Reliability analysis of seismic bearing capacity of strip footing by stochastic slip lines method

Computers and Geotechnics 91 (2017) 203–217

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Research Paper

Reliability analysis of seismic bearing capacity of strip footing by stochastic slip lines method A. Johari a,⇑, S.M. Hosseini a, A. Keshavarz b a b

Department of Civil and Environmental Engineering, Shiraz University of Technology, Shiraz, Iran School of Engineering, Persian Gulf University, Bushehr, Iran

a r t i c l e

i n f o

Article history: Received 16 November 2016 Received in revised form 3 May 2017 Accepted 21 July 2017

Keywords: Seismic bearing capacity Random field Stochastic slip lines method Monte Carlo simulation

a b s t r a c t In this research, the reliability analysis of seismic ultimate bearing capacity of strip footing is assessed with implementing slip lines method coupled with random field theory. The probability density functions of seismic and static bearing capacities which are log-normal and nearly normal distribution respectively are compared to each other. The predicted Probability Density Function (PDF) of the seismic bearing capacity by slip line method is verified, with those of the Terzaghi equation and Monte Carlo simulation (MCs). For uncertainties analysis by Terzaghi equation the Nc, Nq and Nc are assessed stochastically. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Determination of seismic bearing capacity of foundations is an important problem for the safe design in the seismic zone. Due to seismic loading, foundations may experience a reduction in bearing capacity and increase in settlement. In the last years the seismic effect has increased in many national design codes of foundation according to recent records. A rigorous seismic design of foundations can be done using seismic soil-structure interaction, which is capable to consider nonlinear soil behavior under dynamic loading. However, it is costly and time-consuming and is suitable only for important projects. Routine analysis of bearing capacity and the seismic response of the superstructure are decoupled. Base on this simplification, the methods for calculating the seismic bearing capacity of the strip footing can be classified into four major categories: limit equilibrium method [1–3], limit analysis [4–6], numerical methods [7,8] and the method of characteristics [9–12]. These methods with related important researches are explained briefly in subsections. 1.1. Limit equilibrium method Limit equilibrium is the most common analysis method of bearing capacity. In this method, at the first a sliding surface is assumed. Then, to obtain the ultimate limit load, the equilibrium ⇑ Corresponding author. E-mail address: [email protected] (A. Johari). http://dx.doi.org/10.1016/j.compgeo.2017.07.019 0266-352X/Ó 2017 Elsevier Ltd. All rights reserved.

equations are solved. A major limitation of this method is caused by the absence of a stress strain relationship. This method is used usually on the soil slope stability analysis and is followed by many researchers to analyze the bearing capacity of foundations. Sarma and Iossifelis [1] determined the seismic bearing capacity factors using the limit equilibrium technique of slope stability analysis with inclined slices. Their analysis showed that the factor Nq is dependent on the inertia of the surcharge load; the relationship between NC and Nq given in the literature was found to be incorrect for inclined loads and the inertia of the soil mass certainly has an effect on Nc. They showed the results of analysis in graphical forms as functions of the horizontal acceleration factor and of the angle of internal friction of the soil. Richards [2] based on limit analysis and using coulomb mechanism including inertial forces in the soil and on the footing gave expressions for seismic bearing capacity factors that are directly related to their static counterparts. They found that reduction in foundation capacity was due to both the seismic degradation of soil strength and the lateral inertial forces transmitted by shear to the foundation through the structure and any surcharge. A straightforward sliding-block procedure with examples was also presented for computing these settlements due to loss of bearing capacity for short time periods. Sarma and Iossifelis [1] and Richards et al. [2], had examined the reduction of static bearing capacity of cohesionless soils for horizontal accelerations of an earthquake. Budhu and Al-Karni [3] proposed the seismic bearing capacity factors with consideration of vertical acceleration and soil cohesion. They derived the seismic

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bearing capacity factors for a c-/ soil within the framework of the Mohr-Coulomb theory. The horizontal and vertical accelerations, the effects of inertia forces of the soil below and above the footing and applied loads, were considered in that research. 1.2. Limit analysis The basis of limit analysis rests upon two upper bound and lower bound theorems. The correct answer is revealed by identical upper and lower bounds. In lower bound theorem any stress system in which the applied forces are just sufficient to cause yielding. Similarly, in upper bound any velocity field that can operate is associated with an upper bound solution. In limit analysis, ultimate bearing capacity of foundation is calculated using the stress-strain relationship and failure mechanism. Among important contributions are the following researches. Soubra [4] calculated the seismic bearing capacity factors of shallow strip footings using the upper bound method of limit analysis. The pseudo-static approach was considered by taking into account static inertia forces. The solutions obtained were rigorous upper-bound ones in the framework of the limit analysis theory for an associated flow rule Coulomb material. Soubra [5] investigated the static and seismic bearing capacity problem of shallow strip footings. Two kinematically admissible failure mechanisms separately for both static and seismic conditions in the framework of the upper bound theorem were considered. The numerical results of the static and seismic bearing capacity factors in the form of design charts for practical use in geotechnical engineering were presented. Ghosh [6] by considering the pseudo-dynamic approach, examined the effect of soil friction angle, horizontal and vertical seismic accelerations, soil amplification, shear and primary wave velocities travelling through the soil layer during earthquake on the seismic bearing capacity factor Nc for a surface to very shallow strip footing. The result showed that the magnitude of Nc decreases with the increase in soil amplification, shear and primary wave velocities, which cannot be predicted by the existing pseudo-static approach. 1.3. Numerical methods The numerical methods such as Finite Difference Method (FDM), Finite Element Method (FEM) and Discrete Element Method (DEM) are used as conventional methods in geotechnical problems. These methods need more time to analyze but also offer the advantage that problems with complex geometries or complex constitutive models can be solved. Shafiee and Jahanandish [7] used the finite element method to estimate the seismic bearing capacity of strip footings for a wide range of friction angles and seismic coefficients. Also, they presented curves relating seismic bearing capacity factors to earthquake acceleration. Pane et al. [8] performed a finite difference numerical analysis aimed at evaluating the seismic effects on the ultimate bearing capacity of shallow strip foundations. Both structure inertia and soil inertia effects and possibility of superposition of these inertia effects are investigated. As results of their research, it was found that in some cases the soil inertia may play a significant role in the seismic capacity of the system, and that simple one-constant equations can be readily used in foundation design. 1.4. Method of characteristics The method of characteristics avoids the assumption of arbitrary slip surfaces, and produces zones within which equilibrium and plastic yield are simultaneously satisfied for given boundary

