Dimensioning of Isolated Footing Submitted to the under Biaxial Bending Considering the Low Concrete Consumption

Transcription

1 International Journal Materials Engineering 2017, 7(1): 1-11 DOI: /j.ijme Dimensioning Isolated Footing Submitted to te under Biaxial Bending Considering te Low Concrete Consumption Wanderson Luiz de França Filo 1, Roberto Cust Carvalo 1, AndréLuís Cristoro 1,*, Francisco Antonio Rocco Lar 2 1 Department Civil Engineering (DECiv), Federal University São Carlos (UFSCar) - São Carlos, Brazil 2 Department Structural Engineering, São Paulo University (EESC/USP), São Carlos, Brazil Abstract Tis researc presents te development and implementation area algoritms for geometric dimensioning rigid-foot-isolated reinforced subjected to te under biaxial bending. Te area algoritms are valid for square footing, omotetic to te pillar wit equal Bludgers and a rectangular dimension unrelated to te pillar. In te development algoritms, it is considered tat te load is positioned witin te central core footing inertia, i.e., applicable only in cases small eccentricities. Tis issue is relevant wen considering tat normally te geometry footing subjected to under biaxial bending wit poor accuracy is calculated, resulting in inaccurate dimensions terefore may increase te cost te element. In addition to te assumptions and analyzes used in te development algoritms, several numerical examples are sown to demonstrate teir efficiency. Examples were prepared considering te consumption for eac geometric type footing. Ultimately, a comparison is made between te results, setting te most economical type geometry in terms consumption. Keywords Footing, Reinforced, Under biaxial bending 1. Introduction Foundations are structural elements, usually in reinforced, wose function is to ensure te stability te work and transmit te loads acting on te structure resistant layer soil. Tey can be classified as superficial or deep [1]. Te coice te type foundation to use for a particular building sould only be made after consideration some elements, suc as te study adjacent buildings as well as teir type foundation and state te same, te caracteristics and te mecanical properties te soil on-site, te groundwater level, te values te loads to be transmitted to te foundation, type material tat makes up te superstructure, te cost, some tecnical aspects for carrying out te work, te types foundation on te market and oters factors [1]. Direct foundation is te first type foundation to be searced and just presenting advantage wen te area occupied by te Foundation cover between 50% to 70% te available area. However, tis type foundation sould not be used in cases non-compacted embankment, st clay, st sand and very st, collapsible * Corresponding autor: alcristoro@gmail.com (AndréLuís Cristoro) Publised online at ttp://journal.sapub.org/ijme Copyrigt 2017 Scientific & Academic Publising. All Rigts Reserved soils, tere is water were te lowering te water table is not justified [1]. Te footing is a type sallow foundation and can be te type associated, individually or race. It is intended to receive all active actions in te structure (pillars, walls or walls) and distribute tem directly into te ground so as not to cause differential settlements, wic may adversely affect te structural system, and not to break te ground. Its main advantage is tat tey are fast running (wic is not te case caissons) and do not require te use specific equipment and transport (suc as cuttings). Tis type foundation is recommended especially wen te ground is omogeneous [2]. As a structural element, te footing works by transmitting all actions te superstructure troug te stress distribution under te foundation base to te ground. Terefore, te elements te structure footing sould provide adequate resistance requests and sould be properly sized. Te geometric sape will depend on te design data, essentially te load capacity te soil and te load to be transmitted [2]. Among isolated footings, one can present various geometries and, in some cases, tey can be related to te dimensions te pillar. For most cases, it adopts te following types geometries in plant:

