Seismic Design of Shallow Foundations: Principles, Design Methodologies and Current Indian Practices

Abstract

The purpose of a foundation/substructure is to transfer the loads from the superstructure safely to the underlying soil. Safe and economical design of foundations to withstand various probable loading conditions is the prime role of a geotechnical engineer. The design of an earthquake-resistant foundation is highly challenging, as seismic loads are quite complex in nature. Design of a foundation against seismic loading requires not only a thorough understanding of the behaviour of the founding soil, response of the structure and interaction of the founding soil and the structure under earthquake loading but also a proper understanding of the basis for the recommendations of the relevant design standards of the country. An attempt is made here to discuss the basis for the recommendations of the Indian geotechnical design standards for shallow foundation design which deal with fixing the plan area and the founding depth (location from the ground surface), the prime focus being demystifying the design philosophy of IS 1893 for the seismic design of shallow foundations. In summary, a clear step-by-step design procedure that is to be adopted while designing a shallow foundation as per Indian Standards is presented.

3.1 Introduction

Seismic risk is defined as the convolution of hazard, vulnerability and exposure [1–4]. Seismic risk reduction is possible only through either lowering exposure or decreasing the vulnerability, since minimising seismic hazard is not possible. Seismic risk reduction is done either through planning interventions such as avoiding high hazard areas for infrastructure development, or engineering interventions such as designing and constructing infrastructure meeting the performance levels for a given target seismic hazard or through the deployment of early warning systems to reduce human losses. Under the ambit of lowering exposure, economic losses can be minimised primarily through proper urban planning and human losses can be minimised by using reliable early warning systems, whereas decreasing vulnerability is done through designing and constructing the infrastructure to desired performance levels. However, to either lower exposure or decrease vulnerability, accurate estimation of seismic hazard is essential.

Seismic codes aim to accomplish seismic risk reduction through publishing hazard maps, codifying simplified procedures for the approximate estimation of hazard, giving generic guidelines to lower exposure and specific guidelines to reduce the vulnerability of the structures/foundations to be designed and constructed for the estimated hazard. Towards lowering exposure, seismic codes recommend avoiding the construction of structures/foundations near the vicinity of active seismic faults, unstable slopes, landfills of hazardous waste sites, dormant or active mine or cavernous lime stones, flood plains and loose to medium dense fine sands (SP) located adjacent to deep rivers/active seismic regions. As land is premium and avoiding sites is to be done judiciously, seismic codes recommend carrying out site-specific studies in high seismicity regions to assess the ground stability under the design seismic action, considering the effect of local soil and site conditions on site amplification. This chapter deals with the specific guidelines of the Indian geotechnical design standards to reduce the vulnerability of the shallow foundations to be designed and constructed.

Foundations are broadly classified into shallow and deep foundations. Selection of a suitable type of foundation depends on subsurface soil characteristics, magnitude of loads from the superstructure and the requirements of the superstructure. If a shallow foundation, i.e. a Strip/Isolated/Combined/Raft foundation is selected, then the length, width, and founding depth of the selected foundation need to be specified. In case of a deep foundation like a Pile/Pier/Well/Pile Group, the length, diameter, and the number of piles, as the case may be, must be specified. A foundation is usually considered as shallow if its founding depth is lower than its width. Shallow foundations derive resistance from the bearing soils. They are used when soil at shallow depths is strong enough to withstand the stresses imparted by the superstructure. Shallow foundations should usually be the first choice, as they are economical, do not need special construction techniques or special equipment for drilling and provide better quality control. However, isolated foundations are not recommended when excessive settlements and/or large lateral soil movements (lateral spreading) are expected under the design earthquake. Mat foundations are preferred on soft or loose soils. They perform better than isolated, continuous and tied foundations by minimising differential soil movements due to earthquakes by bridging over loose pockets of soil. Deep foundations are preferred when loads from the superstructure are too high, water table is shallow, there are restrictions over open excavation or to bypass liquefiable soils, if competent soil (dense sands, stiff clays or rocks) exists at shallow depths (say, 10–20 m).

The design of a shallow foundation involves fixing the plan area and founding depth of the foundation, which comes under the purview of the geotechnical design of foundations and the structural design of the foundation element which comes under the purview of the structural design of foundations. The discussion here is limited to the geotechnical design of foundations. Across the world, the geotechnical design of foundations is done through either the Working Stress Design (WSD)/Allowable Stress Design (ASD) or the Limit State Design (LSD)/Load and Resistance Factor Design (LRFD). The difference between LSD and LRFD design philosophies is graphically shown in Fig. 3.1 [5]. The partial safety factors of the LSD method which was adopted by the Eurocode in 1993 were calibrated against the global factors of safety of the ASD philosophy to produce comparable design results [6]. The differences in philosophies of geotechnical design of foundations are treated at length in [5, 6]. The North American building codes like the UBC/IBC and the loading standard ASCE 7 permit the use of both ASD and LRFD, whereas the Indian codes of practice permit the use of only the ASD philosophy for the geotechnical design of foundations. The geotechnical design problems are traditionally dealt with under two distinct categories, the stability and the elasticity problems. LSD deals with the stability problems under the ultimate limit state and the elasticity problems under the serviceability limit state [6].

Fig. 3.1

Difference between LSD and LRFD [7]

Various stages in the seismic design and construction of a foundation are depicted in the flowchart shown in Fig. 3.2. Apart from ‘Hazard Estimation’ and ‘Seismic Design’, the rest of the stages are more or less similar to the design of foundations subjected to gravity loads and done the same way world over.

Fig. 3.2

Stages in the seismic design and construction of foundations

Shallow foundations are subjected to varying combinations of vertical load, V, horizontal load, H, and moment, M. Many design standards and codes of practice [8, 9], while estimating the ultimate bearing capacity of shallow foundations, consider the effect of H using correction factors and M using reduced foundation dimensions. These approaches to account for the influence of H and M on the ultimate bearing capacity were developed independently, without considering the interaction between the vertical load, horizontal load and the moment. Some design standards/codes/guidelines [10–13] do provide V–H–M capacity envelopes derived through experimental studies [14, 15–20] or analytical studies [21–27] to check the safety of shallow foundations [28, 29]. Apart from these standards, in recent decades, some literature was published on the interaction surface approach for the design of shallow foundations resting on cohesive [30–44] or cohesionless soils [14, 16, 45, 46]. However, all the above studies are focussed on the design of shallow foundations resting on level ground and consider either exclusively cohesive or cohesionless soils only. Also, only a few of them include seismic loading.

