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Design Charts for Seismic Bearing Capacity of Shallow Footings Awad Ali Al-Karni King Saud University, College of Engineering, Civil Engineering Department P.O. Box 800, Riyadh, 11421 Saudi Arabia Abstract Seismic loading could reduce the soil bearing capacity and cause large settlement or failure of the supported foundation. Mathematical analysis was developed in literature on calculating the seismic bearing capacity. The developed analysis used horizontal and vertical acceleration coefficients that applied by earthquake shaking in evaluating the seismic bearing capacity of dry soils. In this paper, design charts based on previous analysis of seismic bearing capacity by the author were developed. These charts will be used to evaluate whether the design based on the calculated static bearing capacity is adequate or not. If not, a new suggested design should be tried and checked again. Detail description of the procedures of using these design charts was given. Example problem was provided to show how to implement these procedures. The study shows that the most effective parameters on seismic bearing capacity are the horizontal acceleration coefficient (kh), the vertical acceleration coefficient (kv), the static safety factor (FsS), the factor D(=c/H), and the footing depth of embedment (Df), and the soil angle of friction (). Introduction Several theoretical studies on the seismic bearing capacity of soils supporting shallow footings have been presented (Sarma and Issofelis, 1990; Richards et. al., 1991, Richards et. al., 1993; Budhu and Al-Karni, 1993, Al-Karni, 1993). These studies used a limit equilibrium analysis with various assumptions on the shape of the failure surface. Richards et. al. (1991) used a simple Coulomb type planar failure surface, while Sarma and Issofelis (1990), and Budhu and Al-Karni (1993) and Al-Karni (1993) used logarithmic spiral failure surfaces. Budhu and Al-Karni (1993) presented the differences between the seismic bearing capacities calculated from these theories. The intention of this contribution is to present a graphical method for checking the stability of designed shallow footings when subjected to seismic loads and to suggest the required minimum safety factor for new design. The developed charts in here are based on the seismic bearing capacity presented by Budhu and Al-Karni (1993). Seismic Bearing Capacity Equation The seismic bearing capacity factors described by Budhu and Al-Karni (1993) can be used to modify popular static bearing capacity equations (Terzaghi, 1943, Meyerhof, 1963, Hansen, 1970, Vesic, 1973). For example, the Meyerhof bearing capacity 1 equation for vertical load can be modified to become a general equation to include seismic effects as follows. quE CNc S scdcec q f NqS sqdqeq 0.5BNS s d e (1) where quE is the ultimate seismic bearing capacity, C is soil cohesion, NcS, NqS, NS, are static bearing capacity factors, s and d are shape and depth factors respectively, q f is the overburden pressure, B is the footing width, is the unit weight of the soil, and ec, eq, and e are the seismic factors calculated from Budhu and Al-karni (1993) as 1 ec exp 4.3k h D (2) (3) 5.3k 1.2 h eq (1 k v ) exp 1 k v 9k 1.1 2 e (1 k v ) exp h 3 1 k v (4) where kh is the horizontal acceleration coefficient, kv is the vertical acceleration coefficient, D = C/(H) and H is the depth of the failure zone from the ground surface given as H 0.5B exp tan D f 2 cos 4 2 (5) where Df is the depth of embedment of the footing and is the angle of friction of the soil Critical Acceleration The critical acceleration is the horizontal acceleration that applied by the seismic shaking at which the footing will start to move down inside the soil mass. At the critical acceleration the allowable seismic bearing capacity (qaE) reduces to the allowable static bearing capacity (qaS), or qaE qaS (6) By denoting the static safety factor with FsS, and