stresses. Furthermore, the effect of seismic forces on the bearing capacity of foundations can be entered into method of characteristics. Among important contributions are the research of Kumar and Mohan Rao [9], Keshavarz and Jahanandish [10]. Kumar and Mohan Rao [9] assessed the effect of horizontal earthquake body forces on the bearing capacity of foundations. They also investigated changes of the bearing capacity factors Nc, Nq and Nc as functions of earthquake acceleration coefficient for different values of soil friction angle. Keshavarz and Jahanandish [10] analyzed the seismic bearing capacity of reinforced soil slopes. For this purpose, the earthquake effect using horizontal and vertical pseudo-static seismic coefficients was considered. A number of graphs regarding critical load distribution for a uniformly reinforced slope, also a slope with linearly increasing reinforcement and a slope with linearly decreasing reinforcement in terms of variations in horizontal seismic coefficient were proposed. Keshavarz et al. [11] analyzed the seismic bearing capacity of strip foundations situated on reinforced soils. They showed the ultimate bearing capacity increases due to reinforcement by introducing another bearing capacity factor, Nt. Cascone and Casablanca [12] carried out the evaluation of static and seismic bearing capacity factors for a shallow strip footing using the method of characteristics, which was extended to the seismic condition by means of the pseudo-static approach. The results, for both smooth and rough foundations, were checked against those obtained through finite element analysis. Vo and Russell [13] studied the bearing capacity of strip footings on unsaturated soils using slip line theory. The suction profiles was considered are non-uniform with depth and was correspond to vertical flow of water by infiltration or evaporation and suction influences was included using the effective stress concept. This paper showed the similar and independent effects of cohesion and the contribution of suction to the effective stress in the governing equations. It showed that the influence of a nonuniform suction profile on bearing capacity is significant, and the depth to the ground water table and the footing width have significant roles in how much suction influences the bearing capacity. Charts were presented that permit assessment of bearing capacity changes that may occur when changes to suction are expected, due to seasonal fluctuations of soil moisture, drought or flooding. Analysis of seismic bearing capacity of strip footing is usually implemented with the assumption of homogeneous or averaged soil and earthquake properties. Therefore, the methods for calculating the bearing capacity of the shallow strip footing are restricted by the use of single valued parameters. Reliability analysis provides a means of evaluating the combined effects of uncertainties and offers a logical framework for choosing bearing capacity that are appropriate for the degree of uncertainty and the consequences of failure. Thus, as an alternative the deterministic assessment, a reliability assessment of bearing capacity would be useful in providing better engineering decisions. Since three decades ago, many probabilistic methods have been devised for analysis of bearing capacity of strip footing. Fenton and Griffiths [14] modeled soil with spatially varying shear strengths using random field theory and elasto-plastic finite element method to evaluate bearing capacity. Theoretical predictions of the mean and standard deviation of bearing capacity were derived with independency of c and / and using a geometric averaging model and then verified via Monte Carlo simulation. Przewłócki [15] applied the method of characteristics to a strip footing based on the stochastic subsoil. The investigation was limited only for the special case of a purely cohesive material. It allowed determination of the influence of the spatial variability of cohesion on the variance of the collapse load of the bearing

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capacity problem. The obtained results showed that the variance vanishes for decreasing correlation of the cohesion. Popescu et al. [16] examined the effect of random heterogeneity of soil properties on bearing capacity and followed a MCs in combination with non-linear finite element analysis. They demonstrated that the inherent spatial variability of soil shear strength can drastically modify the basic form of the failure mechanism in this bearing capacity problem. Consequently, there is no average failure mechanism (surface) in this problem, leading to the conclusion that MCs is the only methodology capable of providing a solution to this geomechanics problem. Soubra et al. [17] studied the effect of the spatial variability of the soil properties on the ultimate bearing capacity of a vertically loaded shallow strip footing in static case. The deterministic model used is based on numerical simulations using the Lagrangian explicit finite difference code FLAC3D. Their results showed that the average bearing capacity of a spatially random soil is lower than the deterministic value obtained for a homogeneous soil. A critical case appears when the autocorrelation distances are equal to the footing breadth. For this case, the mean value of the footing load reaches a minimum. Cho and Park [18] presented a numerical procedure for a probabilistic analysis that considers the spatial variability of crosscorrelated soil properties and applied to study the bearing capacity of spatially random soil with different autocorrelation distances in the vertical and horizontal directions. The approach integrates a commercial finite difference method and random field theory into the framework of a probabilistic analysis. It is found that, a nonsymmetric failure mechanism caused by the spatial heterogeneity, is not manifested in the deterministic. Puła and Chwała [19] presented a new approach to the reliability analysis of shallow foundations in which soil strength parameters were considered as random fields. These fields were averaged along the slip lines that resulted from Prandtl’s mechanism [20]. An efficient algorithm for evaluating reliability measures when the bearing capacity along kinematically admissible slip lines investigated. Motra et al. [21] evaluated the uncertainties and quality of bearing capacity factor prediction models of shallow foundations. Sixty models with different modeling approaches through a statistical framework that aids in uncertainty quantification and model quality evaluation were conducted. Their results showed that the more inaccurate the input parameters are, the more uncertain the quality of the estimated model prediction becomes. With increasing model uncertainty, the quality of the model also decreases. The purpose of this paper is reliability analysis of the seismic bearing capacity of the strip footing under the Prandtl’s failure mechanism [20]. The analyses are carried out by coded computer program, which implement slip lines method coupled with random field theory. Random fluctuations of soil parameters and earthquake cause random changes in the slip lines locations which in turn changing the seismic bearing capacity. For this purpose, the most effective parameters theory is used to implement the spatial variability of soil properties. The stochastic output of the slip lines method is verified by random variable in terms of Monte Carlo simulation and Terzaghi equation. Furthermore, an example is presented to illustrate the efficiency of the proposed method in reliability analysis of seismic bearing capacity of strip footing.