2 2 Wanderson Luiz de França Filo et al.: Dimensioning Isolated Footing Submitted to te under Biaxial Bending Considering te Low Concrete Consumption Circular: Square (Fig. 11); Rectangular: Wit equal overangs (Figure 9); Homotetic to te pillar (Figure 8); Unrelated to te pillar (Figure 7). For elevation, [3] and [4] mention tat te footing can ave a constant section (straigt) or variable eigt (pyramidal), as sown in Figure 1a and 1b, respectively. pressures at te base te foundation can be admitted as evenly distributed, except for supported foundations on rock. Figure 2. Distribution pressure on rigid or stiffness footing Figure 1. Types isolated footing [5] mention tat, by teir geometric caracteristics, pyramidal footings can make a great transmission load acting on te columns to te ground. everteless, tis solution is more expensive, especially regarding te use forms, and requires wit dry consistency, making its density tends to be deficient. [2] suggests tat te inclination α must be around 30 - wic corresponds to te angle internal friction te average compactness, because tere is no slip te, allowing te use side forms eigt 0 (Figure 1a). Tis solution is cost-effective, wereas te pyramidal sape te footing generates considerable savings in te consumption. Terefore, tis researc evaluate te pyramidal footing structural model. 2. Stress Distribution in te Soil [3] reported tat te stress distribution in te soil can be affected by factors suc as te stiffness -related foundation, soil properties, te intensity and caracteristics applied loads, and especially te area te footing base. Te form te stress distribution in te soil caused by a footing subjected to a centered axial load is directly dependent on several factors suc as: footing classification as stiffness, soil type (clay or sandy), load capacity te soil and intensity applied load. Bot rigid (Figure 2) or flexible footing (Figure 3), te real stress distribution is not uniform and te bending moment varies along te proper footing geometry, but it is recommended for structural calculation purposes, te Figure 3. Distribution pressure on flexible footing 3. Used Criteria Tis researc used te rigid footing, tus te eigt () te footing, sown in Figure 4, is conditioned wit te factor rigidity. For tis, we used te following criteria stiffness: A a p 3 Were: = eigt te footing; A= dimension te footing in one direction; a p = is te dimension te Pillar in te same direction. Wen cecking te expression above, in bot directions, te footing is considered rigid. Oterwise, te footing is considered flexible. Figure 4. Isolated pyramidal footing To determine te eigt 0, sown in Figure 4, [4] recommends te following criteria: cm

3 International Journal Materials Engineering 2017, 7(1): To calculate te volume, uses te truncated pyramid expression (V), expressed by Equation 1. V = 0 3. A. B + AB. a p. b p + a p b p + 0. AB (1) 4. Development Algoritms Te algoritms are designed so tat te entire base te footing becomes compressed, or in cases small eccentricities. In suc cases, te eccentricity is suc tat te load is positioned witin te central core inertia te footing, ten te neutral axis passes out te element. For tis, must satisfy te two conditions: e x A 6 M y A 6 e y B 6 M x B 6 In te case large eccentricities tat is, were te neutral axis passes troug te footing, tis solution is not valid since for te analysis te stresses is necessary to use oter teories, as sown in te researcers developed by [6-9]. Figure 5 sows an area subjected to any under biaxial bending, were stress are different in te four corners te contact surface. In tis case, in wic te neutral axis passes out te element, consider te ypoteses te general teory bending, tat is, te stress distribution vary linearly. So: σ (x,y) = A ± M x I x. C y ± M y I y. C x (2) Figure 6. Rectangular area submitted to under biaxial bending Were: S = A. B; C x = A/2; C y = B/2; I y = A³. B/12; I x = A. B 3 /12. Substituting in Equation 2: σ 1 = + 6.M x + 6.M y A.B A.B² A².B σ 2 = + 6.M x 6.M y A.B A.B² A 2.B σ 3 = 6.M x + 6.M y A.B A.B² A².B σ 4 = 6.M