Design of Slab on Grade Using Modulus of Subgrade Reaction

Subgrade Reaction

The subgrade reaction coefficient method is a method of applying the interaction calculation method to the seismic response analysis of the cross section of the underground structure and can be applicable to buried or semiburied underground structures.

From: Shield Tunnel Engineering , 2021

Structure type and design of shield tunnel lining

Shuying Wang , ... Junsheng Yang , in Shield Tunnel Engineering, 2021

5.8.4 Subgrade reaction coefficient method

The subgrade reaction coefficient method is a method of applying the interaction calculation method to the seismic response analysis of the cross section of the underground structure and can be applicable to buried or semiburied underground structures. The effect of the surrounding stratum medium is simulated by multipoint compression springs and shear springs, and the structure can be simulated by beam elements. The method includes three basic steps: (1) calculating the spring constant of the surrounding stratum medium, (2) calculating the seismic displacement of the surrounding stratum, and (3) calculating the seismic response of the underground structure. The static finite element method is used for approximate calculation of the resistance spring constant of the surrounding stratum, and a piecewise one-dimensional model or a plane finite element model is used for approximate calculation of the seismic displacement of the surrounding stratum.

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Design of Underground Structures

Bai Yun , in Underground Engineering, 2019

3.1.2.1 Load Types

The loads that usually play an important part in the design of a tunnel lining include the vertical and lateral ground pressure, water pressure, self-weight of the lining, surcharge, and subgrade reaction.

Vertical ground pressure (P _V ) is induced by the self-weight of ground layers above the crown of the tunnel. For a shallow tunnel, the vertical earth pressure at the crown of the lining ( $P_V$ ) equals the overburden pressure of the above layers, as indicated in Fig. 3.13 and Formula 3.4 (where ${\gamma }_i$ and ${\gamma }_j$ are the unit weight of soil stratum of the layer i and j located, respectively, above and below groundwater level, and P _o is the surcharge).

Figure 3.13. Section of tunnel and surrounding ground.

(3.4) $P_V=P_o+\sum {\gamma }_i{H}_i+\sum {\gamma }_j{H}_j$

For a deep tunnel, the vertical ground pressure is reduced due to the arch effect or soil fraction. This reduced earth pressure (P _e1) can be calculated from Terzaghi's formula (see Formula 3.7).

Horizontal ground pressure (P _H ) is the difference between earth pressure inside and outside the tunnel. This is similar to an earth-retaining wall where earth at the back of the retaining wall exerts a lateral force on it. To calculate the horizontal ground pressure of tunnels, the vertical ground pressure $P_V$ is thus multiplied by the coefficient of lateral earth pressure K ₀ at rest.

The combined effect of water pressure acting on the peripheral of the lining is the uplifting buoyancy. It should be compared with the total downward load. If the former is larger, the tunnel will float.

Self-weight of the lining is considered a dead load acting on the central line of the lining cross section. In other words, it is calculated as the surface weight in kN/m². Formulas (3.5) and (3.6) show the calculation of the self-weight ( $P_g$ ) of the lining as a function of the weight of the concrete linings per unit length (W), the radius of the central line of the lining cross section ${(R}_c)$ , the unit weight of the concrete ( ${\gamma }_c$ ), and the thickness of the lining (t) (Fig. 3.14).

Figure 3.14. Self-weight of tunnel lining.

(3.5) $P_g=\frac{W}{2\pi R_c}\left(\mathrm{circlular\; section}\right)$

(3.6) $P_g={\gamma }_c\times t\left(\mathrm{rectangular\; section}\right)$

The surcharge is the result of traffic or structures above the tunnel. The Chinese design code identifies three types of surcharge: human load, road traffic load, and railway traffic load. Since they are live loads, both static and dynamic effects should be considered.

The subgrade reaction is the resistance from peripheral ground layers. It is a reaction force which depends on the ground displacement. It is assumed to be proportional to ground displacement with a proportionality factor that depends on the ground stiffness and lining dimensions (Fig. 3.15).

Figure 3.15. Subgrade reaction (soil reaction).

To understand this mechanism, the subgrade can be viewed as springs that are connected to the lining. When a circular tunnel section deforms under external loads and becomes an ellipse, the subgrade reaction occurs. However, it is important to note that the subgrade reaction and the ground pressure introduced at the beginning of this section are different load types. Ground pressure can be said to be active since it pushes a tunnel structure, whereas subgrade reaction is said to be passive as it is the result of a structure pushing the ground.

In addition to the abovementioned load types, other effects potentially also need to be considered, such as construction loads, seismic loads, the effect of an adjacent tunnel, etc.

