the plate and membrane forces.
For example, with the sign notations and conventions of annex LL, for the breakdown of stresses nEdx and mEdx
applied on the face with perpendicular axis x in two membranous stresses nEdxs and nEdxi, the equilibrium equations are:
nEdxs+nEdxi = nEdx
ys×nEdxs - yi×nEdxi = mEdx
The equations (LL137) and (LL138) are simply deduced from them.
If the plate is cracked by transverse stresses, the additional longitudinal stresses given by (LL126) should be added. The expressions (LL143) and (LL144) are deduced from them.
Finally, a special treatment should be given to the case (frequent) where the layers of steel are not centered in the outer membranes. This is the subject of the equations (LL149) and (LL150), which allow correction of the membranous stresses to take account of this eccentricity.
IV.4.5.Summary
In summary, the complete justification process of a cracked plate is as follows:
- choice of layer thicknesses (2 parameters)
- possible choice of θ if the plate is cracked under the transverse shear stress (1 parameter)
- for each membrane, choice of the inclination of the longitudinal shear stress struts (2 parameters)
If there is a combination of these 5 parameters such that the stresses in the concrete and the steels in the three layers are satisfied, then the plate is justified.
IV.5.Conclusion
The method called "sandwich" is a general method that appears satisfactory from a theoretical point of view, but whose implementation requires many calculations. In current cases, when it is possible, the simplified combination rules as described above will be favored. In more complex cases, for plates subjected to multiple stresses, the method in annex LL gives an overall, consistent answer to the question of dimensioning of the concrete plates and can help to define optimum quantities of reinforcement.