{"id":386,"date":"2025-02-25T04:15:25","date_gmt":"2025-02-25T04:15:25","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/5-5-bearing-capacity-of-footings-on-layered-soils\/"},"modified":"2026-03-16T14:04:30","modified_gmt":"2026-03-16T14:04:30","slug":"5-5-bearing-capacity-of-footings-on-layered-soils","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/5-5-bearing-capacity-of-footings-on-layered-soils\/","title":{"raw":"5.5 Bearing capacity of footings on layered soils","rendered":"5.5 Bearing capacity of footings on layered soils"},"content":{"raw":"<h2>5.5.1 Critical thickness of the surficial soil layer<\/h2>\nOne of the key assumptions of the common analytical solutions presented in <a href=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/5-4-common-bearing-capacity-equations-and-practical-considerations\/\">Chapter 5.4<\/a> is that of uniform soil conditions, which is rarely the case in practice. When the geotechnical investigation reveals a non-uniform soil profile, we have to estimate whether the thickness of the top soil layer, measured below the embedment depth, is enough so that the failure surface will develop entirely inside it (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.36-definition-of-critical-thickness.png\">Figure 5.36<\/a>).\n\n[caption id=\"attachment_385\" align=\"aligncenter\" width=\"1024\"]<img class=\"wp-image-379 size-large\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2025\/03\/5.36-definition-of-critical-thickness-1024x330.png\" alt=\"The figure on the left presents a footing of width B embedded in soil at depth Df. The top soil layer has thickness H_cr, measured from the elevation of the footing's base. Underneath the top soil layer there is a different bottom soil layer. A generalised failure mechanism develops underneath the footing inside the top layer, and the mechanism extends up to a depth H_cr, measured from the elevation of the footing's base. The figure on the right presents the variation of the parameter H_cr\/B as function of the friction angle of the top soil layer. \" width=\"1024\" height=\"330\"> Figure 5.36. Definition of the critical thickness, <em>H<sub>cr<\/sub><\/em> for failure to develop within the surficial soil layer.[\/caption]\n\nIt is reasonable to assume that if the thickness of the soil layer below the foundation, <em>H<sub>cr<\/sub><\/em> is less than the depth that the Prandtl mechanism penetrates into the soil (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.26-hr.png\">Figure 5.26<\/a>) the failure surface will be entirely contained in the top soil layer, and the subsoil can be considered as homogeneous for bearing capacity calculations.\n\n<strong>(5.35)<\/strong> [latex]{H_{cr}} = \\dfrac{B}{{2\\cos \\left( {{{45}^o} + \\dfrac{{\\varphi '}}{2}} \\right)}}{e^{\\left( {A\\tan \\varphi '} \\right)}}[\/latex]\n\nWhere <em>A<\/em> = (45\u00b0+<em>\u03c6\u2032<\/em>\/2) in rad.\n\nAlternatively, the critical thickness <em>H<sub>cr<\/sub><\/em> can be calculated as (Budhu 2011):\n\n<strong>(5.36)<\/strong> [latex]{H_{cr}} = \\dfrac{{3B\\ln \\left( {\\dfrac{{{q_{top}}}}{{{q_{bottom}}}}} \\right)}}{{2\\left( {1 + \\dfrac{B}{L}} \\right)}}[\/latex]\n\nwhere:\n<ul>\n \t<li><em>q<sub>top <\/sub><\/em>is the bearing capacity of the footing with the particular dimensions, resting on the surface of an infinitely deep layer with the properties of the <em>top<\/em> soil layer, and<\/li>\n \t<li><em>q<sub>bottom<\/sub><\/em> is the bearing capacity of the footing with the particular dimensions, resting on the surface of an infinitely deep layer with the properties of the <em>bottom<\/em> soil layer.<\/li>\n<\/ul>\nIf the thickness of the top soil layer measured below the embedment depth, <em>H<sub>top <\/sub><\/em>is less than the critical thickness, <em>H<sub>cr<\/sub>, <\/em>there is no general analytical method to estimate the bearing capacity, except of course numerical methods. Some characteristic cases are examined in the following paragraphs.\n\n<hr>\n\n<h2>5.5.2 Footing on a soft clay layer overlying a stiff soil formation<\/h2>\nIn the case where a thin soft clay formation is found near the ground surface, good engineering practice suggests excavating the soft clay layer, and replacing it with well-compacted coarse-grained fill material. Founding structures directly on soft clays with shallow foundation should be avoided, except light structures such as one-storey buildings.\n\nExperimental results have shown that in the case of a footing resting on soft clay overlying a stiffer stratum, the mechanism of bearing capacity consists of lateral squeezing of the soft soil, as the footing \u201csinks\u201d into the top layer. Tomlinson and Boorman (1995) proposed the following expressions to estimate bearing capacity in that case:\n\n[caption id=\"attachment_385\" align=\"aligncenter\" width=\"500\"]<img class=\"wp-image-380 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/5.37-HR-e1743552136286.png\" alt=\"Schematic of a footing embedded in a soft clay layer. The width of the footing is B. The footing is loaded with a pressure qf. The thickness of the soft clay layer is z, measured from the footing's base. Underneath the soft clay layer there is a stiff soil layer.\" width=\"500\" height=\"181\"> Figure 5.37. Footing on a soft clay layer overlying a stiff soil formation.[\/caption]\n\nFor circular\/square footings:\n\n<strong>(5.37)<\/strong> [latex]{q_f} = \\left( {\\dfrac{B}{{2z}} + \\pi + 1} \\right){S_u}{\\rm{ \\:for \\:}}\\dfrac{B}{z} \\ge 2[\/latex]\n\nFor strip footings:\n\n<strong>(5.38)<\/strong> [latex]{q_f} = \\left( {\\dfrac{B}{{3z}} + \\pi + 1} \\right){S_u}{\\rm{ \\:for\\: }}\\dfrac{B}{z} \\ge 6[\/latex]\n\nwhere <em>z<\/em> is defined in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.37-HR.png\">Figure 5.37<\/a> above. When <em>B\/z <\/em>&lt; 2 for circular\/strip footings or <em>B\/z <\/em>&lt; 6 for strip footings (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.37-HR.png\">Figure 5.37<\/a>), the expressions for homogeneous soil should be rather used.