“LATERAL EARTH PRESSURE” AN ACTIVE LEARNING ASSIGNMENT ON “LATERAL EARTH PRESSURE” PREPARED BY: 130180106044 JAISWAR ANUJ 130180106039 GOSWAMI JAY 130180106013 CHAUDHARY MUKESH 130180106033 GHOTI MAHIPAL GUIDED BY : PROF. B. M. PUROHIT 2015 SOIL MECHANICS DEPARTMENT OF CIVIL ENGINEERING GOVERNMENT ENGINEERING COLLEGE, DAHOD

CONTENT 1.0 Lateral earth pressure 2.0 Types of lateral earth pressure 2.1 Active earth pressure 2.2 Passive earth pressure 2.3 Earth pressure at rest 3.0 Plastic equilibrium in soil 4.0 Example

1.0 LATERAL EARTH PRESSURE The pressure exerted by earth backfill on the back of the wall is called Lateral pressure. At the top of the wall earth pressure is zero. At the base of the wall pressure acting on the wall is, 𝑝 𝑎 = 𝑘 𝑎 𝛾 𝐻 Total force acting on the back of the wall is, 𝑃 𝑎 = 1 2 𝑘 𝑎 𝛾 𝐻 2 This force is equal to the area of the pressure triangle. This total force acts at height 𝐻 3 from the base.

Fig. 1 Lateral earth pressure H = O H 𝑃 𝑎 𝐻 3 𝑝 𝑎 = 𝑘 𝑎 𝛾 𝐻 Fig. 1 Lateral earth pressure

2.0 TYPES OF LATERAL EARTH PRESSURE As per Rankine theory (1860), the lateral earth pressure depends upon the following Forces : The flexibility of the wall The properties of soil The drainage condition Mode of movement of the wall 2.0 TYPES OF LATERAL EARTH PRESSURE There are three types of lateral earth pressure : Active earth pressure Passive earth pressure Earth pressure at rest.

2.1 ACTIVE EARTH PRESSURE When the retaining wall moves away from the backfill, earth pressure on the back of the wall decreased and becomes minimum. With further movement of wall the pressure does not decreases. This minimum pressure is known as active earth pressure. A state of active earth pressure occurs when the soil mass yield in such a way that it tends to stretch horizontally. It is a state of plastic equilibrium as the entire soil mass is on the verge of failure. The failure plan makes an angle 45 ° + ∅ 2 with horizontal.

Fig. 2 Active earth pressure 𝜎 𝑉 = 𝜎 1 FAILURE PLANE 𝜎 3 𝜎 ℎ = 𝜎 3 RESISTING FORCE 𝜎 1 𝑃 𝑎 DOWNWARD MOMENT 45 ° + ∅ 2 AWAY FROM THE FILL Fig. 2 Active earth pressure

2.2 PASSIVE EARTH PRESSURE When the retaining wall moves towards the backfill, earth pressure on the back of the wall increased and becomes maximum. With further movement of the wall the pressure does not increases. This maximum pressure is known as passive earth pressure. A state of passive earth pressure exists when the movement of the wall is such that the soil tends to compress horizontally. The failure plan makes an angle 45 ° − ∅ 2 with horizontal.

Fig. 3 Passive earth pressure 𝜎 𝑉 = 𝜎 3 FAILURE PLANE 𝜎 1 𝜎 ℎ = 𝜎 1 UPWARD MOMENT 𝜎 3 𝑃 𝑃 RESISTING FORCE 45 ° − ∅ 2 TOWARDS FILL Fig. 3 Passive earth pressure

2.3 EARTH PRESSURE AT REST When the soil mass is not subjected to any lateral yielding or movement, the lateral pressure is called the earth pressure at rest. This case occurs when the retaining wall is firmly fixed at its top and is not allowed to rotate or move laterally. The at rest condition is known as the elastic equilibrium. 𝑝 0 = 𝑘 0 𝛾 𝐻 𝑃 0 = earth pressure at rest 𝑘 0 may be computed approximately as given by jaky 𝑘 0 =1 − sin 𝑓 𝑘 0 = 𝜇 1− 𝜇 = Coefficient of earth pressure at rest

Fig. 4 Variation in earth pressure with moment of wall C B Passive earth pressure (𝑃 𝑃 ) earth pressure at rest (𝑃 𝑟 ) A Passive earth Pressure (𝑃 𝑎 ) moment away from fill no moment moment towards fill Fig. 4 Variation in earth pressure with moment of wall