x 6.M y A.B A.B² A².B 4.1. Footing wit Unrelated to te Column Te matematical model for tis geometry (Figure 7) was developed by [10]. Tis type footing as C a C b but unlike omotetic footing, and ave no relation wit te column (te dimensions are arbitrary plants). (3) (4) (5) (6) Figure 5. submitted to te composed oblique bending. Adapted from [10] From Equation 2, te meaning te variables are: A= Contact Área; = Axial Load; M x = Moment in direction axis X; M y = Moment in te direction axis Y; C x = Distance in direction X from te axis Y; C y = Distance in direction Y from te axis X; I y = Moment inertia around te axis Y; I x = Moment inertia around te axis X. Figure 6 sows a rectangular area subjected to te under biaxial, were stress are different in te four corners te contact surface. Figure 7. Footing wit any relation in plant

4 4 Wanderson Luiz de França Filo et al.: Dimensioning Isolated Footing Submitted to te under Biaxial Bending Considering te Low Concrete Consumption Were: C a and C b are te overangs te footing; A is te biggest dimension te footing; B is te smallest dimension te footing; a p is te biggest dimension te column; b p is te smallest dimension te column. For tis, considering te following limits for te eccentricities: A A 1ª A 2ª e B B 1ª B 2ª 4.2. Homotetics Footing to te Column Te omotetic footing, sown in Figure 8, are tose in wic te relation between te sides te base is te same between te sides te pillar section, tat is: e x = M y = B 6 = 6. M y B e y = M x = A 6 = 6. M x A Equating, we ave: Ten: 6. M x A = 6. M y B B = A.M y M x (7) 1 t condition Te minimun stress is zero, so: Substituting (7) in (8): 0 =. A. M x M y Analogously: σ 4 = σ min = 0 0 = A. B 6. M x A. B² 6. M y A 2. B 0 =. A. B 6. M x. A 6. M y. B (8). A 6. M x. A 6. M y. A. M x M y A 1ª = 12.M y B 1ª = 12.M x (10) 2 t condition - Te maximum stress is equal to te permissible tension te soil, ten: σ 1 = σ max = σ Adm σ Adm = A. B + 6. M x A. B² + 6. M y A². B σ Adm. A 2. B 2 =. A. B + 6. M x. A + 6. M y. B (9) σ Adm. M x. A 3. M y. A 12. M y 2 = 0 (11) Solving Equation 11 gives te value A. Ten, te value B is given by: B 2ª = A 2ª.M x M y (12) Solving Equations 11 and 12 find out te values A and B, respectively, a rectangular footing, wen te maximum stresses is equal to te load capacity te soil. Terefore, te final dimensions te footing are found, as follows: Figure 8. Homotetics Footing to te pillar Were: C a e C b is te overangs te footing. A is te biggest dimension te footing; B is te smallest dimension te footing; a p is te biggest dimension te column; b p is te smallest dimension to te column. Te relation (r) tat determines te sides te footing is: Ten: r = A B = a p b p A = B. r B = A r 1 t condition - Te minimum stress is zero, ten: (13) σ 4 = σ min = 0 0 = A. B 6. M x A. B² 6. M y A². B 0 =. A. B 6. M x. A 6. M y. B (14) Replacing (13) in (14), is obtained: 0 =. A. A r 6. M x. A 6. M y. A r A 1ª = 6. M x.r+m y From Equation 15, obtains te values A. Ten: (15) B 1ª = A 1ª (16) r Solving Equations 15 and 16 we found te values A and B, respectively, a omotetic footing wen te minimum stress is zero.

5 International Journal Materials Engineering 2017, 7(1): t condition - Te maximum stress is equal to te permissible tension te soil, ten: σ 1 = σ max = σ Adm σ Adm = A. B + 6. M x A. B² + 6. M y A². B σ Adm. A 2. B 2 =. A. B + 6. M x. A + 6. M y. B (17) Replacing (13) in (17), we obtain: σ Adm. A 2. A r 2 =. A. A r + 6. M x. A + 6. M y. A r 0 =. A. r + 6. M x. r M y. r σ Adm. A 3 (18) Solving Equation 18 gives te value A. Ten, te value B is given by: B 2ª = A 2ª (19) r Solving Equations 18 and 19 finds te values A and B, respectively, a omotetic footing, wen te maximum stress is equal to te load capacity te soil. Terefore, te final dimensions te footing are found as following: A A 1ª A 2ª e B B 1ª B 2ª 4.3. Footing wit Equal Overangs Te footing wit equal overangs, sown in Figure 9, is one te most used types geometric because te armature is approximately equal in bot directions. Figure 9. Footing wit equal overangs Were: C a e C b are te overangs te footing; A is te