3.2 Shallow Foundation Resisting Mechanisms, Earthquake Loading and Possible Failure Modes, Design Approaches for Various Loads

3.2.1 Resisting Mechanisms of the Founding Soil

The way a shallow foundation withstands stresses is different based on the nature of the applied loading. Figure 3.3 presents the load resisting mechanisms of the founding soil against different types of foundation loads.

Fig. 3.3

Resisting mechanisms of the founding soil against different types of loads: a Soil bearing pressure; b Dead weight of the foundation and backfill; c Friction resistance or adhesion of soil and lateral earth pressure; and d Redistribution of bearing pressure

Earthquake loading is quite complex and the analysis of foundations under earthquake loading is highly challenging. Table 3.1 in Section 3.2.3 lists various possible modes of failures of shallow foundations under earthquake loading, the understanding of which is essential for the design of safe and economic foundations.

3.2.2 Components of Earthquake Loading

Earthquakes generate the following types of loads on foundations.

• Alternating Horizontal loads,

• Alternating Vertical loads and

• Alternating Moments.

In addition to the generation of the above type of loads, earthquakes reduce the strength of the soils due to the development of excess pore water pressures and/or liquefaction. Excess pore water pressures can also cause ground failures such as sand boils and lateral spreading.

3.2.3 Possible Shallow Foundation Failures

Shallow foundations experience different modes of failure based on the type and intensity of loading and the type of foundation soil. Broadly, foundation failures can be classified as shown in Table 3.1 [47].

Table 3.1 Foundation failures, corresponding causative loads and soil failures

Seismic design of a shallow foundation requires the consideration of all the above possible modes of failure, their causative loads and failure mechanisms. While calculating the design seismic loads, it is important to include the effects of site amplification. The design approach to be adopted to ensure safety under the action of the causative loads is described in brief in the following sections.

3.2.4 Design Against Vertical Loads

World over, the basic approach for the geotechnical design of foundations for a concentrically vertical load remains more or less same. However, the approach for design against the horizontal and the moment loads vary significantly among countries and between structural and geotechnical engineers.

3.2.5 Design Against Horizontal Loads

Table 1806.2 of the International Building Code 2018 [48] apart from containing the settlement-limited presumptive vertical foundation pressure values also includes presumptive values of lateral bearing pressure and lateral sliding resistance which are used in the check against lateral loading. The effect of horizontal loads on the vertical load carrying capacity is not considered. On the other hand, Eurocode 7 [8] and the Indian standard IS 6403 [9] have adopted different versions of the bearing capacity equation containing the inclination factors to consider the effect of horizontal loads on the vertical load carrying capacity of the foundation. Although the AASHTO LRFD Bridge Design specifications [49] includes a version of the bearing capacity equation containing the inclination factors, as stated in the commentary of the code, the usual practice is to ignore the inclination factors. The reason for this practice is attributed to the fact that the information on the design vertical and horizontal loads is missing at the time of geotechnical testing and report preparation. The code further suggests omitting the use of inclination factors in the case of foundations with modest embedment, citing two reasons. The first reason given is that the tests by Meyerhof which led to the inclusion of inclination factors have shown that for foundations with the depth of embedment ratio close to 1, the effect of load inclination on bearing resistance is relatively small. The second reason given is that the resistance factors provided in Article 10.5.5.2.2 of the code were derived for vertical loads alone without considering the effect of horizontal loads.

3.2.6 Design Against Moments

The geotechnical design for moments is done through either Peck’s approach (elastic stress method) or Meyerhof’s approach (equivalent area method). When using Peck’s approach, the maximum value of the elastic bearing stress is compared with the value of the allowable bearing capacity value estimated for the case of a concentric vertical load without accounting for the effect of moment loading. Many methods and charts based on Peck’s approach were proposed over the years. A comprehensive list of such methods is presented in [50]. While using the equivalent area method, codes prescribe to use reduced foundation dimensions while calculating the capacity as well as the demand. However, the usual practice is to calculate the demand using reduced dimensions, while the capacity is calculated using the original foundation dimensions since the values of design moments are unknown at the time of the preparation of the geotechnical design recommendations report.

3.3 Shallow Foundation Design Alternatives

Seismic design of a shallow foundation, in a broad sense, can be done in one of the following two ways as depicted in Fig. 3.4:

Fig. 3.4

Possible shallow foundation responses to seismic loading [51]—With permission from John Wiley & Sons, Inc. (a) Elastic Soil and Hinged Column; (b) Elastic Column and Inelastic Soil; (c) Elastic Column and Elastic Soil, Inelastic Foundation Element; and (d) Permanently deformed Foundation Element due to Cyclic loading

• 1. Allow inelastic deformations in the structural system and keep the founding soil elastic: Allowing inelastic deformations in the structural system alone can be done in one of the following two ways:

  • (i) Capacity design of the foundation, such that the founding soil and the foundation (structural) remain elastic, while the column undergoes inelastic deformations.

  • (ii) Not applying the capacity design of the foundation, resulting in the column and the founding soil remaining elastic, while the foundation (structural) undergoes inelastic deformations. Such a design alternative is not desirable.

• 2. Allow inelastic deformations in the founding soil and keep the structural system elastic: Allowing inelastic deformations in the founding soil alone can be achieved through relying upon either foundation rocking or/and foundation sliding. However, care should be taken to ensure that the inelastic deformations in the founding soil do not lead to the following kind of foundation failures.

  • (i) Excessive permanent tilting of foundation,

  • (ii) Excessive permanent sliding of foundation and

  • (iii) Excessive permanent settlement of foundation.

