Step function method used in calculating continuous footing foundation placed of elastic and discrete soil.

Marin, Cornel ; Hadar, Anton ; Grigoras, Stefan 等

1. INTRODUCTION

This paper presents an alternative method of expressing the variation of the bending moment [M.sub.iy](x) and translations w(x) of elastic beam placed on elastic and discrete soil using the MathCAD STEP FUNCTION [PHI](x-a), (Marin, 2007). This method is an alternative to WINKLER method for elastic beams placed of elastic and continuous soil (Manoliu, 1986).

The practical application is the design of a continuous footing foundation placed under four columns. The axial compressive force in the columns is P. The layout of the continuous footing loading is represented in fig.1.

The properties of soil under the footings lead to a loading model having as reactions the distributed loads [q.sub.0] depending on translation (Manoliu, 1986, Gruia & Haida, 1990).

For this loading model, the condition between the internal soil reactions [V.sub.k] and the corresponding translations [w.sub.k] will become (Marin, 2007):

[FIGURE 1 OMITTED]

2. ANALYTICAL EXPRESSIONS

One considers a beam subjected to bending. The beam is characterized by length L and constant bending stiffness Ely. There are 4 different load types, presented in fig. 2 (Marin & Marin, 2006, Constantinescu et al., 2005):

--Bending moment N, at distance a from the left end of the beam;

--Concentrated force P, at distance b from the left end of the beam;

--Uniform distributed load [q.sub.0] which acts on a beam segment delimitated by the distances e and f

The differential equation of translations w(x) and rotations [[phi].sub.y](x) corresponding to a cross-section is (Marin, 2006):

[FIGURE 2 OMITTED]

[d.sup.2]w/[dx.sup.2] = - [M.sub.iy]/[EI.sub.y] (2)

Integrating twice the differential equation (2), one obtains after the first step, the rotations function [[phi].sub.y](x) and then the translations function w(x):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The analytical expressions for the cross sectional resultants considering the loading types from fig. 2, using the step function [PHI] are:

For shear force:

[T.sub.z](x) = -P x [PHI](x -b) - [q.sub.0] x (x - e) x [PHI] (x - e) + + [q.sub.0] x (x - f) x [PHI] (x - f) (4)

For bending moment:

[M.sub.iy] (x) = -N x [PHI](x - a) - P x (x - b) x [PHI] (x - b) - - [q.sub.0] x [(x - e).sup.2]/2 x [PHI] (x - e) + [q.sub.0] x [(x - f).sup.2]/2 x [PHI] (x - f) (5)

For the cross-sectional rotations functions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

For the cross-sectional displacements functions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

3. NUMERICAL APPLICATION FOR COMPUTING DIAGRAMS

The application's task is to obtain the expressions of required diagrams for the continuous footing foundation represented by the model in fig. 1 using the step function. The required diagrams will be:

--shear force diagrams [T.sub.z](x);

--bending moment diagrams [M.sub.iy](x);

--cross-sectional rotations distribution [[phi].sub.y](x);

--cross-sectional displacements distribution w(x) (settlements of the soil under the foundation).

The unknown reactions [V.sub.1], [V.sub.2], ..., [V.sub.7] will be obtain writing the following equations of equilibrium and deflections (Marin, 2006):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where: [r.sub.k] (k=1, ... 7) is distance from the left end of the beam to reaction [V.sub.k];

W(x) is the function defined by relation (7) for this particular case.

3.1 Hypothesis 1-low stiffness soil

Replacing in MathCAD the parameters values: P=10kN; b=1m; a=2 m; c=1m; k = [10.sup.7]N/m; EI = [10.sup.6] N x [m.sup.2] the following values of reactions and variation diagrams for the foundation (fig. 3) are obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

3.2 Hypothesis 2-high stiffness soil

Replacing in MathCAD the same parameters values: excepting k=108N/m the following values of reactions and variation diagrams for the foundation (fig. 4) are obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

4. CONCLUSIONS

This method is well suited for the constructive optimization of the structure, using the simulated values and varying different parameters: in the current application, the stiffness k was changed for optimization purposes.

As noticed by authors, the fact that the step function method uses simple operation expressions, it's obvious that the numerical applications are being solved fast with a minimum number of computational cycles. Because traditional methods use integral expressions which usually are solved by means of numerical methods, they imply a large number of computational cycles, causing slower results.

5. REFERENCES

Constantinescu, I.N., Picu, C, Hadar, A., Gheorghiu, H. (2006) Rezistenta materialelor (Strength of materials), BREN Publishing House, Bucharest (in Romanian).

Gruia, A. & Haida, V. (1990), Geotehnica si fundatii (Geotechnics and foundations), "Traian Vuia" Politechnic Institute, Timisoara (in Romanian).

Manoliu, I. (1986), Geotehnica si fundapi (Geotechnics and foundations), Technical Publishing House, Bucharest (in Romanian).

Marin, C. (2006), Rezistenta materialelor si elemente de teoria elasticitatii (Strength of materials and elements of theory of elasticity), Biblioteca Publishing House, http://fsim.valahia.ro/cursuri.html (in Romanian).

Marin, C. (2007), Aplicatii ale teoriei elasticitatii in inginerie (Engineering applications of theory of elasticity), Biblioteca Publishing House http://fsim.valahia.ro/cursuri.html (in Romanian).

Marin, C., Marin, A. (2006), Metoda analitica pentru calculul deplasarilor si rotirilor barelor drepte supuse la incovoiere (Analytical method for calculus of slope and deflection in straigth beams), SIMEC 2006, UTCB Bucharest (in Romanian).