{"id":403,"date":"2025-02-26T02:56:31","date_gmt":"2025-02-26T02:56:31","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/5-6-numerical-modelling-of-footings-resting-on-the-surface-of-drained-soil\/"},"modified":"2026-03-16T14:05:07","modified_gmt":"2026-03-16T14:05:07","slug":"5-6-numerical-modelling-of-footings-resting-on-the-surface-of-drained-soil","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/5-6-numerical-modelling-of-footings-resting-on-the-surface-of-drained-soil\/","title":{"raw":"5.6 Numerical modelling of footings resting on the surface of drained soil","rendered":"5.6 Numerical modelling of footings resting on the surface of drained soil"},"content":{"raw":"The numerical analysis results presented in <a href=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/example-5-5\/\">Example 5.5<\/a> demonstrate that the use of equivalent strength parameters together with an associative flow rule provides reasonable estimates of the bearing capacity of foundations, while avoiding numerical instabilities that stem from using as input the actual friction angle and dilation angle of drained materials, and consequently of non-associative flow rule.\n\nHowever, this is not the only potential source of numerical instabilities in simulations where we seek to estimate the bearing capacity of shallow foundations. Unlike undrained soils modelled with the Mohr-Coulomb <em>Undrained<\/em> material model, where shear strength does not depend on stress levels, drained soils modelled with the Mohr-Coulomb <em>Drained<\/em> material model will feature zero shear strength near the model surface if i) Their effective cohesion is set to be zero (see <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.4-mohr-coulomb-circle-at-failure.png\">Figure 5.4<\/a>), and ii) The footing is not embedded as it is in <a href=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/example-5-5\/\">Example 5.5<\/a>. Finite element codes have difficulty handling materials of zero strength, and the simulation may end prematurely (before the bearing capacity is reached) due to poor convergence. While using as input a nominal effective cohesion <em>c\u2032 \u2248<\/em>\u00a01 kPa to avoid modelling a material of zero strength near the surface where geostatic stresses are zero will certainly improve numerical stability, this is not always sufficient.\n\nAn alternative modelling strategy is illustrated in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.49-HR.png\">Figure 5.49<\/a>: Instead of applying a prescribed displacement directly on the model surface (Model A) to obtain the force-displacement curve, as we did e.g., in <a href=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/example-5-2-estimation-of-the-short-term-undrained-collapse-load-with-numerical-methods\/\">Example 5.2<\/a>, we apply the prescribed displacement on the top of a geometry cluster representing the footing (Model B). A weightless linear elastic material of very high stiffness is assigned to that cluster, to model a rigid footing. On the model surface we introduce a weightless thin crust, as another separate geometry cluster. To this cluster we assign a material of arbitrarily high cohesion (\u201cinfinite\u201d strength) but also of very low stiffness, so that the stresses that will develop in this \u201cdummy\u201d crust will be extremely low, and will not contribute to a fictitious increase in the predicted bearing capacity. Notice that we still use Davis equivalent parameters <em>\u03c6<\/em><em>*<\/em>, <em>\u03c8<\/em><em>*<\/em> for the actual soil, but we have considered its effective cohesion <em>c\u2032<\/em>\u00a0to be zero.\n\nResults of the two model variants are presented in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.49-HR.png\">Figure 5.49<\/a>, in terms of the predicted force-displacement curve. Observe that Model A failed to reach the collapse load, and the analysis ended prematurely with an error message. That is despite the fact that arc-length control was used to solve the system of non-linear equations, which can provide solution to problems where softening is expected, i.e., when the stiffness matrix is not always a positive definite through the analysis. On the other hand, Model B\u2019s analysis was completed successfully, and the collapse load can be inferred from the force-displacement curve. Notice also that, up to the analysis stage where Model A\u2019s analysis failed, both models produced identical results. This suggests that the introduced technique facilitates convergence, at no expense of model accuracy.