![\"Diagram](\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/01\/3.9-rectangular-pressure-1-e1743558919273.png\")
Figure 3.9. Rectangular pressure acting on the surface of a homogeneous elastic half-space.[\/caption]\n\nIf we assume 2:1 distribution of stresses in the half-space ( Figure 3.9. Rectangular pressure acting on the surface of a homogeneous elastic half-space.[\/caption]\n\nIf we assume 2:1 distribution of stresses in the half-space (Figure 3.9<\/a>) we can obtain an approximation of the additional vertical stress \u0394\u03c3<\/em>z<\/sub> at any depth z<\/em>, as:\n\n(3.15)<\/strong>\u00a0 [latex]\\Delta {\\sigma _z} = \\dfrac{{{q_{ext}}BL}}{{\\left( {B + z} \\right)\\left( {L + z} \\right)}}[\/latex]\n\nThe above Eq. 3.15, albeit approximate, will provide reasonably accurate results for depths z <\/em>> B<\/em>.\n\nVertical stress increments under the corner of a rectangular pressure can be calculated using the so-called Fadum charts. Such a chart providing the factor Iqr <\/sub><\/em>= \u0394\u03c3<\/em>z<\/sub>\/qext<\/sub><\/em> is presented in Figure 3.10<\/a>. The use of this chart for calculating vertical stresses for geometries other than that illustrated in Figure 3.10<\/a> is discussed later in <\/a>Chapter 3.7<\/a>.\n\n[caption id=\"attachment_185\" align=\"aligncenter\" width=\"775\"]<\/a> Figure 3.10. Values of influence factor Iqr<\/sub><\/em> to calculate vertical stress increments underneath the edge of a rectangular area of dimensions nz<\/em> x mz<\/em> on which a vertical pressure qext<\/sub><\/em> is applied (after Fadum 1948).[\/caption]","rendered":"The common case of a rectangular footing transferring vertical loads from the structure to the soil is neither a plane-strain nor an axisymmetric problem, as the ones discussed above (Figure 3.9<\/a>). Therefore, finding the exact distribution of additional stresses with depth resulting from a rectangular pressure acting on the surface of a homogeneous elastic half-space requires some perhaps cumbersome mathematical manipulations.<\/p>\n If we assume 2:1 distribution of stresses in the half-space (Figure 3.9<\/a>) we can obtain an approximation of the additional vertical stress \u0394\u03c3<\/em>z<\/sub> at any depth z<\/em>, as:<\/p>\n (3.15)<\/strong>\u00a0 [latex]\\Delta {\\sigma _z} = \\dfrac{{{q_{ext}}BL}}{{\\left( {B + z} \\right)\\left( {L + z} \\right)}}[\/latex]<\/p>\n The above Eq. 3.15, albeit approximate, will provide reasonably accurate results for depths z <\/em>> B<\/em>.<\/p>\n Vertical stress increments under the corner of a rectangular pressure can be calculated using the so-called Fadum charts. Such a chart providing the factor Iqr <\/sub><\/em>= \u0394\u03c3<\/em>z<\/sub>\/qext<\/sub><\/em> is presented in Figure 3.10<\/a>. The use of this chart for calculating vertical stresses for geometries other than that illustrated in Figure 3.10<\/a> is discussed later in <\/a>Chapter 3.7<\/a>.<\/p>\n<\/a>
![Figure 3.10<\/a>. The use of this chart for calculating vertical stresses for geometries other than that illustrated in](\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/01\/3.10-values-of-influence-1.png\"){kind=link}