Analytical Chem Soil HW Solutions 27

Analytical Chem Soil HW Solutions 27 - the given value of...

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Unformatted text preview: the given value of φ'apparent. Remember that the rotation on the Mohr circle must be divided by 2 to give the actual rotation in the physical plane. φ’apparent τ φ’ = 18° P T S ω4 φ’apparent O t ω1 2θ1 90° + φ’apparent 90° - φ’apparent 18° s’ σ’ C R Q Figure Q2.8a: Mohr circle of stress The orientations θ of the clay laminations are given by the angles clockwise from the horizontal plane θ1, θ2, θ3 and θ4, corresponding to the points P, Q, R and S respectively on Figure Q2.8a. From triangle OTC, t/s' = sinφ'apparent From triangle OPC, angle OCP = 180° - ω1 - 18° and angle OCP = 2θ1 + (90° - φ'apparent) Applying the sine rule to triangle OPC, s'/sinω1 = t/sin18° ⇒ sinω1 = sin18°/(t/s') or sinω1 = sin18°/sinφ'apparent (note ω1 is acute, ie less than 90°) Applying the sine rule to triangle OSC, s'/sinω4 = t/sin18° ⇒ sinω4 = sin18°/(t/s') or sinω4 = sin18°/sinφ'apparent (note ω4 is obtuse, ie greater than 90°) By considering the geometry of the Mohr circle shown in Figure Q2.8a, the values of θ1 to θ4 may be determined as follows. 2θ1 = (90° + φ'apparent) - (ω1 + 18° ) ⇒ θ1 = 0.5 × (72° - ω1 + φ'apparent) 27 ...
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