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BEM for Other Plate Problems

John T. Katsikadelis , in The Boundary Element Method for Plate Analysis, 2014

3.4.2 AEM solution

The AEM provides an efficient solution method for Eq. (3.82) for any function ${\mathit{p}}_{\mathit{s}}={\mathit{p}}_{\mathit{s}}\left(\mathit{x},\mathit{y},\mathit{w},\mathit{w}{,}_{\mathit{x}},\mathit{w}{,}_{\mathit{y}},\mathit{w}{,}_{\mathit{xy}},\dots \right)$ representing the subgrade reaction, which may depend linearly or nonlinearly on the deflection $\mathit{w}$ and its derivatives.

The application of the AEM as described in Section 3.3.2.1 converts Eq. (3.82) into the equation

(3.89) $\mathit{D}\mathbf{b}+{\mathbf{p}}_{\mathit{s}}\left(\mathbf{w},\mathbf{w}{,}_{\mathit{x}},\mathbf{w}{,}_{\mathit{y}},\mathbf{w}{,}_{\mathit{xy}},\dots \right)=\mathbf{f}$

which by virtue of Eq. (3.46) results in an equation of the form

(3.90) $\mathit{D}\mathbf{b}+{\mathbf{p}}_{\mathit{s}}\left(\mathbf{b}\right)=\mathbf{f}$

Equation (3.90) is a linear or nonlinear algebraic equation depending on the function (3.83), whose solution gives the vector $\mathbf{b}$ of the nodal values of the fictitious load. Evidently, once $\mathbf{b}$ is known, the deflections and the derivatives are computed from Eq. (3.46). Special care should be given in deriving the boundary conditions for free edges, when the mechanical model includes shear or bending layer [28].

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Design of a Foundation Under Vibrating Equipment

Mohamed A. El-Reedy PhD , in Onshore Structural Design Calculations, 2017

7.3.2 Soil Parameter

The allowable soil-bearing or allowable pile capacity for foundations for equipment designed for dynamic loads shall be a maximum of half of the normal allowable for static loads. Noting that, the allowable soil bearing will be provided from the geotechnical consultant office, which will perform a soil investigation test.

The maximum eccentricity between the center of gravity of the combined weight of the foundation and machinery and the bearing surface shall be 5% in each direction.

Structures and foundations that support vibrating equipment shall have a natural frequency that is outside the range of 0.80 to 1.20 times the exciting frequency.

For this type of foundation, there are some soil parameters that are required in case of vibrating equipment.

•

The allowable bearing capacity

•

The soil density: ρ

•

The shear modulus: G

•

The shear wave velocity shall be calculated from the following formula:

$V_s=\sqrt{\frac{G}{\rho }}$

•

Dynamic coefficient of subgrade reaction for vertical vibration ( K _z ).

For a rigid rectangular footing of plan area A on a semi-infinite elastic half-space (Barkan, 1962; Richart et al., 1970):

$K_z=\frac{B_z}{\sqrt{A}}\left(\frac{G}{(1-\nu )}\right)$

where:

V=Poisson's ratio and equal to 0.3 as a typical value for the sand,

G=shear modulus,

B _z =the ratio of the length to breadth of footing of about 1.40.

Barkan suggests that for allowable static bearing capacity of about 0.15   N/mm², K _z should not be less than 0.03   N/mm³.

•

Coefficient of subgrade reaction for horizontal vibrations: K _x

$K_x=0.5K_z$

•

Coefficient of subgrade reaction for rocking vibrations: K _ϕ

$K_{\varphi }=2K_z$

•

Coefficient of subgrade reaction for torsion vibrations: K _θ

$K_{\theta }=0.75K_z$

Table 7.2 provides the equations to calculate the effective radius of footing in case of horizontal rocking, vertical rocking, and torsional vibration (Table 7.3).

Table 7.2. Embedment coefficient for spring

Mode of vibration	Effective radius of footing, r o	Embedment coefficient
Horizontal vibration	$\sqrt{BL/\pi }$	η x =1+0.6(1−ν)(h/r o )
Vertical vibration	$\sqrt{BL/\pi }$	η z =1+0.55(2−ν)(h/r o )
Rocking	$\sqrt[4]{(B{L}^3)/3\pi }$	η ϕ=1+1.2(1−ν)(h/r o )+0.2(2−ν)(h/r o )3
Torsional	$\sqrt[4]{\frac{BL(B^2+L^2)}{6\pi }}$	–

B and L are the width and length of the foundation, respectively. h is the foundation depth embedment below the grade.

