{"id":195,"date":"2025-01-28T23:55:02","date_gmt":"2025-01-28T23:55:02","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/3-7-discussion\/"},"modified":"2026-03-16T13:57:47","modified_gmt":"2026-03-16T13:57:47","slug":"3-7-discussion","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/3-7-discussion\/","title":{"raw":"3.7 Discussion","rendered":"3.7 Discussion"},"content":{"raw":"While applying the expressions presented in the above paragraphs, one should keep in mind that stress increments due to external loadings are total<\/em> stresses, and initially will be resisted by both the pore water pressure and the soil skeleton. As discussed in Part 4, settlement in soils is due to effective<\/em> stress changes only.\n\nApplication of these solutions can be extended to treat more complex loadings, under specific conditions: The Saint-Venant principle (1855) suggests that \u201cthe difference between the effects of two dissimilar but statically equivalent loadings becomes very small at sufficiently large distances from the loading\u201d. <\/em>Consider for example the different stress distributions presented in Figure 3.11<\/a>, and assume we are interested in estimating soil stresses at a depth equal to about the diameter of the loaded area 2r<\/em>0<\/sub> (Figure 3.11<\/a>). From that depth and below, stresses in soil depend on the magnitude of the resultant of the loading, and not on its distribution. In other words, in certain cases we can use elasticity theory expressions even if the actual distribution of the loading is not the same as the idealised uniform distribution, provided that the resultant is the same.\n\n[caption id=\"attachment_192\" align=\"aligncenter\" width=\"400\"]\"Diagram Figure 3.11. Vertical increment stress distribution below the axis of a circular area where different pressure distributions of equal resultant are applied.[\/caption]\n\nThe expressions provided in the previous paragraphs for the calculation of vertical stress increments with depth due to various loading types can be applied for the determination of the influence depth <\/em>of the loading, defined as the depth where additional vertical stress \u0394\u03c3<\/em>z<\/sub> is reduced to 10% of the mean stress applied on the soil surface (<\/em>\u0394\u03c3<\/em>z<\/sub>\/<\/em>qext<\/sub> = <\/em>0.1).<\/em>\n\nPlotting the distribution of the additional normalised (against the pressure applied on the surface qext<\/sub><\/em>) stress with depth below the axis of symmetry of a circular area, and of a strip pressure with equivalent width (Figure 3.12a<\/a>), we determine the influence depth to be:\n\n(3.16) <\/strong>[latex]{\\left( {\\dfrac{z}{{{r_0}}}} \\right)_{\\tfrac{{\\Delta {\\sigma _z}}}{{{q_{ext}}}} = 0.1}} = {\\rm{4}}[\/latex]\u00a0 \u00a0 \u00a0 for uniform pressure on a circular area of radius r<\/em>0<\/sub>\n\n(3.17) <\/strong>[latex]{\\left( {\\dfrac{z}{{{r_0}}}} \\right)_{\\tfrac{{\\Delta {\\sigma _z}}}{{{q_{ext}}}} = 0.1}} = {\\rm{13}}[\/latex]\u00a0 \u00a0 \u00a0 for strip pressure of width B<\/em> = 2r<\/em>0<\/sub>\n\nSimilarly, plotting in Figure 3.12b<\/a> the distribution of the additional normalised vertical stress with depth below the axis (r<\/em> = 0, Figure 3.5<\/a>) of a point load, provided from the expression:\n\n(3.18)<\/strong> [latex] \\Delta {\\sigma _z}\\left( {r = 0} \\right) = \\dfrac{3}{2}\\dfrac{{{Q_{ext}}}}{\\pi }\\dfrac{1}{{{z^2}}}[\/latex]\n\nand below the axis (x<\/em> = 0, Figure 3.6<\/a>) of a line load, provided from the expression:\n\n(3.19)<\/strong> [latex]\\Delta {\\sigma _z}\\left( {x = 0} \\right) = \\dfrac{{2{Q_{ext}}}}{\\pi }\\dfrac{1}{z}[\/latex]\n\nwe determine the influence depth to be 2 m for the point load, and 6.5 m for the line load.\n\nThe above findings may prove to be very useful for the preparation of a finite element model, for determining the extent of the geometry that must be simulated to accurately consider the effects of an external loading.\n\n[caption id=\"attachment_192\" align=\"aligncenter\" width=\"800\"]\"The Figure 3.12. Vertical stress increment distribution below the axis of (left) a circular and a strip pressure of equal width, and (right) a point load and a line load of equivalent magnitude.