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Analytical Formulas for Highly-Eccentric Rigid Rectangular Footings on Elastic Soils

A.S. Bezmalinovica,∗

a

Universidad de Santiago de Chile, Av. Ecuador 3659, Santiago, Chile.

Abstract

This paper is concerned with analytically determining the linear contact pressure distribution under rigid rectangularfootings subjected to high two-way eccentricity. Four cases are considered that describe all potential scenarios for thegiven geometry.

For each case, recalling the equations of static equilibrium and the usual assumptions for rigid footings resting on anelastic soil, relationships between coefficients of a general plane and the loads and dimensions of a rectangular shallowfoundation, are found.

Solutions for the presented system of nonlinear equations can be reached using any conventional numerical scheme.Several case studies are selected to validate the proposed formulation and to establish its transcendency over graphical,tabular and iterative methods.

Keywords: shallow rectangular foundation, biaxial loading, two-way eccentricity, linear pressure distribution, activecompression zone, Newton-Raphson method2016 MSC: 00-01, 99-00

1. Introduction

. Due to the typical service conditions of engineering struc-tures, the distinctive load state of foundations has a biax-ial nature. From a Geotechnical Design perspective, the

soil-structure interaction defines the maximum compres-5sion stress pmax, transmitted to the supporting material.This value is then contrasted to the ultimate capacity puof the bearing soil.

A footing is considered rigid if v ≤ 2h [calavera], whereh is the footing height and v the free distance between10a supported column and the footing edge. If the mate-rial properties of the bearing soil are conceived elastic,the ratio of stress to settlement is constant. Since a rigidfoundation remains undeformed as it settles, the verticaldisplacement and the bearing pressure must be linearly15distributed beneath the base.

Depending on the position of a concentrated load P ,equivalent to the total soil pressure, the footing base maybe in full-compression or a fraction of it may lift. Un-der the latter event, an Active Compression Zone (ACZ)20appears, wherein the soil is allowed to react.

The condition p = 0 defines the neutral axis, while| p| ≥ 0 specifies a zone of the footing base-area called “ker-nel”, that bounds the position of P for the full-compressioncase occurrence. The full-compression has been studied by25Jarquio [1], Vitone and Valsangkar [2], Algin [3], Highter

∗Corresponding author

Email address: [email protected](A.S. Bezmalinovic)

[10] and others, giving formulas and charts for dimension-ing the footing area.

Footings with high two-way eccentricity have been ad-dressed by Meyerhof [?] and Teng [1], showing graphical30

methods, charts and the related equations. Roark [2] pro-vides tables, and Peck [3] mentions an iterative method.Yet, these procedures yield approximate solutions and areinefficient for designing structures subjected to several loadcombinations.35

Therefore, the objective of the work is to provide ana-lytical formulas for the linear contact pressure distributionunder rigid rectangular footings, subjected to high biax-ial eccentricity. The method is presented as systems of nonlinear equations that describe four eccentricity cases.40Solutions for several case-studies are reached using theNewton-Raphson scheme...

2. Equilibrium equations of two-way eccentric rigid

footings

Theory. Consider describing the geometry of a rigid foot-45ing using a cartesian reference (x,y,z), centered at thecenter of gravity of the element base-area region. Thestructure is subjected to a concentric load Nz, and torquesMx,My. Here, Nz is considered positive in compression,while Mx and My have a sign convention given by the50right-hand rule (see fig.?).

Defining the load eccentricities as ex = My/Nz andey = Mx/Nz, the previous biaxial load state is equivalent

to a single load Nz, placed at the point (ex, ey, 0) (fig?).

Preprint submitted to Computers and Geotechnics May 18, 2016


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Let p(x, y) be the bearing stress under a rigid founda-tion.

Nz =

Ω

p(x, y) dΩ

, (1)

Mx = Ω

(−y) p(x, y) dΩ

, (2)

My =

Ω

x p(x, y) dΩ

, (3)

where Ω

is the region described by the Active Compres-55sion Zone.