Table 7.3. Effect of depth of embedment on damping ratio

Mode of vibration	Damping ratio embedment factor
Vertical	${\alpha }_{\epsilon }=\frac{1+1.9(1-v)(h/{\gamma }_0)}{\sqrt{{\eta }_{\epsilon }}}$
Horizontal	${\alpha }_{\gamma }=\frac{1+1.9(2-v)(h/{\gamma }_0)}{\sqrt{{\eta }_{\gamma }}}$
Rocking	${\alpha }_{\varphi }=\frac{1+0.7(1-v)(h/r_0)+0.6(2-v){(h/r_0)}^3}{\sqrt{{\eta }_{\varphi }}}$

Source: Ary et al., 1984.

${\eta }_s=1+0.6(1-v)\left(\frac{h}{r_0}\right)$

${\eta }_x=1+0.55(2-v)\left(\frac{h}{r_0}\right)$

${\eta }_{\varphi }=1+1.2(1-v)\left(\frac{h}{r_0}\right)+0.2(2-v){\left(\frac{h}{r_0}\right)}^3$

Table 7.4 provides a summary of the calculation of the mass ratio, the damping ratio, the spring constant, and the natural frequency for different modes of vibration based on the foundation dimensions and the geotechnical data from the soil investigation tests.

Table 7.4. Main parameters

Vibration mode	Mass ratio (B)	Damping parameters	Damping ratio	Spring constant (K)	Natural frequency, f n
Vertical	$B_z=\frac{(1-\nu )m}{4\rho r_o^3}$	$c_z=\frac{3.4r_o\sqrt{\rho G}}{1-\nu }$	$D_z=\frac{0.425}{\sqrt{B_z}}$	$k_z=\frac{4G{r}_o}{1-\nu }$	$f_{nz}=\frac{1}{2\pi }\sqrt{\frac{k_z}{m}}$
Horizontal	$B_x=\frac{(7-8\nu )m}{32(1-\nu )\rho r_o^3}$	$c_x=\frac{18.4(1-\nu )\sqrt{\rho G}}{7-8\nu }$	$D_x=\frac{0.288}{\sqrt{B_z}}$	$k_x=\frac{32(1-\nu )G{r}_o}{7-8\upsilon }$	$f_{nx}=\frac{1}{2\pi }\sqrt{\frac{k_x}{m}}$
Rocking	$B_{\psi }=\frac{3(1-\nu )m}{8\rho r_o^3}$	$c_{\psi }=\frac{0.8r_o^4\sqrt{\rho G}}{(1-\nu )(1+B_{\psi })}$	$D_{\psi }=\frac{0.15}{(1+B_{\psi })\sqrt{B_{\psi }}}$	$k_{\psi }=\frac{8G{r}_o^3}{3(1-\nu )}$	$f_{n\psi }=\frac{1}{2\pi }\sqrt{\frac{k_{\psi }}{I_{\psi }}}$
Torsional	$B_{\theta }=\frac{I_{\theta }}{\rho r_o^5}$	$c_{\theta }=\frac{4\sqrt{B_{\theta }\rho G}}{1+2B_{\theta }}$	$D_{\theta }=\frac{0.50}{1+2B_{\theta }}$	$k_{\theta }=\frac{16}{3}G{r}_o^3$	$f_{n\theta }=\frac{1}{2\pi }\sqrt{\frac{k_{\theta }}{I_{\theta }}}$

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Laterally loaded piles

Ruwan Rajapakse , in Pile Design and Construction Rules of Thumb (Second Edition), 2016

17.2 Lateral loading analysis: simple procedure

17.2.1 Design methodology of laterally loaded piles

It is not possible to conduct a p–y curve analysis without a computer. The following method is much more simple and can be computed manually.

•

It is assumed that a pile is being held by springs as shown in Fig. 17.6 . The spring constant or the coefficient of subgrade reaction varies with the depth. In most cases, coefficient of subgrade reaction increases with depth.

Figure 17.6. Design methodology.

•

A simplified analysis of lateral loads on piles can be conducted by assuming the coefficient of subgrade reaction to be a constant with depth. For most cases the error induced by this assumption is not significant.

•

When a pile is subjected to a horizontal load, it would try to deflect.

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The surrounding soil would generate a resistance against deflection.

•

The resistance provided by the soil is represented by a series of springs. The spring constant is taken as the coefficient of subgrade reaction (k).

•

In reality (k) changes with depth.

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Simplified analysis is conducted assuming (k) to be a constant.

•

For most cases this assumption does not produce a significant error.

