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The horizontal Soil Subgrade Reaction in Pile Foundations is often determined using the Brom’s method (Broms, 1964). The method is widely used as it takes into consideration the length of the piles (short or long), the type of the soil (cohesive or cohesionless), and the boundary condition at the pile head (free-head or fixed-head).

The method introduces certain simplifications such as a rigid-perfectly plastic behavior for the pile-soil interaction, and the dependence of the subgrade reaction on the soil type. Its main disadvantage is that it cannot be applied in a layered substratum.

According to Broms (1964), the first step of the analysis is the definition of the pile length (short, intermediate or long). The characterization depends on the mechanical and shape characteristics of the pile as well as on the mechanical properties of the ground.

Accordingly, the response of the pile is classified as shown **Table 1**. Indicative values of **k _{0}** and

**Table 1**: Classification of the pile response based on Broms (1964).

**Table 2**: Suggested values of the Winker’s modulus of subgrade reaction for overconsolidated clays obtained from tests on a default square plate with width B=0.305m (Kavvadas, 2008).

**Table 3**: Suggested values of the n_{h} coefficient for sands and normally consolidated clays (Kavvadas, 2008).

Based on the derived response of a pile (short, intermediate or long), the type of soil and whether the pile is defined as free-head or fixed-head, Broms (1964) categorized certain cases regarding the subgrade reaction and the developed bending moment. These cases are analyzed below.

According to Broms (1964), a short pile fails when the applied horizontal load exceeds the ultimate lateral resistance of the soil.

When a pile is subjected to a horizontal load, the soil is forced to develop its maximum lateral resistance which is known as the passive state lateral pressure. The maximum value of coefficient **K** is denoted as **K _{P}**. This coefficient is a function of the soil friction angle (

The maximum developed earth pressure at the bottom of the pile is equal to:

Broms (1964) derived the ultimate horizontal load **N _{u}** that a pile can carry before soil failure as:

Where **B** and **L** are the width (m) and length (m) of the pile, respectively, **γ** refers to the soil unit weight (kN/m^{3}), and **e** is the distance between the free end of the pile and the surface of the ground (m), as depicted in **Figure 1**.

where:

The derived bending moment can then be compared to the moment bearing capacity of the pile.

The bending moment also has a different distribution and its maximum value is located at the point where the pile meets the surface of the ground beneath its fixed-end (**Figure 2**).

In cohesive soils, the undrained shear strength **c _{u}** is used to obtain the maximum lateral subgrade reaction.

To obtain the maximum horizontal load **N _{u}**, integration of shear forces acting on the pile is required. The integral of the shear forces at the upper part of the pile up to the maximization of the bending moment yields:

Accordingly, the corresponding integral for the shear forces at lower part of the pile yields:

where **M _{max,Upper} = M_{max,Lower}**. Moreover, based on the geometry of the problem, the following correlation can also be considered:

Finally, the maximum horizontal load **N _{u}** is correlated with

Based on **equations [10],** **[11]**, **[12]** and **[13]**, the maximum horizontal load **N _{u}** and the maximum bending moment

The maximum soil subgrade pressure is the same as in the case of a free-end pile and is calculated using **equation [9]**. The corresponding ultimate horizontal load and the maximum bending moment that the pile is subjected to, are determined through **equations [14]** and **[15]**.

Unlike short piles, when long piles are subjected to a horizontal load, they fail when their maximum bending capacity is exceeded. The pile deformation tends to be concentrated within a certain length **L'** (**L’ ). **

** **

The depth,** f**, where the plastic hinge forms is estimated as:

As discussed earlier, the maximum horizontal load** N _{u}** is derived via the bending capacity of the pile

In **cohesive **soils, the behavior of a **long**, **free-head** pile is similar. Failure typically occurs when the pile bending capacity is exceeded. As discussed in the cases of short piles, lateral earth pressures are derived using the undrained shear strength **c _{u}** of the soil.

**N _{u}** is derived as a function of

**f **refers to the point where the plastic hinge forms and is calculated as:

In **cohesive **soils, a **long**, **fixed-head** pile deforms in a similar manner as in cohesionless soils. The lateral earth pressure and bending moment change, however, and the distribution of bending moment will slightly be affected as depicted in **Figure 8**.

Based on the value of the undrained shear strength of the soil, the suggested value of **k _{0}** (

Therefore:

Accordingly, based on the characteristics of the pile and the properties of the soil, the pile can be characterized as ** short**.

The methodology that will be used is the one described for a __short__, __fixed-head__ pile in __cohesive__ soils.

The corresponding ultimate horizontal load and the maximum bending moment are calculated through **equations [14]** and **[15]**.

As a result, the maximum horizontal load that the soil may carry without experiencing failure is **22,950kN**. Under this horizontal load, the moment capacity of the pile must be greater than **M _{u}=131,962.5 kNm**.

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