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# Calculating Lateral Earth Pressure on a Retaining Wall

One of the main loads acting on a retaining wall is the lateral earth pressure. そんな理由で, doing a good estimation of its magnitude and distribution is decisive in the design of a Concrete Retaining Wall. 一般に, there are three different types of lateral earth pressure depending on the direction that the wall tends to move:

• At-rest earth pressure: When the wall is completely restrained from moving
• Active earth pressure: When the wall may tilt away from the retained soil
• パッシブ土圧: When the wall may be pushed into the retained soil

## Lateral earth pressure distribution

$$\sigma_h = K_* cdot (\ガンマ z)$$

どこ $$K_*) の値を取る \(K_o$$ 静止圧の場合, $$K_a$$ アクティブな圧力の場合, そして $$K_p$$ 受動的圧力の場合.

からの横方向の地下応力の与えられた式を統合する $$0$$ に $$H$$ に $$z), the resultant turns out to be: \(P_*=\frac{1}{2} K_* \cdot \gamma \cdot H^2$$

This resultant’s line of action is located $$\フラク{2}{3}H$$ from the surface. The lateral earth pressure distribution, its resultant, and location described above are illustrated in the following picture: It is important to mention that the presented distribution and resultant calculation approach only applies to soil pressures acting on a vertical backface. In the case of a retaining wall with an inclined backface (like the one in the picture above), the surface where the soil pressure acts is still considered vertical since it is assumed to act in a vertical plane located where the heel ends.

さらに, when the backfill is inclined at some angle $$\alpha$$ with respect to the horizontal, the pressure distribution and its resultant are inclined at that same angle $$\alpha$$ as illustrated in the following: Correctly estimating the lateral earth pressure distribution and its resultant is a crucial step in the Retaining Wall Design Process. この側方土圧が擁壁設計プロセスにどのように含まれているかについての詳細は、, 記事参照 ここに. Let’s now dive into the formulas for calculating the resultant lateral earth pressure exerted on a Retaining Wall by the soil in different conditions.

## At-rest lateral earth pressure

This approach for calculating the lateral earth pressure against a Retaining Wall can only be used if the wall is completely at rest and is not allowed to move either away from the soil or into the soil, this condition ensures that the horizontal strain in the soil is zero. この場合, the coefficient of at-rest pressure ($$K_o$$) is the one that replaces $$K_*) in the previous equations. That coefficient is the only unknown for calculating the pressure distribution and its resultant. For normally consolidated soil, the relation for \(K_o$$ です:

$$K_o = 1-sin(\ファイ)$$

どこ $$\ファイ) is the effective angle of friction of the soil in consideration. For overconsolidated soil, the coefficient may be calculated using the following expression: \(K_o = (1-それなし(\ファイ))\cdot OCR^{それなし(\ファイ)}$$

どこ $$\ファイ) is the effective angle of friction, そして \(OCR$$ the overconsolidation ratio of the soil in consideration.

Plugging this coefficient into the expression for calculating the resultant force from the lateral earth pressure at-rest yields to:

$$P_o=\frac{1}{2} \gamma \cdot H^2 \cdot K_o$$

## Active lateral earth pressure

The previous approach can be used when the wall does not yield at all, しかしながら, if a wall tends to move away from the soil, the soil pressure on the wall at any depth will decrease. この場合, the coefficient of at-rest pressure ($$K_a$$) is the one that replaces $$K_*) in the initial equations. Using Rankine’s approach for a granular backfill, and assuming that the pressure acts in a vertical backface, the active earth-pressure coefficient may be calculated using the equation: \(K_a=cos(\アルファ) \フラク{cos(\アルファ) – \平方根{cos^2(\アルファ) – cos^2(\ファイ)}}{cos(\アルファ) + \平方根{cos^2(\アルファ) – cos^2(\ファイ)}}$$

どこ $$\ファイ) is the angle of friction of the soil in consideration and \(\alpha$$ is the angle of inclination of the backfill surface with respect to the horizontal.

Plugging this coefficient into the expression for calculating the resultant force from the lateral earth pressure at active condition yields to:

$$P_a=frac{1}{2} \ガンマ cdot H^2 cdot K_a) The Rankine active pressure calculations presented before are based on the assumption that the wall is frictionless. ## Passive lateral earth pressure The lateral earth pressure acting on a retaining wall is considered passive when the wall is pushed into the soil mass, in that condition, the horizontal stress will increase with respect to the at-rest condition. この場合, the coefficient of at-rest pressure (\(K_p$$) is the one that replaces $$K_*) in the initial equations. Using Rankine’s approach for a granular backfill, and assuming that the pressure acts in a vertical backface, the passive earth-pressure coefficient may be calculated using the expressions: When the backfill is completely horizontal \(K_p = tan^2(45º+\frac{\ファイ}{2})$$

When the backfill is inclined a certain angle with respect to the horizontal

$$K_p=cos(\アルファ) \フラク{cos(\アルファ) + \平方根{cos^2(\アルファ) – cos^2(\ファイ)}}{cos(\アルファ) – \平方根{cos^2(\アルファ) – cos^2(\ファイ)}}$$

どこ $$\ファイ) is the angle of friction of the soil in consideration and \(\alpha$$ is the angle of inclination of the backfill surface with respect to the horizontal.

Plugging this coefficient into the expression for calculating the resultant force from the lateral earth pressure at passive condition yields to:

$$P_p=\frac{1}{2} \gamma \cdot H^2 \cdot K_p$$

## 参考文献

それか, B. M. (2010). 基礎工学の原則, SI版. 章 7 横土圧. センゲージ ラーニング.

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