\n\n<hr>\n\n<h2>5.5.3 Footing on stiff soil overlying a soft clay layer<\/h2>\nThis case could well correspond to the common problem of a footing resting on a foundation improvement layer made of well-compacted coarse-grained fill, overlying a soft subgrade. A <em>\u201cpunching\u201d <\/em>failure mechanism may develop under these circumstances (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.38a-HR.png\">Figure 5.38a<\/a>).\n\nA simplified method for estimating the bearing capacity of a footing of any shape resting on the surface of, or embedded in a stiff soil formation overlying a soft clay layer consists of the following steps: First, we estimate the bearing capacity of the footing while considering uniform soil with the properties of the top stiff soil layer, which must not be critical for the design.\n\nAccordingly, we estimate using Eq. 5.24 the net bearing capacity <em>q<sub>eq<\/sub><\/em> of a hypothetical equivalent footing with the same shape and dimensions, embedded at a depth <em>D<sub>f<\/sub>,<sub>eq <\/sub><\/em>= <em>D<sub>f<\/sub>+z<\/em> equal to the thickness of the top stiff soil layer (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.38a-HR.png\">Figure 5.38<\/a>). Bearing capacity failure of the actual footing will be reached when the vertical stress \u0394<em>\u03c3<\/em><sub>z<\/sub> transferred from the actual footing to the interface of the stiff soil-soft clay layer (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.38b-HR.png\">Figure 5.38b<\/a>) becomes equal to <em>q<sub>eq<\/sub><\/em> i.e., \u0394<em>\u03c3<\/em><sub>z<\/sub> = <em>q<sub>eq<\/sub><\/em>. The following approximate expressions can be used to calculate \u0394<em>\u03c3<\/em><sub>z<\/sub>, while assuming 2:1 stress distribution (see <a href=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/3-6-stresses-in-the-soil-due-to-a-rectangular-pressure\/\">Chapter 3.6<\/a>):\n\nFor a rectangular footing:\n\n<strong>(5.39<\/strong><strong>)<\/strong> [latex]\\Delta {\\sigma _z} = q\\left[ {\\dfrac{{BL}}{{\\left( {B + z} \\right)\\left( {L + z} \\right)}}} \\right][\/latex]\n\nFor a square\/circular footing:\n\n<strong>(5.40)<\/strong> [latex]\\Delta {\\sigma _z} = q{\\left[ {\\dfrac{B}{{\\left( {B + z} \\right)}}} \\right]^2}[\/latex]\n\nFor a strip footing:\n\n<strong>(5.41)\u00a0<\/strong>[latex]\\Delta {\\sigma _z} = q\\left[ {\\dfrac{B}{{\\left( {B + z} \\right)}}} \\right][\/latex]\n\nin the above expressions <em>q<\/em> is the net bearing capacity of the actual footing. Therefore substituting \u0394<em>\u03c3<\/em><sub>z <\/sub>= <em>q<sub>eq<\/sub><\/em> in the Eqs. 5.39 to 5.41 that corresponds to the actual footing shape will provide its net bearing capacity <em>q<\/em>. If the actual footing is embedded in soil, its bearing capacity will be <em>q<sub>f<\/sub><\/em> = <em>q<\/em> + <em>\u03b3<\/em><em>D<sub>f<\/sub><\/em>.\n\n[caption id=\"attachment_385\" align=\"aligncenter\" width=\"600\"]<img class=\"wp-image-381 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/5.38a-HR-e1743552156863.png\" alt=\"Schematic of a footing embedded in stiff soil, at depth Df, underlaid by soft clay. The width of the footing is B. The footing is loaded with a pressure qf. The thickness of the stiff soil layer is z, measured from the footing's base. Vertical stresses from the footing are distributed with a 2:1 distribution, and a composite failure mechanism develops consisting of two shear bands in the stiff soil and a generalised failure surface in the soft clay.\" width=\"600\" height=\"260\"> Figure 5.38a. Punching of a footing into the underlying soft clay layer.[\/caption]\n\n&nbsp;\n\n[caption id=\"attachment_385\" align=\"aligncenter\" width=\"600\"]<img class=\"wp-image-382 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/5.38b-HR-e1743552172643.png\" alt=\"Schematic of an equivalent footing embedded in stiff soil, at depth Df,eq=Df+z, underlaid by soft clay. The width of the footing is B. The footing is loaded with a pressure qeq. The thickness of the stiff soil layer is Df+z, measured from the surface. A generalised failure mechanism develops in the soft clay.\" width=\"600\" height=\"206\"> Figure 5.38b. Concept for estimating the bearing capacity accounting for the development of the failure surface inside the soft clay layer.[\/caption]\n\nNote that the methodology described above conservatively ignores the shear resistance of the top layer, and the energy dissipated along the shear planes located with it. Another simplifying assumption is that the inclination of the shear planes relative to the vertical in the top layer is ignored i.e., the width of the equivalent, hypothetical footing is taken equal to the width of the real one. There are more refined methods in the literature that do not require introducing these assumptions, however their range of application is limited to specific footing geometries and soil types. Such a method has been developed by Salimi Eshkevari <em>et al.<\/em> (2019a) for estimating the bearing capacity of strip footings resting on the surface of a sand layer overlying a soft clay, which is relevant to the design of working platforms for tracked plants. Salimi Eshkevari <em>et al.