[\/caption]\n\nAnother advantage arising from the consideration of the soil as a linear elastic material is that the principle of superposition<\/em> applies. This suggests that stress increments due to multiple loadings acting simultaneously can be calculated as the sum of stress increments due to each one of the loads or pressures considered separately as (Figure 3.13<\/a>):\n\n(3.20)<\/strong>\u00a0[latex]\\Delta {\\sigma _z}\\left( {{Q_{ext}} + {q_{ext,1}} + {q_{ext,2}}} \\right) = \\Delta {\\sigma _z}\\left( {{Q_{ext}}} \\right) + \\Delta {\\sigma _z}\\left( {{q_{ext,1}}} \\right) + \\Delta {\\sigma _z}\\left( {{q_{ext,2}}} \\right)[\/latex]\n\n[caption id=\"attachment_192\" align=\"aligncenter\" width=\"350\"]\"Decomposition Figure 3.13. Estimation of total stress increments due to multiple loads and pressures acting simultaneously, by applying the principle of superposition.[\/caption]\n\nStresses resulting from other, more complex loading distributions can be also calculated, if \"negative\", tensile loadings are considered together with compressive loadings (Figure 3.14<\/a>). This concept is demonstrated in Example 3.1<\/a>.\n\n(3.21)<\/strong> [latex]\\Delta {\\sigma _z}\\left( {{q_{ext}}} \\right) = \\Delta {\\sigma _z}\\left( {{q_{ext,1}}} \\right) - \\Delta {\\sigma _z}\\left( {{q_{ext,2}}} \\right)[\/latex]\n\n[caption id=\"attachment_192\" align=\"aligncenter\" width=\"350\"]\"Illustration Figure 3.14. Estimation of stress increments due to complex pressure distributions, by taking advantage of the principle of superposition.[\/caption]\n\nFigure 3.10<\/a> also can be used in tandem with the principle of superposition to obtain vertical stress increments underneath complex pressure distributions. For example, Figure 3.15a<\/a> demonstrates how we can calculate vertical stress increments underneath point A, located midway from two rectangular areas of dimensions H<\/em> \u00d7 B<\/em> separated by distance Y<\/em> on which pressure qext<\/sub><\/em> is applied e.g., underneath the tracks of a piling rig.<\/em> The vertical stress increment at any depth at point A due to the two rectangular pressures shown in Figure 3.15a<\/a> is equal to four (4) times the vertical stress increment at the edge of a rectangular pressure qext<\/sub><\/em> of dimensions (Y<\/em>\/2+H<\/em>) \u00d7 B<\/em>\/2 minus<\/em> four (4) times the vertical stress increment at the edge of a rectangular pressure qext<\/sub><\/em> of dimensions (Y<\/em>\/2+H\/<\/em>2) \u00d7 B<\/em>\/2. Vertical stress increments at the edge of rectangular pressures are calculated directly from Figure 3.10<\/a>.\n\nSimilarly, Figure 3.15b<\/a> demonstrates how we can calculate vertical stress increments underneath point B, located at the centre of a rectangular area of dimensions H<\/em> \u00d7 B <\/em>on which pressure qext<\/sub><\/em> is applied e.g., underneath the centre of a rectangular footing. The vertical stress increment at any depth at the centre B of a rectangular pressure qext<\/sub><\/em> of dimensions H<\/em> \u00d7 B<\/em> is equal to four (4) times the vertical stress increment at the edge of a rectangular pressure qext<\/sub><\/em> of dimensions Y<\/em>\/2 \u00d7 B<\/em>\/2, calculated according to Figure 3.10<\/a>.\n\n[caption id=\"attachment_192\" align=\"aligncenter\" width=\"500\"]\"Diagram<\/a> Figure 3.15a. Estimation of stress increments due to complex pressure distributions, by taking advantage of the principle of superposition and using Fadum\u2019s chart (Figure 3.10).[\/caption]\n\n[caption id=\"attachment_192\" align=\"aligncenter\" width=\"400\"]\"Diagram<\/a> Figure 3.15b. Estimation of stress increments due to complex pressure distributions, by taking advantage of the principle of superposition and using Fadum\u2019s chart (Figure 3.10).[\/caption]\n\nTo end with, it should be noted that all the solutions presented above were derived for \u201cflexible\u201d loads and pressures, and do not account for the rigidity of the foundation. If the foundation is rigid, stress increments for the same loading on the surface would be generally lower by 15% to 30%. However, usually we do not correct stress distribution for the rigidity of the footing since, as soil non-linearity is not taken into account, we prefer the \u201cerror\u201d to be on the conservative side.","rendered":"

