### Transcription

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