# PRINCIPAL STRESSES AND SHEAR STRESS

- The vertical stress intensity is not the maximum direct stress intensity within the dam section.
- It is therefore necessary to determine the intensity of the maximum normal stress.
- The maximum normal stress occurs on the major principal plane.
- The D/S face of the dam which is inclined at an angle α to the vertical is a principal plane, as no shear stress acts on it.
- If any tail water is there it contributes a pressure p’ entirely normal to the plane and is one of the principal stresses.
- The second principal plane will be inclined at right angles to the face of the dam and hence at an angle α to the horizontal.
- The stress acting on such plane is the major principal stress Pmax.
- All the stresses can be represented by Mohr’s circle. See Fig. 13.6.
- The vertical stress Pn at toe of the dam in terms of principal stresses may be given as follows.

\[P_{n}= P+(P_{max}-P)(1+cos2\Theta )\]

\[P_{n}= P+(P_{max}-P)cos^{2}\Theta\]

\[P_{max}= \frac{P_{n}}{cos^{2}\Theta }-\frac{P(1-cos^{2}\Theta )}{cos^{2}\Theta }\]

\[= P_{n}sec^{2}\Theta -p.tan^{2}\Theta\]

For maximum value of compressive stress, the tail water depth is assumed as zero.

Hence p=0

\[P_{max}= p_{n}sec^{2}\Theta\]

Shear stress pt from Mohr’s circle is given by

\[P_{t}= \left ( \frac{P_{max}-P}{2} \right )sin2\alpha\]

\[= (P_{max}-p)sin\Theta cos\Theta\]

Putting

\[p= 0-and-p_{max}= p_{n}sec^{2}\Theta -in-this-equation-we-get\]

\[p_{t}= p_{n}sin\alpha cos\Theta sec^{2}\Theta\]

\[= p_{n}tan\Theta\]

**Equation (1) **

- gives the maximum intensity of direct stress which will occur near the D/S toe and which will act on a plane inclined at an angle θ to the horizontal.
- Equation (2) given intensity of shear stress on a horizontal plane near Toe.
- The maximum intensity of stress at heel can also be similarly computed. For U/S face let

\[P_{max}\]

Major principal stress on a plane inclined at βº with the horizontal. (Maximum intensity of direct stress).