rz, sxz. Within the failure zone, the three stresses are related by three equations: two equilibrium equations and one yield equation. It is assumed that the soil is at yield and only need two variables (p, w) to introduce stress state. The angle w is measured by the direction of the major principal stress (r1) with respect to the positive x-axis as shown in Fig. 1. Stress p is called the average stress and equals to (Fig. 2):

p ¼ ðrxx þ rzz Þ=2

ð1Þ

It can be rewritten with respect to r1 and r3

p ¼ ðr1 þ r3 Þ=2

ð2Þ

where

rxx = Normal stress in the x direction rzz = Normal stress in the z direction r3 = Minor principal stress According to Fig. 3, equations of equilibrium under plane strain conditions are established [10]:

@ rxx @ sxz þ ¼ fx @x @z

ð3Þ

@ sxz @ rzz þ ¼ fz @x @z

ð4Þ

Fig. 1. Orientation of characteristic lines in the plane strain problem.

2. Seismic bearing capacity by slip lines method The bearing capacity of the shallow strip footing is a type of two-dimensional problem concerning plane plastic flow for which zones of the material are in a failure condition. For these plane problems there will be three unknown components of stress: rx,

205

Fig. 2. Stress Mohr circle for c-/ soil in the plane strain problem.

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2.1. Finite difference method Eqs. (8)–(11) can be solved using the finite difference method [8]. Starting from two points A and B with known solutions, the results for the intersection point C can be obtained (Fig. 4). Based on the boundary conditions, an iterative process is needed to obtain the values of xC, zC, wC and pC. Using the finite difference method Eqs. (8) and (10) can be rewritten as follow:

for b characteristics :

zC  zB ¼ tmp xC  xB

ð12Þ

for a characteristics :

zC  zA ¼ t mm xC  xA

ð13Þ

where

tmp ¼

tanðwC þ lC Þ þ tanðwB þ lB Þ 2

ð14Þ

tmm ¼

tanðwC  lC Þ þ tanðwA  lA Þ 2

ð15Þ

Fig. 3. Stress state on an element in the plane strain problem.

where fx = cKh; Body force in x direction fz = c(1-Kv); Body force in z direction Kh = Horizontal seismic coefficient Kv = Vertical seismic coefficient sxz = Shear stress c = Soil unit weight

By solving Eqs. (12)–(15) simultaneously, xC and zC are derived as follow:

xC ¼

zA  zB  xA tmm þ xB t mp tmp  t mm

zC ¼ ðxC  xB Þtmp þ zB

Assuming the material is in a state of failure and the stresses obey the Mohr–Coulomb yield condition, according to Fig. 2 the stresses can also be expressed as follows:

rxx ¼ pð1 þ sin / cos 2wÞ þ c cos / cos 2w

ð5Þ

rzz ¼ pð1 þ sin / cos 2wÞ  c cos / cos 2w

ð6Þ

sxz ¼ ðp sin / þ c cos /Þ sin 2w

ð7Þ

ð16Þ ð17Þ

For calculating wC and pC, Eqs. (9) and (11) should be written in form of finite difference.

for b characteristics :  sin 2lðpC  pB Þ þ Bmp ðwC  wB Þ ¼ A1 ð18Þ for a characteristics : sin 2lðpC  pA Þ þ Bmm ðwC  wA Þ ¼ A2

ð19Þ

where

ð8Þ

dz ¼ tanðw  lÞ dx

þ f z ½cos 2lðxC  xA Þ  sin 2lðzC  zA Þ Bmp ¼ sin /ðpC þ pB Þ þ 2c cos /

ð22Þ

Bmm ¼ sin /ðpC þ pA Þ þ 2c cos /

ð23Þ

ð9Þ

wC ¼

A3 A4

ð24Þ

ð10Þ

 cos /dp þ 2ðp sin / þ c cos /Þdw ¼ ðcos /dx þ sin /dzÞf x þ ðsin /dx  cos /dzÞf z

ð21Þ

By solving Eqs. (18) and (19) simultaneously, wC and pC are derived as follow:

cos /dp þ 2ðp sin / þ c cos /Þdw ¼ ðcos /dx  sin /dzÞf x þ ðsin /dx þ cos /dzÞf z

ð20Þ

A2 ¼ f x ½sin 2lðxC  xA Þ þ cos 2lðzC  zA Þ

By substituting Eqs. (5)–(7) into Eqs. (3) and (4) and simplification the following governing equations are obtained as follow:

dz ¼ tanðw þ lÞ dx

A1 ¼ f x ½sin 2lðxC  xB Þ  cos 2lðzC  zB Þ þ f z ½cos 2lðxC  xB Þ þ sin 2lðzC  zB Þ

where c = Cohesion / = Internal friction angle

ð11Þ

where

l ¼ p=4  /=2 In this paper, the w + l characteristics (b characteristics) are the lines which make a clockwise acute angle with the direction of r1 and the w-l characteristics (a characteristics) which make a counter-clockwise acute angle with the direction of r1 (Fig. 1).