<\/em> analysed the results of a series of Finite Element Limit Analyses and concluded to the following expression that provides the bearing capacity of a strip footing on a layered profile of sand over clay <em>q<sub>f<\/sub>,<sub>l<\/sub><\/em>:\n\n<strong>(5.42)<\/strong> [latex]{q_{f,l}}B = \\gamma {H^2}{K_{sr}}\\tan \\varphi ' + {N_{cu}}{S_u}\\left[ {B + 2H\\tan \\theta } \\right] + \\gamma {H^2}\\tan \\theta \\le qB[\/latex]\n\nwhere <em>B<\/em> is the footing width, <em>H<\/em> is the thickness of the sand layer, <em>\u03b3<\/em> is the unit weight of sand, <em>\u03c6\u2032<\/em>\u00a0is the friction angle of sand, <em>S<sub>u<\/sub><\/em> is the undrained shear strength of the clay layer, <em>N<sub>cu <\/sub><\/em>= 5.14 is the bearing capacity factor for strip footings on undrained soil, <em>q<\/em> is the bearing capacity of a footing of width <em>B<\/em> resting on the surface of uniform sand, calculated according to Eq. 5.25. The angle <em>\u03b8<\/em> that provides the inclination of the shear planes that will develop in the sand layer can be calculated as:\n\n<strong>(5.43)<\/strong> [latex]\\theta ({\\rm{rad}}) = \\alpha \\ln \\left( {\\dfrac{{{S_u}}}{{\\gamma H}}} \\right) + \\beta[\/latex]\n\n<strong>(5.44)\u00a0<\/strong>[latex]\\alpha = 0.039\\ln \\left( {\\tan \\varphi '} \\right) - 0.164[\/latex]\n\n<strong>(5.45)\u00a0<\/strong>[latex]\\beta = 0.597\\ln \\left( {\\tan \\varphi '} \\right) - 0.051[\/latex]\n\nNote that angle <em>\u03b8<\/em> can be positive or negative. Thus, unlike the approximate method described above, the width <em>B<\/em>+2<em>H<\/em>tan<em>\u03b8<\/em> of the equivalent footing in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.38a-HR.png\">Figure 5.38<\/a> can be greater, or less than the width of the actual footing <em>B<\/em>. Finally, the punching shear coefficient <em>K<sub>sr<\/sub><\/em>, that is correlated with the normal stress acting on the shear planes that will develop in the sand layer, is calculated as:\n\n<strong>(5.46)\u00a0<\/strong>[latex]{K_{sr}} = \\delta \\left( {\\dfrac{{{S_u}}}{{\\gamma H}}} \\right) + 2[\/latex]\n\n<strong>(5.47)\u00a0<\/strong>[latex]\\delta = - 3.45\\left( {\\tan \\varphi '} \\right) + 8.693[\/latex]\n\n<hr>\n\n<h2>5.5.4 Strip footings on layered sands<\/h2>\nEstimation of the bearing capacity of strip footings resting on the surface of a relatively thin layer of dense sand overlaying a weak layer of loose sand underpins the design of working platforms for heavily loaded tracked piling rigs and cranes, on sites which loose sand deposits are found at the ground surface. Hanna (1981) was among the first to study this problem, and concluded to the following expression for estimating the bearing capacity in terms of stress when a punching failure mechanism develops:\n\n<strong>(5.48)\u00a0<\/strong>[latex]{q_{f,l}} = {q_b} + \\gamma {H^2}\\dfrac{{{K_s}\\tan {{\\varphi '}_1}}}{B} - {\\gamma _1}H \\le q[\/latex]\n\nwhere <em>B<\/em> is the footing width, <em>H<\/em> is the thickness of the top (dense) sand layer, <em>\u03b3<\/em><sub>1<\/sub> is the unit weight of the dense sand layer, <em>\u03c6<\/em><em>\u2032<sub>1<\/sub><\/em>\u00a0is the friction angle of the dense sand layer, <em>q<sub>b<\/sub><\/em> is the bearing capacity of a footing of width <em>B <\/em>embedded at depth <em>H<\/em> in a uniform sand layer with the properties of the loose sand, calculated as:\n\n<strong>(5.49)\u00a0<\/strong>[latex]{q_b} = 0.5{\\gamma _2}B{N_{\\gamma 2}} + {\\gamma _1}H{N_{q2}}[\/latex]\n\nwhere <em>\u03b3<\/em><sub>2<\/sub> is the unit weight of the loose sand layer, while <em>\u039d<sub>\u03b3<\/sub><\/em><sub>2<\/sub> and <em>\u039d<\/em><sub>q2<\/sub> are the bearing capacity factors of the loose sand layer, featuring friction angle <em>\u03c6<\/em><em>\u2032<sub>2<\/sub><\/em>, calculated according to Eq. 5.18 and <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.31-HR.png\">Figure 5.31<\/a>, respectively. Moreover, <em>q<\/em> in Eq. 5.48 is the bearing capacity of a footing of width <em>B<\/em> resting on the surface of uniform dense sand with friction angle <em>\u03c6<\/em><em>\u2032<sub>1<\/sub><\/em>\u00a0, calculated according to Eq. 5.25, and <em>K<sub>s<\/sub><\/em> is a punching shear resistance coefficient that depends on the friction angles of the two sand layers, and is provided in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.39-HR.png\">Figure 5.39<\/a>.\n\n&nbsp;\n\n[caption id=\"attachment_385\" align=\"aligncenter\" width=\"500\"]<img class=\"wp-image-383 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/5.39-HR-e1743552193902.png\" alt=\"Graph showing the variation of the coefficient of punching shear resistance K_s, with the friction angle of the bottom sand layer \u03c62. Different curves corresponds to different friction angles of the top sand layer \u03c61, ranging from 30 deg to 50 deg.\" width=\"500\" height=\"479\"> Figure 5.39. Determination of the coefficient of punching shear resistance <em>K<sub>s<\/sub><\/em> for strip footings on layered sand (after Hanna 1981).[\/caption]\n\n&nbsp;\n\n[caption id=\"attachment_385\" align=\"aligncenter\" width=\"1024\"]<img class=\"wp-image-384 size-large\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/5.40-hr-1024x799.png\" alt=\"Four graphs depicting zones corresponding to different failure modes (punching, transitional, general shear and shallow transitional failure) depending on the combination of friction angle \u03c61 and the parameter H\/B. Each one of the graphs corresponds to a different value of the friction angle \u03c62.\" width=\"1024\" height=\"799\"> Figure 5.40. Failure mechanisms for strip footings on layered sand for different friction angles and dimensionless thickness <em>H\/B<\/em> combinations (Salimi Eshkevari et al. 2019b).[\/caption]\n\nAn important assumption underlying Hanna\u2019s Eq. 5.48 is that a punching failure mechanism develops, irrespective of the relative strength of the two layers. Salimi Eshkevari <em>et al.