3.0 PLASTIC EQUILIBRIUM IN SOIL A soil mass is said to be in a state of plastic equilibrium if every point of it is on the Verge of failure. Rankine theory considers the stress in a soil mass when it reaches a state of plastic equilibrium, that is, when shear failure is imminent at every point within a soil mass. minor principal stress major principal stress 45 ° − ∅ 2 45 ° + ∅ 2 major minor 𝑘 𝛾 𝐻 (b) Passive case slip lines (a) Active case slip lines

Fig. 5 Active and Passive states of Plastic Equilibrium Failure envelope 𝜏= 𝜎 tan ∅ 𝐹 2 Slip line 𝐹 1 Slip line 𝜃 𝜃 𝐴 𝜎 𝑃 1 𝑃 𝑎 𝜃 𝜃 Slip line Slip line 𝐹 ′ 1 Active state Mohr circle Passive state Mohr circle 𝐹 ′ 2 𝜃 =45 ° − ∅ 2 θ = 45 ° + ∅ 2 ( c) Mohr circles Fig. 5 Active and Passive states of Plastic Equilibrium

Fig. 5 shows a soil element at a depth Z subjected to a verticle stress 𝜎 𝑣 and horizontal stress 𝜎 ℎ . Since no shear stress act on the horizontal and vertical planes, 𝜎 𝑣 𝑎𝑛𝑑 𝜎 ℎ are principal stresses. The Mohr's stress circle representing active and passive state are shown in figure. If the wall movies away from the backfill, the soil element expands and the value of 𝜎 ℎ decreases and becomes minimum, so that the state of plastic equilibrium is developed. The Mohr circle touches the strength envelope at 𝐹 1 , and the sand mass is said to be in the active Rankine state. For active state, 𝜎 𝑉 = 𝜎 1 = major principal stress 𝜎 ℎ = 𝜎 3 = minor principal stress

𝜎 1 𝜎 3 = 𝑡𝑎𝑛 2 45°+ ∅ 2 =𝑁∅ 𝑘 𝑎 = coefficient of active earth pressure 𝑘 𝑎 = 𝜎 ℎ 𝜎 𝑣 = 𝜎 3 𝜎 1 = 1 𝑡𝑎𝑛 2 45°+ ∅ 2 = 𝑐𝑜𝑡 2 45°+ ∅ 2 = 𝑡𝑎𝑛 2 45°− ∅ 2 = 1 − sin ∅ 1+ sin ∅ The minimum values of 𝜎 ℎ is defined as active pressure ( 𝑝 𝑎 ) . 𝑝 𝑎 = 𝑘 𝑎 𝜎 𝑣 = 𝑘 𝑎 𝛾 𝑧 Circle I represent an active state, in whuich the pole P1 corresponding to the minor principal stress 𝜎 3 and point A corresponds to the major principal stress 𝜎 1 .

𝑃 1 𝐹 1 and 𝑃 1 𝐹 ′ 1 shows the directions of failure planes ( slip lines ) inclined at angle 𝜃 With horizontal. θ = 45 ° + ∅ 2 On the other hand, if the wall moves towards the backfill, there will be uniform compression in the horizontal direction. This leads to increase in the value of 𝜎 ℎ from its original value, while the value of 𝜎 𝑣 remains constant. 𝜎 ℎ value increase and becomes maximum and will thus constitute the major principal stress 𝜎 1 , while 𝜎 𝑣 becomes minor principal stress 𝜎 3 . 𝜎 𝑉 = 𝜎 3 𝜎 ℎ = 𝜎 1

𝑘 𝑝 = 𝜎 ℎ 𝜎 𝑣 = 𝜎 1 𝜎 3 = 𝑡𝑎𝑛 2 45°+ ∅ 2 = 1+ sin ∅ 1− sin ∅ The maximum value of 𝜎 1 is reached when the Mohr circle touches the strength envelope at 𝐹 2 . The soil is then said to be in the Passive Rankine State, and corresponding lateral pressure is called passive earth pressure. 𝑝 𝑝 = 𝑘 𝑝 𝛾 𝑧 Circle II represent passive state, in which failure plan makes an angle 𝜃 with horizontally. 𝜃 =45 ° − ∅ 2

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