biggest dimension te footing; B is te smallest dimension te footing; a p is te biggest dimension te column; b p is te smallest dimension te column. In tis case C a = C b, ten: A B = a p b p A = B + a p b p B = A a p b p A = B + a p b p B = A a p b p Denominating: = a p b p A = B + B = A (20) 1 t condition - Te minimum voltage is zero, ten: σ 4 = σ min = 0 0 = A. B 6. M x A. B² 6. M y A². B 0 =. A. B 6. M x. A 6. M y. B (21) Substituting (20) in (21), we ave: 0 =. A. A 6. M x. A 6. M y. A 0 =. A 2. A. 6. M x. A + M y. A M y. (22) Extracting te root te Equation 22 gives te value A. Ten, from Equation 20, one obtains te value B. Hence: B 1ª = A 1ª (23) Solving Equations 22 and 23 are te values A and B, respectively, a footing wit te same balance, wen te minimum voltage is zero. 2 t condition - Te maximum voltage is equal to te load capacity te soil, ten: σ 1 = σ max = σ Adm σ Adm = A. B + 6. M x A. B² + 6. M y A². B σ Adm. A 2. B 2 =. A. B + 6. M x. A + 6. M y. B (24) Substituting (20) in (24), obtains: σ Adm A 2 A 2 = A A + 6M x A + 6M y A (25) 0 = A A + 6M x A + 6M y A σ Adm A 2 A 2 Extracting te root Equation 25, we obtains te value A. Ten, from Equation 20, we obtains te value B. Hence: B 2ª = A 2ª (26) Solving Equations 25 and 26 results in te values A and B, respectively, a footing wit te same balance, wen te minimum stress is zero. Terefore, te final dimensions te footing are found as follows: 4.4. Square Footing A A 1ª A 2ª e B B 1ª B 2ª Te matematical model for tis type geometry was also developed by [10]. Figure 10 sows a square area L sides submitted under biaxial bending, were tensions are different in te four corners te contact surface.

6 6 Wanderson Luiz de França Filo et al.: Dimensioning Isolated Footing Submitted to te under Biaxial Bending Considering te Low Concrete Consumption Were: A = L²; C x = L/2; C y = L/2;; I y = L 4 /12; I x = L 4 /12. Terefore, te stress stays: σ 1 = L² + 6.M x σ 2 = L² + 6.M x σ 3 = L² 6.M x σ 4 = L² 6.M x L M y 6.M y + 6.M y (27) (28) (29) 6.M y L 3 (30) For te above conditions, te maximum eccentricities sould be: e x = M y = L 6 e y = M x = L 6 a p is te biggest dimension te column; b p is te smallest dimension te column. 1 t condition - Te minimum stress is zero, ten: σ 4 = σ min = 0 0 = L² 6. M x 6. M y 0 =. L 6. M x 6. M y 0 =. L 6. M x 6. M y L = 6.M x +6.M y (31) Solving Equation 31 we find L value a square footing wen te minimum stress is zero. 2 t condition - Te maximum stress is equal to te load capacity te soil, ten: σ 1 = σ max = σ Adm σ Adm = L² + 6. M x + 6. M y σ Adm. L 3. L 6. (M x + M y ) = 0 (32) Extracting te root Equation 32, L value a square footing is obtained wen te maximum stress is equal to te load capacity te soil. Te final value te L side sall be te greater te values obtained in te 1st and 2nd condition. So: L L 1st L 2st Figure 10. Square area submitted under biaxial bending. Adapted from [10] Figure 11. Squares footing Were: L a = L b = footing overangs; L is te side te footing 5. Application Algoritms Te algoritms developed in te previous section determines te minimum area required for te entire base te footing becomes compressed, tus brings te possibility to obtain a more economical geometric dimensioning. [10] sows tat for te same requests, footings wit circular section ave advantage in terms contact area over te footings a square or rectangular section. However, te work presented by [10] presents a comparison, only in terms contact area, excluding te specific consumption different types geometry. Terefore, we present a comparative study for cases rectangular (no relation to te column, omotetic and equal Bludgers) and square footings. Te study was done considering tree cases geometric dimensioning for te four different geometries footings and te results are sown in Table 1 (wit equal balance), Table 2 (omotetic), Table 3 (witout relation to te abutment) and Table 4 (square). Te comparison was done by analyzing not only te area in plan, as sown in Table 5, but also te consumption. It uses te same at te end requests and te results are compared, as sown in Figures 12 and 13.