<\/em> (2019b) have shown that the failure mechanism that will develop depends on the friction angles of the two layers, as well as the ratio of the thickness of the top dense sand layer over the width of the footing <em>H<\/em>\/<em>B<\/em>. Using <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.40-hr.png\">Figure 5.40<\/a> one can determine which failure mechanism will develop. Accordingly, Salimi Eshkevari <em>et al.<\/em> (2019b) proposed the following expression for calculating the bearing capacity of a strip footing resting on dense sand over loose sand, for the cases where a transitional (or shallow transitional) mechanism is expected to develop:\n\n<strong>(5.50)<\/strong> [latex]{q_{f,l}} = 0.5{\\gamma _1}BN_\\gamma ^ * \\le q[\/latex]\n\nwhere the modified bearing capacity factor <em>N<\/em><sub>\u03b3<\/sub>* is given in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.41-hr.png\">Figure 5.41<\/a> for different values of <em>\u03c6<\/em><em>\u2032<\/em><sub>1<\/sub>, <em>\u03c6<\/em><em>\u2032<\/em><sub>2<\/sub> and <em>H<\/em>\/<em>B<\/em>. If <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.40-hr.png\">Figure 5.40<\/a> suggests that a punching failure mechanism will develop, Salimi Eshkevari <em>et al.<\/em> recommended using Hanna\u2019s expression (Eq. 5.48) to determine the bearing capacity. Bearing capacity factor values in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.41-hr.png\">Figure 5.41<\/a> are applicable to the case where <em>\u03b3<\/em><sub>1<\/sub>\/<em>\u03b3<\/em><sub>2<\/sub> = 1.2, which is probably a reasonable approximation for most practical cases. Note also that <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.41-hr.png\">Figure 5.41<\/a> covers only specific <em>H<\/em>\/<em>B<\/em> ranges for which a transitional mechanism will develop. Finally, the upper bound to the bearing capacity factor <em>N<\/em><sub>\u03b3<\/sub>* (dashed lines in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.41-hr.png\">Figure 5.41<\/a>) is the conventional bearing capacity factor <em>\u039d<sub>\u03b3<\/sub><\/em>, and indicates cases where the thickness of the top dense sand layer exceeds the critical thickness, thus a general shear failure mechanism will develop and <em>q<sub>f<\/sub>,<sub>l<\/sub><\/em> = <em>q<\/em>.\n\n&nbsp;\n\n[caption id=\"attachment_385\" align=\"aligncenter\" width=\"1024\"]<img class=\"wp-image-385 size-large\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/5.41-hr-1024x971.png\" alt=\"Five graphs presenting the variation of the factor N*\u03b3 as function of the parameter H\/B. Each one of the graphs corresponds to a different value of the friction angle \u03c61. Each graph contains multiple curves, with each curve corresponding to a different \u03c62 value. The upper bound of the factor N*\u03b3 is indicated in each graph with a dashed line. The figure on the bottom right presents the value of the upper bound N\u03b3 as function of the friction angle \u03c61. The expression that provides the upper bound is N\u03b3=(\u039dq-1)tan(1.4\u03c61')\" width=\"1024\" height=\"971\"> Figure 5.41. Modified bearing capacity factor <em>N<sub>\u03b3<\/sub>*<\/em> for different values of <em>\u03c6\u2032<sub>1<\/sub><\/em>, <em>\u03c6\u2032<sub>2<\/sub><\/em>\u00a0and <em>\u0397\/\u0392<\/em> (Salimi Eshkevari et al. 2019b) and upper bound bearing capacity factor <em>N<sub>\u03b3<\/sub><\/em> for generalised failure according to Meyerhoff.[\/caption]\n\n<hr>\n\n<h2>5.5.5 Footings on multi-layered soil profile<\/h2>\nA lower-bound estimate of the bearing capacity of a footing resting on a multi-layered soil profile can be found by simply considering a homogeneous soil with the properties of the weakest layer, an approach that may lead to perhaps over-conservative results. Numerical methods are recommended for the treatment of such problems.","rendered":"<h2>5.5.1 Critical thickness of the surficial soil layer<\/h2>\n<p>One of the key assumptions of the common analytical solutions presented in <a href=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/5-4-common-bearing-capacity-equations-and-practical-considerations\/\">Chapter 5.4<\/a> is that of uniform soil conditions, which is rarely the case in practice. When the geotechnical investigation reveals a non-uniform soil profile, we have to estimate whether the thickness of the top soil layer, measured below the embedment depth, is enough so that the failure surface will develop entirely inside it (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.36-definition-of-critical-thickness.png\">Figure 5.36<\/a>).<\/p>\n<figure id=\"attachment_385\" aria-describedby=\"caption-attachment-385\" style=\"width: 1024px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" class=\"wp-image-379 size-large\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2025\/03\/5.36-definition-of-critical-thickness-1024x330.png\" alt=\"The figure on the left presents a footing of width B embedded in soil at depth Df. The top soil layer has thickness H_cr, measured from the elevation of the footing's base. Underneath the top soil layer there is a different bottom soil layer. A generalised failure mechanism develops underneath the footing inside the top layer, and the mechanism extends up to a depth H_cr, measured from the elevation of the footing's base. The figure on the right presents the variation of the parameter H_cr\/B as function of the friction angle of the top soil layer.