The mechanism shown in Fig. 3.5 needs to be ensured when capacity design is adopted for the seismic design of the structure. Such mechanisms are only possible when the foundation is capacity designed. Eurocode 8 recommends capacity design of foundation as the guiding principle for the seismic design of shallow foundations. The American and Indian codes currently do not recommend the capacity design of foundation. A foundation designed as per Eurocode 8 will likely have the plastic hinge in the column while the foundation (structural) and the founding soil remain elastic, whereas a foundation designed as per the American/Indian code is more likely to have the inelastic deformations either in the foundation (structural) or the founding soil since the foundation is not capacity designed. So, many foundations designed as per the current American/Indian code are likely to undergo rocking. However, the level of inelastic deformations in the founding soil cannot be estimated using the current code-based procedures. Increasingly, there is a call from the Geotechnical Earthquake Engineering fraternity to adopt unconventional seismic responses like foundation rocking, sliding and seismic isolation of surface foundations exploiting the properties of natural liquefiable soil, to minimise damage in the superstructure. By not adopting capacity design of foundation, as in the case of current American/Indian code, one is highly likely to end up designing a rocking/sliding foundation, although not intending to do so.

Fig. 3.5

Mechanism to be ensured through Capacity Design [52]

A simplified analysis, assuming a pinned base, shows that the founding soil is highly likely to enter the inelastic range when subjected to a rare event. It is important to note that estimation of seismic forces will have a bearing on the seismic design of shallow foundations. Hence, it is very important to understand the assumptions in computing the seismic forces. The computation of seismic forces depends on the period estimation, hazard estimation (seismic zone) and the shape of the design response spectra. Capacity design needs to be adopted for safety–critical structures. However, it would be uneconomical to adopt for normal structures. For all normal structures, designing using either the current practice or rocking principles is desirable. The problem with the use of current practice is, though it allows rocking unconsciously, it may lead to foundation failures because of the lack of foundation design considering actual rocking. Hence, it is highly desirable to come up with guidelines/design procedures allowing rocking within acceptable limits.

3.4 Seismic Settlements

Settlement calculations are done using Terzaghi's Consolidation theory by which the compression of the foundation soils under the applied building loads is determined from laboratory consolidation curves. The time rate of consolidation is dependent upon the rate at which water and air can escape from the voids of the soil subjected to increased pressures from building loads. This rate is relatively fast for sands and slow for saturated clays.

If we consider the effect of seismic loads on the settlement behaviour of a typical building as shown in Fig. 3.6, we observe that the earthquake loads result in an increase in the vertical loads on the exterior column foundations. In considering how the foundations should be designed for the increased vertical loads due to these lateral forces, it might seem obvious that the exterior foundations should be made larger to accommodate the combined vertical and seismic loads. If the size of the exterior foundations is increased because of seismic design considerations while leaving the interior foundations at the same size as required for support of dead loads, the settlement of the exterior foundations under dead loads would be less than previously considered for the smaller foundations and the differential settlements between exterior and interior columns would be increased.

Fig. 3.6

Settlements under gravity and seismic loads [53]

Thus, an apparent conflict exists in the design of foundations for vertical static and lateral seismic loads. To resolve this conflict, it is necessary to consider the difference in loading conditions for the vertical and lateral loads, and the behaviour of the foundation soils under these different conditions. First, it should be realized that the time duration of loading is very important in predicting the response of foundation soils to loading. Design wind loads may be imposed frequently or for a sufficient length of time so that they can be considered similar to other live loads to which the structure is subjected. However, short time wind, seismic or blast loads require a different concept of foundation design than used for conventional live loads.

One approach to the design of shallow foundations for these transient loads is to permit an increase in the allowable bearing pressures. However, soil behaviour under transient loads is very much dependent on its physical properties. Generally speaking, sands are quite sensitive to the transient loads, while clays are relatively insensitive. To examine this difference between sandy and clayey soils and to evaluate its effect on foundation design, the load deflection characteristics of each soil type will be considered.

Figure 3.7 presents a typical load–settlement curve for an isolated foundation on sandy soils. The solid line represents the deflection of the foundation under permanently applied (dead) loads. Because of the relatively fast drainage characteristics of sand, these settlements will occur quite rapidly. If instrumentation were provided to record the load–settlement response of the foundation to transient loads, a pattern similar to that indicated by the dashed lines in Fig. 3.7 would be obtained. The magnitude of the settlement is predominantly dependent upon the soil structure. For dense sandy soils, the settlement is comparatively small. As the density of the sand deposit decreases, the settlement increases. In the case of saturated sands, if the density is less than a certain critical value, the soil structure might collapse during an earthquake, transmitting the load to the pore water, resulting in a flow failure known as liquefaction. If this behaviour is anticipated, either the loose sandy soils should be stabilised or the building should be supported independently of this stratum.

Fig. 3.7

Load–Settlement patterns in sandy soils due to Seismic Load [53]

In establishing allowable bearing pressures and in estimating foundation settlements for short-term loads, a general knowledge of the behaviour of the foundation soils must be relied upon. Foundations founded on loose sands may yield or fail during seismic disturbances, even though their behaviour under permanently applied vertical loads may have been satisfactory. Buildings founded on a dense sand formation may suffer no damage during an earthquake, even though the short time vertical loads may be much higher than those imposed permanently. Thus, some increase in bearing pressures for foundations founded on dense sandy soils may be permitted for transient seismic loads without forfeiting a significant safety margin against failure.

Usually, clays are less preferred as foundation materials than sands owing to their greater compressibility. However, clays do possess some advantages over sands vis-à-vis their behaviour under transient loads. Figure 3.8 presents a typical load–settlement curve for an isolated foundation on clayey soils. The settlement curve for permanent loads is similar to that of sandy soils, although the magnitude of the settlement and the time rate of settlement are quite different. Also, the curve indicates more abrupt soil yielding at loads beyond the design range. As illustrated in Fig. 3.8, a foundation resting on clay would not exhibit substantial settlement during an earthquake, because the intergranular structure of the clay cannot adjust to the change of load during this short time. Despite the development of high pore pressures in the clay by the increased loads imposed during the earthquake, the time lag for consolidation of the clay will prevent appreciable deflections from occurring. Thus, foundations on clayey soils may be designed for comparatively high, transient loads without excessive settlement of the foundations. The allowable bearing pressures depend largely upon the physical characteristics of the soil and to a certain extent upon the type of foundation and the structure.

Fig. 3.8

Load–Settlement patterns in clayey soils due to Seismic Load [53]

3.5 Seismic Design of Shallow Foundations as per IS 1904

IS 1904 [54] considers seismic load as a transient load and due to the reasons mentioned in Sect. 3.4, the stresses in the founding soil due to seismic loads are allowed a certain percentage increase over the allowable values.