Fig. 4. The a and b characteristics in the slip lines method.

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pC ¼ pB þ

A1  Bmp ðwC  wB Þ Amp

ð25Þ

where

Under static conditions Kh = KV = 0 and w0 = 0, mean stress p0 result:

p0 ¼


 A1 þ wB Bmp  Bmm wA  A2 A3 ¼  sin 2l pB  pA þ sin 2l

ð26Þ

A4 ¼ ðBmp þ Bmm Þ

ð27Þ

q þ c cos / 1  sin /

ð33Þ

2.2.2. Footing-soil interface boundary In this boundary (OA in Fig. 5), z is only known parameter. The stress values in this boundary are:

2.2. Boundary conditions

rf ¼ qf ð1  K V Þ

ð34Þ

The simple slip lines field is shown on Fig. 5. It is composed of a Cauchy zone OCD, a Goursat logarithmic spiral zone OBC and a mixed zone OAB [15]. OD is ground surface boundary and surcharge q is applied on it vertically. OA is strip foundation width (B). After determining the state of stress, along the last (wl) characteristics (OA), the failure load on the footing is obtained by considering the overall vertical force equilibrium during edge OA.

sf ¼ qf K h

ð35Þ

2.2.1. Ground surface boundary In this boundary (OD in Fig. 5), x and z are known parameters. The stress values in this boundary are:

r0 ¼ qð1  K V Þ

ð26Þ

s0 ¼ qK h

ð27Þ

With applying above stresses, w0 in this border can be derived as follow:



  p0 sin d 1 w0 ¼ 0:5 sin d p0 sin / þ c cos / where

d ¼ tan1




Kh 1  KV

ð28Þ

ð29Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðp0  r0 Þ2 þ s20 ¼ R

ð30Þ

R ¼ p0 sin / þ c cos /

ð31Þ

By solving Eqs. (30) and (31), p0 is derived as follow:

r0 þ c cos / sin / cos2 /

wf ¼ 0:5

p  sin1

! ! pf sin d d pf sin / þ c cos /

ð36Þ

where pf is the mean stress for footing-soil interface points. 2.2.3. Singularity point A difficulty in net solving is associated with the high stress gradients developed at the singularity at the footing edge (Point O in Fig. 5). Just outside the footing edge, the major principal stress acts horizontally and w is zero; below the footing base, however, the major principal stress acts vertically and w is p/2. Therefore, the principal directions change abruptly at the edge of the footing and this creates computational difficulties. The stresses being determined only by the surface load. At this point dx = dz = 0; therefore, the Eq. (11) can be simplified to:

dp  2ðp tan / þ cÞdw ¼ 0



Using Mohr circle (Fig. 2) the relation between p0 and radius of Mohr circle (R) is as follow:

p0 ¼

By satisfying the failure criterion and the above condition, the value of w along the footing surface (wf) can be derived as following formula:

þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðr0 sin / þ c cos /Þ  ðs0 cos /Þ2 cos2 /

ð32Þ

ð37Þ

By solving Eq. (37), the average stress, p, at the singularity point obtains as follow:



p ¼ c cot / þ ðp0 þ c cot /Þ exp ½2 tan /ðw  w0 Þ if

/–0

p ¼ p0 þ 2cðw  w0 Þ

/¼0

if

ð38Þ 3. Seismic bearing capacity factors The seismic bearing capacity can be evaluated using the formula introduced by Terzaghi [22] for a strip footing resting on a homogeneous dry soil subjected to a vertical and uniformly distributed load:

Fig. 5. A typical net of characteristic lines.

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qult ¼ cN c þ qN q þ 0:5cBNc

ð39Þ

where qult = Ultimate seismic bearing capacity; Nc, Nq and Nc = Seismic bearing capacity factors; In this study, the evaluation of seismic bearing capacity factors is carried out for a shallow strip foundation on a Mohr–Coulomb material under drained conditions. The superposition principle is used to evaluate the bearing capacity factors. Zhu et al. [23] showed that the maximum error that results from conventional superposition approximation is up to 5–9% on the conservative side in most cases, being generally less than 10%. If the soil be homogeneous, bearing capacity factors can be calculated using the superposition of the effects principle considering effect of surcharge, unit weight and cohesion separately. It is known that the seismic bearing capacity factors not only depend on the soil friction angle, but also depend on the seismic coefficient [7]. Nc and Nq can be obtained without solving the whole slip lines network. In order to evaluate the bearing capacity factor Nc, the values of surcharge and unit weight are assumed to be zero (Fig. 6a). This factor is expressed as [24].

pf ð1  sin / cos 2wf Þ  c cos w cos 2wf Nc ¼ 1  Kv where

wf ¼ 0:5

" 1

p  d  sin

ð40Þ

!# pf sin d pf sin / þ c cos /

8   < c cot / þ c cos / þ c cot / expð2w tan /Þ if f 1sin / pf ¼ : cð1 þ 2w Þ if f

