Cosec ip sinip j
Solution A: Soil surface horizontal
Assuming that 6 = 0.50' = 17.5° (see page 203), substituting <f>' = 35°, ¡3 = 0°, 6 = 17.5° and 0 = 90° into the formula for Ka:
This value is inclined at 17.5° to the normal to the back of the wall so that the total horizontal active thrust according to Coulomb, is 58.43 x cos 17.5° = 55.7 kN.
Note If 6 had been assumed equal to 0° the calculated value of total horizontal thrust would have been the same as that obtained by the Rankine theory of Example 6.1.
Solution B: Soil surface sloping at 35°
Substituting <f>' = 35°, ¡3 = 35°, 6 = 17.5° and i,6 = 90° into the formula gives Ka = 0.704. Hence
Total active thrust = 0.5 x 0.704 x 19 x 52 = 167.2 kN Total horizontal thrust = 167.2 x cos 17.5° = 159.5 kN Increase in horizontal thrust = 159.5  55.7= 104 kN
The Culmann line construction
When the surface of the retained soil is irregular Coulomb's analytical solution becomes difficult to apply and it is generally simpler to make use of a graphical method proposed by Culmann in 1866, known as the Culmann line construction. Besides being able to cope with irregular soil surfaces the method can also deal with irregular combinations of uniform and line loads.
The procedure is to select a series of trial wedges and find the one that exerts the greatest thrust on the wall. A wedge is acted upon by three forces:
W, the weight of the wedge;
Pa, the reaction from the wall;
R, the reaction on the plane of failure.
At failure, the reaction on the failure plane will be inclined at maximum obliquity, <p, to the normal to the plane. If the angle of wall friction is 6 then the reaction from the wall will be inclined at 6 to the normal to the wall (6 cannot be greater than 0). As active pressures are being developed the wedge is tending to move downwards, and both R and Pa will consequently be on the downward sides of the normals (Fig. 6.5b). W is of known magnitude (area ABD x unit weight) and direction (vertical) and R and Pa are both of known direction, so the triangle of forces can be completed and the magnitude of Pa found (Fig. 6.5c). The value of the angle of wall friction, 6, can be obtained from tests, but if test values are not available S is usually assumed as 0.5 to 0.750.
In Fig. 6.7 the total thrust on the wall due to earth pressure is to be evaluated, four trial wedges having been selected with failure surfaces BC, BD, BE and BF. At some point along each failure surface a line normal to it is drawn, after which a second line is constructed at 0 to the normal. The resulting four lines give the lines of action of the reactions on each of the trial planes of failure. The direction of the wall reaction is similarly obtained by drawing a line normal to the wall and then another line at angle 6 to it.
The weight of each trial slice is next obtained, and starting at a point X these weights are set off vertically upwards as points di, d2, etc. such that Xdi represents the weight of slice 1 to some scale, Xd2 represents the weight of slice 2 + slice 1, and so on.
(a) Space diagram
(b) Force diagram Fig. 6.7 Culmann line construction for a cohesionless soil.
(a) Space diagram
(b) Force diagram Fig. 6.7 Culmann line construction for a cohesionless soil.
A separate triangle of forces is now completed for each of the four wedges, the directions of the corresponding reaction on the failure plane and of Pa being obtained from the space diagram. The point of intersection of R and Pa is given the symbol e with a suffix that tallies with the wedge analysed, e.g. the point ei represents the intersection of Pal and R,.
The maximum thrust on the wall is obviously represented by the maximum value of the length ed. To obtain this length a smooth curve (the Culmann line) is drawn through the points elt e2, e3 and e4. A tangent to the Culmann line which is parallel to Xd4 will cut the line at point e; hence the line ed can be drawn on the force diagram and the length ed represents the thrust on the back of the wall due to the soil.
If required, the position of the actual failure plane can be plotted on the space diagram, the angle e3Xe2 on the force diagram equalling the angle EBD on the space diagram whilst the angle eXe2 similarly equals the angle GBD where BG = failure plane.