•

The equation for lateral load analysis is given as follows (Matlock and Reese, 1960):

$u=2^{\text{1/2}}\frac{H}{k}\times {\left(\frac{l_{\text{c}}}{4}\right)}^{-1}\text{+}\frac{M}{k}\times {\left(\frac{l_{\text{c}}}{4}\right)}^{-2}$

where u, lateral deflection; H, applied lateral load on the pile (normally due to wind or earth pressure); k, coefficient of subgrade reaction (assumed to be a constant with depth); M, moment induced due to lateral forces (when the lateral load is acting at a height above the ground level, then moment induced also should be taken into consideration); and l _c, critical pile length (below this length, the pile is acting as an infinitely long pile).

Now, l _c is obtained by using the following equation:

$l_{\text{c}}=4{\left[\frac{{(EI)}_{\text{p}}}{k}\right]}^{1/4}$

(EI)_p, Young's modulus and moment of inertia of the pile. In the case of wind loading, the moment of inertia should be taken against the axis, which has the minimum moment of inertia, since wind load could act from any direction.

In the case of soil pressure and water pressure, the direction of the lateral load does not change. In these situations, the moment of inertia should be taken against the axis of bending.

•

A similar equation is obtained for the rotational angle (θ) at the top of the pile.

$\theta =\frac{H}{k}\times {\left(\frac{l_c}{4}\right)}^{-2}+2^{1/2}\frac{M}{k}\times {\left(\frac{l_{\text{c}}}{4}\right)}^{-3}$

Derivation of this equation is provided by Matlock and Reese (1960).

ϕ′ value of sandy soil can be calculated using the following equation:

ϕ′ = 53.881 − 27.6034 e^−0.0147N (Peck et al., 1974), where (N, average SPT value of the strata).

•

Note: coefficient of subgrade reaction (k) can be obtained using Table 17.1.

Table 17.1. Coefficient of subgrade reaction (k) versus N (SPT, Johnson and Kavanaugh, 1968)

SPT (N)	8	10	15	20	30
k (kN/m3)	2.67 E−6	4.08 E−6	7.38 E−6	9.74 E−6	1.45 E−6

Similarly soil parameters for other strata also need to be provided.

17.2.2 Soil parameters for clayey soils

Soil parameters required for clayey soils:

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S _u (undrained shear strength. "S _u" is obtained by conducting unconfined compressive strength tests).

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ɛ _c (strain corresponding to 50% of the ultimate stress. If the ultimate stress is 3 tsi, then "ɛ _c" is the strain at 1.5 tsi).

•

k _s (coefficient of subgrade reaction).

Coefficient of subgrade reaction for clay soils is obtained from Table 17.2.

Table 17.2. Coefficient of subgrade reaction versus undrained shear strength (Reese, 1975)

	Average undrained shear strength (tsf)
k s (lb/in3)	0.5–1	1–2	2–4
Static	500	1000	2000
Cyclic	200	400	800

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Design, construction and installation of support structures for offshore wind energy systems

K. Lesny , W. Richwien , in Wind Energy Systems, 2011

Monopiles

With a monopile foundation the design parameters are the diameter, the embedment length and the wall thickness. The standard design method is the p-y method which is based on the model of the beam on elastic foundation, idealizing the soil as a series of independent springs (see Fig. 16.3). In this model the nonlinear soil behaviour is considered by a nonlinear spring stiffness defined by the so-called p-y curve which relates the subgrade reaction p to the horizontal displacement y in a certain depth z along the pile.

The p-y curves have been developed for various types of soil usually from a small number of field tests on piles with diameters of less than 1   m. Fig. 16.13 shows exemplarily the p-y curves developed by Reese et al. (1974) for sand.

16.13. Qualitative progression of p-y curves for sand in various depths z according to Reese et al. (1974) (Lesny, 2010)

The initial gradient k  · z shown in Fig. 16.13 is related to the stiffness of the soil by the relationship k  = E_S/D where E_S is the oedometer modulus of the soil and D the pile diameter (Terzaghi, 1955). As the initial gradient increases linearly with depth z, a linear increasing soil stiffness is therefore assumed as well. The factor k is given, for example, in DNV (1992) as a function of the relative density of the soil and the corresponding angle of internal friction (see Fig. 16.14).

16.14. Factor k of the initial increase of the p-y curve for sand as a function of the angle of internal friction and the relative density (DNV, 1992).

The corrected maximum subgrade reaction p _u ^corr (z) of the p-y curves shown in Fig. 16.13 is derived from the theoretical maximum subgrade reaction p_u (z) which itself is based on two failure modes depending on the depth z along the pile (Parker and Reese, 1971; Reese et al., 1974). In the near-surface area a three-dimensional earth pressure model, as shown in Fig. 16.15, is used.

16.15. Failure model for calculating the maximum subgrade reaction for sand in the near-surface area according to Reese et al. (1974) (Lesny, 2010).