Shallow foundations shall be proportioned for the following load combinations:

• (1) Dead load + Live load and

• (2) Dead load + Live load + Seismic load,

where

• (i) Dead load includes the weight of the superstructure, the substructure and the overlying fill.

• (ii) Live loads from the superstructure, in accordance with IS 875 Part 2 1987 shall be taken for proportioning and designing the foundations.

• (iii) If seismic load is less than 25 percent of the load combination 1, then load combination 2 may be neglected in design and the load combination 1 shall be compared with the safe bearing load to satisfy allowable bearing pressure.

• (iv) If seismic load is more than 25% of the load combination 1, foundations shall be checked for load combination 2. The safe bearing capacity shall be increased in accordance with Table 1 of IS 1893 Part 1. In cohesionless soils, liquefaction analysis and computation of seismic settlements shall be made as well.

3.5.1 Design Against Settlements, as per IS 1904

For foundations resting on cohesionless soils, the settlements shall be computed corresponding to Dead load + live load + seismic load, since in such types of soils, settlements occur near instantaneously.

For cohesive soils, the settlements shall be computed corresponding to permanent loads alone. Permanent loads include dead load, weight of all fixed equipment and one half of the design live load.

The distinction between permanent and temporary loads largely depends upon the judgement of the engineer-of-record of the project. The design engineer should strive to minimise the differential settlements due to the live load variation by ensuring equal bearing pressure for all the foundations under the service load.

3.6 Seismic Effects to be Considered as Per IS 1893 Part 1

3.6.1 Influence of Soil Type on Intensity of Shaking:

• (i) Influence of soil type on the intensity of shaking (horizontal) is accounted for, in the calculation of horizontal seismic coefficient.

$${\alpha }_{h}={A}_{h}=\frac{\left(\frac{Z}{2}\right)\left(\frac{{S}_{a}}{g}\right)}{\left(\frac{R}{I}\right)}$$

where Z = Zone Factor (Range: 0.36, 0.24, 0.16 and 0.10). An appropriate zone factor is to be selected based on the location of the project site.

I = Importance Factor (Range: 1.0–1.5)

R = Response Reduction Factor (Range: 3.0–5.0 for RC Buildings)

$\frac{{S}_{a}}{g}$ = Avg. spectral acceleration coefficient (Fig. 3.9), where the effect of soil type is considered (Maximum value is 2.5).

• (ii) Influence of soil type on the vertical shaking is also considered indirectly.

$${\alpha }_{v}={A}_{v}=\frac{2}{3}\cdot {\alpha }_{h}=\frac{\left(\frac{2}{3}\times \frac{Z}{2}\right)\left(\frac{{S}_{a}}{g}\right)}{\left(\frac{R}{I}\right)}$$

Note: The value of $\frac{{S}_{a}}{g}$ in the above equation is taken as 2.5, for buildings governed by IS 1893 Part 1 and liquid retaining tanks governed by IS 1893 Part 2.

3.6.2 Increase in Allowable Bearing Pressures in Soils

IS 1893 Part 1 permits an increase in allowable bearing pressures in soils where earthquake forces are not expected to cause significant settlement, to avoid undesirable differential settlements that can take place prior to earthquake occurrence. This provision drew inspiration from [53] to have different (increasing) factors for different soils, unlike the UBC/IBC codes, which use the same increasing factor irrespective of the type of soil. When earthquake forces are included, the allowable bearing pressure in soils shall be increased as per Table 1 of IS 1893 Part 1, depending upon the type of soil.

3.6.3 Accounting for Liquefiable Soils

• (i) In soil deposits consisting of submerged loose sands and soils falling under classification SP with corrected standard penetration N-values less than 15 in seismic Zones III, IV, V and less than 10 in seismic Zone II, within the top 5 m, the vibration caused by an earthquake may cause liquefaction or excessive total and differential settlements. Such sites should preferably be avoided while locating new structures and should be avoided for locating structures of important projects.

  Table 3.2 Desirable N-values (corrected values)

• (ii) If N-values (corrected values) at the project site are lower than the desirable N-values listed in Table 3.2 and if there is no option to avoid the site, appropriate site improvement techniques (such as improving compaction or stabilisation) should be adopted to achieve suitable N-values.

• (iii) Alternatively, a deep pile foundation may be provided and taken to depths well into the layer which is not likely to liquefy.

• (iv) Marine clays and other sensitive clays are also known to liquefy due to the collapse of soil structure and will need special treatment according to site condition.

3.6.4 Other Guidelines for the Seismic Design of Shallow Foundations as per IS 1893 Part 1

• (i) The allowable bearing pressure shall be determined in accordance with IS 6403 [47] or IS 1988 [55].

• (ii) If any increase in bearing pressure has already been permitted for forces other than seismic forces, the total increase in allowable bearing pressure when seismic force is also included shall not exceed the limits specified in Table 1 of IS 1893 Part 1.

• (iii) Isolated R.C.C. foundations without tie beams or unreinforced strip foundations shall not be permitted in soft soils with N < 10.

3.7 Seismic Design of Shallow Foundations Using Pseudostatic Method as per IS 1893 Part 1

Foundations subjected to earthquake loading can be analysed/checked for their safety by using pseudostatic analysis. Various steps involved in the application of pseudostatic analysis are given below:

• I. Estimation of Earthquake Loading,

• II. Conversion of the Earthquake Load into an Equivalent Static Load and

• III. Apply conventional Bearing Capacity theories to obtain the Safe Bearing Capacity.

3.7.1 Estimation of Earthquake Loading

• (i) Horizontal Dynamic Load Calculation

Horizontal seismic coefficient for the Design Basis Earthquake (DBE) is estimated using

$${\alpha }_{h}={A}_{h}=\frac{\left(\frac{Z}{2}\right)\left(\frac{{S}_{a}}{g}\right)}{\left(\frac{R}{I}\right)}$$

Horizontal load due to earthquake

Fig. 3.9

Response spectra for obtaining (S_a/g) based on the fundamental period of the structure (IS 1893 Part 1) [56]

$${H}_{e}={\alpha }_{h}W$$

• (ii) Vertical Dynamic Load Calculation

Vertical seismic coefficient is taken as

$${\alpha }_{v}={A}_{v}=\frac{2}{3}\cdot {\alpha }_{h}=\frac{\left(\frac{2}{3}\times \frac{Z}{2}\right)\left(\frac{{S}_{a}}{g}\right)}{\left(\frac{R}{I}\right)}$$

Note: The value of $\frac{{S}_{a}}{g}$ in the above equation is taken as 2.5, for buildings governed by IS 1893 Part 1 and liquid retaining tanks governed by IS 1893 part 2.

