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A comparative study for the design of rectangular and circularisolated footings using new modelsArnulfo Luévanos-RojasJuarez University of Durango State, Gómez Palacio, Durango, México. arnulfol [email protected]: June 4th, 2015. Received in revised form: January 18th, 2016. Accepted: January 22th, 2016.AbstractThis paper presents a comparative study for the design of reinforced concrete isolated footings that are rectangular or circular in shape andsubjected to axial load and moments in two directions using new models to obtain the most economical footing. The new models take intoaccount the real soil pressure acting on contact surface of the footing and this pressure is different in all the contact area, with a linearvariation, this pressure is presented in terms of the axial load, the larger moment around the “X” axis and the smaller moment around the“Y” axis, where the centroidal axes are “X” and “Y” of the footing. The main part of this research is to show the differences between thetwo models. Results show that the circular footings are more economical compared to the rectangular footings. Therefore, the new modelfor the design of circular isolated footings should be used, and complies with real conditions.Keywords: rectangular footings design; circular footings design; bending moments; bending shear; punching shear.Un estudio comparativo para diseño de zapatas aisladas de formarectangular y circular usando nuevos modelosResumenEste trabajo presenta un estudio comparativo para diseño de zapatas aisladas de concreto reforzado de forma rectangular y circular sometidas acarga axial y momentos en dos direcciones usando nuevos modelos para obtener la zapata más económica. Los nuevos modelos consideran lapresión real del suelo actuando sobre la superficie de contacto de la zapata y esta presión es diferente en toda el área de contacto, con una variaciónlineal, esta presión se presenta en términos de la carga axial, el momento mayor alrededor del eje “X” y el momento menor alrededor del eje “Y”,donde los ejes centroidales son “X” e “Y” de la zapata. La parte principal de esta investigación es mostrar las diferencias de los dos modelos.Los resultados muestran que las zapatas circulares son más económicas con respecto a las zapatas rectangulares. Por lo tanto, el nuevo modelopara diseño de zapatas aisladas circulares se debe utilizar, y cumple con las condiciones reales.Palabras clave: diseño de zapatas rectangulares; diseño de zapatas circulares; momentos flexionantes; cortante por flexión; cortante porpenetración.1. IntroductionA foundation is a part of the structure which transfers theloads to the soil. Foundations are classified into superficialand deep foundations. There are important differencesbetween the two: depending on geometry, type of soil, andstructural functionality, and its constructive systems [1-11].A superficial foundation is a structural member where thedimensions of the cross section are large in comparison to theheight. The function of this type of foundation is to transferthe loads of a building to the soil at relatively shallow depths,less than 4 m approximately in relation to the level of thenatural ground surface [1-12].The distribution of soil pressure under a footing is afunction of the type of soil, the relative rigidity of the soil andthe footing, and the depth of foundation at the level of contactbetween the footing and the soil. A concrete footing on sandunder concentric loading will have a pressure distributionsimilar to Fig. 1a. When a rigid footing is resting on sandysoil, the sand near the edges of the footing tends to displacelaterally when the footing is loaded. This tends to decrease insoil pressure near the edges, whereas soil at a distance from The author; licensee Universidad Nacional de Colombia.DYNA 83 (196), pp. 149-158. April, 2016 Medellín. ISSN 0012-7353 Printed, ISSN 2346-2183 OnlineDOI: http://dx.doi.org/10.15446/dyna.v83n196.51056

