1.4 Surface loading
1.4.3 Trapezoidal load on a rectangular area
For more than half a century, the integration of Boussinesq’s elastic equation for a trapezoidal surface loading has eluded investigators on this subject. As a result, graphical and finite-element procedures are currently being utilized to solve this problem. The following derivations show the integration of the elastic equation for total vertical stress due to a trapezoidal loading by using a superposition method and the derived equations for the rectangular and triangular surface loading.
Figure 1.41 Vertical stress distribution.
Figure 1.42 Total vertical stress distribution.
0 50 100 150 200 250 300
Vertical Stress Distribution
0 20 40 60 80 100 120
Depth in feet
Vertical stress in psf
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Total Vertical Stress Distribution
0 20 40 60 100 80 120
Depth in feet
Total vertical stress in lb/ft.
Table 1.10 Tabulated Values of Vertical Stresses
Vertical Stress Values Total Vertical Stress Values
P Po h in ft. Pq in psf PT POT h in ft. PqT in lb/ft.
1.4.3.1 Derivation
Figure 1.43 shows a trapezoidal loading common to roadway embankments.
Using the principle of superposition, the vertical stress under the center of the embankment at a point h is derived as follows:
Due to MON = Equation 1.193, in which q is replaced by bq/(b − c) (1.194) Due to RQN = Equation 1.193, in which q is replaced by cq/(b − c) (1.195)
Hence,
PzT = 2(MON − RQN) = {Equation 1.194 − Equation 1.195} (1.196) Similarly, the total vertical stress under point M can be derived by super-position of three triangular loads (q1 − q2 − q) using Equation 1.191, in which
q1 = 2bq/(b − c) with b1 = 2b, q2 = {q(b + c)/(b − c)} with b2 = b + c and q with b = b − c
1.4.3.2 Application
The derived formulas can determine the exact average stress valve to use in the standard settlement formula due to a surface load of a roadway embankment. Moreover, the magnitude of the transmitted surface load on the roof of any underground structure can be precisely calculated. The principle
Figure 1.43 Vertical stress under a trapezoidal loading intensity.
b Symmetrical about
centerline c
N R Q
q
M X
S O
h
PzT
Point Z
of superposition must be employed in the solution for the dispersion of this trapezoidal surface load. From these values the total vertical stress on this rectangular area can be calculated to any desired degree of accuracy, and hence, an accurate prediction on the soil settlement and accurate estimate of the transmitted vertical stress on the roof of underground structures can be determined.
Example 1.12
In Figure 1.43 the embankment is 600 ft. long and has the values b = 50 ft., c = 30 ft., and q = 1 ksf. Determine the vertical stress values under the center of the embankment.
Solution: Using the Excel program for uniform and triangular loading in Table 1.11 and Figures 1.44 and 1.45 is the required solution to this problem. The superpositioning of triangular load-ings are as follows:
For triangle MNO the parameters are a = 300 ft., b = 50 ft., and q = 2.50 ksf.
For triangle RQN the parameters are a = 300 ft., b = 30 ft., and q = 1.50 ksf. The vertical stresses under the center are obtained from 4(MNO − RQN) loadings.
Hence, two sets of Excel programs for triangular loading have to be set up before the final tabulation and plots of vertical stress distribution can be obtained.
In Figure 1.46, at any point between A and B, the vertical pressure at a depth under point G is determined from two sets of triangular loading; that is,
(OEG − AEF) + (CDG − BDF) (1.197)
Figure 1.44 Vertical stress distribution.
0 200 400 600 800 1000 1200
Vertical Stress Distribution
0 20 40 60 80 100 120
Depth in feet
Vertical stress in psf
in which q in each triangle is determined from the relationships such that q1 = qx/(b − c) in triangle OEG; q2 = q[x − (b − c)]/(b − c) in triangle AEF; q3 = q(2b − x)/
(b − c) in triangle CDG; q4 = q[(b + c) − x]/(b − c) in triangle BDF. Hence, q = (q1 − q2) + (q3 − q4) (1.198) Table 1.11 Tabulated Values of Vertical Stresses
M N O - R Q N
Figure 1.45 Total vertical stress distribution.
Figure 1.46 Point (between A and B) inside the loaded area.
0 10000 20000 30000 40000 50000 60000
Total Vertical Stress Distribution
0 20 40 60 100 80 120
Depth in feet
Total vertical stress in lb/ft.
