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• Waldemar Cepeda Murillo

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Virginia Polytechnic Institute And State University The Charles E. Via, Jr. Department of Civil and Environmental Engineering CENTER FOR GEOTECHNICAL PRACTICE AND RESEARCH

ZSTRESS 2.0: A Computer Program for Calculation of Vertical Stresses Due to Surface Loads By Bingzhi Yang and J. Michael Duncan

Center for Geotechnical Practice and Research 200 Patton Hall, Virginia Tech Blacksburg, Virginia 24061-0105

July 2002

INTRODUCTION This computer program is an updated version of a program with the same name written in 1987 by Rick Allen and Mike Duncan. The original program, which operated under DOS, has become outdated and inconvenient by today’s standards. This new version, which is Windows-based, is easier to use. The program (called “Zee-Stress”) computes vertical stresses due to surface loads, which are useful in computing settlements. The program calculates stresses due to point loads and loads distributed over rectangular areas, using the Boussinesq and the Westergaard equations. ZSTRESS 2.0 was written in Microsoft Visual Basic 6.0. The program is interactive, and has facilities for creating, storing and editing data files, and for printing results. METHOD OF ANALYSIS ZSTRESS 2.0 calculates changes in vertical stresses due to surface loads using both the Boussinesq and the Westergaard solutions for point loads and for loads distributed uniformly over rectangular areas. The equations used within the program are: Boussinesq Equations (Poulos and Davis, 1974) 3Q z 3 Point Load: ∆σ z = 2π R 5 Where: ∆σ z = Change in vertical stress Q = magnitude of point load z = depth of stress point

(stress) (force) (length)

(length) R = x2 + y2 + z2 x = distance from load to stress point in x-direction (length) y = distance from load to stress point in y-direction (length) Rectangular Load:

Where:

∆σ z =

q 2π

 −1  BL   BLz  1 1   2 + 2   +   tan  r2   zr3   r3  r1 

∆σ z = change in vertical stress beneath corner of rectangular loaded area (stress) q = magnitude of surface pressure (stress) B = width of rectangle (length) L = length of rectangle (length) z = depth of stress point (length)

r1 = B 2 + z 2

(length)

r2 = L2 + z 2

(length)

r3 = B 2 + L2 + z 2

(length)

1

Westergaard Equations (Fadum, 1948) Q z Point Load: ∆σ z = 2 2 1.5 2 2π (r + 0.5 z )

Where r = x 2 + y 2 (length) The other terms are the same as for the Boussinesq point load equation.  q  −1  2 BL   tan  2 2 2 0.5   2π  z B L z ( 0 . 5 ) + +   Where the terms are the same as for Boussinesq rectangular load equation.

Rectangular Load:

∆σ z =

Superposition is used to compute stresses due to multiple loads, and to compute stresses at points that are not beneath the corners of distributed loads. The Boussinesq equations apply for any value of Poisson’s ratio. The Westergaard equations are for Poisson’s ratio equal to zero; it has become standard practice in geotechnical engineering to use these forms of the Westergaard equations. The stresses calculated using these Westergaard equations are smaller than those calculated using the Boussinesq equations. The equations for the Westergaard solution for Poisson’s ratio larger than zero lead to even smaller stresses, and are seldom used. The positions of the loads and the locations of the points at which the stresses are calculated are specified using an x-y-z coordinate system, as shown in Figure 1. The coordinates x and y are horizontal, whereas z is vertical and positive downward. The origin of coordinates can be placed at any convenient location, and the value of x, y and z for loads and stress points can be positive or negative. Loads can be applied at any value of z. However, the equations used in calculating the stresses were derived for loads applied at the surface of an elastic half-space. The program is therefore best suited for calculating stresses due to loads applied at the surface or near the surface. The formulas for stresses due to distributed loads used in ZSTRESS 2.0 are for the vertical change in stress beneath the corner of a uniformly loaded rectangular area. The program uses superposition techniques to calculate changes in stress at other positions. “Stress points” are the points where stresses are calculated. As seen in plan, the stress point can be inside, outside, or on the boundary of the loaded area. Stress points can be located at any depth. If a stress point coincides with the position of a point load, the stress is theoretically infinite, and no value is given. If a stress point falls within a rectangular loaded area and is at the same depth as the load, the calculated stress is equal to the applied load (q). If a stress point falls on the edge of a loaded area (at the same depth), the change in stress is 0.5q. If a stress point falls on the corner (at the same depth), the change in stress is 0.25q.

2

Point loads at any locations. Load magnitude = Q

(Horizontal) y

Uniformly loaded rectangular areas at any locations. Load intensity = q

L B

x (Horizontal)

∆

z

Vertical change in stresses at any location = combined effect of all loads.

z = depth, positive down (vertical)

(a) Coordinate system

y L

L

(x,y) x>0 y>0

(x,y) B x0

Origin of coordinates

L (x,y) x