substituting of qaS by quS/ FsS, where quS is the ultimate bearing capacity, equation (6) becomes, qaE quS Fs S (7) and by replacing qaE by quE/FsE, where quE is the ultimate seismic bearing capacity and FsE is the seismic safety factor, then quE q uS FsE Fs S (8) where FsE is equal to one at the critical acceleration and by rearranging Eq. (8), then 2 FsE quE Fs s quS (9) Parametric Study The parameters that used in calculating the bearing capacity for static and seismic case include the angle of friction , the factor D (c/H), the ratio Df/B and L/B, kv, kh, and the static safety factor Fss. The effect of each of these parameters will be presented below. The object of this parametric study is to show how significant each of these parameters will affect the seismic safety factor. Some of these parameters may be excluded due to its insignificant effect. Effect of Angle of Friction, Equation (9) has been solved numerically for different angle of friction and a sample of the results is shown by Fig. 1. This figure shows a difference of about 0.06 of kh at FsE=1 for =0 =40. Such a result makes no significant effect of the angle of friction on the seismic safety factor on most cases. Based on this result, the effect of angle of friction was noticed to be significant when both c> and Df>0. So, the charts were developed for angle of friction equal to zero (for c> and Df>0) and angle of friction grater than zero (for c> and Df>0 soil). 2.4 2.0 1.6 f=0 f=10 f=20 f=30 f=40 Df/B=1.0, kv=0.4 D=0.4, FsS=3 FsE 1.2 0.8 0.4 0.0 0.0 0.2 0.4 kh 0.6 0.8 1.0 Fig. 1. Variation of FsE with kh at different values of angle of friction, . Effect of D The value of D is representing mainly the effect of soil cohesion on the seismic bearing capacity. By assuming that the angle of friction and Df/B are equal to zero to reduce the bearing capacity equation to one term, the seismic safety factor (FsE ) can be calculated as Fs E e 4.3kh (1 D ) Fs s (10) 3 By plotting this relationship, the variation of FsE with kh can be shown as in Fig. 2a for different values of D at static safety factor of 3. This figure shows a significant effect of D on the seismic safety factor, where the critical acceleration changes from 0.4 at D=0.4 to 0.8 at D=3.2. Therefore, for soil with zero angle of friction, the most significant parameters that affecting the soil bearing capacity in seismic environment is the value of D and the static safety factor. The effect of D was found to be significant also when Df/B>0 as shown in Fig. 2b. Therefore, the design charts will be developed for different values of D in the range of 0 to 2.0. 3.0 2.5 2.0 D=0.4 D=0.8 D=1.6 D=3.2 D=6.4 3.0 2.5 2.0 D=0.4 D=0.8 D=1.6 D=3.2 D=6.4 Df/B=3, kv=0 FsS=3, =30 FsE FsE 1.5 1.0 0.5 0.0 0.0 0.3 0.6 kh Df/B=0, kv=0 FsS=3, =0 1.5 1.0 0.5 0.0 0.9 1.2 0.0 0.2 0.4 kh 0.6 0.8 1.0 (a) (b) Fig. 2. Variation of FsE with kh at different values of D (a) at Df/B=0, (b) at Df/B=3. Effect of Df/B Fig. 3 shows the effect of Df/B is significant for Df/B between 0 and 1. For example, at D=0.8, the critical acceleration at FsE equal to one is equal to 0.36 at Df/B=0.0 and equal to 0.28 at Df/B=1. However, there is no significant changes when Df/B becomes >1. For example, the difference in the critical acceleration values at Df/B =1 and Df/B =2 is about 0.03. This resulted in development of design charts considering only Df/B =0 and Df/B1. Effect of L/B An example of the results of the variation of FsE with kh at different values of L/B is shown in Fig. 4. This Figure shows no significant effect of the L/B on the value of the seismic safety factor. The parameter L/B will be excluded during the development of the design charts. 