ð41Þ

/–0 /¼0 ð42Þ

For calculating the bearing capacity factor Nq, the effects of cohesion and unit weight are removed (Fig. 6b). This factor is expressed as [24]:

Nq ¼

1  sin / cos 2wf expð2 tan /ðwf  w0 Þ 1  sin / cos 2w0

ð43Þ

where 1

wf ¼ 0:5ðp  sin ðsin d= sin /Þ  dÞ 1

w0 ¼ 0:5ðsin ðsin d= sin /Þ  dÞ

ð44Þ ð45Þ

The capacity factor Nc should be calculated by solving the slip lines network. For this bearing capacity factor, both values of surcharge and cohesion cannot be assumed to be zero, because in this case the singularity point will not be solved. For these purpose, a small value should be considered for surcharge or cohesion (Fig. 6c). Using Eq. (39) this factor is expressed as:

Nc ¼

quc 0:5cB

ð46Þ

where quc must be obtained by net solving. Using the Eqs. (40)–(46), bearing capacity factors were obtained for different values of the friction angle and horizontal earthquake coefficient. The results are shown for Nc, Nq and Nc in Figs. 7–9 respectively. It can be seen that the bearing capacity factors increases with increasing friction angle or decreasing horizontal seismic coefficient. Utilizing these figures, bearing capacity factors are achieved easily for both static and seismic conditions.

4. Verification To verify the accuracy of the proposed factors including Nc, Nq and Nc, the obtained results are compared with different methods from literature in Figs. 10–15 for friction angles 30° and 40°. Literature solution techniques for determining the bearing capacity factors are given in Table 1. It can be seen that for Nc and Nq there is reasonable agreement between the results by the proposed method and different methods from literature specially by Kumar and Mohan Rao [9]. While the results of the proposed method for Nc aren’t very close to Kumar and Mohan Rao [9] results. Because in proposed method, similar to several conducted researches [1– 3,5,24], one-sided failure mechanism is assumed whereas Kumar and Mohan Rao [9] considered a two-sided failure mechanism. In this study, single-side failure mechanism is used because: (i) the single-side failure mechanism requires less computational time especially for stochastic analyses; (ii) both-sides failure mechanism is applicable for the computation of Nc with smaller values of Kh [9]. Furthermore, as previously indicated, the values of Nc and Nq are almost the same for the single-side and two-sides failure mechanism; (iii) as previously mentioned, all researchers who have calculated the ultimate bearing capacity of strip footing for seismic case, except Kumar and Mohan Rao [9], have adopted single-side mechanism. A comparison between the seismic ultimate bearing capacities using several methods is presented in Table 3. For this purpose the selected constant parameters are given in Table 2.

Fig. 6. Loading conditions for determining the seismic bearing capacity factors.

A. Johari et al. / Computers and Geotechnics 91 (2017) 203–217

209

Fig. 10. Comparison between Nc for / = 30° by different methods with respect to Kh. Fig. 7. Variation of the Nc for different friction angles with respect to Kh.

Fig. 11. Comparison between Nc for / = 40° by different methods with respect to Kh.

Fig. 8. Variation of the Nq for different friction angles with respect to Kh.

Fig. 12. Comparison between Nq for / = 30° by different methods with respect to Kh.

Fig. 9. Variation of the Nc for different friction angles with respect to Kh.

layers of soil deposits. Consequently, a probabilistic approach seems quite appropriate when investigating geotechnical problems such as bearing capacity analysis.

5. Reliability analysis 5.1. Random field theory The spatial variability of soil properties distinguishes geotechnical engineering from other areas of civil engineering. Physical and mechanical parameters vary randomly within even homogeneous

Soil composition and properties vary from one location to another, even within homogeneous layers. The variability is attrib-

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A. Johari et al. / Computers and Geotechnics 91 (2017) 203–217 Table 1 Solution techniques for determining the bearing capacity factors. Researcher

Technique

Sarma and Iossifelis [1] Richards et al. [2] Budhu and Al-Karni [3] Soubra [4] Kumar and Mohan Rao [9] Proposed method

Limit equilibrium Limit equilibrium Limit equilibrium Upper bound limit analysis Method of characteristics Method of characteristics

by Vanmarcke in his fundamental research [26]. A random field can be described by mean, standard deviation and correlation function.

Fig. 13. Comparison between Nq for / = 40° by different methods with respect to Kh.

5.1.1. Correlation function The spatial correlation of soil parameter is considered by correlation function. The correlation function of a given soil parameter can be estimated from measured data of the parameter at different locations [27]. Correlation function between two soil parameters represents the degree of dependence between these parameters. Some researchers proposed several ranges for correlation coefficient between shear strength parameters c and /, qc,/, from 0.70 to 0.24 (e.g. [28–31]). In this study, qc,/ = 0.5 is considered. The correlation between two different random variables x1 and x2 is measured by a correlation function q defined as follows [32]:

qðx1 ; x2 Þ ¼

Fig. 14. Comparison between Nc for / = 30° by different methods with respect to Kh.

1

rx1 rx2

E½ðx1  lx1 Þðx2  lx2 Þ

ð47Þ

In which x1 and x2 might be the values of two different properties or the values of the same property at two different locations. rxi and lxi are respectively the standard deviation and the mean value of the variable xi (i = 1, 2). In this study, Markov correlation function [19] is used because it is the most conservative function. It is a squared exponential 2-D correlation function with different correlation distances in the horizontal and vertical directions as follows [19]:




qðx; x0 Þ ¼ exp 

jx  x0 j jz  z0 j  lx lz

 ð48Þ

where x and x0 are spatial coordinates, lx and lz are correlation lengths in horizontal and vertical directions, respectively. 5.1.2. Correlation length While geotechnical data vary in space, the magnitudes of a property at two adjacent locations are likely to be strongly correlated. As the distance between the two locations increases, the correlation weakens until it vanishes. This distance is called correlation length. Property measurements within a radius l are likely to be strongly correlated, while measurements more than l apart are weakly or no longer correlated [33].