\" width=\"1024\" height=\"330\" srcset=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/03\/5.36-definition-of-critical-thickness-1024x330.png 1024w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/03\/5.36-definition-of-critical-thickness-300x97.png 300w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/03\/5.36-definition-of-critical-thickness-768x247.png 768w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/03\/5.36-definition-of-critical-thickness-1536x495.png 1536w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/03\/5.36-definition-of-critical-thickness-2048x659.png 2048w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/03\/5.36-definition-of-critical-thickness-65x21.png 65w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/03\/5.36-definition-of-critical-thickness-225x72.png 225w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2025\/03\/5.36-definition-of-critical-thickness-350x113.png 350w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><figcaption id=\"caption-attachment-385\" class=\"wp-caption-text\">Figure 5.36. Definition of the critical thickness, <em>H<sub>cr<\/sub><\/em> for failure to develop within the surficial soil layer.<\/figcaption><\/figure>\n<p>It is reasonable to assume that if the thickness of the soil layer below the foundation, <em>H<sub>cr<\/sub><\/em> is less than the depth that the Prandtl mechanism penetrates into the soil (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.26-hr.png\">Figure 5.26<\/a>) the failure surface will be entirely contained in the top soil layer, and the subsoil can be considered as homogeneous for bearing capacity calculations.<\/p>\n<p><strong>(5.35)<\/strong> [latex]{H_{cr}} = \\dfrac{B}{{2\\cos \\left( {{{45}^o} + \\dfrac{{\\varphi '}}{2}} \\right)}}{e^{\\left( {A\\tan \\varphi '} \\right)}}[\/latex]<\/p>\n<p>Where <em>A<\/em> = (45\u00b0+<em>\u03c6\u2032<\/em>\/2) in rad.<\/p>\n<p>Alternatively, the critical thickness <em>H<sub>cr<\/sub><\/em> can be calculated as (Budhu 2011):<\/p>\n<p><strong>(5.36)<\/strong> [latex]{H_{cr}} = \\dfrac{{3B\\ln \\left( {\\dfrac{{{q_{top}}}}{{{q_{bottom}}}}} \\right)}}{{2\\left( {1 + \\dfrac{B}{L}} \\right)}}[\/latex]<\/p>\n<p>where:<\/p>\n<ul>\n<li><em>q<sub>top <\/sub><\/em>is the bearing capacity of the footing with the particular dimensions, resting on the surface of an infinitely deep layer with the properties of the <em>top<\/em> soil layer, and<\/li>\n<li><em>q<sub>bottom<\/sub><\/em> is the bearing capacity of the footing with the particular dimensions, resting on the surface of an infinitely deep layer with the properties of the <em>bottom<\/em> soil layer.<\/li>\n<\/ul>\n<p>If the thickness of the top soil layer measured below the embedment depth, <em>H<sub>top <\/sub><\/em>is less than the critical thickness, <em>H<sub>cr<\/sub>, <\/em>there is no general analytical method to estimate the bearing capacity, except of course numerical methods. Some characteristic cases are examined in the following paragraphs.<\/p>\n<hr \/>\n<h2>5.5.2 Footing on a soft clay layer overlying a stiff soil formation<\/h2>\n<p>In the case where a thin soft clay formation is found near the ground surface, good engineering practice suggests excavating the soft clay layer, and replacing it with well-compacted coarse-grained fill material. Founding structures directly on soft clays with shallow foundation should be avoided, except light structures such as one-storey buildings.<\/p>\n<p>Experimental results have shown that in the case of a footing resting on soft clay overlying a stiffer stratum, the mechanism of bearing capacity consists of lateral squeezing of the soft soil, as the footing \u201csinks\u201d into the top layer. Tomlinson and Boorman (1995) proposed the following expressions to estimate bearing capacity in that case:<\/p>\n<figure id=\"attachment_385\" aria-describedby=\"caption-attachment-385\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" class=\"wp-image-380 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/5.37-HR-e1743552136286.png\" alt=\"Schematic of a footing embedded in a soft clay layer. The width of the footing is B. The footing is loaded with a pressure qf. The thickness of the soft clay layer is z, measured from the footing's base. Underneath the soft clay layer there is a stiff soil layer.\" width=\"500\" height=\"181\" srcset=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.37-HR-e1743552136286.png 500w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.37-HR-e1743552136286-300x109.png 300w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.37-HR-e1743552136286-65x24.png 65w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.37-HR-e1743552136286-225x81.png 225w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.37-HR-e1743552136286-350x127.png 350w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><figcaption id=\"caption-attachment-385\" class=\"wp-caption-text\">Figure 5.37. Footing on a soft clay layer overlying a stiff soil formation.<\/figcaption><\/figure>\n<p>For circular\/square footings:<\/p>\n<p><strong>(5.37)<\/strong> [latex]{q_f} = \\left( {\\dfrac{B}{{2z}} + \\pi + 1} \\right){S_u}{\\rm{ \\:for \\:}}\\dfrac{B}{z} \\ge 2[\/latex]<\/p>\n<p>For strip footings:<\/p>\n<p><strong>(5.38)<\/strong> [latex]{q_f} = \\left( {\\dfrac{B}{{3z}} + \\pi + 1} \\right){S_u}{\\rm{ \\:for\\: }}\\dfrac{B}{z} \\ge 6[\/latex]<\/p>\n<p>where <em>z<\/em> is defined in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.37-HR.png\">Figure 5.37<\/a> above. When <em>B\/z <\/em>&lt; 2 for circular\/strip footings or <em>B\/z <\/em>&lt; 6 for strip footings (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.37-HR.png\">Figure 5.37<\/a>), the expressions for homogeneous soil should be rather used.<\/p>\n<hr \/>\n<h2>5.5.3 Footing on stiff soil overlying a soft clay layer<\/h2>\n<p>This case could well correspond to the common problem of a footing resting on a foundation improvement layer made of well-compacted coarse-grained fill, overlying a soft subgrade. A <em>\u201cpunching\u201d <\/em>failure mechanism may develop under these circumstances (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.38a-HR.png\">Figure 5.38a<\/a>).