6.3.4 Point of application of the total active thrust
With either the Rankine or the Coulomb analytical methods the total active thrust, Pa is given by the expression:
Pa=i7H2Ka where Ka is the respective value of the coefficient of active earth pressure, H = height of wall and 7 = unit weight of retained soil.
The position of the centre of pressure on the back of the wall, i.e. the point of application of Pa, is largely indeterminate. Locations suitable for design purposes are given in Fig. 6.8 and are based on the Rankine theory (with its assumption of a triangular distribution of pressure). For most practical purposes these locations of Pa can also be used in conjunction with Pa values obtained from a Coulomb analytical solution.
When using the Culmann line construction, the magnitude of Pa is obtained directly from the force diagram. Its point of application may be assumed to be where a line drawn through the centroid of the failure wedge, and parallel to the failure plane, intersects the back of the wall. (See Fig. 6.20.)
6.4 Surcharges
The extra loading carried by a retaining wall is known as a surcharge and can be a uniform load (roadway, stacked goods, etc.), a line load (trains running parallel to a wall), an isolated load (column footing), or a dynamic load (traffic).
Uniform load
In the analytical solution the load is considered as equivalent to an extra height of soil.
Equivalent height is given by the expression:
7 = unit weight of soil ws = intensity of uniform load/unit area ip = angle of back of wall to horizontal (3 — angle of inclination of retained soil.
The surcharge can therefore be regarded as an extra height of soil, he, placed on the top of the wall.
Pressure due to the surcharge, pu = Ka7he, is distributed uniformly over the back of the wall with its centre of pressure acting at half the wall's height (Fig. 6.9). When the surface of the fill is horizontal, ¡3 = 0 and he = ws/7.
In Fig. 6.9 Pu — thrust on wall due to surcharge and Pa = thrust on wall due to earth pressure; Pa and Pu can be combined to give the magnitude and point of application of the resultant thrust.
With the Culmann line construction the weight of surcharge on each slice is merely added to the weight of the slice. The weight of each wedge plus its surcharge is plotted as Xd]? Xd2, etc. and the procedure is as described before.
Even when a retaining wall is not intended to support a uniform surcharge it should be remembered that it may be subjected to surface loadings due to ws sin ip 7 sin(t/> + /?)
where
K"
u u plant movement during its construction. It is at this time that the wall will be at its weakest state and BS8002: 1994 Code of practice for earth retaining structures recommends that walls be designed to carry a uniform surcharge of 10kN/m2 (Chapter 7).
Line load
The lateral thrust acting on the back of the wall as a result of a line load surcharge is best estimated by plastic analysis, as described in BS 8002: 1994, Code of practice for earth retaining structures. An approximate mathematical solution was given in the predecessor to this code, Civil Engineering Code of Practice No. 2, Earth retaining structures (1951).
In Fig. 6.10 the line load, WL, affects the wall as if it were a horizontal force of magnitude KaWL Its point of application is obtained from the procedure shown in the illustration, which applies whether the back of the wall is vertical or is sloping.
With the Culmann line construction the weight of WL is simply added to the trial wedges affected by it (Fig. 6.11). The Culmann line is first constructed as before, ignoring the line load. On this basis the failure plane would be BC and Pa would have a value 'ed' to some force scale.
Slip occurring on BCi and all planes further from the wall will be due to the wedge weight plus WL. For plane BCi, set off (Wi + WL) from X to d, and continue the construction of the Culmann line as before (i.e. for every trial wedge to the right of plane BC], add WL to its weight). The Culmann line jumps from ei to and then continues to follow a similar curve.
The wall thrust is again determined from the maximum ed value by drawing a tangent, the maximum value of ed being in this case e'j d,. If WL is located far enough back from the wall it may be that ed is still greater than ejdj; in this case WL is taken as having no effect on the wall.
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