Design of slab-on-grades supported with soil reinforced by rigid inclusions

J. Racinais1 and C. Plomteux MENARD, Nozay, France

ABSTRACT

The design of industrial and logistic building’s slab-on-grades is a complex exercise. The design needs to consider the different loading types and configurations (uniform or alternated loading, racks, live loadings…) together with the relative positions fromthe hinged constructions joints to the loads, whose position and intensity can vary during the life of the structure. The non-uniform stress reaction distribution in the soil reinforced with rigid inclusions creates an additional stress in the slab with adifferent pattern than the ones of the loads and of the joints. The optimization of the design of the slab becomes a complex problem with three different intertwined patterns (loading, joints, and rigid inclusions) that can move relative to one anotherwith usually no typical symmetry conditions. Existing code of practice dedicated to slab-on-grades are only able to consider uniform soil conditions and the typical size of those structures forbid the modelling of the full extent of the slab. Through thedecomposition of this complex problem into the sum of three unit variable-separated problems, this paper presents a simple and comprehensive method to take into account all the parameters of the equation. This method is a powerful solution which is easyto use while allowing for the precise optimization of the design of slab-on-grades. The approach has been validated and calibrated with an extensive number of finite element calculations and has been integrated in the French ASIRI national researchprogram in France. Keywords: design, slab, rigid inclusions, subgrade reaction, load transfer platform, bending moment

1 MENARD, 2 rue Gutenberg, 91620 Nozay, France. jerome.racinais@menard-mail.com

1 INTRODUCTION

Slab-on-grades are used in industrial and logistic buildings for heavy storage which usually consists of several loading types and configurations from uniform (noted Load Case 1 as LC1 in the following) and alternated loading (noted LC2) to storage racks (noted LC3) inducing multiple punctual loadings.

While usually keeping an equivalent maximum contractual average value (typically 3 to 8 t/m²), the position and intensity of the loads, coming from temporary storage, can vary greatly during the life (and change of user) of the

structure. Their relative position from the hinged construction joints thus varies, highly influencing the stress distribution in the slab.

The design of those slabs is typically based on the soil profile, which is represented either by an equivalent vertical reaction coefficient KZ or an equivalent deformation modulus ES.

When potential settlement of the slab is a problem, an efficient and cost-effective solution consists of reinforcing the soil with a dense grid of vertical rigid or semi-rigid inclusions. In this scenario, a granular Load Transfer Platform is installed between the slab and the rigid inclusions.

The reduction of settlement provided by the reinforcement induces a significant reduction of the bending moment in the slab. However, the non-uniform distribution of stress reaction in a reinforced soil creates an additional stress in the slab on a different pattern from the stress reaction distribution of the loads and of the joints.

It should be noted that slab-on-grades are the only concrete structure in a building where tension stresses are allowed and thus require a precise estimation of the stresses.

The optimization of the design of the slab becomes a complex problem with three intertwined patterns (loading, joints, and rigid inclusions) that can move relatively to one another.

2 SEPARATION OF VARIABLES

In terms of deformation, the soil reinforced with rigid inclusions can easily be represented by an equivalent soil profile, each layer being affected by an equivalent Young modulus E* and a Poisson ratio ν (see Figure 1).

This equivalent soil profile can be deducted from an elementary axial-symmetrical calculation, centered on one single inclusion (see figure 2), where an equivalent average uniform load is applied.

Figure 1. Equivalent uniform soil profile

By defining, for any of the possible type of

loading, the following parameters:

- ES(NJ): bending moments distribution in a continuous slab (without any hinged joints) over an equivalent homogeneous soil profile

- ES(JT): bending moments distribution in a slab with hinged joints over an equivalent homogeneous soil profile

- RI(NJ): bending moments distribution in a continuous slab over a soil reinforced with rigid inclusions,

- RI(JT): bending moments distribution in a slab with hinged joints over a soil reinforced with rigid inclusions.

We can combine those parameters in order to decompose this complex problem (consisting of calculating RI(JT) for any possible configuration) in three separated-variable problems:

RI(JT) = [ma] + [mb] + [mc]

2.1 Parameter [ma]

Parameter [ma] = ES(JT) represents the impact of the loadings configuration on a slab with joints, without any impact from the non uniform reaction of the inclusions. This parameter directly is calculated by the structural engineer for all the configurations and relative positions between the loads and the hinged construction joints according to applicable codes of practice and regulation.

2.2 Parameter [mb]

Parameter [mb] = [RI(NJ)−ES(NJ)] represents the impact of the rigid inclusions on a slab without joints and doesn’t depend on the loading distribution. [mb] has typically the same value for LC1, LC2 and LC3 as long as their average surface loading are the same. As deducting ES(NJ) from RI(NJ) allows us to remove the effect of the distribution of the loads over the surface of the slab, this parameter only depends on the equivalent average loading and doesn’t depend on its type and configuration (uniform, alternate, punctual) (see 3.2).

As a consequence, [mb] can be estimated from an elementary axial-symmetrical calculation, centered on one single inclusion,

where an equivalent average uniform load is applied.