(a − x) I a
2b X
Point (below) G
D
E
A B
x O II
F 2c
G q O
Z y
Y Y
C 2b − y
(b − c)
Point 2b y − (b − c)
The point under consideration will divide the plan of the loaded area into two parts designated as I and II, whose area is 2b(a − x) and 2bx, respectively. Substitute Equation 1.197 into Equation 1.198 to obtain the total pressure under point G for areas I and II.
Example 1.13
Using the dimensions in Example 1.12, calculate the vertical stresses under the point in Figure 1.46 when x = 200 ft., y = 30 ft., and q = 1 ksf.
Solution: First construct Tables 1.12 and 1.13 for area I and area II. Then use the Excel program set up in previous example to obtain Table 1.14 and Figures 1.47 and 1.48 as the required solu-tion to the problem.
Table 1.12 Breakdown of Areas for Superposition
Table 1.13 Assigned Values for Computation
Area I Area II
Triangle Dimensions Dimensions
a b a b
OEG (a - x) y x y
AEF (a - x) y - (b - c) x y - (b - c)
CDG (a - x) 2b - y x 2b - y
BDF (a - x) 2c - [y - (b - c)] x 2c - [y - (b - c)]
Area I Area II
Triangle a b q a b q
OEG 400 30 1.5 200 30 1.5
AEF 400 10 0.5 200 10 0.5
CDG 400 70 3.5 200 70 3.5
BDF 400 50 2.5 200 50 2.5
In Figure 1.49, at any point between O and F, the vertical stress under point G involves the relationships of the triangles CDG, ABD, AEF, and EGO such that
CDG − (ABD + AEF) + EGO (1.199)
in which q for each triangle is determined as follows: q1 = q(2b − y)/(b − c) in triangle CDG; q2 = q(b + c − y)/(b − c) in triangle ABD; q3 = qy/(b − c) in triangle EGO; q4 = q(b − c − y)/(b − c) in triangle AEF.
Figure 1.47 Vertical stress distribution.
Figure 1.48 Total vertical stress distribution.
0 200 400 600 800 1000 1200
Vertical Stress Distribution
0 20 40 60 80 100 120
Depth in feet
Vertical stress in psf
0 10000 20000 30000 40000 50000 60000
Total Vertical Stress Distribution
0 20 40 60 100 80 120
Depth in feet
Total vertical stress in lb/ft.
Table 1.14 Breakdown of Areas for Superposition
Example 1.14
In Figure 1.49 determine the vertical stresses under the point where x = 100 ft., y = 10 ft., and other dimensions are the same as Example 1.13.
Solution: Construct Table 1.14 and use the same Excel set up as in the previous example. Table 1.15 and Figures 1.50 and 1.51 are the required solutions.
Figure 1.49 Point (between O and A) inside the loaded area.
(a − x) a
I 2b X
Point (below)
G
D
A E
G q
F B
II x
O
2c
O
Z y
Y Y
C 2b − y
(b − c)
Point 2b
In Figure 1.52, the vertical stress at a point outside the loaded area is determined by methods outlined above except that the vertical stress is obtained using superposition such that areas represented by
BOD + EGO − (ABC + AFG) (1.200)
will determine the value of the vertical stress under point O.
The value of q for each triangular loading is obtained as follows: q1 = (2b + y) q/(b − c) in triangle BOD; q2 = (b + c + y)q/(b − c) in triangle ABC; q3 = (y + b − Figure 1.50 Vertical stress distribution.
Figure 1.51 Total vertical stress distribution.
0 100 200 300 400 500 600
Vertical Stress Distribution
0 20 40 60 80
Depth in feet
Vertical stress in psf
0 5000 10000 15000 20000 25000 30000 35000
Total Vertical Stress Distribution
0 20 40 60 80
Depth in feet
Total vertical stress in lb/ft.
Table 1.15 Tabulated Values of Vertical Stresses
c)q/(b − c) in triangle AFG; q4 = qy/(b − c) in triangle EGO. Table 1.16 shows the breakdown of areas for computing the vertical stress.
Figure 1.53 shows the diagram for calculating the vertical stress at a point outside adjacent sides of the loaded area. For this case the breakdown of areas is shown in Table 1.17.
Figure 1.52 Point outside the loaded area.
(a – x) I a
2b X
Point (below)
G
q E G
F A
x II
y
O B
O D
Z y
Y Y
C
2b + y Point
2b 2c