# Seismic Design of Shallow Foundations

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1 Shallow Foundations Page 1 Seismic Design of Shallow Foundations Reading Assignment Lecture Notes Other Materials Ch. 9 FHWA manual Foundations_vibrations.pdf Homework Assignment The factored forces for the design of a sign post foundation are: B = 2 feet L = 2.6 feet D =? feet (you determine this) Vertical static = 12 kips Vertical dynamic = 2.4 kips (upward or downward, most critical) Horizontal dynamic = 4 kips (in X direction = longest footing dimension) Moment about y axis = 9 kip feet From this information, calculate the following: D for adequate FS against bearing capacity failure (15 points) Maximum soil pressure (5 points) D for FS against sliding (neglect passive pressure) (10 points) 2. Complete CVEEN 7330 Modeling Exercise 5 (40 points)

2 Shallow Foundations Page 2 Introduction All ground response consider thus far has not considered the effect of the structure on ground response. The presence of a structure, either buried or at the surface, changes the free-field motion. In a manner similar to evaluation of seismic stability of slopes, earthquake effects on foundations can be modeled using either pseudo-static approach or a dynamic response approach. a. b. In the pseudo-static analysis, the effects of the dynamic earthquakeinduced loads on the foundation are represented using static forces and moments. Typically, the pseudo-static forces and moments are calculated by applying a horizontal force equal to the weight of the structure times a seismic coefficient through the center of gravity of the structure. The seismic coefficient is generally a fraction of the peak ground acceleration for the design earthquake and may also be dependent upon the response characteristics of the structure, the behavior of the foundation soils, and the ability of the structure to accommodate permanent seismic displacement. In a dynamic response analysis, the dynamic stiffness and damping of the foundation is incorporated into a numerical model of the structure to evaluate the overall seismic response of the system and the interaction between the soil, foundation and structure.

4 Shallow Foundations Page 4 Dynamic Response Analysis Approach The dynamic stiffness of the foundation is incorporated into an analytical model of the superstructure to evaluate the overall seismic response of the system. The foundation of a structure typically has six degrees of freedom (modes of motion) (Fig. 66) a. horizontal sliding (two orthogonal directions) b. vertical motion c. rocking about two orthogonal axis d. torsion (rotation) about the vertical axis. The response of the foundation to the above modes of motion is thus described by a 6 x 6 stiffness matrix, having 36 stiffness coefficients (Fig. 66). Similarly, a 6 x 6 matrix is needed to described the damping of the foundation. a. Internal damping of the soil is commonly incorporated in the site response model used to calculate design ground motions, and not in the foundation model.

5 Shallow Foundations Page 5

6 Shallow Foundations Page 6 Dynamic Response Analysis Approach (cont.) Typically, the geotechnical engineer provides the values of the stiffness and damping matrix to the structural engineer for use in the dynamic response analysis of the structure. Based on the results of the analysis, the structural engineer should then provide the peak dynamic loads and deformations of the foundation elements back to the geotechnical engineer. The geotechnical engineer then compares the dynamic loads and deformations to acceptable values to ascertain if the seismic performance of the foundation is acceptable. This sometimes is an iterative process to achieve a satisfactory design. If a dynamic response of the structure-foundation is performed, the bearing capacity, sliding, overturning and settlement of the shallow foundation should be evaluated using pseudo-static limit equilibrium analysis.

7 Shallow Foundations Page 7 Dynamic Response Analysis (cont.) Dynamic response analyses incorporate the foundation system into the general dynamic model of the structure. The combined analysis is commonly referred to as the soil-structure-interaction, SSI analysis. In SSI analyses, the foundation system can either be represented by a system of springs (classical approach), or by a foundation stiffness (and damping) matrix. The latter approach, commonly used for SSI analyses of highway facilities, is commonly referred to as the stiffness matrix method approach. The general form of the stiffness matrix for a rigid footing was presented in figure 66. The 6 x 6 stiffness matrix can be incorporated in most structural engineering programs for dynamic response analysis to account for the foundation stiffness in evaluating the dynamic response of the structural system. The diagonal terms of the stiffness matrix represent the direct response of a mode of motion to excitation in that mode while the off diagonal terms represent the coupled response. Many of the off diagonal terms are zero or close to zero, signifying that the two corresponding modes are uncoupled (e.g., torsion and vertical motion) and therefore may be neglected. In fact, for symmetric foundations loaded centrically, rocking and sliding (horizontal translation) are the only coupled modes of motion considered in a dynamic analysis. Often, all of the off-diagonal (coupling) terms are neglected for two reasons : (1) the values of these off-diagonal terms are small, especially for shallow footings; and (2) they are difficult to compute. However, the coupling of the two components of horizontal translation to the two degrees of freedom of rocking (tilting) rotation may be significant in some cases. For instance, coupled rocking and sliding may be important for deeply embedded footings where the ratio of the depth of embedment to the equivalent footing diameter is greater than five. The reader is referred to Lam and Martin (1986) for more guidance on this issue. The stiffness matrix, K, of an irregularly shaped and/or embedded footing can be expressed by the following general equation: where KECF is the stiffness matrix of an equivalent circular surface footing, is the foundation shape correction factor, and is the foundation embedment factor.