Figure 2. General principle of axial-symmetrical models In term of bending moment in the slab, this

calculation results in a positive bending moment +Msup at the vertical of the inclusion and a negative bending moment −Minf in the middle of the grid (see figure 3).

For the range of applied loading of this type of structure, the bending moments Msup and Minf can be considered proportional to the loading.

Figure 3. bending moment in the slab resulting from

elementary axial-symmetrical calculation As a consequence, parameter [mb] is always

included in the interval [+Msup; −Minf].

2.3 Parameter [mc]

Parameter [mc] = [RI(JT)−ES(JT)] − [RI(NJ) −ES(NJ)] represents the interaction of the rigid inclusions with the joints, without any impact from the distribution of loading or from the non

uniform reaction of the inclusions. Deducting [RI(NJ)−ES(NJ)] that represents the impact of the rigid inclusions on a continuous slab from [RI(JT)−ES(JT)] that represents the impact of the rigid inclusions on a slab with hinged joints, allows us to isolate the sole impact of the relative position of the joints from the inclusions and remove the effect of both the surface distribution of the loads, and the direct impact of the inclusion on the slab itself. This combination gives the same results for any loading configuration with same surface average value (see 3.3) and depends only on the geometry of the joints and of the inclusions.

By construction, hinged joints cannot transmit bending moments but can only transmit shear forces. The effect of the joints is thus to bring the bending moment in the slab to zero at position of the joint and to “shift” the bending moment curve around the joint by the corresponding value (see figure 4 around x = 3 m and x = 15 m)).

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Figure 4. comparison of bending moment in the slab with and without joints (position of joints are represented by bold

green lines) As a consequence, when we look at the

interaction between the joints and the inclusions (represented by parameter [mc]), the maximum effect of the joint can only be to bring the bending moment created by the inclusions to zero at position of the joints. The bending moment in the slab around the joints is thus “shifted” by the corresponding value. The bending moment created by the inclusion being [mb] = [+Msup; −Minf], the maximum impact of parameter [mc], representing the interaction

between the joints and the inclusion is [mc] = −[mb] = [+Minf; −Msup].

2.4 Conclusion

The main principle of the additional bending moment method proposed in this article is then to decompose the problem in three separated-variable problems, namely [ma], [mb] and [mc], that can be easily calculated separately.

Since that RI(JT) = [ma] + [mb] + [mc], that [mb] is always included in the interval [+Msup; −Minf] and that [mc] is always included in the interval [+Minf; −Msup], the provided method allows the two correcting terms [mb] and [mc] to be added to parameter [ma] normally calculated by the structural engineer:

RI(JT) = ES(JT) + [(Msup+Minf); −(Msup+Minf)]

The following paragraphs illustrate how these correcting terms can be easily assessed with the help of simple FEM calculation and how the precision of those intervals can be minimize provided information on the geometry of the joints and of the rigid inclusions.

3 FINITE ELEMENT MODELISATION

3.1 Preliminaries

In the course of the French ASIRI research program, extensive FEM calculation have been carried out with several FEM codes (PLAXIS 3D, FLAC 3D, COSMOS/M) in order to validate the proposed calculation method.

The 3D FEM model that has been used for those calculations is 15 m x 15 m, with 6 m spacing between the joints and 2.5 m spacing between the inclusions.

Different load cases have been considered, but for comparison, those load cases have been chosen with the same average surface loading: - Load Case 1 (LC1) : uniform load at 15.6

kPa - Load Case 2 (LC2) : alternated loading with

2.3 m wide loaded strips (34 kPa) and 2.7 m wide unloaded alleys

- Load Case 3 (LC3) : storage racks (59 kN point loads) in 1 m x 3 m spacing and 2.7 m wide unloaded alleys

Figure 5. top view of 3D FEM model with dimensions Analysis of the bending moment is made in

both principal directions from cross-sections A to H (in X direction) and 1 to 6 (in Y direction). Simple stress distribution consideration confirms that those directions are the most critical in terms of bending moments.

By convention, positive bending moment tends to create tension in the upper fiber of the slab and negative bending moment tends to create tension in the lower fiber of the slab.

3.2 Parameter [mb]

Parameter [mb] corresponds to the additional bending moment induced by the non-uniform reaction created by the rigid inclusions on a continuous slab (without any hinged joints).

In terms of bending moment in the slab, elementary axial-symmetrical calculation where an equivalent average uniform load (15.6 kPa) is applied results in a positive bending moment of +Msup = +5.7 kN.m/m at the vertical of the inclusion and a negative bending moment −Minf = −3.0 kN.m/m in the middle of the grid.

Figure 6. 3D FEM model for calculation of RI(JT) for load

case LC3 Figure 7 shows, for each load case (LC1 to

LC3) the corresponding bending moment distribution calculated with the 3D model from ASIRI program for: - ES(NJ): equivalent soil profile without any

hinged joints - RI(NJ): rigid inclusions without any hinged

joints - [mb]=[RI(NJ)−ES(NJ)] Although the 3 load cases create very different bending moment RI(NJ) in the slab in terms of intensity and distribution, the [mb] parameter is very similar for all cases and the curves show a common profile with a maximal value +Msup at the location of a rigid inclusion and a minimal value −Minf at the centre of a grid mesh. Values of the bending moments are completely similar to the elementary cell results under equivalent average load q = 15.6 kPa.

