Introduction to Stresses - Stability Analysis of Gravity Dams, Strength of Materials | Strength of Material Notes - Agricultural Engg - Agricultural Engineering PDF Download

Introduction:  In this lesson we will derive expressions for the base pressure and stresses developed in a gravity dam. 

Module 5 Lesson 31 Fig.31.1

Fig. 31.1.

 In the above figure let R be the resultant force cutting the base at a distance \[\bar x\] from the toe of the dam. The components of R in x and y direction are obtained as,

\[{R_x}={F_H}\]                                (31.1)

\[{R_y}=W + {F_V} - U\]        (31.2)

\[\bar x\] is obtained as,

\[\bar x={{{M_{toe}}} \over {R{}_y}}\]                        (31.3)

The eccentricity of from the centre of the base is given by, \[e = {b {\left/2} - \bar x\]   .The nominal stress at any point on the base is the sum of direct stress and bending stress.

The direct stress is always compressive and given by,

\[{\sigma _{cc}} = {{{R_y}} \over b}\]      [per unit length of the dam]     (31.4)

Bending moment about the centre of the base is, \[M={R_y} \times e\]  . Corresponding bending stress at a distance x from the centre of the base is given by,

\[{\sigma _{bc}}=\pm {{Mx} \over I}\]            (31.5)

Where, I is the second moment of area of the base per unit length of the dam. I is given by,

\[I={{1 \times {b^3}} \over {12}}={{{b^3}} \over {12}}\]          (31.6)

Therefore total normal stress at a distance x from the centre of the base is,

\[{p_n}={\sigma _{cc}} + {\sigma _{bc}}={{{R_y}} \over b} \pm {{Mx} \over I} = {{{R_y}} \over b} \pm {{12Mx} \over {{b^3}}}\]         (31.7)

The resulting moment produces tension at heel and compression at toe.

Therefore,

\[{p_{nheel}}={{{R_y}} \over b} - {{12\left( {{R_y} \times e} \right)\left( {{b {\left/2}})} \over {{b^3}}}={{{R_y}} \over b}\left({1-{{6e} \over b}} \right)\]             (31.8)

\[{p_{toe}}={{{R_y}} \over b} + {{12\left( {{R_y} \times e} \right)\left( {{b{\left/2}})} \over {{b^3}}}={{{R_y}} \over b}\left( {1 + {{6e} \over b}} \right)\]            (31.9)

The distributions of normal stress at the base of the dam for three different situations are shown in Figure 31.2.

Fig. 31.2.