Derive principal and shear stress equations at toe/heel

Consider a gravity dam section with a u/s face making an angle $\phi_u$ and a d/s face making an angle $\phi_d$ with the vertical. Let $\sigma_z$ be the vertical normal stress on a horizontal plane and $\tau$ the shear stress on that plane. Compression is taken as positive. Per unit length of dam.

1. Toe (with tail water and earthquake)

At the toe the d/s face is loaded by the tail water pressure $p'=\gamma_wh_t$ and, in an earthquake, by the hydrodynamic pressure $p_e'$ of the tail water. In the critical case (the dam accelerating towards the upstream, so that the reservoir pressure increases), the dam moves away from the tail water and the pressure on the d/s face falls. The face then carries only a normal stress $p''=p'-p_e'$ and no shear stress (water cannot carry shear). So the face is a principal plane with $\sigma_3=p''$.

The unit normal to the d/s face makes an angle $\phi_d$ with the horizontal: $n=(\cos\phi_d,\ \sin\phi_d)$. The traction on the face must be purely normal, $\sigma\cdot n=p''n$. With the stress components $\sigma_z$ (vertical), $\sigma_x$ (horizontal) and $\tau$:

$$\tau\cos\phi_d+\sigma_z\sin\phi_d=p''\sin\phi_d\ \Rightarrow\ \tau=-(\sigma_z-p'')\tan\phi_d$$ $$\sigma_x\cos\phi_d+\tau\sin\phi_d=p''\cos\phi_d\ \Rightarrow\ \sigma_x=p''+(\sigma_z-p'')\tan^2\phi_d$$

The minus sign only shows that the shear on the horizontal plane acts in the opposite sense to the one assumed. Taking the shear at the toe as positive (the usual textbook convention):

Shear stress at the toe:

$$\boxed{\tau_{toe}=(\sigma_z-p'')\tan\phi_d}$$

The sum of normal stresses is invariant: $\sigma_1+\sigma_3=\sigma_x+\sigma_z$, with $\sigma_3=p''$:

$$\sigma_1=\sigma_x+\sigma_z-p''=\sigma_z(1+\tan^2\phi_d)-p''\tan^2\phi_d$$

Major principal stress at the toe:

$$\boxed{\sigma_1=\sigma_z\sec^2\phi_d-p''\tan^2\phi_d},\qquad p''=p'-p_e'$$

(When the tail water is absent, $p''=0$ and $\sigma_1=\sigma_z\sec^2\phi_d$, $\tau=\sigma_z\tan\phi_d$.)

2. Heel (reservoir pressure and earthquake)

On the u/s face the normal pressure is $p''=p+p_e$, where $p=\gamma_wh$ is the hydrostatic pressure at the heel and $p_e$ the hydrodynamic pressure intensity (Westergaard/Zangar, $p_e=C\alpha_h\gamma_wh$). In the same way, with the face normal pointing upstream at angle $\phi_u$ to the horizontal, the shear has the opposite sense to that at the toe:

$$\tau_{heel}=-(\sigma_z-p'')\tan\phi_u=(p''-\sigma_{z})\tan\phi_u,\qquad \sigma_1=\sigma_z\sec^2\phi_u-p''\tan^2\phi_u$$

For a vertical u/s face $\phi_u=0$: $\tau=0$ and the principal stresses are $\sigma_z$ and $p''$.

3. Vertical stress $\sigma_z$

$\sigma_z$ comes from the combined vertical load and moment, including the tail water weight and the earthquake effect:

$$\sigma_z=\frac{\Sigma V}{B}\left(1\pm\frac{6e}{B}\right),\qquad e=\frac{B}{2}-\frac{\Sigma M_R-\Sigma M_O}{\Sigma V}$$

where $\Sigma M_O$ includes the moment of the hydrodynamic thrust $P_e$ (acting at about $0.4h$ above the base) and the inertia force $\alpha_hW$ at the centre of gravity, and $\Sigma V$ includes the weight of the tail water on the d/s slope. The plus sign gives the toe stress and the minus sign the heel stress for the reservoir-full case.

4. Conditions

• The shear stress and principal stress are compared with the permissible values: $\sigma_1\le\sigma_{perm}$ and $\tau\le\tau_{perm}$.
• At the toe, tail water reduces both $\tau$ and $\sigma_1$ (because $p''>0$), and the earthquake reduces this relief ($p''=p'-p_e'$). At the heel the hydrodynamic pressure increases $p''$, which reduces $\sigma_1$ and, when $p''>\sigma_z$, increases $\tau$.