Fig. 15. Comparison between Nc for / = 40° by different methods with respect to Kh.

uted to factors such as variations in mineralogical composition, conditions during deposition, stress history, and physical and mechanical decomposition processes. The spatial variability of soil properties is a major source of uncertainty [25]. From a probabilistic view point, this fact means that a given soil property could be adequately described by a random field. Important progress in the application of mathematical random field theory was achieved

5.1.3. Discretization of random field In most of the geotechnical problems, the random fields are assumed to be weakly homogeneous which specified by the mean and variance value. Based on this consideration, the discretization procedure is necessary to reduce a continuous random field to a finite set of random variables with particular care to the variance value. In this study, covariance matrix decomposition approach [14,34] is used for discretization of random field. 5.1.4. Covariance matrix decomposition approach The discretization of random field by covariance matrix decomposition approach is conducted based on following steps [14,34]:

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A. Johari et al. / Computers and Geotechnics 91 (2017) 203–217 Table 2 The selected constant parameters for determining the bearing capacity. Parameter

c (kN/m2)

q (kN/m2)

c (kN/m3)

B (m)

/ (Degree)

Kh

Value

20

20

18

1

30

0.2

Table 3 Comparison of the seismic ultimate bearing capacity. Researcher

Nc

Nq

Nc

qu (kN/m2)

Sarma and Iossifelis [1] Richards et al. [2] Budhu and Al-Karni [3] Soubra [4] Kumar and Mohan Rao [9] Proposed method

19.15 15.42 18.74 20.20 20.42 20.42

10.38 9.80 8.80 10.60 10.40 10.40

9.62 9.26 6.42 7.49 4.82 5.00

677.18 587.74 608.58 683.41 659.78 661.40

1. Calculation of the correlation matrix q(x1, x2), using Eq. (48) 2. Computation of the Cholesky decomposition of q(x1, x2) by lower triangular matrix (A) as follow:

AAT ¼ qðx1 ; x2 Þ

(3) Obtaining the PDF of seismic ultimate bearing capacity for strip footing by bearing capacity factors using superposition (Terzaghi equation).

ð49Þ 7. Illustrative example

3. Generating two independent standard normally distributed random fields as follow:

Gi ¼ AZ i

i ¼ 1; 2

ð50Þ

where Z = Standard normal distribution function 4. If two random variables correlate then it is necessary the correlated Gi should be determined by steps 4 and 5:

"

1

qc;u

qc;u

1

T

LL ¼

# ð51Þ

5. random c and / fields as follow:  Obtaining  the cross-correlated

 

Gc

G/

¼

L11

0:0

G1

L21

L22

G2

ð52Þ

The Gc and G/ should be utilized instead of G1 and G2 in Eq. (53) 6. Using mean and standard deviation for each random parameter the realization can be performed as follow:

X i ¼ lx ðxi Þ þ rx ðxi ÞGi

ð53Þ

where Xi = Correlated randomly parameter lx = Mean rx = Standard deviation of parameter Xi 6. Computer program

To demonstrate the efficiency and accuracy of the proposed method in determining PDF and reliability assessment of the seismic ultimate bearing capacity of strip footing an example problem with arbitrary parameters is presented. A schematic representation of bearing capacity problem of the strip footing is presented in Fig. 16. The strip footing is subjected to a centrally located vertical load that gradually increases and finally induces the Prandtl’s failure mechanism in the neighborhood of the strip footing. The overburden or surcharge pressure q is considered at the level of the footing. The value of it is determined based on the footing embedded depth and the unit weight of the soil above the footing base. The stochastic parameters with truncated normal and exponential distributions are given in Tables 4 and 5, respectively. Furthermore, the deterministic parameters are given in Table 6. In order to implementation of soil parameters uncertainty based on random field, at the first using the mean values of random parameters the network of slip lines constructed and the bearing capacity problem solved deterministically. Using the considered uncertain parameters (Tables 4 and 5) and the obtained network (discretized random field), the stochastic parameters are generated 500,000 for each element. A typical realization of random field for lx = 3 m and lz = 3 m is shown in Figs. 17a–17e. The soil heterogeneity and effect of correlation coefficient, qc,/ is clearly visible in this figure. Figs. 17a–17c show spatial variability of cohesion, friction angle and unit weight, respectively. Figs. 17d and 17e show spatial variability of cohesion and friction angle when qc,/ = 0.5. It can be seen more compatibility between cohesion and friction angle in Figs. 17d and 17e

To determine the seismic bearing capacity of foundation based on random field theory, a program was codded in MATLAB. MCs is used to verify the coded program. It is capable to consider the uncertainties of soil parameters and horizontal earthquake acceleration coefficient in stochastic analysis. The major capability of the programs is as follows: (1) Implementation of soil parameters uncertainty based on MCs and random field. (2) Developing the PDF of seismic ultimate bearing capacity for strip footing by stochastic slip line method without using superposition.

Fig. 16. Bearing capacity problem of the strip footing.

