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Vertical Stress Increases
Chapter 8
Point Load
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Point Load
Point Load
32 ⁄
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Point Load
32
1⁄
⁄ 1 ⁄
‐4 ‐3 ‐2 ‐1 0 1 2 3 4
Point Load
I1
r/z
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ExampleA vertical point load of 25 kN is applied at the ground surface. What is the stressincrease in the ground 2 m below the ground surface directly beneath the load?
∆252
0.4775 2.98 ⁄
r = 0, z = 2
ExampleA vertical point load of 25 kN is applied at the ground surface. What is the stressincrease in the ground 2 m below the ground surface and 2 m to the side?
∆252
0.0844 0.5275 ⁄
r = 2, z = 2
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Line Load
NOTE:qhasunitsofkN/m
Line Load
2 1⁄ 1
2⁄⁄ 1
∆⁄
2⁄⁄ 1
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‐4 ‐3 ‐2 ‐1 0 1 2 3 4
I
x/z
2⁄⁄ 1
Line Load
ExampleA vertical line load of 8 kN/m is applied at the ground surface. What is the stressincrease in the ground 4 m below the ground surface directly beneath the load?
∆8 /4
0.637 1.274 ⁄
x = 0, z = 4
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ExampleA vertical line load of 8 kN/m is applied at the ground surface. What is the stressincrease in the ground 4 m below the ground surface and 2 m off to the side?
∆8 /4
0.407 0.814 ⁄
x = 2, z = 4
Circular Loaded Area
NOTE:qhasunitsofkN/m2 orkPa
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Circular Loaded Area
∆1
1⁄ 1 /
Circular Loaded Area
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.2 0.4 0.6 0.8 1 1.2
/q
z/R
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ExampleA uniform load of 80 kPa is applied at the ground surface over a circular areawith a diameter of 8 m. What is the stress increase directly beneath the centerof the loaded area at a depth of 8 m?
∆ 80 0.2845 22.76
z = 8, R = 4
Circular Loaded Area
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ExampleA uniform load of 80 kPa is applied at the ground surface over a circular areawith a diameter of 8 m. What is the stress increase 2 m below the ground surfaceat a radial distance of 4 m from the load centerline?
∆ 80 0.417 33.36
z = 8, R = 4
r = 4 = R
Rectangular Loaded Area
∆
2 1, 2 1
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ExampleA uniform load of 80 kPa is applied at the ground surface over a rectangular areawith a length of 5 m and a width of 2 m. What is the stress increase beneath thecorner of the loaded area at a depth of 10 m?
210
0.2
510
0.5
∆ 80 0.04 3.2
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ExampleA uniform load of 80 kPa is applied at the ground surface over a rectangular areawith a length of 5 m and a width of 2 m. What is the stress increase beneath thecorner of the loaded area at a depth of 10 m?
510
0.5
210
0.2
∆ 80 0.04 3.2
L
B
Rectangular Loaded Area
A'
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L
B
Rectangular Loaded Area
A'
L/2 L/2
B/2
B/2
= L/B
z/(B/2) =
∆
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Example
Example
44
1.0
24
0.5
∆ 150 0.12 18
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Example
14
2 1 0.5 1 0.25 11 0.25 0.25 1
1 0.25 21 0.25 1
tan2 1 0.5 1 0.25 11 0.25 0.25 1
0.120175
′ 1 1 ′ 0.5 0.25
1 0.5 0.5
′ ′ 1 0.25 0.25
Example
14
0.25
24
0.5
∆ 150 0.048 7.2
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Example
∆ 18.0 7.2 10.8