Vertical load due to earthquake

$${V}_{e}={\alpha }_{v}W$$

• (iii) Calculation of Dynamic Moment

Moment may be calculated as the product of the maximum dynamic horizontal load estimated in step (i) and the centre of gravity of the structure or obtained through structural analysis.

3.7.2 Conversion of Earthquake Loading into Equivalent Static Load

• Calculate total vertical, horizontal and moment loads

$$H={H}_{g}+{H}_{e}$$

$$V={V}_{g}+{V}_{e}$$

$$M={M}_{g}+{M}_{e}$$

• The combined effect of V and H can be replicated using an equivalent inclined load.

• The combined effect of M and V can be replicated using an equivalent eccentric load.

• The combined effect of V, H and M can be replicated using an equivalent eccentrically inclined load. (i.e. foundation is now an eccentrically obliquely loaded foundation).

3.7.3 Estimation of Bearing Capacity Under Earthquake Loading

• (i) Find out effective dimensions of the foundation due to eccentric loading (L′ and B′):

$${e}_{b}=\frac{{M}_{b}}{V}$$

$$B^{\prime} = B-2{e}_{b}$$

$${e}_{l}=\frac{{M}_{l}}{V}$$

$$L^{\prime} = L-2{e}_{l}$$

• (ii) Calculate obliquity of loading ‘i’:

  $$i=( H/V)$$

• (iii) Use Generalised Bearing Capacity (GBC) equation with inclination factors to find ${q}_{ult}$.

$${q}_{ult}=c{N}_{c}({s}_{c}{d}_{c}{i}_{c})+{q}^{{\prime}}{N}_{q}({s}_{q}{d}_{q}{i}_{q})+0.5\gamma B{N}_{\gamma }({s}_{\gamma }{d}_{\gamma }{i}_{\gamma }){R}_{w}^{{\prime}}$$

where ${q}_{ult}$= Ultimate Bearing Capacity of soil,

${N}_{c},{N}_{q},{N}_{\gamma }$ are the bearing capacity factors,

${s}_{c},{s}_{q},{s}_{\gamma }$ are shape factors,

${i}_{c},{i}_{q},{i}_{\gamma }$ are inclination factors,

${d}_{c},{d}_{q},{d}_{\gamma }$ are depth factors.

Bearing capacity factors and other factors can be considered from any of the GBC theories or using the IS 6403 formula which is in fact based on Hansen, Vesic and other methods.

• (i) From q_ult, calculate q_netsafe

• (ii) Increase q_netsafe value, as per Table 1 of IS 1893 Part 1.

• (iii) Estimate, load carrying capacity, Q.

• (iv) Compare Q with V:

  If Q > V, foundation is safe.

  Otherwise, redesign the foundation with new dimensions.

• (v) Estimate settlements of underlying soils due to earthquake loading and then estimate total settlements. See whether this value is within the limits or not. If not, redesigning of foundation is necessary.

3.7.4 Accounting for Soil Strength Reduction

If soil is expected to liquefy or develop significant excess pore water pressures, it is required to carry out an effective stress analysis considering the increase in the pore water pressures during earthquake shaking. For this purpose, it is essential to estimate the liquefaction potential of the site by following the IS 1893 procedure, which has been adopted from [57]. For liquefied soil, use residual/steady-state strength of the soil in analysis.

3.7.5 Estimation of Sliding Failure

Calculate lateral resistance and sliding resistance offered by the soil and then compare it with the lateral force. Use earth pressure theories for calculation of lateral resistance of soils. This probably is not going to govern in the case of foundations resting on level ground. It is important only in the case of foundations resting on slopes where significant later sliding is expected.

3.7.6 Seismic Design Procedure as Per Indian Standards

The seismic coefficient $\frac{\left(\frac{Z}{2}\right)\times \left(\frac{{S}_{a}}{g}\right)}{\left(\frac{R}{I}\right)}$ represents inelastic design for DBE-level earthquake, whereas the seismic coefficient $\frac{Z\times \left(\frac{{S}_{a}}{g}\right)}{\left(\frac{R}{I}\right)}$ represents inelastic design for MCE-level earthquake.

Shallow Foundation Design as per IS codes (Geotechnical) is summarised below.

• Foundation Design is done using Working Stress Design.

• q_ns = q_nu/Factor of Safety | Usually, Factor of Safety = 3.

• q_a = min(q_ns,q_np).

where q_nu = Net Ultimate Bearing Capacity,

q_ns = Net Safe Bearing Capacity,

q_np = Net Soil Pressure for Specified Settlement,

q_a = Allowable Bearing Capacity.

The following discussion assumes that q_a= q_ns in the design case under consideration. Under such assumption, seismic design of a shallow foundation for DBE is as follows:

• For Type A soil, q_a* = 1.5q_a=1.5q_ns = 1.5q_nu/3 = q_nu/2

• DL + LL → q_ns

• DL + LL + EQ (DBE) → 1.5q_ns

• Therefore, EQ (DBE) ~ 0.5q_ns and EQ (MCE) ~ 1.0q_ns

• Therefore, DL + LL + EQ (MCE) → 2.5q_ns < q_nu,

• where q_a* = Allowable Bearing Capacity for earthquake load combination

However, the above procedure is strictly valid if and only if the applied loads are vertical only. An attempt is made here using a case study (see Sect. 3.7.7) to see if the foundation design as per the above procedure is valid when the foundation is loaded by horizontal forces and moments in addition to the vertical load.

Foundation design method used in practice is summarised below:

• Inclination factors are not used in the calculation of the bearing capacity. Horizontal loads are not considered. If considered, then it is ensured that Horizontal Load < Vertical Load × Coefficient of Friction

For Axial + Moment,

• When e < B/6, P/A + M_x/Z_x + M_y/Z_y < q_a (Method 1a).