In greater depths a two-dimensional failure model was used, in which the subsoil in the area of the pile is idealized by cubic elements (Fig. 16.16). This model is based on the assumption that only horizontal yielding occurs, but no vertical displacements. More details of the p-y method and an overview of various p-y curves are presented, for example, in Reese and van Impe (2001) and Lesny (2010).

16.16. Failure model for calculating the maximum subgrade reaction for sand in greater depths according to Reese et al. (1974) (Lesny, 2010).

The general applicability of this method for monopile diameters up to 6   m or even more in sandy subsoil as often encountered in the southern North Sea has been verified in a comprehensive finite-element analysis conducted by Wiemann (2007). However, it has also been shown in this study that the assumption of a linearly increasing soil stiffness with depth significantly overestimates the actual stiffness conditions around the tip of the monopile. As a consequence the pile length determined with the p-y method does not provide a sufficient restraint of the pile in the soil. Wiemann showed that with a simple modification of the factor k a full restraint can be achieved:

[16.3] $k^{*}\left(D\right)=k_{\mathit{ref}}\cdot {\left(\frac{D_{\mathit{ref}}}{D}\right)}^{\frac{4\left(1-a\right)}{4+a}}$

Empirical values for the exponent a are a  ≈   0.6 for medium dense natural sands and a  ≈   0.5 for dense natural sands. As a reference pile (with the parameter D_ref and k_ref )the pile tested by Reese et al. (1974) with a diameter of D_ref   =   0.61   m was adopted. the pile in the soil. Wiemann showed that with a simple modification of the factor k a full restraint can be achieved:

However, the pile head displacements are not necessarily reduced by this modification and are still underestimated using the p-y method compared to the finite-element results (see Wiemann, 2007). Therefore, this method should be used with great care until further validation, especially by prototype measurements, is achieved.

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Use of Explosion in Soil Improvement Projects

Shuwang Yan , Jian Chu , in Ground Improvement Case Histories, 2015

18.3.3 Verification

Boreholes were drilled to examine the depth of the crushed stones after soil improvement. One of the borehole logs is shown in Fig. 18.8. The stones were found to be present up to 9   m, in which the top 5 or 6   m was densely packed, whereas the remaining 4 or 5   m was embedded in clay. Below the crushed stone layer were a silty gravel layer and the weathered sandstone layer.

Figure 18.8. Borehole log of the replaced soil layer.

Plate load tests were conducted using a 1.0   ×   1.0-m square plate. The load was applied via a hydraulic jack reacted against a steel beam that was counterbalanced by dead weight. The plate was placed on the ground surface before the 6   m of embankment was built. The results of a typical plate load test are shown in Fig. 18.9. The results indicate that the improved ground had adequate bearing capacity. Using the load versus settlement curve shown in Fig. 18.9 , the modulus of subgrade reaction, k _s, which is used for pavement design (American Association of State Highway and Transportation Officials, 1993), can be determined as the secant modulus for a specified point on the curve (Bowles, 1996). The modulus of subgrade reaction determined from the initial linear portion of the curve was 120   MPa (see Fig. 18.9). Note that the plate load test results only reflected the condition of the upper layer of 1.5–2   m deep in the compacted stone layer. The critical area for settlement would be the deeper zone where the stone was mixed with soft clay, which was not significantly stressed by the plate load tests. Therefore, the plate load test results gave an optimistic picture of the load settlement behavior. The settlement of improved foundation soil measured 3 months after the opening of the highway was more than the maximum settlement shown in Fig. 18.9 but less than 30   mm. The total settlement of the highway measured at the same time was less than 100   mm. The total allowable settlement as specified by the Ministry of Transport for design of expressways in China was 300   mm.

Figure 18.9. Results of plate load test.

Ground-penetrating radar (GPR) was used to detect the distribution of the crushed stones in the soft clay. The radar system transmits repetitive, short-time electromagnetic waves into the ground from a broad bandwidth antenna. Some of the waves are reflected when they hit discontinuities in the subsurface, whereas some are absorbed or refracted by the materials to which they come into contact. The reflected waves are picked up by a receiver, and the elapsed time between wave transmission and reception is automatically recorded. This method is explained in detail in Koerner (1984).

The GPR system used in this project adopted a frequency of 100   MHz. This frequency was chosen to suit the depth of the crushed stone layer. GPR tests were conducted along six lines of a total length of 417   m. Two lines were along the longitudinal direction and four lines along the transverse direction of the highway. One scanned profile is shown in Fig. 18.10. The crushed stones in the top 5   m of the soil profile were detected. Soft clay pockets within this layer could also be identified from the image, as indicated by arrows in Fig. 18.10. However, the stones in the deeper layer could not be identified clearly from the image. This may be because the radar wave became much less ineffective when it penetrated the layer of stones embedded in clay.