7 International Journal Materials Engineering 2017, 7(1): Load capacity te soil (K/m² Table 1. Results for footing wit overangs EQUAL OVERHAGS Mx My A B Tmin Tmax A B Tmin Tmax A1 245,25 686,7 98,10 147,15 2,47 2,17 10,53 245,25 2,50 2,20 5,50 12,00 237,71 0,70 0,25 2,33 B1 196,20 686,7 68,67 98,10 2,53 2,23 47,75 196,20 2,55 2,25 5,74 47,54 191,83 0,70 0,25 2,42 C1 147,15 490,5 98,10 147,15 3,13 2,83 0,00 110,95 3,15 2,85 8,98 0,41 108,86 0,90 0,30 4,70 D1 98,10 490,5 68,67 98,10 3,07 2,77 17,64 98,10 3,10 2,80 8,68 17,68 95,34 0,90 0,30 4,55 A2 245,25 686,7 98,10 147,15 2,44 2,19 10,70 245,25 2,45 2,20 5,39 10,91 243,90 0,65 0,20 2,03 B2 196,20 686,7 68,67 98,10 2,50 2,25 47,66 196,20 2,50 2,25 5,63 47,66 196,20 0,70 0,25 2,40 C2 147,15 490,5 98,10 147,15 3,11 2,86 0,00 110,66 3,15 2,90 9,14 0,79 106,60 0,90 0,30 4,81 D2 98,10 490,5 68,67 98,10 3,04 2,79 17,59 98,10 3,10 2,80 8,68 17,68 95,34 0,90 0,30 4,57 A3 245,25 686,7 98,10 147,15 2,52 2,12 11,12 245,25 2,55 2,15 5,48 12,17 238,34 0,75 0,25 2,42 B3 196,20 686,7 68,67 98,10 2,58 2,18 47,92 196,20 2,60 2,20 5,72 47,73 192,37 0,75 0,25 2,53 C3 147,15 490,5 98,10 147,15 3,17 2,77 0,00 111,49 3,20 2,80 8,96 0,49 109,00 0,95 0,30 4,86 D3 98,10 490,5 68,67 98,10 3,12 2,72 17,71 98,10 3,15 2,75 8,66 17,76 95,49 0,95 0,30 4,70 Load capacity te soil (K/m² IPUT DATA Axial load (K) Moments (K.m) CALCULATED VALUES Minimum dimensions (K/m²) Case 1 (Column 20x50 cm) Case 2 (Column 25x50 cm) Case 3 (Column 25x40 cm) Table 2. Results for omotetic footing VALUES OF DESIG (K/m²) IPUT DATA CALCULATED VALUES VALUES OF DESIG Axial load (K) Moments (K.m) Minimum dimensions (K/m²) HOMOTHETIC Mx My A B Tmin Tmax A B Tmin Tmax Case 1 (Column 20x50 cm) 0 A1 245,25 686,7 98,10 147,15 3,68 1,47 8,39 245,25 3,70 1,50 5,55 10,03 237,43 1,10 0,40 3,71 B1 196,20 686,7 68,67 98,10 3,77 1,51 45,26 196,20 3,80 1,55 5,89 45,16 188,02 1,10 0,40 3,93 C1 147,15 490,5 98,10 147,15 4,80 1,92 0,00 106,45 4,80 1,95 9,36 0,50 104,30 1,45 0,50 7,98 D1 98,10 490,5 68,67 98,10 4,63 1,85 16,43 98,10 4,65 1,85 8,60 16,41 97,62 1,40 0,50 7,19 Case 2 (Column 25x50 cm) A2 245,25 686,7 98,10 147,15 3,28 1,64 10,77 245,25 3,30 1,65 5,45 11,47 240,77 0,95 0,30 3,02 B2 196,20 686,7 68,67 98,10 3,36 1,68 47,17 196,20 3,40 1,70 5,78 46,92 190,69 1,00 0,35 3,49 C2 147,15 490,5 98,10 147,15 4,20 2,10 0,00 111,22 4,20 2,10 8,82 0,00 111,22 1,25 0,40 6,36 D2 98,10 490,5 68,67 98,10 4,12 2,06 17,40 98,10 4,15 2,10 8,72 17,49 95,07 1,25 0,40 6,29 Case 3 (Column 25x40 cm) (K/m²) 0 A3 245,25 686,7 98,10 147,15 4,05 1,35 5,97 245,25 4,05 1,35 5,47 5,98 245,21 1,25 0,40 3,97 B3 196,20 686,7 68,67 98,10 4,15 1,38 43,03 196,20 4,15 1,40 5,81 43,13 193,26 1,25 0,40 4,21 C3 147,15 490,5 98,10 147,15 5,40 1,80 0,00 100,93 5,40 1,80 9,72 0,00 100,93 1,70 0,60 9,79 D3 98,10 490,5 68,67 98,10 5,09 1,70 15,30 98,10 5,10 1,70 8,67 15,31 97,84 1,60 0,55 8,16

