deformation analysis - seoul national university

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Advanced Soil Mechanics I SNU Geotechnical and Geoenvironmental Engineering Lab. 2-1 Chapter II Deformation Analysis 2.1 Stress Distribution in Soil ex) What is settlement caused by embankment loading? σ (Applied Stress) γ = h v σ = ( stress induced by σ ) 1-D loading h If NC clays, consolidation settlement, 0 0 0 ' ' log 1 v v c c H e C S σ σ σ + + = When B/H 1, then 1-D loading (under center of structure) is valid. ( vσ σ =∆ ) When B/H<1, then we must calculate stress distribution throughout soil mass.(∆σ ∆σ v ) P v σ A Sand Clay A P / = σ B ∆σ = γh NC clay Sand v σ 0 ' v σ H

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Page 1: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-1

Chapter II

Deformation Analysis

2.1 Stress Distribution in Soil

ex) What is settlement caused by embankment loading?

σ∆ ≡ (Applied Stress) γ= h

vσ∆= ( ≡stress induced by σ∆ )

→ 1-D loading

h

If NC clays,

consolidation settlement,

0

0

0 '

'log

1 v

vcc H

e

CS

σσσ ∆+

+=

When B/H ≥ 1, then 1-D loading (under center of structure) is valid. ( vσ σ∆ = ∆ )

When B/H<1, then we must calculate stress distribution throughout soil

mass.(∆σ ≠∆σv)

P

vσ∆

A

Sand

Clay

AP /=∆σ

B

∆σ = γh

NC clay

Sand

vσ∆

0'vσ

H

Page 2: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-2

� Notes

- In case that 1-D loading condition is no longer valid,

ⅰ) σσ ∆≠∆ v

ⅱ) vu σ∆≠∆

ⅲ) 0≠ε∆ h

� Use elasticity to calculate the stress distribution.

⇒ Boussinesq approach.

Assumptions

1. Soil is homogeneous and isotropic.

2. Soil is linear elastic.

3. Semi-infinite soil mass (No rigid base nearby).

4. Perfectly flexible footing.

Can get 1. good estimate of vσ∆ .

2. but poor estimate of hσ∆ (unless plane strain condition)

L/B ≥ 5

(→Generally consolidation settlement is estimated by vu σ∆=∆ , and 0=hε )

Page 3: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-3

� Stress Distributions

- Point load : depth ≥ 3 times of width(diameter) of square ft (circular ft).

- Line load : depth ≥ 3 times of width of strip ft.

1. Fig. 2-1 (NAVFAC DM 7.1~165) (p2-5, 2-6)

Formulas for stresses

2. Fig. 2-2 (NAVFAC DM 7.1~167) (p2-7)

Difference between square and strip footings

- z I pσ = ×

3. Fig. 2-3 (NAVFAC DM 7.1~168) (p2-8)

Vertical stress beneath a corner of a uniformly loaded rectangular area

4. Fig. 2-4 (NAVFAC DM 7.1~169) (p2-9)

Vertical stress under uniformly loaded circular area

5. Fig. 2-5 (NAVFAC DM 7.1~170) (p2-10)

Vertical stress under embankment load of infinite length

6. Fig. 2-6 (NAVFAC DM 7.1~171) (p2-11)

Vertical stress under corner of triangular load

Page 4: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-4

- Comments on charts

i) Stresses penetrate further for larger loads.

ii) If the size of footing increases, stresses penetrate further.

iii) Stresses for strip footing penetrate further than stresses for square or

circular footing.

iv) For square or rectangular footing, stresses other than corner can be

found by superposition.

- Rule of thumb to find critical depth

critical depth : depth at which soil compression contributes significantly

to surface settlements

i) Sands

Depth at which vσ∆ is 20% of the in situ, effective stresses( vo'σ )

ii) Clays

≥∆ vσ 10% of vo'σ

at center 4×=∆ PIcenter σ

at corner of

� PIPI BCDGABCDEFc −=∆σ

A B C

F E

G D

Page 5: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-5

Fig. 2-1 Formulas for stresses in Semi-Infinite Elastic Foundation

(NAVFAC DM 7.1-165)

Page 6: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-6

Fig. 2-1 (continued) Formulas for Stresses in Semi-Infinite Elastic Foundation

(NAVFAC DM 7.1-165)

Page 7: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-7

Fig. 2-2 Stress Contours and Their Application (NAVFAC DM 7.1-167)

Page 8: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-8

Fig. 2-3 Influence Value for Vertical Stress Beneath a Corner of a Uniformly

Loaded Rectangular Area (Boussinesq Case) (NAVFAC DM 7.1-168)

Page 9: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-9

Fig. 2-4 Influence Value for Vertical Stress Under Uniformly Loaded Circular Area

(Boussinesq Case) (NAVFAC DM 7.1-169)

Page 10: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-10

Fig. 2-5 Influence Value for Vertical Stress Under Embankment Load of Infinite

Length (Boussinesq Case) (NAVFAC DM 7.1-170)

Page 11: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-11

Fig. 2-6 Influence Value for Vertical Stress Beneath Triangular Load (Boussinesq

Case) (NAVFAC DM 7.1-171)

Page 12: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-12

� Newmark charts

(�useful for irregular loaded area)

1. Determine location and depth(z), where stress increment is desired to

obtain.

2. Adopt a scale such that the distance OQ(=1 inch) in Fig. is equal to the

depth z.

(i.e. if z=30ft, scale is 30ft)

3. Draw the plane of loaded area to scale determined in (2).

4. Place the plane on Newmark chart with point under consideration over

the center.

5. Count the number of blocks, N, of the influence chart which fall inside

the plane. 6. Calculate vσ∆ as

qFNv =∆σ

where, q = applied stress and,

F = influence value of charts (=0.001)

Page 13: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-13

Fig. 2-7 Influence chart for vertical stress ( )z vσ σ=∆ (Newmark, 1942)

Page 14: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-14

•••• Comments on Stress Distributions 1. Use superposition for areas with different applied pressures.

2. For embedded structures,

�Conservative (i.e. higher loads), because shear resistance of soil at

boundary between embedded structure and soil is neglected.

3. Vertical stresses are affected by layering, if soils have much different E

values.

4. Stiff layer at ground surface dissipates the induced stresses rapidly.

(Hand out Fig 6-4 in p2-15, 7.1-179 in p2-16)

→ Use this to get ∆σv

� superposition

-(∆σv)1 ↓

+(∆σv) 2

∆σv = (∆σv) 2 - (∆σv) 1

D γ P � Vertical load

at depth, D is (P- γD)

or

γD + P

At ground

surface

At depth, D

Page 15: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-15

Fig. 2-8 Basic pattern of Burmister Two-Layer Stress Influence Curves

(Strip footing)

Page 16: Deformation Analysis - Seoul National University

Advanced Soil Mechanics I

SNU Geotechnical and Geoenvironmental Engineering Lab.

2-16

Fig. 2-9 Influence Values for Vertical Stresses Beneath Uniformly Loaded

Circular Area (Two-Layer Foundation) (NAVFAC DM 7.1-179)