<\/p>\n<p>A simplified method for estimating the bearing capacity of a footing of any shape resting on the surface of, or embedded in a stiff soil formation overlying a soft clay layer consists of the following steps: First, we estimate the bearing capacity of the footing while considering uniform soil with the properties of the top stiff soil layer, which must not be critical for the design.<\/p>\n<p>Accordingly, we estimate using Eq. 5.24 the net bearing capacity <em>q<sub>eq<\/sub><\/em> of a hypothetical equivalent footing with the same shape and dimensions, embedded at a depth <em>D<sub>f<\/sub>,<sub>eq <\/sub><\/em>= <em>D<sub>f<\/sub>+z<\/em> equal to the thickness of the top stiff soil layer (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.38a-HR.png\">Figure 5.38<\/a>). Bearing capacity failure of the actual footing will be reached when the vertical stress \u0394<em>\u03c3<\/em><sub>z<\/sub> transferred from the actual footing to the interface of the stiff soil-soft clay layer (<a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.38b-HR.png\">Figure 5.38b<\/a>) becomes equal to <em>q<sub>eq<\/sub><\/em> i.e., \u0394<em>\u03c3<\/em><sub>z<\/sub> = <em>q<sub>eq<\/sub><\/em>. The following approximate expressions can be used to calculate \u0394<em>\u03c3<\/em><sub>z<\/sub>, while assuming 2:1 stress distribution (see <a href=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/3-6-stresses-in-the-soil-due-to-a-rectangular-pressure\/\">Chapter 3.6<\/a>):<\/p>\n<p>For a rectangular footing:<\/p>\n<p><strong>(5.39<\/strong><strong>)<\/strong> [latex]\\Delta {\\sigma _z} = q\\left[ {\\dfrac{{BL}}{{\\left( {B + z} \\right)\\left( {L + z} \\right)}}} \\right][\/latex]<\/p>\n<p>For a square\/circular footing:<\/p>\n<p><strong>(5.40)<\/strong> [latex]\\Delta {\\sigma _z} = q{\\left[ {\\dfrac{B}{{\\left( {B + z} \\right)}}} \\right]^2}[\/latex]<\/p>\n<p>For a strip footing:<\/p>\n<p><strong>(5.41)\u00a0<\/strong>[latex]\\Delta {\\sigma _z} = q\\left[ {\\dfrac{B}{{\\left( {B + z} \\right)}}} \\right][\/latex]<\/p>\n<p>in the above expressions <em>q<\/em> is the net bearing capacity of the actual footing. Therefore substituting \u0394<em>\u03c3<\/em><sub>z <\/sub>= <em>q<sub>eq<\/sub><\/em> in the Eqs. 5.39 to 5.41 that corresponds to the actual footing shape will provide its net bearing capacity <em>q<\/em>. If the actual footing is embedded in soil, its bearing capacity will be <em>q<sub>f<\/sub><\/em> = <em>q<\/em> + <em>\u03b3<\/em><em>D<sub>f<\/sub><\/em>.<\/p>\n<figure id=\"attachment_385\" aria-describedby=\"caption-attachment-385\" style=\"width: 600px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" class=\"wp-image-381 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/5.38a-HR-e1743552156863.png\" alt=\"Schematic of a footing embedded in stiff soil, at depth Df, underlaid by soft clay. The width of the footing is B. The footing is loaded with a pressure qf. The thickness of the stiff soil layer is z, measured from the footing's base. Vertical stresses from the footing are distributed with a 2:1 distribution, and a composite failure mechanism develops consisting of two shear bands in the stiff soil and a generalised failure surface in the soft clay.\" width=\"600\" height=\"260\" srcset=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.38a-HR-e1743552156863.png 600w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.38a-HR-e1743552156863-300x130.png 300w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.38a-HR-e1743552156863-65x28.png 65w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.38a-HR-e1743552156863-225x98.png 225w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.38a-HR-e1743552156863-350x152.png 350w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><figcaption id=\"caption-attachment-385\" class=\"wp-caption-text\">Figure 5.38a. Punching of a footing into the underlying soft clay layer.<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_385\" aria-describedby=\"caption-attachment-385\" style=\"width: 600px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" class=\"wp-image-382 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/5.38b-HR-e1743552172643.png\" alt=\"Schematic of an equivalent footing embedded in stiff soil, at depth Df,eq=Df+z, underlaid by soft clay. The width of the footing is B. The footing is loaded with a pressure qeq. The thickness of the stiff soil layer is Df+z, measured from the surface. A generalised failure mechanism develops in the soft clay.\" width=\"600\" height=\"206\" srcset=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.38b-HR-e1743552172643.png 600w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.38b-HR-e1743552172643-300x103.png 300w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.38b-HR-e1743552172643-65x22.png 65w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.38b-HR-e1743552172643-225x77.png 225w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.38b-HR-e1743552172643-350x120.png 350w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><figcaption id=\"caption-attachment-385\" class=\"wp-caption-text\">Figure 5.38b. Concept for estimating the bearing capacity accounting for the development of the failure surface inside the soft clay layer.<\/figcaption><\/figure>\n<p>Note that the methodology described above conservatively ignores the shear resistance of the top layer, and the energy dissipated along the shear planes located with it. Another simplifying assumption is that the inclination of the shear planes relative to the vertical in the top layer is ignored i.e., the width of the equivalent, hypothetical footing is taken equal to the width of the real one. There are more refined methods in the literature that do not require introducing these assumptions, however their range of application is limited to specific footing geometries and soil types. Such a method has been developed by Salimi Eshkevari <em>et al.