• When e > B/6, 2P/(3L(0.5B-e)) < q_a (Method 1b).

Foundation design method as per IS 6403 is summarised below:

• Inclination factors are used in the calculation of the bearing capacity to account for the influence of horizontal loads on bearing capacity.

For Axial + Moment,

– P/(L’ × B’) < q’_a (Method 2),

where L′ = L−2e_L, B′ = B−2e_B and q′_nu is calculated using B′.

3.7.7 Case study

A building with the specifications mentioned below and the plan and the elevation as shown in Figs. 3.10 and 3.11 is modelled and analysed in ETABS. The analysis results for the central column, an edge column and a corner column are reported in Tables 3.3, 3.4 and 3.5. Surface foundations for the central column, the edge column and the corner column are designed as per Methods 1 and 2 mentioned above and the same has been validated using OptumG3.

Fig. 3.10

Plan of the building

Fig. 3.11

Elevation of the building

Table 3.3 Results for central column from ETABS

Table 3.4 Results for edge column from ETABS

Table 3.5 Results for corner column from ETABS

The following data has been considered:

Seismic Zone V, Type A soil,

Z = 0.36 for DBE, Z = 0.72 for MCE, R = 5.

All Columns: 0.45 m × 0.45 m.

All Beams: 0.25 m × 0.5 m.

Dead Load: 12 kN/m², 10 kN/m² (on roof).

Live Load: 4 kN/m², 1.5 kN/m² (on roof).

For φ = 38°, N > 30 → Type A soil → 50% increase in q_na.

Hence, c = 0 and φ = 38° are taken as soil parameters in OptumG3.

Three surface foundations (individual models) of size 2 m × 2 m, 1.8 m × 1.8 m and 1.6 m × 1.6 m with a vertical multiplier load at the centroid of the foundation are analysed using finite element limit analysis. The results from the analyses are presented below.

Vertical Load Multiplier for 2m × 2m: 4734.8 kN

Safe Load for 2m × 2m: 1578.3 kN

SBC for 2m × 2m: 394.6 kN/m²

Vertical Load Multiplier for 1.8m × 1.8m: 3802.8 kN

Safe Load for 1.8m × 1.8m: 1267.6 kN

SBC for 1.8m × 1.8m: 391.2 kN/m²

Vertical Load Multiplier for 1.6m × 1.6m: 2575.7 kN

Safe Load for 1.6m × 1.6m: 858.6 kN

SBC for 1.6m × 1.6m: 335.4 kN/m²

3.7.8 Central Column

Method 1a:

$$\begin{aligned} & {\text{DL}} + {\text{LL}} \to {{1522} / 4} = 380.5 < 394.6\,{\text{KN}}/{\text{m}}^2 \left( {q_{ns} } \right) \\ & {\text{DL}} + {\text{LL + EQX}}\left( {{\text{DBE}}} \right) \to {P / {A\left( {{{1 + 6e_{BDBE} } / B}} \right)}} \\ & = 380.5 \times \left( {1 + 6 \times 0.110/2} \right) = 506.1 < 391.2\,{\text{KN}}/{\text{m}}^2 \left( {1.5q_{ns} } \right) \\ & {\text{DL}} + {\text{LL + EQX}}\left( {{\text{MCE}}} \right) \to {P / {A\left( {{{1 + 6e_{BMCE} } / B}} \right)}} \\ & = 380.5 \times \left( {1 + 6 \times 0.221/2} \right) = 632.8 < 1183.8\,{\text{KN}}/{\text{m}}^2 \left( {3q_{ns} } \right) \\ \end{aligned}$$

Method 2:

$$\begin{aligned} & {\text{DL}} + {\text{LL}} \to {{1522} / 4} = 380.5 < 394.6\,{\text{KN}}/{\text{m}}^2 \left( {q_{ns} } \right) \\ & {\text{DL}} + {\text{LL + EQX}}\left( {{\text{DBE}}} \right) \to {P / {L\left( {B - 2e_{BDBE} } \right),\,B^{\prime}_{DBE} = 1.78\,{\text{m}}}} \\ & = {{1522} / 2} \times \left( {2 - 2 \times 0.110} \right) = 427.5 < 526.8\,{\text{KN}}/{\text{m}}^2 \left( {1.5q^{\prime}_{ns} } \right) \\ & {\text{DL}} + {\text{LL + EQX}}\left( {{\text{MCE}}} \right) \to {P / {L\left( {B - 2e_{BMCE} } \right),\,B^{\prime}_{MCE} = 1.558\,{\text{m}}}} \\ & = {{1522} / 2} \times \left( {2 - 2 \times 0.221} \right) = 448.5 < 922.2\,{\text{KN}}/{\text{m}}^2 \left( {3q^{\prime}_{ns} } \right) \\ \end{aligned}$$

A 2 m × 2 m surface foundation is modelled in OptumG3 and checked for the MCE condition. The foundation is safe under the MCE loads.

3.7.9 Edge Column

Method 1a:

$$\begin{aligned} & {\text{DL}} + {\text{LL}} \to {P / {A\left( {{{1 + 6e_{BDE} } / B}} \right)}} \\ & = 213 \times \left( {1 + 6 \times 0.014/1.8} \right) = 223 < 391.2\,{\text{KN}}/{\text{m}}^2 \left( {q_{ns} } \right) \\ & {\text{DL}} + {\text{LL + EQX(DBE)}} \to {P / {A\left( {{{1 + 6e_{BDBE} } / B}} \right)}} \\ & = 213.1 \times \left( {1 + 6 \times 0.191/1.8} \right) = 446.8 < 586.8\,{\text{KN}}/{\text{m}}^2 \left( {1.5q_{ns} } \right) \\ & {\text{DL}} + {\text{LL + EQX(MCE)}} \to {P / {A\left( {{{1 + 6e_{BDBE} } / B}} \right)}} \\ & = 333.2 \times \left( {1 + 6 \times 0.304/1.8} \right) = 670.6 < 1173.7\,{\text{KN}}/{\text{m}}^2 \left( {3q_{ns} } \right) \\ \end{aligned}$$

