Bridge Return Section Calculator

For Bridges specifically in case of Culverts (upto 6m) or Minor Bridges (upto 30m) most of the time height of approaches section is less than 8m-10m, due to ease of construction PCC sections are preferred I such cases. Here I have given some standard section calculations by height and expected batter inputs.

Left return

Note:- Input Values in red only

 Right return 

Note:- Input Values in red only

 Natural or Induced footing

    When a load is applied over an area on an elastic solid, some deformation occurs at the surface, which induces compressive stress together with shear and tensile stresses at the periphery of the area affected; the volume included within the spheres of equal compressive stress is compressed, becoming more resistant thereby, until a ball of greater density and strength is involved which acts as natural footing to spread the load over an area so great that the intensity of pressure does not sensibly deform the adjacent material.

    In a similar manner , when a load is applied to soil more or less devoid of ability to withstand tension, the pressure compresses a spheroid of soil into a shape somewhat resembling a huge onion , of such density and size as to afford a natural footing. This condensed spheroid serves to distribute the pressure over sufficient area of soil to reduce the intensity to a degree that will not deform the soil on which it rests beyond its elastic strength.

The term "Bulb of Pressure" has been applied to this compressed spheroid of soil. The spheroid is compressed by the expulsion of air and moisture, thereby becoming a denser unit capable of action as a natural earth footing.

Spread and depth of pressure soils

When a load is applied on footing on soil stratum, there occurs a settlement, which by means of shear along the periphery is distributed to an ever increasing area with the depth. 

    Let the angle of spread is ⍺  which is equivalent to angle of internal friction of soil

    Assume side of square footing is b

    The area over which the load is spread at any depth y is 

"A" = (b + 2y tan ⍺)²                                    ..................................(1)

    This Area A is projected area of the bulb of pressure on a horizontal plane, one can conclude that pressure fades to zero at edge of this area it may approximately considered for purpose of calculations as equivalent to uniform intensity over an area 

"a" = (b + y tan ⍺)²                                    ..................................(2)

Above expression shows that the intensity of pressure on horizontal plane diminishes with the square of the depth and consequently, soon is reduced to an intensity within the elastic resistance of the soil, when no further displacement of soil particles occurs.

Bearing under the eccentric loads

if the modulus of compressibility of soil were constant, the pressure under a footing eccentrically loaded would vary directly along the foundation from heel to toe, since the deformation under rigid foundation must vary, the most common assumption is that, within the range of plasticity, the deformation is proportional to load, the assumption is approximately true for following strata

1) Strata containing well-bedded granular materials

2) Strata containing very stiff clay like hardpan

3) Strata containing shale , rock etc.

Important note:-The eccentricity "e" is the distance from middle to the point where resultant of all loads cuts the base. The resultant of total soil reaction is necessary for equilibrium collinear with resultant of all vertical loads.

For more plastic soils and loosely compacted sand the modulus decreases with the load in some relation. If the decrease is direct the variation of intensity varies as the ordinates to a Parabola, the practical effect of this fact is to reduce the resisting moment of a soil reaction against overturning i.e Kern becomes the middle quarter instead of middle third and resisting moment for "e" about heel is $$ {5\over 8}.Wb $$ instead of  $$ {2\over 3}.Wb $$  as for straight line distribution of pressure.

CASE 1) Eccentricity lies between mid 1/3rd of footing

For straight line variation of soil pressure

 $$ P_1= {W\over A}.(1+{6e\over b}) $$

For Parabolic variation of soil pressure

$$ P_1= {W\over A}.(1+{4e\over b}) $$

CASE 2) Eccentricity lies on 1/3rd of footing on toe side

For straight line variation of soil pressure

$$ P_1= {2W\over A} $$

For parabolic variation of soil pressure

$$ P_1= {3W\over 2A} $$

CASE 3) Eccentricity lies in 1/3rd of footing on toe side (not in mid 1/3rd)

For straight line variation of soil pressure

 $$ P_1= {2W\over 3({b\over 2}-e)} $$

For Parabolic variation of soil pressure

$$ P_1= {9W\over 16({b\over 2}-e)} $$

in progress.................