The 3D analysis thus confirms that [mb] is independent of the loading configuration and depends only on the equivalent average load q. This parameter is thus always included in the interval:

[mb] = [+Msup; −Minf]

3.3 Parameter [mc]

Figure 8 shows, for each load case (LC1 to LC3) the corresponding bending moment distribution calculated with the 3D model from ASIRI program for: - RI(NJ) : rigid inclusions without any - hinged joints - RI(JT) : rigid inclusions with hinged joints - [mc]=[RI(JT)- ES(JT)]-[RI(NJ)- ES(NJ)] Positions of hinged joints are indicated by bold green lines.

Once again, as expected, although the three load cases create very different bending moment RI(JT) in the slab in terms of intensity and distribution, the [mc] parameter is exactly the same for all cases and the curves show a common profile with a maximal value +Minf and a minimal value –Msup located at the position of the joints.

Those finite element models show that parameter [mc] does not depend on the loading configuration and moreover, calculations leads to the conclusion that [mc] is always included in the interval:

[mc] = -[mb] = [+Minf ; −Msup]

We can look in details at the value of [mb] and [mc] compared to the relative position of the joints and of the inclusions (see figures 7 and 8 −LC1−RI(NJ), LC1−RI(JT) and LC1−[mc]): - at x = 15 m, the inclusions are positioned

under the joint, the bending moment without joint was equal to +Msup and the addition of joint bring it to 0. At this location, the interaction between rigid inclusion and joint is given by [mc] = –Msup

- at x = 9 m, the joint is positioned exactly in between 2 inclusions. The bending moment without joint was equal to −Minf and the addition of the joint bring it to 0. At this location, the interaction between rigid inclusion and joint is given by [mc] = +Minf

Figure 7. Parameter [mb]

Figure 8. Parameter [mc]

- at x = 3 m, the closest rigid inclusion is located at 0.50 m from the joint. At this location, [mb]= +0.9 kN.m/m. When we look at the bending moment curve from the elementary cell calculation (see figure 9), and in particular the value of the bending moment at 0.5 m from the center of the inclusion, the value is also +0.9 kN.m/m which is normal since [mb] represent the effect of the inclusion on a continuous slab which can be directly calculated from the elementary cell (see §3.2). At this location also, [mc]= −0.9 kN.m/m= −[mb].

Figure 9. Bending moment profile from elementary cell Depending on the distance between rigid

inclusions and joints, we are able to determine [mc].

Figure 10. Parameter [mc]

Finally, as illustrated in Figure 10, the method

allows for different levels of precision depending on the knowledge of the designer concerning the relative position between rigid inclusions and hinged joints: 1. If the position of hinged joints is not known,

there is no possibility of optimization. The most conservative values +Minf and –Msup have to be considered. This range covers all

the possible positions of the hinged joints and rigid inclusions.

2. If the position of the hinged joints is known but not the position of the rigid inclusions, it is possible to consider the range limited by the upper bound curve and the lower bound curve. Range of [mc] is well-known at the location of a joint [+Minf ; −Msup] and the effect of a joint is 0 at a sufficient distance ; we can draw a hyperbolic curve between two consecutive hinged joints. These two curves take into account all the relative positions possible of the rigid inclusions with respect to the known locations of the hinged joints.

3. The knowledge of the relative position between rigid inclusions and joints allows one to draw the actual curve mc from the exploitation of the bending moment distribution deduced from the elementary cell.

4 CONCLUSIONS

Design of slabs-on-grade supported with soil reinforced by rigid inclusions can be completed by adding the results of: - [ma] model over an equivalent soil profile - [mb]: effect of the rigid inclusions on a

continuous slab bounded by the range [+Msup ; −Minf]

- [mc]: interaction between rigid inclusions and hinged joints bounded by the range [+Minf ; −Msup]

In the general case, the effect of the rigid inclusions interacting with joints is bounded by :

[(Msup+Minf) ; −(Msup+Minf)]

We can note that parameter [mc] is not needed if there is no hinged construction joints in the slab.

For a defined load intensity, the terms [mb] and [mc] do not depend on the loads configuration, contrary to the term [ma]. Parameters [mb] and [mc] can be easily deduced from an elementary cell analysis subjected to an equivalent average load q.

The proposed approach has been well tested using detailed 3D finite element analyses confirming its accuracy. The developed approach is conservative and simple as compared to the completion of full 3D FEM analyses.

Finally, this approach allows for the beneficial separation of the geotechnical design and the structural design, which greatly optimizes all aspects of the design phase. Geotechnical design consists in providing an equivalent soil profile and the values of [mb] and [mc]. Structural design consists of calculating the bending moments distribution in the slab over the equivalent soil profile (classical calculation based on existing code of practice) and adding at the end the values of [mb] and [mc] to take into account the rigid inclusions interacting with joints.