If a linear pressure distribution p(x, y) = Ax + By + C is assumed, then (1), (2) and (3) yield the following systemof equations:

Nz = Ω

(Ax − By + C ) dx dy, (4)

Mx = −

Ω

(Axy − B y2 + Cy) dx dy, (5)

My =

Ω

(Ax2 − Bxy + Cx) dx dy, (6)

and the line Ax + By + C = 0 identifies the neutral axis.Rewriting (4), (5) and (6):

Nz =A S

y− B S

x + C Ω

, (7)

Mx = − A I

xy + B I

x− C S

x, (8)

My =A I

y− B I

xy + C S

y, (9)

where S

x,S

y and I

x, I

y are the first and second moments

of area, respectively.All comma-denoted symbols are “effective”, since they60

are referred to the ACZ of the foundation. These proper-ties are all nonlinear functions of A, B and C . If the ACZholds a degree of symmetry, the product of inertia I

xy = 0.

3. Formulas for two-way highly-eccentric rectan-

gular rigid footings65

Introduction. In contrast, a Calculation section representsa practical development from a theoretical basis.

In the following, the right-hand-side of (7), (8) and (9)are solved on a rectangular footing of base-area dimensionsLx,Ly. Analytical nonlinear expressions on the form f =70...... are derived, noticing four possible configurations forthe ACZ (namely, I, II, III and IV), depending on theneutral axis position.

The sign of loads is (to be) taken positive, since includ-ing the orientation of torques only gives the quadrant on75the xy-plane where footi... (but affects the characteriza-

tion of the ACZ), and tensile loads relax the demand of soil bearing capacity...

On a rectangular footing, the kernel is: |ex/Lx + ey/Ly| ≤1/6.80

3.1. Case I

For relatively small eccentricities (when exactly?)(...).

The neutral axis intersects the lines x = −Lx/2 and y =Ly/2.

Ω

1 =[−Lx/2, Lx/2] × [−Lx/2,−(1/B)(−ALx/2 + C )](10)

Ω

2 =[−(1/A)(BLy/2 + C ), Lx/2] × [−(1/B)(−ALx/2 + C ), Ly/2](11)

Ω

3 =[−Lx/2,−(1/A)(BLy/2 + C )] × [−(1/B)(−ALx/2 + C ),−(1/(12)

Nz = Lx (LyB + LxA − 2C ) (−LyB + LxA + 2C )

8B

+ Lx (LyB + LxA + 2C ) (LyB − LxA + 2C )

8B

+ (LyB − LxA + 2C )

3

48AB (13)

−Mx =Lx

L3y

B3 − 3L2y

B2C + L3x

A3 − 3L2x

A2C + 4C 3

24B2

+ 1

192AB2

(LyB − LxA + 2C ) (LyB + 3LxA + 2C )

L2yB2 + 4LyAC + 4LyLxA2B + 7L2xA2 − 16LxAC + 4C 2+

(LyB − LxA + 2C )3

(LyB + 3LxA − 6C )

384AB2 (14)

My = L3

xA (LyB + LxA − 2C )

24B

+(LyB + LxA + 2C ) (LyB − LxA + 2

192A2B(15)

L2y

B2 + 4LyA(C − LxB) + LxA(7L

192A2B(16)

− (LyB − LxA + 2C )

3 (3LyB + Lx384A2B

(17)

3.2. Case II 85

The neutral axis intersects the lines y = ±Ly/2.

Nz =Ly

L2y

B2 + 3L2x

A2 + 12LxAC + 12LxC 2

24A , (18)

−Mx =L3y

B (LxA + 2C )

24A , (19)

My = Ly

L3xA3 − L2yB2C + 3L2xA2C − 4C 3

24A2 (20)

2


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3.3. Case III

The neutral axis intersects the lines x = ±Lx/2.

Nz =Lx L

2x

A2 + 3L2y

B2 − 12LyBC + 12LxC 2

24B

, (21)

−Mx =Lx

L3y

B3 − 3L2y

B2C + L2x

A2C + 4C 3

24B2 , (22)

My = L3

xA (2C − LyB)

24B (23)

3.4. Case VI

The neutral axis intersects the lines x = Lx/2 and y =90−Ly/2.

