Geotechnical Engineering

DHARM

STRESS DISTRIBUTION IN SOIL 371

B

z

q/unit areaL

m= B/z

n = L/z

Fig. 10.15 Vertical stress at the corner of a uniformly loaded rectangular area

σz =

q mn m n

mn mn

mn

mn

mn m n

(^4) mn mn
21
1
2
1
21
1
22
22 22
22
22
1
22
π^2222
()
()
.
()
()
tan
(
++
+++
++
++

• ++
  ++−
  L
  N
  M
  M
  O
  Q
  P
  P
  −
  ...(Eq. 10.45)
  where m = B/z and n = L/z.
  The second term within the brackets is an angle in radians. It is of interest to note that
  the above expressions do not contain the dimension z; thus, for any magnitude of z, the under-
  ground stress depends only on the ratios m and n and the surface load intensity. Since these
  equations are symmetrical in m and n, the values of m and n are interchangeable.
  Equation 10.45 may be written in the form:
  σz = q. Iσ ...(Eq. 10.46)
  where Iσ = Influence value
  = (/ )^14. tan
  21
  1
  2
  1
  21
  1
  22
  22 22
  22
  22
  1
  22
  π 22 22
  mn m n
  mn mn
  mn
  mn
  mn m n
  mn mn
  ++
  +++
  F
  H
  G
  G
  I
  K
  J
  J
  ++
  ++
  F
  HG
  I
  KJ


• 
  

  ++
  ++−
  F
  H
  G
  G
  I
  K
  J
  J
  L
  N
  M
  M
  O
  Q
  P
  P
  −
  ...(Eq. 10.47)
  Based on this equation, Fadum (1941) has prepared a chart for the influence values for
  sets of values for m and n, as shown in Fig. 10.16.
  Steinbrenner (1934) has given another form of chart for this purpose, which is shown in
  Fig. 10.17, plotting the influence values Iσ on the horizontal axis and 1/m (= z/B) on the vertical
  axis, for different values of L/B (= n/m).
  The vertical stress at a point beneath the centre of a uniformly loaded rectangular area
  may be found using the influence value for a corner by the principle of superposition, dividing
  the rectangle into four equal parts by lines parallel to the sides and passing through the centre.
  In fact, if we derive the expression for the vertical stress beneath the centre of the rectangle,
  we may obtain that beneath a corner due to load from one fourth of this area by just dividing it