\n\n[caption id=\"attachment_402\" align=\"aligncenter\" width=\"745\"]<img class=\"wp-image-402 size-large\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2025\/03\/5.49-HR-745x1024.png\" alt=\"The figure on the top presents the analysis model A, where the footing is modelled as prescribed displacement acting on the surface of the model. The width of the prescribed displacement is 0.5 m. The properties of the soil are; M-C (Drained) material model, \u03b3 = 9.75 kN\/m^3, E' = 30 MPa, v' = 0.2, \u03c6* = \u03c8* = 32 deg, c' = 0. Two force-displacement curves are presented in an inset figure, the force-displacement curve from Model A does not reach an asymptote and it is noted that the analysis terminated due to error. The force-displacement curve from another Model B does reach an asymptote, and is identical to the curve from Model A up to the point where the later stops. That model B is shown in the figure below. The footing is modelled as a separate geometry cluster, with a prescribed displacement on its top. The properties of the footing are linear elastic material model, \u03b3 = 0, \u0395' = 30000 GPa, v' =0. A dummy crust material is placed on the surface of model B, and that geometry cluster has thickness 0.1m. The properties of the crust material are: M-C (Drained) material model, \u03b3 = 0 kN\/m^3, E' = 0.1 kPa, v' = 0.2, \u03c6* = \u03c8* = 0 deg, c' = 1 MPa. \" width=\"745\" height=\"1024\"> Figure 5.49. Modelling of a rigid and rough strip footing resting on the surface of drained soil. Model A: Applying a prescribed displacement directly on the model\u2019s surface; Model B: Applying a prescribed displacement on top of a \u201crigid\u201d weightless cluster representing the footing, and introducing a \u201cdummy\u201d thin crust featuring very high cohesion and very low stiffness.[\/caption]","rendered":"<p>The numerical analysis results presented in <a href=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/example-5-5\/\">Example 5.5<\/a> demonstrate that the use of equivalent strength parameters together with an associative flow rule provides reasonable estimates of the bearing capacity of foundations, while avoiding numerical instabilities that stem from using as input the actual friction angle and dilation angle of drained materials, and consequently of non-associative flow rule.<\/p>\n<p>However, this is not the only potential source of numerical instabilities in simulations where we seek to estimate the bearing capacity of shallow foundations. Unlike undrained soils modelled with the Mohr-Coulomb <em>Undrained<\/em> material model, where shear strength does not depend on stress levels, drained soils modelled with the Mohr-Coulomb <em>Drained<\/em> material model will feature zero shear strength near the model surface if i) Their effective cohesion is set to be zero (see <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.4-mohr-coulomb-circle-at-failure.png\">Figure 5.4<\/a>), and ii) The footing is not embedded as it is in <a href=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/example-5-5\/\">Example 5.5<\/a>. Finite element codes have difficulty handling materials of zero strength, and the simulation may end prematurely (before the bearing capacity is reached) due to poor convergence. While using as input a nominal effective cohesion <em>c\u2032 \u2248<\/em>\u00a01 kPa to avoid modelling a material of zero strength near the surface where geostatic stresses are zero will certainly improve numerical stability, this is not always sufficient.<\/p>\n<p>An alternative modelling strategy is illustrated in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.49-HR.png\">Figure 5.49<\/a>: Instead of applying a prescribed displacement directly on the model surface (Model A) to obtain the force-displacement curve, as we did e.g., in <a href=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/example-5-2-estimation-of-the-short-term-undrained-collapse-load-with-numerical-methods\/\">Example 5.2<\/a>, we apply the prescribed displacement on the top of a geometry cluster representing the footing (Model B). A weightless linear elastic material of very high stiffness is assigned to that cluster, to model a rigid footing. On the model surface we introduce a weightless thin crust, as another separate geometry cluster. To this cluster we assign a material of arbitrarily high cohesion (\u201cinfinite\u201d strength) but also of very low stiffness, so that the stresses that will develop in this \u201cdummy\u201d crust will be extremely low, and will not contribute to a fictitious increase in the predicted bearing capacity. Notice that we still use Davis equivalent parameters <em>\u03c6<\/em><em>*<\/em>, <em>\u03c8<\/em><em>*<\/em> for the actual soil, but we have considered its effective cohesion <em>c\u2032<\/em>\u00a0to be zero.