## Proportioning and design of cantilever and counterfort walls

**PROPORTIONING AND DESIGN OF CANTILEVER AND COUNTERFORT WALLS**

Prior to carrying out a detailed analysis and design of the retaining wall structure, it is necessary to assume preliminary dimensions of the various elements of the structure using certain approximations. Subsequently, these dimensions may be suitably revised, if so required by design considerations

**Position of Stem on Base Slab for Economical Design**

An important consideration in the design of cantilever and counterfort walls is the position of the vertical stem on the base slab. An economical design of the retaining wall can be obtained by proportioning the base slab so as to align the vertical soil reaction R at the base with the front face of the wall (stem). For this derivation, let us consider the typical case of a level backfill. The location of the resultant soil reaction, R, is dependent on the magnitude and location of the resultant vertical load, W, which in turn depends on the dimension X (i.e., the length of heel slab, inclusive of the stem thickness). For convenience in the derivation, X may be expressed as a fraction, *α _{x}*, of the full width L of the base slab (X = α

_{x}L). Assuming an average unit weight γ

_{e}for all material (earth plus concrete) behind the front face of the stem (rectangle abcd), and neglecting entirely the weight of concrete in the toe slab,

For a given location of R corresponding to a chosen value of X, the toe projection of the base slab (and hence its total width, L) can be so selected by the designer as to give any desired distribution of base soil pressure. Thus, representing the distance, L_{R}, from the heel to R as a fraction α_{R} of base width L, , the base pressure will be uniform if L is so selected as to make α_{R} = 0.5. Similarly, for α_{R} = 2/3, the base pressure distribution will be triangular. Thus, for any selected distribution of base pressure, α_{R} is a constant and the required base width

Considering static equilibrium and taking moments about reaction point e, and assuming X_{w} ≈ α_{X}L/2,

For economical proportioning for a given height of wall (h), the length of the base (L) must be minimum, i.e., L/h should be minimum. this implies that

should be maximum. The location of R, and hence the base width for any selected pressure distribution, is dependent on the variable X, i.e., α_{x}. For maximising it,

**Hence, for an economical design, the soil pressure resultant should line up with the front face of the wall**

**Width of Base**

Applying the above principle, an approximate expression for the minimum length of base slab for a given height of wall is obtained from Eq

The effect of surcharge or sloping backfill may be taken into account, approximately, by replacing h with h h_{s} , or h', respectively.

Alternatively, and perhaps more conveniently, using the above principle, the heel slab width may be obtained by equating moments of W and P_{a} about the point d. The required L can then be worked out based on the base pressure distribution desired.

It may be noted that the total height h of the retaining wall is the difference in elevation between the top of the wall and the bottom of base slab. After fixing up the trial width of the heel slab ( = X) for a given height of wall and backfill conditions, the dimension L may be fixed up. Initially, a triangular pressure distribution may be assumed, resulting in