8 Shallow Foundations Page 8 Stiffness The solution for a circular footing rigidly connected to the surface of an elastic half space provides the basic stiffness coefficients for the various modes of foundation displacement, translation, the stiffness coefficient K33 can be expressed as: For horizontal translation, the stiffness coefficients and K22 can be expressed as: For torsional rotation, the stiffness coefficient K can be expressed as: For rocking rotation, the stiffness coefficients K44 and K55 can be expressed as: In these equations, G and v are the dynamic shear modulus and Poisson s ratio for the elastic half space (foundation soil) and R is the radius of the footing. The dynamic shear modulus, G, used to evaluate the foundation stiffness should be based upon the representative, or average, shear strain of the foundation soil. However, there are no practical guidelines for evaluating a representative shear strain for a dynamically loaded shallow foundation. Frequently, the value of G, the shear modulus at very low strain, is used to calculate foundation stiffness. However, this is an artifact of the original development of the above equations for foundation stiffness for the design of machine foundations for vibrations. For earthquake loading, it is recommended that values of G be evaluated at shear strain levels calculated from a seismic site response analysis (i.e., use strain-compatible values of G).

9 Shallow Foundations Page 9 Damping for Circular, Rigid Footings One of the advantages of the stiffness matrix method over the classical approach is that a damping matrix can be included in SSI analysis. The format of the damping matrix is the same as the format of the stiffness matrix shown on figure 66. While coefficients of the damping matrix may represent both an internal (material) damping and a radiation (geometric) damping of the soil, only radiation damping is typically considered in SSI analysis. The internal damping of the soil is predominantly strain dependent and can be relatively accurately represented by the equivalent viscous damping ratio,. At the small strain levels typically associated with foundation response, is on the order of 2 to 5 percent. Radiation damping, i.e., damping that accounts for the energy contained in waves that radiate away from the foundation, is frequency-dependent and, in a SSI analysis, significantly larger than the material damping. Consequently, radiation damping dominates the damping matrix in SSI analyses. The evaluation of damping matrix coefficients is complex and little guidance is available to practicing engineers. Damped vibration theory is usually used to form the initial foundation damping matrix. The theory, commonly used to study (small-strain) foundation vibration problems, assumes that the soil damping can be expressed via a damping ratio, D, defined as the ratio of the damping coefficient of the footing to the critical damping for the six-degree-offreedom system. The damping ratio for a shallow foundation depends upon the mass (or inertia) ratio of the footing. The following table lists the mass ratios and the damping coefficients and damping ratios for the various degrees of freedom of the footing. The damping ratios should be used as shown on figure 66 to develop the damping matrix of the foundation system. It should be noted that this approach only partially accounts for the geometry of the foundations and assumes that small earthquake strains are induced in the soil deposit. For pile foundations or for complex foundation geometry, a more rigorous approach, commonly referred to as the soil-foundation-structure-interaction (SFSI) analysis, may be warranted. SFSI is beyond the scope of this lecture.

10 Shallow Foundations Page 10 Damping (cont.) Damping Table (Circular Footing)

11 Shallow Foundations Page 11 Damping (cont.) Definition of variables on previous page

12 Shallow Foundations Page 12 Damping for Rectangular Footings Application of the foundation stiffness general equation (K = KECF) for rectangular footings involves the following two steps: 1. Calculate the radius of an equivalent circular footing for the various modes of displacement using damping table and Figure 68. For vertical and horizontal (translational) displacements, the equivalent radius, r0, is the radius of a circular footing with the same area as the rectangular footing. For rocking and torsional motions, the calculation of the equivalent radius is more complicated, as it depends on the moment of inertia of the footing. The equivalent radius is then used in the stiffness equations to solve for the baseline stiffness coefficients required in the following formula: K = KECF.

13 Shallow Foundations Page 13 Damping for Rectangular Footings (cont.) 2. Find the shape factor a to be used in (K = KECF) using Figure 69. This figure gives the shape factors for various aspect ratios (LIB) for the various modes of foundation displacement.