<\/p>\n<p>Results of the two model variants are presented in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.49-HR.png\">Figure 5.49<\/a>, in terms of the predicted force-displacement curve. Observe that Model A failed to reach the collapse load, and the analysis ended prematurely with an error message. That is despite the fact that arc-length control was used to solve the system of non-linear equations, which can provide solution to problems where softening is expected, i.e., when the stiffness matrix is not always a positive definite through the analysis. On the other hand, Model B\u2019s analysis was completed successfully, and the collapse load can be inferred from the force-displacement curve. Notice also that, up to the analysis stage where Model A\u2019s analysis failed, both models produced identical results. This suggests that the introduced technique facilitates convergence, at no expense of model accuracy.<\/p>\n<figure id=\"attachment_402\" aria-describedby=\"caption-attachment-402\" style=\"width: 745px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" class=\"wp-image-402 size-large\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2025\/03\/5.49-HR-745x1024.png\" alt=\"The figure on the top presents the analysis model A, where the footing is modelled as prescribed displacement acting on the surface of the model. The width of the prescribed displacement is 0.5 m. The properties of the soil are; M-C (Drained) material model, \u03b3 = 9.75 kN\/m^3, E' = 30 MPa, v' = 0.2, \u03c6* = \u03c8* = 32 deg, c' = 0. Two force-displacement curves are presented in an inset figure, the force-displacement curve from Model A does not reach an asymptote and it is noted that the analysis terminated due to error. The force-displacement curve from another Model B does reach an asymptote, and is identical to the curve from Model A up to the point where the later stops. That model B is shown in the figure below. The footing is modelled as a separate geometry cluster, with a prescribed displacement on its top. The properties of the footing are linear elastic material model, \u03b3 = 0, \u0395' = 30000 GPa, v' =0. A dummy crust material is placed on the surface of model B, and that geometry cluster has thickness 0.1m. The properties of the crust material are: M-C (Drained) material model, \u03b3 = 0 kN\/m^3, E' = 0.1 kPa, v' = 0.2, \u03c6* = \u03c8* = 0 deg, c' = 1 MPa.\" width=\"745\" height=\"1024\" srcset=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/03\/5.49-HR-745x1024.png 745w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/03\/5.49-HR-218x300.png 218w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/03\/5.49-HR-768x1056.png 768w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/03\/5.49-HR-65x89.png 65w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/03\/5.49-HR-225x309.png 225w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/03\/5.49-HR-350x481.png 350w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/03\/5.49-HR.png 886w\" sizes=\"(max-width: 745px) 100vw, 745px\" \/><figcaption id=\"caption-attachment-402\" class=\"wp-caption-text\">Figure 5.49. Modelling of a rigid and rough strip footing resting on the surface of drained soil. Model A: Applying a prescribed displacement directly on the model\u2019s surface; Model B: Applying a prescribed displacement on top of a \u201crigid\u201d weightless cluster representing the footing, and introducing a \u201cdummy\u201d thin crust featuring very high cohesion and very low stiffness.<\/figcaption><\/figure>\n","protected":false},"author":1,"menu_order":11,"template":"","meta":{"pb_show_title":"","pb_short_title":"5.6 Numerical modelling of footings resting on the surface of drained soil","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-403","chapter","type-chapter","status-publish","hentry"],"part":325,"_links":{"self":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/403","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\/403\/revisions"}],"predecessor-version":[{"id":404,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/403\/revisions\/404"}],"part":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/parts\/325"}],"metadata":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/403\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/media?parent=403"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapter-type?post=403"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/contributor?post=403"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/license?post=403"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}