While applying the expressions presented in the above paragraphs, one should keep in mind that stress increments due to external loadings are total<\/em> stresses, and initially will be resisted by both the pore water pressure and the soil skeleton. As discussed in Part 4, settlement in soils is due to effective<\/em> stress changes only.<\/p>\n

Application of these solutions can be extended to treat more complex loadings, under specific conditions: The Saint-Venant principle (1855) suggests that \u201cthe difference between the effects of two dissimilar but statically equivalent loadings becomes very small at sufficiently large distances from the loading\u201d. <\/em>Consider for example the different stress distributions presented in Figure 3.11<\/a>, and assume we are interested in estimating soil stresses at a depth equal to about the diameter of the loaded area 2r<\/em>0<\/sub> (Figure 3.11<\/a>). From that depth and below, stresses in soil depend on the magnitude of the resultant of the loading, and not on its distribution. In other words, in certain cases we can use elasticity theory expressions even if the actual distribution of the loading is not the same as the idealised uniform distribution, provided that the resultant is the same.<\/p>\n

\"Diagram
Figure 3.11. Vertical increment stress distribution below the axis of a circular area where different pressure distributions of equal resultant are applied.<\/figcaption><\/figure>\n

The expressions provided in the previous paragraphs for the calculation of vertical stress increments with depth due to various loading types can be applied for the determination of the influence depth <\/em>of the loading, defined as the depth where additional vertical stress \u0394\u03c3<\/em>z<\/sub> is reduced to 10% of the mean stress applied on the soil surface (<\/em>\u0394\u03c3<\/em>z<\/sub>\/<\/em>qext<\/sub> = <\/em>0.1).<\/em><\/p>\n

Plotting the distribution of the additional normalised (against the pressure applied on the surface qext<\/sub><\/em>) stress with depth below the axis of symmetry of a circular area, and of a strip pressure with equivalent width (Figure 3.12a<\/a>), we determine the influence depth to be:<\/p>\n

(3.16) <\/strong>[latex]{\\left( {\\dfrac{z}{{{r_0}}}} \\right)_{\\tfrac{{\\Delta {\\sigma _z}}}{{{q_{ext}}}} = 0.1}} = {\\rm{4}}[\/latex]\u00a0 \u00a0 \u00a0 for uniform pressure on a circular area of radius r<\/em>0<\/sub><\/p>\n

(3.17) <\/strong>[latex]{\\left( {\\dfrac{z}{{{r_0}}}} \\right)_{\\tfrac{{\\Delta {\\sigma _z}}}{{{q_{ext}}}} = 0.1}} = {\\rm{13}}[\/latex]\u00a0 \u00a0 \u00a0 for strip pressure of width B<\/em> = 2r<\/em>0<\/sub><\/p>\n

Similarly, plotting in Figure 3.12b<\/a> the distribution of the additional normalised vertical stress with depth below the axis (r<\/em> = 0, Figure 3.5<\/a>) of a point load, provided from the expression:<\/p>\n

(3.18)<\/strong> [latex]\\Delta {\\sigma _z}\\left( {r = 0} \\right) = \\dfrac{3}{2}\\dfrac{{{Q_{ext}}}}{\\pi }\\dfrac{1}{{{z^2}}}[\/latex]<\/p>\n

and below the axis (x<\/em> = 0, Figure 3.6<\/a>) of a line load, provided from the expression:<\/p>\n

(3.19)<\/strong> [latex]\\Delta {\\sigma _z}\\left( {x = 0} \\right) = \\dfrac{{2{Q_{ext}}}}{\\pi }\\dfrac{1}{z}[\/latex]<\/p>\n

we determine the influence depth to be 2 m for the point load, and 6.5 m for the line load.<\/p>\n

The above findings may prove to be very useful for the preparation of a finite element model, for determining the extent of the geometry that must be simulated to accurately consider the effects of an external loading.<\/p>\n

\"The
Figure 3.12. Vertical stress increment distribution below the axis of (left) a circular and a strip pressure of equal width, and (right) a point load and a line load of equivalent magnitude.<\/figcaption><\/figure>\n

Another advantage arising from the consideration of the soil as a linear elastic material is that the principle of superposition<\/em> applies. This suggests that stress increments due to multiple loadings acting simultaneously can be calculated as the sum of stress increments due to each one of the loads or pressures considered separately as (Figure 3.13<\/a>):<\/p>\n