Method 2:

$$\begin{aligned} & {\text{DL}} + {\text{LL}} \to {P / {L\left( {B - 2e_{BDL} } \right),\,B^{\prime}_{DL} = 1.772\,{\text{m}}}} \\ & = {{690} / {1.8 \times \left( {1.8 - 2 \times 0.014} \right)}} = 216 < 385.1\,{\text{KN}}/{\text{m}}^2 \left( {q^{\prime}_{ns} } \right) \\ & {\text{DL}} + {\text{LL + EQX(DBE)}} \to {P / {L\left( {B - 2e_{BDBE} } \right),}}\,B^{\prime}_{DBE} = 1.418\,{\text{m}} \\ & = {{885} / {1.8 \times \left( {1.8 - 2 \times 0.191} \right)}} = 346.7 < 462.3\,{\text{KN}}/{\text{m}}^2 \left( {1.5q^{\prime}_{ns} } \right) \\ & {\text{DL}} + {\text{LL + EQX(MCE)}} \to {P / {L\left( {B - 2e_{BMCE} } \right),}}\,B^{\prime}_{MCE} = 1.192\,{\text{m}} \\ & = {{1080} / {1.8 \times \left( {1.8 - 2 \times 0.304} \right)}} = 503.2 < 777.2\,{\text{KN}}/{\text{m}}^2 \left( {3q^{\prime}_{ns} } \right) \\ \end{aligned}$$

A 1.8 m × 1.8 m surface foundation is modelled in OptumG3 and checked for the MCE condition. The foundation is safe under the MCE loads.

3.7.10 Corner Column

Method 1a & 1b:

$$\begin{aligned} & {\text{DL}} + {\text{LL}} \to {P / {A\left( {{{1 + 6e_{BDL} } / B}} \right)}} \\ & = 122.1 \times \left( {1 + 6 \times 0.017/1.6} \right) = 135.5 < 335.4\,{\text{KN}}/{\text{m}}^2 \left( {q_{ns} } \right) \\ & {\text{DL}} + {\text{LL + EQX(DBE)}} \to {{2P} / {3L\left( {0.5B-e_{BDBE} } \right)}} \\ & = {{2 \times 493} / {\left( {3 + 1.6\left( {0.5 \times 1.6 - 0.327} \right)} \right)}} = 434.2 < 503\,{\text{KN}}/{\text{m}}^2 \left( {1.5q_{ns} } \right) \\ & {\text{DL}} + {\text{LL + EQX(MCE)}} \to {{2P} / {3L\left( {0.5B-e_{BMCE} } \right)}} \\ & = {{2 \times 673} / {\left( {3 + 1.6\left( {0.5 \times 1.6 - 0.472} \right)} \right)}} = 855.1 < 1006.1\,{\text{KN}}/{\text{m}}^2 \left( {3q_{ns} } \right) \\ \end{aligned}$$

Method 2:

$$\begin{aligned} & {\text{DL}} + {\text{LL}} \to {P / {L^{\prime}_{DL} B^{\prime}_{DL} ,\,L^{\prime}_{DL} = 1.776\,{\text{m,}}\,{\text{B}}^{\prime}_{DL} = 1.766\,{\text{m}}}} \\ & = {{314} / {1.776}} \times 1.766 = 99.7 < 383.8\,{\text{KN}}/{\text{m}}^2 \left( {q^{\prime}_{ns} } \right) \\ & {\text{DL}} + {\text{LL}} + {{{\text{EQX}}({\text{DBE}}) \to P} / {L^{\prime}_{DBE} B^{\prime}_{DBE} ,\,L^{\prime} = 1.782\,{\text{m,}}\,{\text{B}}^{\prime} = 1.14\,{\text{m}}}} \\ & = {{493} / {1.782}} \times 1.146\, = 241.3 < 373.6\,{\text{KN}}/{\text{m}}^2 \left( {1.5q^{\prime}_{ns} } \right) \\ & {\text{DL}} + {\text{LL}} + {{{\text{EQX}}({\text{DBE}}) \to P} / {L^{\prime}_{MCE} B^{\prime}_{MCE} ,\,L^{\prime} = 1.786\,{\text{m,}}\,{\text{B}}^{\prime} = 0.856\,{\text{m}}}} \\ & = {{673} / {1.786}} \times 0.856\, = 440.2 < 558.2\,{\text{KN}}/{\text{m}}^2 {\text{KN}}/{\text{m}}^2 \left( {3q^{\prime}_{ns} } \right) \\ \end{aligned}$$

2 m × 2 m, 1.8 m × 1.8 m and 1.6 m × 1.6 m surface foundations are modelled in OptumG3 and checked for the MCE condition. Only the 2 m × 2 m foundation is safe under the MCE loads. The 1.6 m × 1.6 m and 1.8 m × 1.8 m foundations designed as per methods 1 and 2, as above can still survive the MCE loads, due to rocking. However, the methods do not provide any way to account for rocking explicitly.

3.7.11 Important Points to Note

The following are some important points to note:

• 1. Current practice of soil testing reports is not suitable as they do not suggest a reduction in bearing capacity due to horizontal loads and moments and they specify bearing capacity based on vertical load alone.

• 2. IS 1893 Part 1 does not explicitly specify which bearing capacity has to be increased. So, users are likely to increase the value of bearing capacity under vertical load alone. Bearing capacity estimated under the combined action of vertical, horizontal and moments is to be considered (see Sect. 7.3).

• 3. Methods 1 and 2 do not provide a way to account for rocking. Foundations designed using these methods are likely to undergo rocking under MCE loads, especially when the load eccentricity is high.

3.8 Calculation of Settlements Under Earthquake Loading

IS codes do not give any guidance on the calculation of settlements under earthquake loading. However, [54] mandates the calculation of settlements under earthquake loading in the case of cohesionless soils. Calculation of settlements under earthquake loading using methods typically used for static loads will be way off from the true settlement values. Hence, a well-established method of estimating settlements under earthquake loading is described below. Earthquakes induce settlements in both saturated and dry soils. The mechanism of earthquake-induced settlements is depicted in Fig. 3.12.