Nz = (−LyB + LxA + 2C )

3

48AB

, (24)

−Mx = (−LyB + LxA + 2C )

3(3LyB + LxA + 2C )

384AB2 ,

(25)

My = (−LyB + LxA + 2C )

3 (LyB + 3LxA − 2C )

384A2B (26)

4. Results

Introduction. Results should be clear and concise.

5. Discussion

Introduction. This should explore the significance of the95results of the work, not repeat them. A combined Re-sults and Discussion section is often appropriate. Avoidextensive citations and discussion of published literature.

6. Conclusions

The main conclusions of the study may be presented100in a short Conclusions section, which may stand alone orform a subsection of a Discussion or Results and Discussionsection.

The presented model allows for an efficiently calcula-tion of the maximum compressive bearing stress.105

Conclusion: in the proposed formulation, a linear pres-sure envelope allows for the footing section properties (thezeroth, first and second moment of area) appear naturally.

max pressure can be calculated using P(x,y) at footingcorners... which one on each case?110

sign of loads: due to the the double symmetry of rect-angles and the usual practice of placing of the referenceframe at the centroid of the shape, changing the torque’ssigns will only affect que quadrant in where the resultant’scentroid will appear. sign could only be used when cal-115culating the footing eccentricities to know . in the for-mulation, the maximum stress value is independent of the

quadrant in which falls the resultant. negative N’s (trac-tion)??

tyhe formulation presented has the particularity of be-120ing more algebraically cumbersome than the other three(more critical) cases.

in the current method, any pressure distribution func-tion and footing shape, can be selected!! The ease to rep-resent rectangular footings125

7. Appendices

If there is more than one appendix, they should beidentified as A, B, etc. Formulae and equations in appen-dices should be given separate numbering: Eq. (A.1), Eq.(A.2), etc.; in a subsequent appendix, Eq. (B.1) and so130on. Similarly for tables and figures: Table A.1; Fig. A.1,etc.

• document style

References

[1] Meyerhoff GG. Some recent research on the bear-135ing capacity of foundations. Can. Geotech. 1963; 1(1):16-26. [2] Bowles JE. Foundation Analysis and Design.3rd ed. New Jersey: McGraw-Hill; 1982. [3] Calavera J.Cálculo de Estructuras de Cimentación. 4th ed. Madrid:INTEMAC; 2000. [4] Teng WC. Foundation Design. New140Jersey: Prentice-Hall Inc; 1979. [5] Young WC, BudynasRG. Roarks Formulas for Stress and Strain. 7th ed. NewJersey: McGraw Hill. 2002. [6] Peck RB, Hanson WE,Thorburn WH. Foundation Engineering. 2nd ed. NewYork: John Wiley and Sons. 1974.145

[7] Jarquio R, Jarquio V. Design of footing area withbiaxial bending. J Geotech Eng-ASCE. 1983; 109(10):1337–1341. [8] Vitone DM, Valsangkar AJ. Stresses fromloads over rectangular areas. J Geotech Eng-ASCE. 1986;112(10): 961–964. [9] Algin HM. Practical formula for150dimensioning a rectangular footing. Eng Struct. 2007;29(6): 1128-1134. [10] Highther WH, Anders JC. Dimen-sioning footings subjected to eccentric loads. J GeotechEng-ASCE. 111(5). 1985; 659–665.

[11] Adrian, I. (2010, November). Pressures distribu-155tion for eccentrically loaded rectangular footings on elasticsoils. In Proceedings of the 2010 international conferenceon Mathematical models for engineering science (pp. 213-216). World Scientific and Engineering Academy and So-ciety (WSEAS).160

Reference style Text: Indicate references by number(s)in square brackets in line with the text. The actual au-thors can be referred to, but the reference number(s) mustalways be given. List: Number the references (numbersin square brackets) in the list in the order in which they165appear in the text.

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