(3.20)<\/strong>\u00a0[latex]\\Delta {\\sigma _z}\\left( {{Q_{ext}} + {q_{ext,1}} + {q_{ext,2}}} \\right) = \\Delta {\\sigma _z}\\left( {{Q_{ext}}} \\right) + \\Delta {\\sigma _z}\\left( {{q_{ext,1}}} \\right) + \\Delta {\\sigma _z}\\left( {{q_{ext,2}}} \\right)[\/latex]<\/p>\n

\"Decomposition
Figure 3.13. Estimation of total stress increments due to multiple loads and pressures acting simultaneously, by applying the principle of superposition.<\/figcaption><\/figure>\n

Stresses resulting from other, more complex loading distributions can be also calculated, if “negative”, tensile loadings are considered together with compressive loadings (Figure 3.14<\/a>). This concept is demonstrated in Example 3.1<\/a>.<\/p>\n

(3.21)<\/strong> [latex]\\Delta {\\sigma _z}\\left( {{q_{ext}}} \\right) = \\Delta {\\sigma _z}\\left( {{q_{ext,1}}} \\right) - \\Delta {\\sigma _z}\\left( {{q_{ext,2}}} \\right)[\/latex]<\/p>\n

\"Illustration
Figure 3.14. Estimation of stress increments due to complex pressure distributions, by taking advantage of the principle of superposition.<\/figcaption><\/figure>\n

Figure 3.10<\/a> also can be used in tandem with the principle of superposition to obtain vertical stress increments underneath complex pressure distributions. For example, Figure 3.15a<\/a> demonstrates how we can calculate vertical stress increments underneath point A, located midway from two rectangular areas of dimensions H<\/em> \u00d7 B<\/em> separated by distance Y<\/em> on which pressure qext<\/sub><\/em> is applied e.g., underneath the tracks of a piling rig.<\/em> The vertical stress increment at any depth at point A due to the two rectangular pressures shown in Figure 3.15a<\/a> is equal to four (4) times the vertical stress increment at the edge of a rectangular pressure qext<\/sub><\/em> of dimensions (Y<\/em>\/2+H<\/em>) \u00d7 B<\/em>\/2 minus<\/em> four (4) times the vertical stress increment at the edge of a rectangular pressure qext<\/sub><\/em> of dimensions (Y<\/em>\/2+H\/<\/em>2) \u00d7 B<\/em>\/2. Vertical stress increments at the edge of rectangular pressures are calculated directly from Figure 3.10<\/a>.<\/p>\n

Similarly, Figure 3.15b<\/a> demonstrates how we can calculate vertical stress increments underneath point B, located at the centre of a rectangular area of dimensions H<\/em> \u00d7 B <\/em>on which pressure qext<\/sub><\/em> is applied e.g., underneath the centre of a rectangular footing. The vertical stress increment at any depth at the centre B of a rectangular pressure qext<\/sub><\/em> of dimensions H<\/em> \u00d7 B<\/em> is equal to four (4) times the vertical stress increment at the edge of a rectangular pressure qext<\/sub><\/em> of dimensions Y<\/em>\/2 \u00d7 B<\/em>\/2, calculated according to Figure 3.10<\/a>.<\/p>\n

\"Diagram<\/a>
Figure 3.15a. Estimation of stress increments due to complex pressure distributions, by taking advantage of the principle of superposition and using Fadum\u2019s chart (Figure 3.10).<\/figcaption><\/figure>\n
\"Diagram<\/a>
Figure 3.15b. Estimation of stress increments due to complex pressure distributions, by taking advantage of the principle of superposition and using Fadum\u2019s chart (Figure 3.10).<\/figcaption><\/figure>\n

To end with, it should be noted that all the solutions presented above were derived for \u201cflexible\u201d loads and pressures, and do not account for the rigidity of the foundation. If the foundation is rigid, stress increments for the same loading on the surface would be generally lower by 15% to 30%. However, usually we do not correct stress distribution for the rigidity of the footing since, as soil non-linearity is not taken into account, we prefer the \u201cerror\u201d to be on the conservative side.<\/p>\n","protected":false},"author":1,"menu_order":7,"template":"","meta":{"pb_show_title":"","pb_short_title":"3.7 Discussion","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-195","chapter","type-chapter","status-publish","hentry"],"part":165,"_links":{"self":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/195","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":1,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/195\/revisions"}],"predecessor-version":[{"id":196,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/195\/revisions\/196"}],"part":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/parts\/165"}],"metadata":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/195\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/media?parent=195"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapter-type?post=195"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/contributor?post=195"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/license?post=195"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}