8 8 Wanderson Luiz de França Filo et al.: Dimensioning Isolated Footing Submitted to te under Biaxial Bending Considering te Low Concrete Consumption Load capacity te soil (K/m² IPUT DATA Axial load (K) Moments (K.m) Table 3. Results to footing unrelated to te pillar O RELATIO WITH COLUM CALCULATED VALUES Minimum dimensions (K/m²) Mx My A B Tmin Tmax A B Tmin Tmax Case 1 (Column 20x50 cm) A1 245,25 686,7 98,10 147,15 2,83 1,89 11,79 245,25 2,85 1,90 5,42 12,40 241,23 0,80 0,25 2,50 B1 196,20 686,7 68,67 98,10 2,83 1,98 48,27 196,20 2,85 2,00 5,70 48,10 192,85 0,80 0,25 2,63 C1 147,15 490,5 98,10 147,15 3,60 2,40 0,00 113,54 3,60 2,40 8,64 0,00 113,54 1,05 0,35 5,28 D1 98,10 490,5 68,67 98,10 3,48 2,43 17,95 98,10 3,50 2,45 8,58 17,98 96,42 1,00 0,35 5,08 Case 2 (Column 25x50 cm) A2 245,25 686,7 98,10 147,15 2,83 1,89 11,79 245,25 2,85 1,90 5,42 12,40 241,23 0,80 0,30 2,68 B2 196,20 686,7 68,67 98,10 2,83 1,98 48,27 196,20 2,85 2,00 5,70 48,10 192,85 0,80 0,30 2,82 C2 147,15 490,5 98,10 147,15 3,60 2,40 0,00 113,54 3,60 2,40 8,64 0,00 113,54 1,05 0,35 5,31 D2 98,10 490,5 68,67 98,10 3,48 2,43 17,95 98,10 3,50 2,45 8,58 17,98 96,42 1,00 0,35 5,11 Case 3 (Column 25x40 cm) VALUES OF DESIG 0 A3 245,25 686,7 98,10 147,15 2,83 1,89 11,79 245,25 2,85 1,90 5,42 12,40 241,23 0,85 0,30 2,77 B3 196,20 686,7 68,67 98,10 2,83 1,98 48,27 196,20 2,85 2,00 5,70 48,10 192,85 0,85 0,30 2,91 C3 147,15 490,5 98,10 147,15 3,60 2,40 0,00 113,54 3,60 2,40 8,64 0,00 113,54 1,10 0,40 5,71 D3 98,10 490,5 68,67 98,10 3,48 2,43 17,95 98,10 3,50 2,45 8,58 17,98 96,42 1,05 0,35 5,24 (K/m²) Load capacity te soil (K/m²) IPUT DATA Axial load (K) Moments (K.m) Table 4. Results for square footing CALCULATED VALUES Minimum dimensions (K/m²) SQUARE Mx My L Tmin Tmáx L Tmin Tmax Case 1 (Column 20x50 cm) 0 A1 245,25 686,7 98,10 147,15 2,32 245,25 9,77 2,35 5,52 239,30 12,53 0,75 0,25 2,44 B1 196,20 686,7 68,67 98,10 2,38 196,20 47,04 2,40 5,76 192,80 48,03 0,75 0,25 2,54 C1 147,15 490,5 98,10 147,15 3,00 109,00 0,00 3,00 9,00 109,00 0,00 0,95 0,35 5,16 D1 98,10 490,5 68,67 98,10 2,92 98,10 17,33 2,95 8,70 96,01 18,06 0,95 0,35 4,99 Case 2 (Column 25x50 cm) A2 245,25 686,7 98,10 147,15 2,32 245,25 9,77 2,35 5,52 239,30 12,53 0,70 0,25 2,35 B2 196,20 686,7 68,67 98,10 2,38 196,20 47,04 2,40 5,76 192,80 48,03 0,75 0,25 2,56 C2 147,15 490,5 98,10 147,15 3,00 109,00 0,00 3,00 9,00 109,00 0,00 0,95 0,30 4,91 D2 98,10 490,5 68,67 98,10 2,92 98,10 17,33 2,95 8,70 96,01 18,06 0,90 0,30 4,58 Case 3 (Column 25x40 cm) VALUES OF DESIG (K/m²) A3 245,25 686,7 98,10 147,15 2,32 245,25 9,77 2,35 5,52 239,30 12,53 0,75 0,25 2,44 B3 196,20 686,7 68,67 98,10 2,38 196,20 47,04 2,40 5,76 192,80 48,03 0,75 0,25 2,54 C3 147,15 490,5 98,10 147,15 3,00 109,00 0,00 3,00 9,00 109,00 0,00 0,95 0,35 5,16 D3 98,10 490,5 68,67 98,10 2,92 98,10 17,33 2,95 8,70 96,01 18,06 0,95 0,30 4,72