Luévanos-Rojas / DYNA 83 (196), pp. 149-158. April, 2016.Figure 1. Pressure distribution under footing: (a) footing on sand; (b) footingon clay; (c) equivalent uniform distribution.Source: [9-12].the edges of a footing is relatively confined. On the otherhand, the pressure distribution under a footing on clay issimilar to Fig. 1b. As the footing is loaded, the soil under thefooting deflects in a bowl-shaped depression, relieving thepressure under the middle of the footing. For designpurposes, it is common to assume the soil pressures arelinearly distributed. The pressure distribution will be uniformif the line of action of the resultant force passes though thecentroid of the footing (see Fig. 1c) [9-12].In the design of superficial foundations, specifically in thecase of isolated footings there are three types of theapplication of loads: 1) Concentric axial load, 2) Axial loadand moment in one direction (uniaxial bending), 3) Axialload and moment in two directions (biaxial bending) [1-14].The hypothesis used in the classical model is to take intoaccount the uniform pressure for the design, i.e., the samepressure at all points of contact of the foundation with thesoil; this design pressure is the maximum value that occurs inan isolated footing [1-14].The classical model for the dimensioning of footings isdeveloped by trial and error, i.e., a dimension is proposed andusing the expression of the bidirectional bending one obtainsthe stresses acting on the contact surface, which must meetthe following conditions: 1) The minimum stress should beequal to or greater than zero, because the soil is not capableof withstanding tensile stresses, 2) The maximum stress mustbe equal to or less than the allowable capacity that the soilcan withstand [1-14].Some papers present the use of load testing onfoundations: Non-destructive load test in pilots [15];Evaluation of the integrity of deep foundations: analysis andin situ verification [16]; Others, show the use of static loadtests in the geotechnical design of foundations [17]; Stabilityof slender columns on an elastic foundation with generalisedend conditions [18]; A novel finite element method fordesigning floor slabs on grade and pavements with loads atedges.Mathematical models that calculate the dimensions ofrectangular, square and circular isolated footings subjected toaxial load and moments in two directions (biaxial bending)were developed [1,3,6], and also a comparative studybetween the rectangular, square and circular footings withrespect to the contact surface on soil was presented [8].Mathematical models for the design of isolated footingsof rectangular and circular shape using a new model werepresented [7,9].This paper presents a comparative study for the design ofreinforced concrete isolated footings that are rectangular orcircular in shape and that support a rectangular columnsubjected to axial load and moments in two directions toobtain the most economical footing. The new models takeinto account the real soil pressure acting on contact surfaceof the footing and this pressure is different in all of the contactarea, with a linear variation, this pressure is presented interms of the axial load, the larger moment around the “X”axis and the smaller moment around the “Y” axis, where thecentroidal axes are “X” and “Y” of the footing. Thecomparison is presented between the two new models interms of: 1) Moment around a a’-a’ axis that is parallel to the“X-X” axis and moment around a b’-b’ axis that is parallel tothe “Y-Y” axis; 2) Bending shear (unidirectional shear force)is localized on a c’-c’ axis that is parallel to the “X-X”; 3)Punching shear (bidirectional shear force); 4) Materials used(reinforcement steel and concrete). This study shows thedifferences between the two models to propose the mosteconomical footing.2. Methodology2.1. General conditionsAccording to Building Code Requirements for StructuralConcrete (ACI 318-13) and Commentary the critical sectionsare: 1) The maximum moment is located on the face of thecolumn, pedestal, or wall, for footings supporting a concretecolumn, pedestal, or wall; 2) Bending shear is presented at adistance “d” (distance from extreme compression fiber tocentroid of longitudinal tension reinforcement) shall bemeasured from face of column, pedestal, or wall for footingssupporting a column, pedestal, or wall; 3) Punching shear islocalized so that its perimeter “bo” is a minimum but need notapproach closer than “d/2” to: (a) Edges or corners ofcolumns, concentrated loads, or reaction areas; and (b)Changes in slab thickness such as edges of capitals, droppanels, or shear caps [7,9,10,20].The general equation for any type of footings subjected tobidirectional bending is [1-14, 21]:(1)where: σ is the stress exerted by the soil on the footing(soil pressure), A is the contact area of the footing, P is theaxial load applied at the center of gravity of the footing, Mxis the moment around the axis “X”, My is the moment aroundthe axis “Y”, Cx is the distance in the direction “X” measuredfrom the axis “Y” up the farthest end, Cy is the distance indirection “Y” measured from the axis “X” up the farthest end,Iy is the moment of inertia around the axis “Y” and Ix is themoment of inertia around the axis “X” [1-14, 21].2.2. A new model for rectangular footingsFig. 2 shows the pressures diagram for a rectangular footingsubjected to axial load and moment in two directions (biaxialbending), where there are different pressures in the four cornersand these vary linearly along the contact surface [7].The stresses anywhere on a rectangular footing subjectedto biaxial bending by equation (1) are found [7]:150