<\/em> (2019a) for estimating the bearing capacity of strip footings resting on the surface of a sand layer overlying a soft clay, which is relevant to the design of working platforms for tracked plants. Salimi Eshkevari <em>et al.<\/em> analysed the results of a series of Finite Element Limit Analyses and concluded to the following expression that provides the bearing capacity of a strip footing on a layered profile of sand over clay <em>q<sub>f<\/sub>,<sub>l<\/sub><\/em>:<\/p>\n<p><strong>(5.42)<\/strong> [latex]{q_{f,l}}B = \\gamma {H^2}{K_{sr}}\\tan \\varphi ' + {N_{cu}}{S_u}\\left[ {B + 2H\\tan \\theta } \\right] + \\gamma {H^2}\\tan \\theta \\le qB[\/latex]<\/p>\n<p>where <em>B<\/em> is the footing width, <em>H<\/em> is the thickness of the sand layer, <em>\u03b3<\/em> is the unit weight of sand, <em>\u03c6\u2032<\/em>\u00a0is the friction angle of sand, <em>S<sub>u<\/sub><\/em> is the undrained shear strength of the clay layer, <em>N<sub>cu <\/sub><\/em>= 5.14 is the bearing capacity factor for strip footings on undrained soil, <em>q<\/em> is the bearing capacity of a footing of width <em>B<\/em> resting on the surface of uniform sand, calculated according to Eq. 5.25. The angle <em>\u03b8<\/em> that provides the inclination of the shear planes that will develop in the sand layer can be calculated as:<\/p>\n<p><strong>(5.43)<\/strong> [latex]\\theta ({\\rm{rad}}) = \\alpha \\ln \\left( {\\dfrac{{{S_u}}}{{\\gamma H}}} \\right) + \\beta[\/latex]<\/p>\n<p><strong>(5.44)\u00a0<\/strong>[latex]\\alpha = 0.039\\ln \\left( {\\tan \\varphi '} \\right) - 0.164[\/latex]<\/p>\n<p><strong>(5.45)\u00a0<\/strong>[latex]\\beta = 0.597\\ln \\left( {\\tan \\varphi '} \\right) - 0.051[\/latex]<\/p>\n<p>Note that angle <em>\u03b8<\/em> can be positive or negative. Thus, unlike the approximate method described above, the width <em>B<\/em>+2<em>H<\/em>tan<em>\u03b8<\/em> of the equivalent footing in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.38a-HR.png\">Figure 5.38<\/a> can be greater, or less than the width of the actual footing <em>B<\/em>. Finally, the punching shear coefficient <em>K<sub>sr<\/sub><\/em>, that is correlated with the normal stress acting on the shear planes that will develop in the sand layer, is calculated as:<\/p>\n<p><strong>(5.46)\u00a0<\/strong>[latex]{K_{sr}} = \\delta \\left( {\\dfrac{{{S_u}}}{{\\gamma H}}} \\right) + 2[\/latex]<\/p>\n<p><strong>(5.47)\u00a0<\/strong>[latex]\\delta = - 3.45\\left( {\\tan \\varphi '} \\right) + 8.693[\/latex]<\/p>\n<hr \/>\n<h2>5.5.4 Strip footings on layered sands<\/h2>\n<p>Estimation of the bearing capacity of strip footings resting on the surface of a relatively thin layer of dense sand overlaying a weak layer of loose sand underpins the design of working platforms for heavily loaded tracked piling rigs and cranes, on sites which loose sand deposits are found at the ground surface. Hanna (1981) was among the first to study this problem, and concluded to the following expression for estimating the bearing capacity in terms of stress when a punching failure mechanism develops:<\/p>\n<p><strong>(5.48)\u00a0<\/strong>[latex]{q_{f,l}} = {q_b} + \\gamma {H^2}\\dfrac{{{K_s}\\tan {{\\varphi '}_1}}}{B} - {\\gamma _1}H \\le q[\/latex]<\/p>\n<p>where <em>B<\/em> is the footing width, <em>H<\/em> is the thickness of the top (dense) sand layer, <em>\u03b3<\/em><sub>1<\/sub> is the unit weight of the dense sand layer, <em>\u03c6<\/em><em>\u2032<sub>1<\/sub><\/em>\u00a0is the friction angle of the dense sand layer, <em>q<sub>b<\/sub><\/em> is the bearing capacity of a footing of width <em>B <\/em>embedded at depth <em>H<\/em> in a uniform sand layer with the properties of the loose sand, calculated as:<\/p>\n<p><strong>(5.49)\u00a0<\/strong>[latex]{q_b} = 0.5{\\gamma _2}B{N_{\\gamma 2}} + {\\gamma _1}H{N_{q2}}[\/latex]<\/p>\n<p>where <em>\u03b3<\/em><sub>2<\/sub> is the unit weight of the loose sand layer, while <em>\u039d<sub>\u03b3<\/sub><\/em><sub>2<\/sub> and <em>\u039d<\/em><sub>q2<\/sub> are the bearing capacity factors of the loose sand layer, featuring friction angle <em>\u03c6<\/em><em>\u2032<sub>2<\/sub><\/em>, calculated according to Eq. 5.18 and <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.31-HR.png\">Figure 5.31<\/a>, respectively. Moreover, <em>q<\/em> in Eq. 5.48 is the bearing capacity of a footing of width <em>B<\/em> resting on the surface of uniform dense sand with friction angle <em>\u03c6<\/em><em>\u2032<sub>1<\/sub><\/em>\u00a0, calculated according to Eq. 5.25, and <em>K<sub>s<\/sub><\/em> is a punching shear resistance coefficient that depends on the friction angles of the two sand layers, and is provided in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.39-HR.png\">Figure 5.39<\/a>.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_385\" aria-describedby=\"caption-attachment-385\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" class=\"wp-image-383 size-full\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/5.39-HR-e1743552193902.png\" alt=\"Graph showing the variation of the coefficient of punching shear resistance K_s, with the friction angle of the bottom sand layer \u03c62. Different curves corresponds to different friction angles of the top sand layer \u03c61, ranging from 30 deg to 50 deg.\" width=\"500\" height=\"479\" srcset=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.39-HR-e1743552193902.png 500w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.39-HR-e1743552193902-300x287.png 300w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.39-HR-e1743552193902-65x62.png 65w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.39-HR-e1743552193902-225x216.png 225w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.39-HR-e1743552193902-350x335.png 350w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><figcaption id=\"caption-attachment-385\" class=\"wp-caption-text\">Figure 5.39. Determination of the coefficient of punching shear resistance <em>K<sub>s<\/sub><\/em> for strip footings on layered sand (after Hanna 1981).