9 International Journal Materials Engineering 2017, 7(1): Table 5. Comparison between all types EQUAL OVERHAGS HOMOTHETIC O RELATIO WITH COLUM A B A B A B Case 1 (Column 20x50 cm) A1 2,50 2,20 5,50 0,70 2,33 3,70 1,50 5,55 1,10 3,71 2,85 1,90 5,42 0,80 2,50 2,35 5,52 0,75 2,44 B1 2,55 2,25 5,74 0,70 2,42 3,80 1,55 5,89 1,10 3,93 2,85 2,00 5,70 0,80 2,63 2,40 5,76 0,75 2,54 C1 3,15 2,85 8,98 0,90 4,70 4,80 1,95 9,36 1,45 7,98 3,60 2,40 8,64 1,05 5,28 3,00 9,00 0,95 5,16 D1 3,10 2,80 8,68 0,90 4,55 4,65 1,85 8,60 1,40 7,19 3,50 2,45 8,58 1,00 5,08 2,95 8,70 0,95 4,99 Case 2 (Column 25x50 cm) A2 2,45 2,20 5,39 0,65 2,03 3,30 1,65 5,45 0,95 3,02 2,85 1,90 5,42 0,80 2,68 2,35 5,52 0,70 2,35 B2 2,50 2,25 5,63 0,70 2,40 3,40 1,70 5,78 1,00 3,49 2,85 2,00 5,70 0,80 2,82 2,40 5,76 0,75 2,56 C2 3,15 2,90 9,14 0,90 4,81 4,20 2,10 8,82 1,25 6,36 3,60 2,40 8,64 1,05 5,31 3,00 9,00 0,95 4,91 D2 3,10 2,80 8,68 0,90 4,57 4,15 2,10 8,72 1,25 6,29 3,50 2,45 8,58 1,00 5,11 2,95 8,70 0,90 4,58 Case 3 (Column 25x40 cm) SQUARE A3 2,55 2,15 5,48 0,75 2,42 4,05 1,35 5,47 1,25 3,97 2,85 1,90 5,42 0,85 2,77 2,35 5,52 0,75 2,44 B3 2,60 2,20 5,72 0,75 2,53 4,15 1,40 5,81 1,25 4,21 2,85 2,00 5,70 0,85 2,91 2,40 5,76 0,75 2,54 C3 3,20 2,80 8,96 0,95 4,86 5,40 1,80 9,72 1,70 9,79 3,60 2,40 8,64 1,10 5,71 3,00 9,00 0,95 5,16 D3 3,15 2,75 8,66 0,95 4,70 5,10 1,70 8,67 1,60 8,16 3,50 2,45 8,58 1,05 5,24 2,95 8,70 0,95 4,72 L Contact area A1 B1 C1 D1 A2 B2 C2 D2 A3 B3 C3 D3 EQUAL OVERHAGS HOMOTHETIC O RELATIO SQUARE Figure 12. Comparison between te required geometrics areas

10 10 Wanderson Luiz de França Filo et al.: Dimensioning Isolated Footing Submitted to te under Biaxial Bending Considering te Low Concrete Consumption Concrete Consumption A1 B1 C1 D1 A2 B2 C2 D2 A3 B3 C3 D3 EQUAL OVERHAGS HOMOTHETIC O RELATIO SQUARE Figure 13. Comparison between te consumption 6. Conclusions Analyzing Figure 12, it is observed tat tere is a sligt variation between te results wic can be neglected due to te adoption design values (multiples 5 cm). Tus, it is clear tat te area in te plant is not te determining factor in establising te type geometry tat as more advantage in terms actual consumption, unless tey are footings wit constant section, sown in Figure 1b. However, Figure 13 sows a greater variation in te results, and may conclude tat, for te examples analyzed, footings wit equal balances ave advantage over oters. Terefore, in terms economy, one can classify