.BOUSSINESQ’S THEORY FOR VERTICAL STRESSES UNDER A CIRCULAR AREA.pdf

2. BOUSSINESQ’S THEORY FOR VERTICAL
STRESSES UNDER A CIRCULAR AREA
• The loads applied to soil surface by footings are not
concentrated loads.
• There are usually spread over a finite area of the footing. It is
generally assumed that the footing is flexible and the contact
pressure is uniform.
• In other words the load is assumed to be uniformly distributed
over the area of the base of footing.

3. • Let us determine the vertical stress at
the point P at depth Z below the centre
of a uniformly loaded circular area.
• Let the intensity of load be q per unit
area and are R be the radius of the
loaded area. Boussinesq’s solution can
be used to determine σ 𝑍
• The load on the elementary ring of
radius r and width dr = q(2πr)dr.
• The load acts at a constant radial
distance r from the point P from fig.

4. ∆σ 𝑍 =
3 𝑞 ∗2πrdr
2π
*
1
𝑧2 *
1
1+
𝑟
𝑧
2 Τ
5
2
The vertical stress due to entire load is given by
σ 𝑍=3qz3
‫׬‬0
𝑅 𝑟𝑑𝑟
𝑟2+𝑧2 5
/
2
……….a
Let 𝑟2 + 𝑧2=U, therefore 2r dr =du
Eq. (a) becomes
σ 𝑍
= 3qz3
‫׬‬𝑧2
(𝑅2+𝑧2) 𝑑𝑢
2𝑢5/2
= 3
2
qz3(−
2
3
) 𝑢−3/2
𝑧
2
𝑅
2 + 𝑧
2

5. = -qz3 1
𝑅2
+𝑧2 3
/
2 −
1
(𝑧2)3/2
= qz3 1
𝑧3 −
1
𝑅2 +𝑍2 3/2
σ 𝑍= q 1 −
1
1+
𝑅
𝑍
2
3
/
2
σ 𝑍
= Ic*q
Ic= is the influence coefficient for the circular area and is given by.
Ic= 1 −
1
1+
𝑅
𝑍
2
3
/
2