4 3.0 2.5 2.0 3.0 Df/B=0 Df/B=1 Df/B=2 Df/B=3 Df/B=4 Fss=3, kv=0.0 D=0.4, =30 2.5 2.0 L/B=0 L/B=1 L/B=2 L/B=3 L/B=4 Fss=3, kv=0.0 D=0.4, = FsE FsE 1.5 1.0 0.5 0.0 0.0 0.2 0.4 0.6 1.5 1.0 0.5 0.0 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 kh kh Fig. 3. Variation of FsE with kh at different values of Df/B. Fig. 4. Variation of FsE with kh at different values of L/B. Effect of kv The effect of kv on seismic bearing capacity was found to be insignificant when Df is equal zero as shown in Fig. 5a. However, the difference could be significant when significant when Df is greater than zero as shown in Fig. 5b. For example, the difference between the values of the critical acceleration at kv =0.4 and kv =0 is about 0.065 at Df=1 and reduces to 0.0 at Df =0. The difference could reach about 0.12 in some cases. It was found that the effect of kv will not change much when Df becomes grater than 1. Therefore, the vertical component of acceleration can not be excluded in developing the design charts for Df>0. Thus the design charts were developed at values of kv in the range of 0.0 to 0.4 in which the common value of kv falls. 3.0 2.5 2.0 3.0 kv=0.0 kv=0.1 kv=0.2 kv=0.3 kv=0.4 D=0.4, =30 FsS=3, Df/B=0 2.5 2.0 kv=0.0 kv=0.1 kv=0.2 kv=0.3 kv=0.4 D=0.4, =30 FsS=3, Df/B=1 FsE 1.5 1.0 0.5 0.0 0.0 0.2 0.4 FsE 1.0 1.5 1.0 0.5 0.0 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 kh kh (a) (b) Fig. 5. Variation of FsE with kh at different values of kv (a) at Df/B=0, (b) at Df/B=1. 5 Effect of Static Safety Factor The value of static safety factor was found to be a major control variable in developing the design charts. The significant effect can be noticed as shown in Fig. 6. From this figure, the critical acceleration increased from 0.28 at FsS= 3 to 0.46 at FsS=7. The design charts were developed at values of static safety factor in the range of 3 to 7. 6.0 5.0 4.0 Fss=3 Fss=4 Fss=5 Fss=6 Fss=7 Df/B=1, kv=0.2 D=0.8, =30 FsE 3.0 2.0 1.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 kh Fig. 6. Example of variation of FsE with kh at different values of static safety factor. Design Charts The proposed design charts represent the variation of FsE with kh for different variables including the angle of friction , the factor D, the ratio of Df/B, kv, and the static safety factor Fss. At certain values of the parameters mentioned above, the seismic safety factor was calculated at different values of kh and a certain value of Fss. This process was repeated for different values of Fss to be able to draw the curves of the variation of FsE with kh for different values of Fss. These procedures will be repeated using different values of the parameters above. Some variables will be eliminated for their little effect on the value of the critical acceleration as discussed above. According to the parametric study above, the design charts were found to be classified into 1. Charts for D>0, Df/B =0, and 0, for any kv. 2. Charts for D0, Df/B 1, and =0, for different kv. 3. Charts for D0, Df/B 1, and >0, for different kv. 4. Charts for D=0, Df/B =0, and >0, for any kv. The design charts were given by Appendix A. Thus, for a given problem, the corresponding charts should be used. If the problem is not fallen in any of the category above (i.e., Df/B=0.25), the closest one should be used since the margin of error is not that significant. Design Procedure 6 The design charts can be used for new design of foundation or for an existing one. It is suggested here to design the foundation first for static case and then check the design using the charts here. If the seismic safety factor is greater than or equal to the required one by the designer, then the static design is safe. If not, a new design should be tried. The following procedure could be followed for using the charts: 1. Design the foundation for static case, or get the design parameters for the existing foundation. 2. Calculate the value of D, and Df/B. 3. Inter to the proper design chart using the value of kh and the design static safety factor. 