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Table 4 Stochastic truncated normal parameters. Parameters

Mean

Standard deviation

Maximum

Minimum

/ (Degree) c (kN/m3) c (kN/m2)

25.0 18.0 20.0

1.5 0.5 1.0

31.0 20.0 24.0

19.0 16.0 16.0

Table 5 Stochastic truncated exponential parameter. Parameter

k

Minimum

Maximum

Mean

Standard deviation

Kh

10

0.2

0.4

0.2687

0.0525

Table 6 Deterministic parameters. Parameters

Value

B (m) q (kN/m2) Kv

1.0 20.0 0.0

Fig. 17d. Spatial variability of cohesion (kPa) with qc,/ = 0.5.

Fig. 17a. Spatial variability of cohesion (kPa) with qc,/ = 0.

Fig. 17e. Spatial variability of friction angle (Deg.) with qc,/ = 0.5.

Fig. 17b. Spatial variability of friction angle (Deg.) with qc,/ = 0.

to other zones. Fig. 18c shows the Cauchy zone has more changes with respect to other zones and finally Fig. 18d shows the distance between the lines of the network increased more than other cases. In other word the network is expanded. In order to compare the results of random field theory with those of the random variable by MCs, the final PDF curves for seismic ultimate bearing capacity of strip footing are determined using the same data and both methods. For this purpose, 500,000 generation points are used for the random field and random variable. The results are shown in Fig. 19. As it can be seen that in this figure, the PDF of the seismic bearing capacity, obtained using the random field is very close to that of the random variable by MCs. Fig. 19 demonstrates the accuracy of the proposed method in determining reliability of the seismic bearing capacity of strip footing. As shown in this figure the PDF of seismic bearing capacity is close to being log-normal.

Fig. 17c. Spatial variability of unit weight (kg/m3) with qc,/ = 0.

with respect to Figs. 17a and 17b (when cohesion increases friction angle decreases and reversely). Random fluctuations of soil parameters cause random changes in the slip lines locations. Four typical realizations which show the changes in slip line network are represented in Figs. 18a– 18d. Fig. 18a shows the effect of soil uncertainties on the mixed zone. It can be seen this zone irregularity more than other zones. Fig. 18b shows the Goursat zone has more changes with respect

Fig. 18a. Typical effect of soil uncertainties on the mixed zone.

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213

Fig. 18b. Typical effect of soil uncertainties on the Goursat zone.

Fig. 20. Comparison of PDF of seismic ultimate bearing capacity by random field with different vertical correlation length and MCs (qc,/ = 0.5). Fig. 18c. Typical effect of soil uncertainties on the Cauchy zone.

Fig. 18d. Typical effect of soil uncertainties on the field.

Fig. 21. Comparison of PDF of seismic ultimate bearing capacity by random field with different horizontal correlation length and MCs (qc,/ = 0.5).

Fig. 19. Comparison of PDF of seismic ultimate bearing capacity by MCs and random field theory (qc,/ = 0, lx = lz = 5 m).

The influence of the correlation length in vertical (lz ) and horizontal direction (lx ) on the PDF of seismic bearing capacity are shown in Figs. 20 and 21 respectively. Comparison of PDF of seismic bearing capacity by MCs and random field with different vertical correlation length and constant horizontal correlation length (lx = 3 m) is shown in Fig. 20. For this analysis the correlation coefficient between cohesion and friction angle qc,/ = 0.5 is considered It can be seen the mean value of seismic bearing capacity

increases with decreasing vertical correlation length. While the standard deviation of seismic bearing capacity decreases with decreasing vertical correlation length that means uncertainty reduction of seismic bearing capacity. Comparison of PDF of seismic bearing capacity by MCs and random field with different horizontal correlation length and constant vertical correlation length (lz = 0.5 m) is shown in Fig. 21. In this analysis the correlation coefficient between cohesion and friction angle qc,/ = 0.5 is considered too. It can be seen no changes occurs in the mean and standard deviation values of seismic bearing capacity. Furthermore, the influence of the correlation coefficient (qc,/) on PDF of seismic bearing capacity is presented in Fig. 22. It can be seen the mean value of seismic bearing capacity increases when correlation coefficient decreases. While the standard deviation of seismic bearing capacity decreases when correlation coefficient decreases. Since the natural soil deposits usually change in the vertical direction and almost no change in the horizontal direction, the lack of change in mean and standard deviation is justified in Fig. 21. To compare the reliability of the bearing capacity of strip footing in seismic and static cases, the PDF of them are determined.

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Fig. 24. Comparison CDF of ultimate bearing capacity in static and seismic cases by MCs (qc,/ = 0.5). Fig. 22. Influence of the correlation coefficient on PDF of seismic ultimate bearing capacity by MCs.

Section 5.1 is used. For stochastic seismic ultimate bearing capacity of strip footing by Terzaghi equation the Nc, Nq and Nc are assessed stochastically. For this purpose, the uncertain parameters in Tables 4 and 5 and deterministic parameters in Table 6 are utilized. The PDF of the Nc, Nq and Nc are shown in Figs. 25–27 respectively. The predicted probability distribution functions of the seismic ultimate bearing capacity by slip line method, Terzaghi equation and Monte Carlo simulation are compared in Fig. 28. It can be seen in this figure, the results obtained using the developed method is very close to that of the Terzaghi equation and Monte Carlo simulation. For determining a smooth PDF of bearing capacity factors those can be modeled by fitted distribution. In this research the PDF of bearing capacity factors were derived from the maximum entropy principle [35,36]. In maximum entropy principle, the unknown PDF f(x) of a random variable X should maximize the entropy S[f (x)] given by [37,38]:

Z S½f ðsÞ ¼ 

þ1

1

f ðxÞ ln½f ðxÞdx

ð54Þ

where Fig. 23. Comparison PDF of ultimate bearing capacity in static and seismic cases by MCs (qc,/ = 0.5).

f ðxÞ ¼ H expðfðxÞÞ In this equation:

fðxÞ ¼ k1 x þ k2 x2 þ . . . þ kN xN Fig. 23 shows the PDF of these cases. As can be seen in this figure, the PDF of seismic bearing capacity has a nearly log-normal while the PDF of static bearing capacity has a nearly normal distribution. In addition, standard deviation of seismic bearing capacity is less than the static case. Also, the mean value of seismic bearing capacity is nearly half of static case. Fig. 24 shows the seismic and static cumulative distribution curve of the strip footing bearing capacity. It can be seen that for the same probability of bearing capacity (e.g. 50%) the seismic and static bearing capacity are about 345.3 kPa and 708.6 kPa respectively.