Figure 18.10. An image of ground-probing radar.

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Analytical modelling of capacity and deformation of single energy piles

Lyesse Laloui , Alessandro F. Rotta Loria , in Analysis and Design of Energy Geostructures, 2020

10.10.1 Background

The load-transfer method is an analysis approach originally proposed by Coyle and Reese (1966) to describe the load-displacement behaviour of conventional piles subjected to only mechanical loads. In recent years, this approach has been extended to describe the response of energy piles subjected to both mechanical and thermal loads by Knellwolf et al. (2011), Pasten and Santamarina (2014), Suryatriyastuti et al. (2014) and Chen and McCartney (2016) and Sutman et al. (2018). This method can solve the axial pile equilibrium equation and thus provides more comprehensive information to account for the influence of both mechanical and thermal loads on the response of single energy piles than the previous approaches resorting to schemes or charts.

The load-transfer analysis approach relies on modelling single piles as being composed of several rigid elements that are linked with unconnected springs (i.e. springs whose response does not depend on the response of the others along the pile) and interact with the surroundings through additional springs. Each of these springs is characterised by a constitutive law that determines the behaviour of the springs upon loading (or unloading) as well as the consequent response of the various pile elements. In the context of load-transfer analyses, the considered constitutive law is typically termed load-transfer function or load-displacement relationship. The discretisation of the pile in a number of elements allows considering various soil layers with distinct properties and the variation of the soil properties with depth. This fact makes the load-transfer method effective to model the response of piles in many practical situations.

Besides the previous capabilities and advantages, however, according to Poulos and Davis (1980) the load-transfer analysis approach should be considered as a fundamentally inferior method to numerical approaches such as the finite element method. Corroborating statements are as follows:

1.

In using the load-transfer relationships, similar to the theory of subgrade reaction ( Winkler, 1867), it is inherently assumed that the movement of the pile at any element is related only to the stress developed at that element and is independent of the stress occurring elsewhere along the pile. No proper account is thus taken of the continuity within the pile and the soil mass, and the previous approach to model pile behaviour may be defined as a 'layer model' (Chow, 1986b; Rotta Loria et al., 2018).

2.

The underlying assumption about the discontinuity of the soil characterising the load-transfer method yields to a number of differences compared to other 'continuous' approaches that do account for the influence of a pile element on the other, such as the equations of Mindlin (1936). Therefore the load-transfer method may be applied with judgement to the analysis of the load-displacement response of piles.

3.

To obtain load-transfer relationships, data obtained from field tests on full-scale instrumented piles or laboratory tests on model-scale piles are needed. To carry out the considered tests, considerably more instrumentation is required on a pile compared than in a more usual pile test. Yet, extrapolation of data from one site to another may not be entirely adequate. Therefore load-transfer analyses may be characterised by drawbacks in situations where piles are embedded in soil deposits for which detailed information is unavailable.

Various load-transfer functions can be employed to characterise the response of piles to loading. Examples of these functions have been proposed by Seed and Chan (1966), Coyle and Reese (1966), Coyle and Sulaiman (1967), Reese et al. (1969), Randolph and Wroth (1978), Frank and Zhao (1982), Armaleh and Desai (1987) and Frank et al. (1991) for piles embedded in both fine- and coarse-grained soil.

According to Poulos and Davis (1980), a number of load-transfer relationships may be required to describe the load-transfer along the whole pile length. In this context, the following aspects should be considered to characterise the axial pile response (Randolph, 2003): the axial capacity of piles markedly depends on (1) the effective stress level and (2) the fabric conditions at the pile–soil interface for any given pile installation technique; in contrast, the axial deformation of piles depends on (1) the soil conditions and (2) the soil properties slightly farther from the pile–soil interface. Typical features of load-transfer functions describing the interaction between a displacement pile and the adjacent soil are reported in Fig. 10.35.

Figure 10.35. Load-transfer functions along a pile.

Modified after Randolph, M.F., 2003. Science and empiricism in pile foundation design. Geotechnique 53 (10), 847–875.

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Processes

Neil Williamson , in Advanced Concrete Technology, 2003

24.5.1.4 Ground bearing slabs

For many years it has been accepted that the work of Westergaard provided a sound basis for ground slab design. His original theoretical work has been modified by experiment, and all of this was brought together in the authoritative Technical Report 550, published in June 1982. In 1988 a more concise document, by Chandler and Neal, BCA Interim Technical Note 11, developed these design principles further, following detailed research into the basic parameters determining loading types and the material characteristics of both the concrete and sub-grade. This latter document contains useful general load tables, and effective guidance on assessment and enhancement of the modulus of sub-grade reaction to use in the slab design. The influence of the sub-grade on ground bearing slab design is taken as that of an elastic medium. Tensile stresses induced in the slab by point loads are, however, quite insensitive to variations in the modulus of sub-grade reaction. See Figure 24.10.