14 Shallow Foundations Page 14 Damping for Rectangular Footings (cont.) Embedment The influence of embedment on the response of a shallow foundation is described in detail in Lam and Martin (1986). The values of the foundation embedment factor from that study are presented in figure 70 for values of D/R less than or equal to 0.5 and in Figure 71 for values of D/R larger than 0.5. For cases where the top of the footing is below the ground surface, it is recommended that the thickness of the ground above the top of the footing be ignored and the thickness of the footing (not the actual depth of embedment Df) be used to calculate the embedment ratio (D/R) in determining the embedment factor.

15 Shallow Foundations Page 15 Damping for Rectangular Footings (cont.) Embedment (cont.)

16 Shallow Foundations Page 16 Load Evaluation - Loads from Dynamic Response Analysis Method 1 - Seismic loads from dynamic response analysis Potential for amplification of ground motion by the structure is included in the peak loads from the dynamic response analysis Combination of loads from dynamic response analysis (vertical and horizontal) for use in bearing capacity, sliding and overturning evaluations. Common Approach for bearing capacity Assume 100% peak vertical (2 cases; 100 percent upward and 100 percent downward) and 40% peak horizontal, applied in the direction that is most critical for stability. Generally 100 percent peak vertical in the downward directions controls the design. Do not forget to apply the static dead loads (both horizontal and vertical) and static moments. These should be added to the seismic loads.

18 Shallow Foundations Page 18 Evaluation Steps - Bearing Capacity Compute the earthquake loads (from Method 1 or Method 2 above) and combine into a single resultant force with an inclination of α and an eccentricity, e (fig 65). For Method 1, use the 100% and 40% of peak inertial force rule to determine the lowest factor of safety. For Method 2, remember that vertical and horizontal earthquake loads are treated separately (do not apply peak horizontal and peak vertical ground acceleration at the same time). Adjust of Bearing Capacity Equation for Eccentric (Moment) Loading Load eccentricity is caused by the applied moment to the foundation Applied moment causes a non-uniform pressure distribution on the bottom of the footing. Equivalent footing width (B') is computed for the footing, where the width of the footing is reduced, to account for load eccentricity Commonly used relations for B' B' = (B-2e) (Meyerhof, 1953) B' = (3B/2-3e) (linear soil pressure distribution) (The calculated values from the above equations tend to be conservative the contact area is usually larger than the calculated values) limit to eccentricity (to prevent uplift) e < B/6 (Hansen, 1953) (for ah < 0.4 g) e < B/4 (Hansen, 1953) (for ah > 0.4 g) Check bearing capacity with loadings from Method 1 or 2. Report the lowest factor of safety that controls the design. Check sliding factor of safety. FHWA guidance

19 Shallow Foundations Page 19

21 Shallow Foundations Page 21 Myerhof's Method Definitions for use of Myerhof's equations Need to use general bearing capacity equation to account for eccentric loads, moments, inclined loads, and different foundation shapes.

22 Shallow Foundations Page 22 Myerhof's Method (cont.) Bearing capacity factors Inclination factors

23 Shallow Foundations Page 23 Myerhof's Method (cont.) Shape factors for L < 6B

24 Shallow Foundations Page 24 Example Calculation Myerhof (Example) - Loading from Dynamic Analysis

25 Shallow Foundations Page 25 Example Calculation Myerhof (Example) - Loading from Dynamic Analysis

26 Shallow Foundations Page 26 Soil Pressure Evert C. Lawton, 2011

27 Shallow Foundations Page 27 Machine Vibrations from Vertical Source (cont.)

28 Shallow Foundations Page 28 Machine Vibrations

29 Shallow Foundations Page 29 Machine Vibrations from Vertical Source

30 Shallow Foundations Page 30

31 Shallow Foundations Page 31 Machine Vibrations from Vertical Source (cont.) Idealization of a system using a spring with a dynamic stiffiness, Kz and a viscous dashpot Cz undergoing a harmonic loading of Pz.

32 Shallow Foundations Page 32 Machine Vibrations from Vertical Source (cont.) Do not need these for FLAC modeling Dynamic stiffness = static stiffness x dynamic stiffness coefficient. See chart A, next page for k(w) values.

33 Shallow Foundations Page 33 Machine Vibrations from Vertical Source (cont.)

34 Shallow Foundations Page 34 FLAC modeling of Machine Vibration (Vertical Source) FLAC Model with 3-D (i.e., radiation) damping

35 Shallow Foundations Page 35 FLAC modeling of Machine Vibration (cont.)

36 Shallow Foundations Page 36 FLAC modeling of Machine Vibration (cont.)

37 Shallow Foundations Page 37 FLAC modeling of Machine Vibration (cont.)

38 Shallow Foundations Page 38 FLAC modeling of Machine Vibration (cont.)

39 Shallow Foundations Page 39 Machine Vibrations from Vertical Source (cont.) FLAC formulation for radiation damping

40 Shallow Foundations Page 40 Blank

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