ð56Þ

where H and ki are coefficients. For Figs. 25–27 the case of N = 2 appears.

8. Comparison of bearing capacity by stochastic slip line method and Terzaghi equation To demonstrate the accuracy of the proposed approach in determining the PDF of the seismic ultimate bearing capacity, the results of the stochastic slip line method are compared with those of Terzaghi equation (Eq. (39)) and MC simulation. In stochastic analysis by slip line method, the random field theory as described in

ð55Þ

Fig. 25. Probability density function of Nc.

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215

Fig. 29. PDF of sensitivity analysis to determine the response of model. Fig. 26. Probability density function of Nq.

Fig. 27. Probability density function of Nc.

carried out. For this purpose, all stochastic parameters’ PDF were increased based on their mean. To assess the influence of changes in PDF of each stochastic parameter, that parameter was increased while the ranges of the other stochastic input parameters were kept constant. The results are shown in Figs. 29 and 30. Fig. 29 shows that with an increase in mean of cohesion, internal friction angle and unit weight the PDF of seismic ultimate bearing capacity (qu) shifts rightwards. Furthermore, with an increase in mean of horizontal seismic coefficient the PDF of seismic ultimate bearing capacity (qu) shifts leftwards, indicating that the negative effect of Kh on seismic ultimate bearing capacity. Fig. 30 shows the effect of changes in mean of each stochastic parameter on Cumulative Distribution Function (CDF) of seismic ultimate bearing capacity. Based on this figure, the amounts of the shift in the curves (change in probability of qu) corresponding to 30% increases in mean of the stochastic parameters with respect to the original PDF for qu equal to 400 kPa are presented in Table 7. This table shows that the internal friction angle is the most effective parameter in qu. The value of probability of original curve is 74.92%. Influence of internal friction angle on the failure zone is shown in Fig. 31. It can be seen that the length of Cauchy zone (L) has near normal distribution when the internal friction angle as stochastic input parameter changes normally. The variation range of L is between 0.00 m and 0.83 m in Fig. 31. Fig. 32 demonstrates the influence of horizontal seismic coefficient on failure zone. It can be seen that L has near exponential distribution. The variation

Fig. 28. Comparison of probability density function of seismic ultimate bearing capacity.

9. Sensitivity analysis To evaluate the seismic bearing capacity model response with respect to changes in input parameters, a sensitivity analysis was

Fig. 30. CDF of sensitivity analysis to determine the response of model and the most effective parameter.

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Fig. 31. Effect of friction angle on failure zone.

Fig. 32. Effect of horizontal seismic coefficient on failure zone.

Table 7 Changes in probability of qu corresponding to 30% increase (shift rightward) in the PDF of input parameters. Stochastic parameter

Shift in /

Shift in c

Shift in c

Shift in Kh

Value of probability (%) Change (%)

1.87 +97.50

43.45 +42.00

71.63 +4.39

98.22 31.10

range of L is between 0.0 m and 0.45 m in Fig. 32. It means the internal friction angle is more effective than horizontal seismic coefficient. These outcomes verify the results of sensitivity analysis in Figs. 29 and 30 and Table 5. 10. Conclusions Seismic ultimate bearing capacity of strip footing is a stochastic problem due to the inherent uncertainties in the geotechnical and earthquake parameters, model performance as well as human uncertainty. In this paper, the random field theory was used to assess the reliability of seismic bearing capacity of strip footing based on the uncertainty in the geotechnical and earthquake properties. For this purpose, a program was codded in MATLAB. The selected stochastic parameters were internal friction angle, cohesion, unit weight and the horizontal seismic coefficient. The spatial variability of stochastic parameters is modeled using theory of random field which is discretized by covariance matrix decomposition approach. Furthermore, the effects of the correlation

length and coefficient are assessed. The width of foundation, surcharge pressure and vertical seismic coefficient were regarded as constant parameters. The results show that the mean value of seismic bearing capacity increases with decreasing correlation length and the standard deviation decreases with decreasing correlation length. It was observed that the mean value of seismic bearing capacity increases with decreasing correlation coefficient (qc,/) and the standard deviation decreases with decreasing correlation coefficient. Comparison of the results of static and seismic reliability assessment of bearing capacity show that the probability density functions of them were close to being log-normal and normal distribution respectively. In addition, standard deviation of seismic bearing capacity was less than the static case. Also, the mean value of seismic bearing capacity was nearly half of static case. The predicted probability distribution functions of the seismic bearing capacity by slip line method, Terzaghi equation and Monte Carlo simulation were compared to each other. The results illustrate obtained PDF using the stochastic slip line method is very close to others. The

A. Johari et al. / Computers and Geotechnics 91 (2017) 203–217

results of sensitivity analysis show that the horizontal seismic coefficient and internal friction angle are the most effective parameters on bearing capacity of the strip foundation.

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