Figure 24.10. Variation of slab thickness with modulus of sub-grade reaction (K) (loading based on BRE IP 19/87 load class: heavy).

CBR test results alone should be considered as too unreliable for slab design, as they reflect only a comparatively shallow stress bulb, and hence do not indicate to what extent the sub-grade is stressed at depth. An adequately defined site investigation (SI) is therefore necessary, particularly in the case of high-bay warehouses with materials handling systems sensitive to floor tolerances which can be disturbed by even very small degrees of consolidation. Further, as loadings on industrial floors have increased, some ground bearing slabs located on plastic soils, designed in accordance with the recognized documents, but based on inadequate soils information, have suffered tensile cracking longitudinally in the aisles as a result of differential consolidation between these unloaded areas and the heavily loaded sections beneath the racks. There is no authoritative guidance to this design problem at this time, but the importance of a site investigation and the assessment of soil plasticity and likely total and differential settlements cannot be over-emphasized. It may be necessary to employ a specialist soils engineer as part of the design team.

Where consolidation of plastic soils is determined to be a potential problem following an assessment of the SI and the actual loading pattern anticipated, a suspended slab is the only effective solution. However, where the ground can be improved using techniques such as vibro-replacement or dynamic compaction, and an adequate granular sub-base is provided (typically 300–450   mm, compacted in 150   mm layers), the slab may be considered to be ground bearing and designed in accordance with the principles for such a slab.

Ground floor slabs are still found with fabric specified in the top and bottom. This is fundamentally wrong, unless the slab has been designed as suspended between piles or beams. The practice of incorporating relatively light fabric to 'span any soft spots' is not consistent with any recognized method of design. Recent research at the University of Greenwich and RMCS (unpublished) has suggested that a layer of mesh reinforcement will increase the load carrying capacity of the slab. However, this does not necessarily result in a theoretically crack-free slab.

Post-tensioned slabs provide an excellent technical solution for both suspended and ground bearing slabs. In ground bearing post-tensioned slabs, random shrinkage cracks can be eliminated, and thinner slabs can result from the elimination of critical corner and edge loading conditions. Similar benefits apply to suspended ground slabs, and here post-tensioned slabs are often much simpler to construct and are frequently shown to be more economic than 'traditional' reinforced concrete design. Post-tensioned slabs have been successfully constructed worldwide for many years, and design is relatively straightforward. As no authoritative guide exists in the UK, it is recommended that reference be made to ACI 360.R-92. This also contains guidance on design of slabs with shrinkage compensating cement.

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Effect of soil–structure interaction and spatial variability of ground motion on seismic risk assessment of bridges

A. Sextos , in Handbook of Seismic Risk Analysis and Management of Civil Infrastructure Systems, 2013

Static loading

Numerous studies have been made on the effect of SSI under static loading. The most commonly adopted engineering method for calculating the pseudo-static interaction between pile and soil is the Winkler model (Matlock and Reese, 1960) in which the soil reaction to pile movement is represented by independent (linear or non-linear) unidirectional translational spring elements distributed along the pile shaft to account for the soil response in the elastic and inelastic range, respectively (Fig. 22.1). Although approximate, Winkler formulations are widely used not only because their predictions are in good agreement with results from more rigorous solutions, but also because the variation of soil properties along the pile length can be relatively easily incorporated. Moreover, they are significantly more efficient in terms of computational time required, especially compared to continuum finite element (FE) or finite difference methods, thus permitting focus to be placed on the structural inelastic response.

22.1. Overview of finite element (FE) modeling of pile groups using uniaxial soil springs.

The mechanical parameters for the spring elements are frequently obtained from experimental results (leading to P–y curves for lateral and T–z curves for axial loading) as well as from simplified models. An expression that is still widely used for the lateral soil resistance–deflection relationship is (API, 1993):

[22.1] $P=0.9p_{\text{u}}\tanh \left[\frac{kH}{0.9P_{\text{u}}}y\right]$

where p _u is the ultimate bearing capacity at depth H, y is the lateral deflection and k is the initial modulus of subgrade reaction which is both depth- and diameter-dependent despite the fact that in many cases ( Pender, 1993) the modulus of the subgrade reaction is assumed to be independent of diameter.