Luévanos-Rojas / DYNA 83 (196), pp. 149-158. April, 2016.32(3)2where: c1 is the dimension of the parallel column to theaxis “Y”, c2 is the dimension of the parallel column to theaxis “X” [7].Now, the gravity center “yc” of the soil pressure is [7]:4124(4)Moment around the axis a’-a’ is [7]:Figure 2. Soil pressures on the rectangular footings.Source: [7].,1212228(5)2.2.1.2. Moment around the b’-b’ axis(2)where: h is the side of the parallel footing the axis “Y”, bis the side of the parallel footing the axis “X”, A bh, Ix bh3/12, Iy hb3/12, Cx x, Cy y [7].The resultant force “FR2” is obtained through the volumeof pressure on the area formed by the axis b'-b’ and corners1 and 4 of the footing, this is [7]:32.2.1. MomentsCritical sections for moments are presented in section a’a’ and b’-b’, as shown in Fig. 3 [7].(6)22Now, the gravity center “xc” of the soil pressure is [7]:2.2.1.1. Moment around the a’-a’ axis4124The resultant force “FR1” is obtained through the volumeof pressure on the area formed by the axis a’-a’ and thecorners 1 and 2 of the footing, this is as follows [7]:(7)Moment around the axis b’-b’ is [7]:228(8)2.2.2. Bending shear (unidirectional shear force)Critical section for the bending shear is obtained at adistance “d” to from the junction of the column with thefooting is presented in section c’- c’ as seen in Fig. 4 [7].The bending shear “Vf” is obtained through the volumeof pressure on the area formed by the axis c’-c’ and corners1 and 2 of the footing [7].Now, the bending shear “Vf” is [7]:3222(9)2.2.3. Punching shear (bidirectional shear force)Critical section for punching shear appears at a distance“d/2” to from the junction of the column with the footing inthe two directions occurs in the rectangular section formedby the points 5, 6, 7 and 8, as shown in Fig. 5 [7].The punching shear acting on the footing “Vp” is obtainedthrough the volume of pressure on the total area minus therectangular area formed by points 5, 6, 7 and 8 [7].Figure 3. Critical sections for moments.Source: [7].151

Luévanos-Rojas / DYNA 83 (196), pp. 149-158. April, 2016.4,4(11)where: R is the radius of the footing, A πR2, Ix πR4/4,Iy πR4/4, Cx x, Cy y [9].2.3.1. MomentsCritical sections for moments are presented in section a’a’ and b’-b’, as shown in Fig. 7 [9].2.3.1.1. Moment around the axis a’-a’The resultant force “FR1” is obtained through the volumeof pressure on the area formed by the semicircle that is abovethe axis a’-a’ of the footing, this is presented as follows [9]:24444Figure 4. Critical sections for the bending shear.Source: [7].2/3Figure 6. Soil pressures on the circular footings.Source: [9].Figure 5. Critical sections for the punching shear supporting a rectangularcolumn.Source: [7].Now, the punching shear “Vp” is as follows [7]:(10)2.3. A new model for circular footingsFig. 6 shows the pressures diagram for a circular footingsubjected to axial load and moment in two directions (biaxialbending), where pressures are presented differently andvarying linearly along the contact surface [9].The stresses anywhere on a circular footing subjected tobidirectional bending by the equation (1) are found [9]:Figure 7. Critical sections for bending moments.Source: [9].152(12)

Luévanos-Rojas / DYNA 83 (196), pp. 149-158. April, 2016.Now, the gravity center “yc” of the soil pressure is [9]:2/43424 282/2 3(13)4 244/4Moment around the axis a’-a’ is [9]:62810124(14)22/24Figure 8. Critical sections for the bending shear.Source: [9].2.3.1.2. Moment around the axis b’-b’Now, the bending shear “Vf” is [9]:The resultant force “FR2” is obtained through the volumeof pressure on the area formed by the semicircle that is on theright side of the axis b’-b’ of the footing, this is [9]:24444/3442(18)/Critical section for punching shear appears at a distance“d/2” to from the junction of the column with the footing inthe two directions occurs in the rectangular section formedby the points 5, 6, 7 and 8, as seen in Fig. 9 [9].The punching shear acting on the footing “Vp” is obtainedthrough the volume of pressure on the total area minus therectangular area formed by points 5, 6, 7 and 8 [9].Now, the punching shear “Vp” is as follows [9]:22(16)4 2422.3.3. Punching shear (bidirectional shear force) 2/2 32234844(15)/Now, the gravity center “xc” of soil pressure is [9]:42223222/4(19)Mome