4. Find the value of FsE and check with the required minimum one. 5. Repeat the design if FsE is below the minimum one. Example Fig. 6 shows a typical shallow foundation carrying a load of 3500 kN. It is required to check the stability of this foundation against earthquake loading with k h=0.25 and kv=0.1. Fig. 6. Typical shallow foundation subject to kh=0.25 and kv=0.1. Solution According to the given soil parameters and foundation dimensions, the ultimate bearing capacity (qu) according to Vesic (1972) procedures is equal to 2195 kN. Since qapp is equal to 700 kN/m2, then the static safety factor is equal to 3.13 (i.e. =2195/700) which is considered as enough for static design. For seismic design, the value of D should be determined first. From Eq. 6, the value of H is equal to 6.95, and then D (=c/H) is equal to 0.2. According to the design chart given by Fig. A.5.2 at (the effect of D in this case is not significant) FsS=3.13, Df/B=1, kh=0.25 and kv=0.1, the seismic 7 factor of safety below one and equal about 0.95. Then the design is not enough to resist the applied seismic load and redesign is recommended. Conclusion Seismic bearing capacity design charts were developed here. These charts were divided into four groups due to the soil properties and the design dimensions of the foundations. These groups include: 1. Charts for D>0, Df/B =0, and 0, for any kv. 2. Charts for D0, Df/B 1, and =0, for different kv. 3. Charts for D0, Df/B 1, and >0, for different kv. 4. Charts for D=0, Df/B =0, and >0, for any kv. Using these chart lead to a quick answer whether the designed foundation is safe or not. In case, of new design, the charts can be used for finding a suggested design value of the static safety factor that satisfy the required minimum value of seismic safety factor. For a given problem, the corresponding charts for the design parameters should be used. If the problem is not fallen in any of the category above (i.e., D f/B=0.25), the closest one should be used since the margin of error is not that significant. In case of more precise results, the calculation should be done according to the given equations above. However, seeking very precise results is not recommended due to the approximation of the method. References Al-Karni, A.A., and Budhu, M. (1994). “ Seismic Settlement of Shallow Footings on Sand”, Vertical and Horizontal Deformations of Foundations and Embankments, Geotechnical Special Publication No. 40, Settlement '94, College Station, USA, pp. 748 - 759. Budhu, M., and Al-Karni, A. A. (1993). "Seismic bearing capacity of soils." Geotechnique, 43 (1) , 181-187. D'Appolonia, D.J., D'Appolonia, E.D. and Brissette, R.F. (1968). "Settlement spread footings on sand." J. SM &. Fnds. Div., ASCE, 94, 1011-1053. of Hansen, J. B. (1970). "A revised and extended formula for bearing capacity." Danish Geotechnical Institute, Bulletin 28, Copenhagen. Meyerhof, G. C., (1963). "Some recent research on the bearing capacity of foundations." Canadian Geotechnical Journal, 1 (1), 16-26. Richards, R., Elms, D.G., and Budhu, M. (1991). "Soil fluidization and foundation behavior." Proc. Second International Conference on Recent Advances in 8 Geotechnical Earthquake Engineering and Soil Dynamics, Rolla,Missouri, I, 719723. Richards, R., Elms, D.G., and Budhu, M. (1993). "Seismic bearing capacity and settlement of foundations." J. Geotech. Engrg., 119 (4), ASCE, 662-674. Sarma, S. K., and Iossifelis, I. S. (1990). "Seismic bearing capacity factors of shallow strip footings." Geotechnique, 40 (2), 265-273. Terzaghi, K., (1943). Theoretical Soil Mechanics, Wiley, New York. Vesic, A.S. (1973). "Analysis of ultimate loads of shallow foundation." J. Geotech. Engrg., ASCE, 99 (1), 43-73. 9 Appendix A The Design Charts for Seismic Bearing Capacity 10 Fig. A.1. Design charts for D>0, Df/B=0, and 0, for any kv. 11 Fig. A. 2. 1. Design charts for D0, Df/B 1, and =0, for kv=0. 12 Fig. A. 2. 2. Design charts for D0, Df/B 1, and =0, for kv=0.2. 13 Fig. A. 2. 3. Design charts for D0, Df/B 1, and =0, for kv=0.4. 14 Fig. A. 3. Design charts for D0, Df/B 1, and >0, for different kv. 15 Fig. A. 4. Design charts for D=0, Df/B =0, and >0, for any kv. 16