Fig. 3.12

Mechanism of earthquake-induced settlements [58]

3.8.1 Dry Sand Settlement

The current Indian code does not have any provision for the calculation of earthquake-induced soil settlements. In clays, it is not very important as clayey soils cannot experience significant settlements under vibrations induced by the earthquake loading. However, sands can experience significant earthquake-induced settlements. Dry sand settlements can be calculated by estimating cyclic shear strains induced on the soil during earthquake loading. Figure 3.13 presents a relation between the induced cyclic shear strain and expected volumetric strain of the soil for different densities of the soil and SPT-N value.

Fig. 3.13

Relationship between volumetric shear strain and cyclic shear strain in terms of a relative density and b standard penetration resistance [59]. (With permission from ASCE)

Cyclic shear strain can either be estimated using detailed ground response analysis or a simplified procedure [59], using Fig. 3.13. Figure 3.13 provides volumetric strain for an earthquake magnitude of 7.5. Soil settlements under any other earthquake magnitude can be obtained using the correction given in Table 3.6.

Table 3.6 Correction for volumetric settlement calculated [59]

$${\gamma }_{cyc}=0.65\frac{{a}_{max}}{g}\frac{{\sigma }_{v}{r}_{d}}{{G(\gamma }_{cyc})}$$

3.8.2 Settlement of Saturated Sands

Settlement of saturated sands can easily be obtained using the chart provided in Fig. 3.14. Figure 3.14 provides a relationship between liquefaction potential of the soil (Factor of safety of soil against liquefaction) and volumetric strain of the soil in terms of relative density and Standard Penetration Value.

Fig. 3.14

Relationship between liquefaction potential of the soil (Factor of safety of soil against liquefaction) and volumetric strain [60]

3.9 Summary and Conclusions

Starts with a discussion on the role of seismic codes in seismic risk reduction and then goes on to explain the criteria for the selection of a suitable type of foundation. Further, it explains the various aspects of the design of a shallow foundation and the design philosophies used worldwide. The section ends with a flowchart depicting various stages in the seismic design and construction of a shallow foundation and a brief note on the various methods used worldwide to consider the combination of vertical, horizontal and moment loading simultaneously.  Section 3.2 lists out the various components of earthquake loading, resisting mechanisms of the founding soil and possible shallow foundation failures when subjected to various components of earthquake loading. It then briefly mentions the various design methods employed for design against vertical loads, design against horizontal loads and design against moments.  Section 3.3 discusses the various shallow foundation design alternatives. Section 3.4 discusses the underlying philosophy of Table 1 of IS 1893 Part 1. Section 3.5 discusses the seismic design philosophy of IS codes. Section 3.6 discusses the specific seismic effects to be considered in the computation of seismic load. Section 3.7 starts with a discussion on the Pseudostatic Method of Seismic Design as per IS 1893 Part 1 and ends with a case study illustrating the application of the IS code guidelines. Section 3.8 briefly discusses the procedures to be adopted for the computation of seismic settlements in dry sands as well as saturated sands.

3.9.1 Steps Involved in the Seismic Design of a Shallow Foundation

As mentioned in Sect. 3.5, shallow foundations shall be proportioned for the following load combinations:

• 1. Dead load + Live load and

• 2. Dead load + Live load + Seismic load.

For a symmetrical building, the minimum number of load combinations a foundation needs to be proportioned are as follows:

• 1. DL + LL.

• 2. DL + LL + EQ in positive X direction (DBE)*.

• 3. DL + LL + EQ in positive Y direction (DBE)*.

For an unsymmetrical building, the minimum number of load combinations* a foundation needs to be proportioned are as follows:

• 1. DL + LL.

• 2. DL + LL + EQ in positive X direction (DBE)*.

• 3. DL + LL + EQ in positive Y direction (DBE)*.

• 4. DL + LL + EQ in negative X direction (DBE)*.

• 5. DL + LL + EQ in negative Y direction (DBE)*.

* Accidental eccentricity must be considered, if significant.

The various steps involved in the seismic design of a shallow foundation are summarised below.

• 1. Select a suitable type of foundation --->  Strip/Individual/Combined/Raft.

• 2. Assume initial/trial dimensions of the selected foundation ----> Length (L), Width (B), Founding depth (D_f).

• 3. As outlined in Sect. 7.3, estimate q_ult (ultimate bearing capacity) using a bearing capacity theory: Terzaghi/Meyerhof/Hansen/Vesic, for each of the above relevant load combinations considering the influence of H_x,H_y,M_x,M_y, as appropriate.

• 4. Obtain q_netsafe and Q_netsafe, for each of the above relevant load combinations.

• 5. Compare Q_netsafe with the vertical loading (V) of the corresponding load combination.

  If Q_netsafe < V -----------> Foundation is unsafe. Repeat steps 2 to 5, with new increased dimensions or increased depth (depending upon the type of soil).

  If Q_netsafe >> V -----------> Designed foundation is uneconomical. Repeat steps ii to v, with decreased dimensions or decreased depth (depending upon the type of soil).

  If Q_netsafe > V -----------> Foundation is Safe. Proceed to the next step

• 6. For the permanent load combination, check whether Q_netsafe is greater than or equal to the safe bearing pressure value. If Q_netsafe is greater than or equal to the safe bearing pressure value, then the foundation needs to be checked for its safety under seismic load combinations. Else, steps from 2 to 5 are to be repeated.

• 7. For safety against seismic load combinations, steps from 3 to 5 are to be repeated with an appropriate increase in the q_netsafe value as per IS 1893 Part 1. The foundation dimensions need to be increased if needed.

• 8. Settlements in case of sands under seismic loading can be evaluated using the guidance available in Sect. 3.8 above. Evaluation of seismic settlements can help in finalising the actual foundation dimensions keeping in mind the consequences of settlement failures on the functionality of the structure in a post-earthquake scenario.

3.9.2 Concluding Remarks

Neither Peck’s method (elastic stress method) nor Meyerhof’s method (effective area method) provides a way to account for rocking. Foundations designed using either of these methods are likely to undergo rocking under MCE loads, especially when the load eccentricity is high.

IS 1904 [54] mandates the calculation of settlements under earthquake loading in case of cohesionless soils. However, IS codes do not give any guidance on the calculation of settlements under earthquake loading. Calculation of settlements under earthquake loading using methods typically used for static loads will be way off from the true settlement values. Hence, an appropriate method of estimating settlements under earthquake loading needs to be used for the computation of seismic settlements.