Stage 5 – Mat Foundation – guide
5 MAT FOUNDATIONS
General
The mat foundation is a type of direct foundation, made as a turned slab. The mat can be extended under the entire
construction surface (general foundation) which ensures a maximum ground support surface of the construction or can be partial,
under certain highly stressed elements of the structure. The general foundation is the recommended solution in a seismic area.
Mat foundations are usually used in the following situations :
soils resistance that require large foundation footing surfaces;
difficult or nonhomogeneous terrain, with risk of differential settlements;
the presence of groundwater;
vertical elements (pillars, walls ) are arranged at small distances which make it difficult to create isolated or continuous
foundations ;
the foundation together with the vertical structural elements of the substructure must create a rigid and resistant box;
high - rise buildings that transmit significant loads to the ground.
Pre-sizing
When establishing the minimum dimensions of the mat, geometric construction criteria are used (interax distances,
column width / length). To establish the minimum height of the mat, 1/8 (1/10) of the maximum interaxle opening will be taken,
as a rule, but not less than 400mm.
It is possible that in certain areas, for technological reasons (slopes , gutters, etc.), the height of the foundation may be
reduced, reducing the capacity of the plain concrete to shear force. In this case, local stirrups and reinforcement for the openings
will be provided . If parts of the foundation are constructed at different elevations, the transition from one elevation to another
will be made through plain concrete steps, respecting the conditions for plain concrete foundations block.
Geotechnical design (Verification at SLU and SLS)
ULS
According to NP112/2014 is needed to check the relation:.
𝑉 ≤𝑅 or Λ = ∙ 100 ≤ 100%
Giving that the most unfavourable design approach is DA3, the checking will be performed only for this design
approach.
where:
𝑉 – vertical concentrated force in the center of the foundation base computed with the values for the ultimate
limit state;
𝑅 – foundation soil resistance for vertical actions, design value.
𝐴’ ∙ 𝑝
𝑅 =
𝛾
where:
𝐴′ –foundation base reduse area.
𝑝 – critical pressure evaluated according to NP112/2014, Annexe F, pg 105)
𝑝 = 0,5 ∙ 𝛾 ∙ 𝐵 ∙ 𝑁 ∙ 𝑏 ∙ 𝑠 ∙ 𝑖 + 𝑞 ∙ 𝑁 ∙ 𝑏 ∙ 𝑠 ∙ 𝑖 + 𝑐′ ∙ 𝑁 ∙ 𝑏 ∙ 𝑠 ∙ 𝑖
𝛾 – partial coefficient for resistances on vertical direction according to the design approach considered.
SLS
The norm NP 112/2014 impose that the effective medium pressure acting on the foundation soil to be less than
the plastic pressure.
Stage 5 – Mat Foundation – guide
𝑝 . ≤𝑝
effective pressure
𝑝 . =
where:
𝑉 – vertical concentrated force in the center of the foundation base evaluated with the values for the
serviceability limit state;
𝐴 – base area.
plastic pressure (according to NP112/2014, pag 117-118)
𝑝 = 𝑚 ∙ (𝛾̅ ∙ 𝐵 ∙ 𝑁 + 𝑞 ∙ 𝑁 + 𝑐 ∙ 𝑁 )
For the SLS we need to verify also if the effective absolute settlement is smaller than the admissible settlement
evaluated according to NP 112-2014.
𝑠 ≤𝑠
Calculation of sectional efforts
Sectional efforts for the design of the foundation are established following the calculation of the infrastructure, which
can have different degrees of complexity. The design of the foundations must consider the compatibility of the ground
deformations with those of the structural elements. The calculation of the sectional efforts ( M , V ) in the characteristic sections
of the foundation are usually obtained with calculation programs that allow modelling the interaction between the foundation
and the ground.
General principles
When calculating the foundation, numerous factors must be considered, the most important being the foundation rigidity
and geometry, the size and distribution of the loads, the foundation soil deformability and strength and the execution stages.
The calculation aims to determine the contact pressures and deformations as well as the bending moments and shear
forces. In the calculations, the foundation can be considered rigid or flexible. The main criteria for assessing the relative rigidity
of the foundation in relation to the foundation soil are presented below.
For general mats having a rectangular shape in plan (L x B) and uniform thickness ( h ), the rigidity is determined with
the expression:
12 ⋅ 𝜋(1 − 𝜈 ) 𝐸 𝐿 𝐵
𝐾 = ⋅ ⋅ ⋅
1−𝜈 𝐸 2ℎ 2ℎ
where:
𝜈– concrete Poisson's ratio;
𝜈 – soil Poisson's ratio;
𝐸– modulus of elasticity for concrete;
𝐸 – linear deformation modul for soil;
𝐵– width;
𝐿– length;
ℎ– height.
The mat can be considered rigid if the following condition is met :
8
𝐾 ≤
𝐿
𝐵
Stage 5 – Mat Foundation – guide
In the case of slabs loaded by concentrated forces from columns arranged equidistantly in both directions and the column
loads do not differ by more than 20% between them, the flexibility coefficient is defined as :
𝑘 𝑏
𝜆=
4𝐸𝐼
where:
𝑏 ; – the width of a strip of foundation
considered between the means of two consecutive
openings between columns, equal to the distance
between two consecutive axes of the columns
(Figure 6);
𝐼 – the moment of inertia of a strip of
foundation considered between the means of two
consecutive openings between columns.