te geometries pyramidal footings in te following sequence: 1º- Rectangular wit overangs; 2º- Square; 3º- Rectangular unrelated to te pillar; 4º- Rectangular omotetic to te pillar. Te difference in consumption is associated wit rigidity te geometry adopted. Tat is, even wit te same area te footing base wit omotetic relationsip requires a greater eigt to acieve te same rigidity value te oter. Regarding te requests, te models presented in tis section can be used for te design footing subjected to centric load wit time in one direction or subjected to moment in bot directions from te neutral axis passes out te element. It is recommended tat tis model is used in soil were it can be considered a linear variation tension being restricted to uses were te neutral line passing troug te element. ACKOWLEDGMETS We are grateful to te Post Graduate Program in Structures and Civil Construction (PPGECiv) te Federal University São Carlos (UFSCar) and te University São Paulo (USP) for te support. REFERECES [1] Alonso, U. R. Exercícios de Fundações. São Paulo: Edgard Blücer, p. [2] Carvalo, R. C.; Pineiro, L. M. Cálculo e Detalamento de Estruturas Usuais de Concreto Armado. 2, 2ªed. São Paulo: PII, p. [3] Velloso, D. A.; Lopes, F. R. Fundações. 1. Rio de Janeiro: Ed. Oficina de Textos, p. [4] Silva, E. L. Análise dos Modelos Estruturais para Determinação dos Esforços Resistentes em Sapatas Isoladas p. Dissertação de Mestrado EESC/USP Escola de Engenaria de São Carlos, São Carlos, SP [5] Moya, F. B. V.; Juárez, J. A. L.; Blázquez, L. B. Apuntes de

11 International Journal Materials Engineering 2017, 7(1): Hormigón Armado. Adaptados a la Instrucción EHE-08. Escuela Politécnica Superior - Universidad de Alicante. San Vicente del Raspeig, España, [6] Özmen, G. Determination Base in Rectangular Footing under Biaxial Bending. DT&E International, Teknik Dergi Vol. 22, o. 4, October [7] Higter, W. H.; Anders, J. C. Dimensioning footing subjected to eccentric loads. ASCE - Journal Geotecnical Engineering, Vol. 111, o. 5, pp , [8] Gutiérrez, J. A. R.; Ocoa, J. D. A. Rigid Spread Footing Resting on Soil Subjected to Axial Load and Biaxial Bending. I: Simplified Analytical Metod. ASCE - Journal Geotecnical Engineering, Vol. 13, o. 2, pp , [9] Bellos, J.; Bakas,. P. Hig computational efficiency troug generic analytical formulation for linear soil pressure distribution rigid spread rectangular footing. ECCOMAS Congress VII European Congress on Computational Metods in Applied Sciences and Engineering, Crete Island, Greece. June [10] Rojas, A. L. A comparative study for dimensioning footing wit respect to te contact surface on soil. International Journal Innovative Computing, Information and Control, vol.10, no.4, pp , 2014.