5. Vertical Stress Explained: Formulas, Examples & How to Calculate
· Stresses induced in a soil mass due to weight of overlying soil and due to applied loads.
· These stresses are required to design a foundation such that the shear stress on any stratum of soil below it does not exceed, after providing Factor of safety for bearing capacity of soil.
· Further the vertical stresses transmitted to the soil layers below the foundation will lead to vertical deformation in the soil, causing foundation settlement.
· This settlement, again should not be allowed to exceed the permissible settlement.
· Hence, the knowledge of distribution of stresses with in a soil mass, induced by loads applied on the surface of soil, is a requisite for a foundation design.
· The stress induced in soil due to applied loads depend upon its stress-strain characteristics. The stress -strain behaviour of soil is extremely complex and it depends upon a large number of factors, such as drainage conditions, water content, void ratio, rate of loading, the load level, and the stress path.
· Generally, the stress strain relationship is assumed to be linear, and fortunately these results are good enough for the problems usually encountered in practise.
· Theory of elasticity is used to determine the stresses in soil mass.
· The main stress- strain parameters required for the application of elastic theories are modulus of Elasticity and Poisson’s ratio.
o Modulus of elasticity can be determined in the laboratory by conducting a triaxial compression test.
o The stress strain curve is plotted between the deviator stress and the axial strain.
· For saturated, cohesive soil: Unconsolidated Undrained (UU) test or unconfined compression test is performed.
· For cohesionless soil: Consolidated drained (CD) test is performed.
The value of modulus of elasticity is generally taken as the secant modulus (1/2 to 1/3) of the peak stress. Sometimes, instead of secant modulus, the initial tangent modulus or the tangent modulus at (1/2 to 1/3) of the peak stress is also used.
o Poisson’s Ratio for elastic material generally Poisson’s ratio varies from 0-0.5.
o For Undrained conditions, the value of Poisson’s ratio is 0.50.
o For drained condition value is less than 0.50.
Case 1: VERTICAL STRESSES DUE TO CONCENTRATED LOAD
Boussinesq Equation: gave theoretical solutions for the stress distribution in an elastic medium subjected to concentrated load on its surface.
Assumptions:
1. Soil mass is elastic.
2. Soil is homogeneous and isotropic.
3. Soil is semi-infinite.
4. Soil is weightless and unstressed before the application of load.
· It should be noted that the vertical stress at the certain depth z is dependent only on the r/z ratio and independent of the material, i.e. Modulus of elasticity and Poisson’s ratio.
· But the solution has been derived assuming the soil is linear elastic.
Case 2: VERTICAL STRESSES DUE TO LINE LOAD
Case 3: VERTICAL STRESSES DUE TO STRIP LOAD
a) Point P below the centre of strip
b) Point B not below the centre of strip
Understanding how vertical stresses are induced and distributed within soil is fundamental to safe and economical foundation design. These stresses, whether due to the weight of the overlying soil or applied surface loads, directly influence settlement and bearing capacity. By applying elasticity theory, supported with parameters such as modulus of elasticity and Poisson’s ratio, engineers can estimate stresses with reasonable accuracy for most practical problems.
So far, we have discussed Boussinesq’s theory and applications for different loading conditions such as concentrated loads, line loads, and strip loads. However, one limitation of Boussinesq’s solution is its assumption of an ideal, homogeneous, isotropic, and perfectly elastic soil mass, which may not always represent real soil behavior. To address this, Westergaard proposed an alternative stress distribution theory, considering a layered soil structure with reinforcement of vertical fibers, leading to a different prediction of stresses beneath applied loads.
Class Activity:
👉 How does Westergaard’s theory of stress distribution differ from Boussinesq’s assumptions, and in what types of soil conditions would it provide a more realistic solution?
Practise Questions:
1. A concentrated load of 2000KN is applied at the ground surface. Determine the vertical stress at a point P which is 6m directly below the load. Also calculate the vertical stress at a point R which is at a depth of 6m but at horizontal distance of 5m from the axis of the load.
2. There is a line load of 120KN/m acting on the ground surface along y-axis. Determine the vertical stress at a point P which has x and z coordinates as 2m and 3.5m, respectively.
ANSWERS: 1. 26.53 KN/m2 , 7.1 KN/m2
2. 12.40 KN/m2
Westergaard’s theory is different from Boussinesq’s because it assumes the soil doesn’t deform sideways as much, so the vertical stress stays more concentrated right under the load. Boussinesq, on the other hand, assumes the soil spreads stress evenly in all directions, which works better for loose or uniform soils.
ReplyDeleteWestergaard’s approach makes more sense when the ground is reinforced, or stiff, like in soils that resist horizontal movement, so it gives a more realistic outcome in those cases.
Westergaard’s theory is different because it assumes the soil has layers and cannot spread sideways easily, so the load goes more straight down. This makes it better for layered or cracked soils. On the other hand, Boussinesq’s theory assumes the soil is the same in all directions, which may not always represent real soil behaviour, so the load spreads out more, which works better for uniform soils like clean sand or gravel.
ReplyDelete1Q A) 26.53N/m2 B) 7.1KN/m2
ReplyDeleteQ) 12.40 KN/m2
1Q A) 26.53N/m2 B) 7.1KN/m2
DeleteQ) 12.40 KN/m2
Westergaard's theory is better for layered or cracked soils because it assumes loads go straight down. Boussinesq's theory works better for uniform soils like sand or gravel, where loads spread out more evenly.