The eraser can be considered flexible if
the following condition is met :
1.75
𝑏 ≥
Approximate methods - Method of reducing loads at the centre of gravity of the mat
Simplified methods are used for the calculation of rigid foundations.
As stages
1) The centre of gravity of the mat surface is determined.
2) The eccentricities due to concentrated loads at the column level are determined
3) The pressures on the base of the mat are calculated:
∑𝑁 𝑒 𝑒
𝑝( ÷ ) = ± 𝑁 𝑦± 𝑁 𝑥
𝐴 𝐼 𝐼
4) The mat is analysed as a whole in each of the two directions parallel to the x and y axes:
- total shear force acting in any section through the foundation is equal to the arithmetic sum of all the loads and contact
pressures to the left of the section considered;
1 2 x
p1 N
p2
ey
ex
4 z 3
y
p4
p3
Figure 7
- the total bending moment acting in the same direction section is equal to the sum of the moments of the same loads and
pressures with respect to the considered section .
The method does not allow the evaluation of the distribution of the total shear force and the total bending moment along
the section. Consequently, it is necessary to introduce some simplifications.
Stage 5 – Mat Foundation – guide
Approximate methods - Method of dividing the mat into strips
When the column loads and column spacings do not differ by more than 20%, the foundation can be divided into
independent design strips . Each design strip is loaded by the forces corresponding to the columns supporting the respective strip.
The contact pressure diagram is determined, assuming a linear Navier -type law of variation .
Obtained values of bending moments and shear forces in significant sections can be used for the evaluation of
foundation reinforcement, although the position of the resultant of the column loads does not coincide with the position of the
centre of gravity of the resultant on the contact pressures.
Exact methods - are methods that consider the interaction between the foundation and the ground. Exact methods differ
depending on the model adopted for the ground. There are 3 categories of models:
1. Discrete environment model – Winkler model : the foundation soil under the foundation is replaced , strictly within
its dimensions, by independent springs.
2. Continuous medium model – Boussinesq model: the foundation soil is a continuous, elastic, homogeneous and
isotropic medium ; the global foundation -soil behaviour is considered over the entire area of influence of the foundation.
3. Hybrid model: the foundation soil is replaced with springs defined by constitutive laws that model the behaviour of
the continuous environment.
Exact methods - Methods based on the Winkler model
Mesh method. The floor is discretized into a number of beams with bending and torsional resistance. The torsional
resistance is defined by the shear modulus G. In finite element terminology, the finite mesh method uses non-conforming
elements because compatibility between element deformations is ensured only at the nodes.
Figure 8
Exact methods - Finite element method
The foundation is modelled by a set of elements interconnected at the nodes, while the soil is modelled by isolated
springs. The discretization can include the foundation and the rest of the structure. The nodes of the structure are assigned a
number of degrees of freedom depending on the type of analysis. Figure 9 shows an example of an analysis in which the
foundation is discretized by a plate element, and the soil by a Winkler medium . In this case, the degrees of freedom are one
translation in the vertical direction (settlement) and two rotations (about the axes in the plane).
Figure 9
Stage 5 – Mat Foundation – guide
Structural design – Mat reinforcements
Longitudinal reinforcement.
The mat is reinforced with horizontal reinforcement on two directions, arranged on the faces of the slab. Also,
reinforcement is required in the middle area of the slab to take up contraction stresses, when the mat has a thickness greater or
equal to 600mm. The reinforcement for intermediate contraction is arranged so that the maximum distance between the
reinforcement meshes does not exceed 500mm, or is determined by calculation.
Transversal reinforcement. Reinforcements for shear or punching force are provided when the relationships in SR EN
1992-1-1:2004 and SR EN 1992-1-1:2004/NB:2008 are not respected, namely:
- for the shear force calculation :
𝑣 ≤ 𝑣
where the width of the element will be taken as one meter (1.0m),
- for the punching calculation:
N Ed
nEd nRd,c
ui d
in which
N Ed - axial punching force in the loading situation considered in the design
u i - is the perimeter of the considered calculation contour
β - coefficient that considers the influence of the bending moment
d - the average useful height of the slab, which can be taken equal to ( d y + d z )/2 with d y , d z the useful heights in
the y and z directions of the calculation section
v DRC = n Rdc is the shear strength of plain concrete
If reinforcements are necessary, the provisions of point II6.1.1(1) of NP 112-2014 will apply.
The transverse reinforcements can be inclined reinforcements, minimum three bars Ф 14 in each direction , or vertical
reinforcements. They are arranged according to point 9.4.3 of SR EN 1992-1-1:2004 and SR EN 1992-1-1:2004/NB:2008.
The minimum reinforcement percentages for the foundation slab is 0.15% for each face and direction and 0.075% for
the intermediate reinforcement. The distance between the bar’s axes will be taken between 150 mm and 400 mm. The minimum
diameter is 14mm for the bars of the grids on the two faces and a minimum of 12mm for the intermediate bars.