The particular expression is also applied both for dynamic or non-linear problems after a simple transformation to a bi-linear relationship by assuming a specific threshold deformation ∆ _y for entering the inelastic range (typically equal to 2.5   cm for cohesionless soils) and a second branch stiffness reduced to a quarter of the initial soil stiffness (Kappos and Sextos, 2001). As an alternative to the above procedure, the static stiffness extrapolated from the complex dynamic stiffness matrix is also used in practice as discussed in the following section.

In addition to the springs attached along the pile shaft, a horizontal inelastic soil spring can be used at the top of the pile to represent the strength and stiffness provided by passive soil resistance against the pile cap, while a vertical, uniaxial, inelastic spring is commonly used at the pile tip to account for downward and upward capacity of the supporting soil (Pender, 1993).

For the case of non-uniform, liquefaction susceptible soil profiles, the lateral subgrade reaction of piles and the maximum reaction force of the laterally spreading soils have to be appropriately reduced at the corresponding locations along the pile length. This reduction factor lies in the range of 0.1–0.2 (Finn, 2005) or 0.05–0.2 (Elgamal et al., 2006; Suzuki et al., 2006). As a rule of thumb, the lateral stiffness provided along the liquefied soil layers is reduced to the level of 10–20% depending on the estimated shear deformation.

Based on the above discussion, from a static response point of view, it can be concluded that the use of lateral soil resistance–deflection curves is a convenient approach for estimating the dynamic characteristics of the bridge–structure system (Kappos and Sextos, 2001). Nevertheless, despite the wide application of the P–y approach for the assessment of the structural response in the design practice, there are certain limitations that have to be stressed:

•

Uncertainty of estimating the parameters involved when load tests are not available (especially of defining p _u and k), is disproportionally high compared to the simplicity of the approach. It is notable that although eq. 22.1 is adopted by both the Multidisciplinary Center for Earthquake Engineering Research and American Technology Council (MCEER/ATC, 2003) as well as the California Department of Transportation (CALTRANS, 2006) guidelines, the proposed sets of the required subgrade moduli differ on average by a factor of 4 (Finn, 2005).

•

Relationships between one-dimensional soil stiffness (expressed in terms of modulus of subgrade reaction k) to two-/three-dimensional soil stiffness, expressed in terms of modulus of elasticity E _s and Poisson's ratio v, has not been established. The verification using 2D and 3D FE analyses is not straightforward. As a result, a set of calibration assumptions is required for establishing a correspondence between the Winkler and plane-strain FE approaches (Kappos and Sextos, 2001) based on the initial formulations proposed (Vesic, 1961):

[22.2] $k=D{k}_h=\frac{0.65E_s}{{\left(1-v^2\right)}^2}\sqrt[12]{\frac{E_s{D}^4}{\left(1-v^2\right)E_p{I}_p}}$

where k_h is the modulus of subgrade reaction and D the pile diameter. Moreover, the transformation from one soil parameter to another is straightforward only in a case where they are assumed constant with depth; a set of additional and case-dependent calibrations is required to obtain agreement in the inelastic range.

•

Pile group effects are essentially neglected. Even if the piles are statically connected using appropriate single valued springs to represent the increased flexibility of a pile group compared to the summation of the stiffness of all individual piles, the estimation of the connecting spring stiffness is highly subjective.

•

Use of the (particular statically based) P–y method often leads to the extension of its application for the case of inelastic dynamic analysis in the time domain. Such an extension, although tempting for special cases of structural design (i.e. performance-based design of new or retrofit of existing important bridges), leads to the misleading perception of modeling refinement without proper understanding and consideration of the complex dynamic nature of SSI phenomena.

It can be concluded that as soil–foundation–structure interaction is a multi-parametric and strongly frequency-dependent phenomenon, it has to be investigated from a dynamic point of view, through a very careful selection of FE models, associated parameters, and modeling assumptions. As a result, the static Winkler spring approach is deemed appropriate only for cases where:

•

bridges are analyzed linearly solely under static loading and the response is assessed accordingly;

•

(standard or modal) pushover analysis is performed in order to quickly assess the inelastic mechanisms that are expected to be developed under earthquake excitation;

•

response spectrum analysis and/or linear time history analysis is performed for a relatively low level of seismic forces;

•

inelastic dynamic analysis is conducted in the time domain but the energy absorption is expected to be mainly concentrated on the superstructure (i.e., through bearings or other dissipating devices) while the material and radiation damping at the soil–foundation interface is a-priori judged of secondary importance (i.e. in cases where the underlying soil formations are stiff and uniform with depth).

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Design of Slab on Grade Using Modulus of Subgrade Reaction

Source: https://www.sciencedirect.com/topics/engineering/subgrade-reaction