<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_385\" aria-describedby=\"caption-attachment-385\" style=\"width: 1024px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" class=\"wp-image-384 size-large\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/5.40-hr-1024x799.png\" alt=\"Four graphs depicting zones corresponding to different failure modes (punching, transitional, general shear and shallow transitional failure) depending on the combination of friction angle \u03c61 and the parameter H\/B. Each one of the graphs corresponds to a different value of the friction angle \u03c62.\" width=\"1024\" height=\"799\" srcset=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.40-hr-1024x799.png 1024w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.40-hr-300x234.png 300w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.40-hr-768x599.png 768w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.40-hr-1536x1199.png 1536w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.40-hr-65x51.png 65w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.40-hr-225x176.png 225w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.40-hr-350x273.png 350w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.40-hr.png 1554w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><figcaption id=\"caption-attachment-385\" class=\"wp-caption-text\">Figure 5.40. Failure mechanisms for strip footings on layered sand for different friction angles and dimensionless thickness <em>H\/B<\/em> combinations (Salimi Eshkevari et al. 2019b).<\/figcaption><\/figure>\n<p>An important assumption underlying Hanna\u2019s Eq. 5.48 is that a punching failure mechanism develops, irrespective of the relative strength of the two layers. Salimi Eshkevari <em>et al.<\/em> (2019b) have shown that the failure mechanism that will develop depends on the friction angles of the two layers, as well as the ratio of the thickness of the top dense sand layer over the width of the footing <em>H<\/em>\/<em>B<\/em>. Using <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.40-hr.png\">Figure 5.40<\/a> one can determine which failure mechanism will develop. Accordingly, Salimi Eshkevari <em>et al.<\/em> (2019b) proposed the following expression for calculating the bearing capacity of a strip footing resting on dense sand over loose sand, for the cases where a transitional (or shallow transitional) mechanism is expected to develop:<\/p>\n<p><strong>(5.50)<\/strong> [latex]{q_{f,l}} = 0.5{\\gamma _1}BN_\\gamma ^ * \\le q[\/latex]<\/p>\n<p>where the modified bearing capacity factor <em>N<\/em><sub>\u03b3<\/sub>* is given in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.41-hr.png\">Figure 5.41<\/a> for different values of <em>\u03c6<\/em><em>\u2032<\/em><sub>1<\/sub>, <em>\u03c6<\/em><em>\u2032<\/em><sub>2<\/sub> and <em>H<\/em>\/<em>B<\/em>. If <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.40-hr.png\">Figure 5.40<\/a> suggests that a punching failure mechanism will develop, Salimi Eshkevari <em>et al.<\/em> recommended using Hanna\u2019s expression (Eq. 5.48) to determine the bearing capacity. Bearing capacity factor values in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.41-hr.png\">Figure 5.41<\/a> are applicable to the case where <em>\u03b3<\/em><sub>1<\/sub>\/<em>\u03b3<\/em><sub>2<\/sub> = 1.2, which is probably a reasonable approximation for most practical cases. Note also that <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.41-hr.png\">Figure 5.41<\/a> covers only specific <em>H<\/em>\/<em>B<\/em> ranges for which a transitional mechanism will develop. Finally, the upper bound to the bearing capacity factor <em>N<\/em><sub>\u03b3<\/sub>* (dashed lines in <a href=\"https:\/\/oercollective.caul.edu.au\/app\/uploads\/sites\/143\/2025\/02\/5.41-hr.png\">Figure 5.41<\/a>) is the conventional bearing capacity factor <em>\u039d<sub>\u03b3<\/sub><\/em>, and indicates cases where the thickness of the top dense sand layer exceeds the critical thickness, thus a general shear failure mechanism will develop and <em>q<sub>f<\/sub>,<sub>l<\/sub><\/em> = <em>q<\/em>.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_385\" aria-describedby=\"caption-attachment-385\" style=\"width: 1024px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" class=\"wp-image-385 size-large\" src=\"https:\/\/libraryresources.nse.org.ng\/wp-content\/uploads\/sites\/9\/2026\/03\/5.41-hr-1024x971.png\" alt=\"Five graphs presenting the variation of the factor N*\u03b3 as function of the parameter H\/B. Each one of the graphs corresponds to a different value of the friction angle \u03c61. Each graph contains multiple curves, with each curve corresponding to a different \u03c62 value. The upper bound of the factor N*\u03b3 is indicated in each graph with a dashed line. The figure on the bottom right presents the value of the upper bound N\u03b3 as function of the friction angle \u03c61. The expression that provides the upper bound is N\u03b3=(\u039dq-1)tan(1.4\u03c61')\" width=\"1024\" height=\"971\" srcset=\"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.41-hr-1024x971.png 1024w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.41-hr-300x284.png 300w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.41-hr-768x728.png 768w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.41-hr-65x62.png 65w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.41-hr-225x213.png 225w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.41-hr-350x332.png 350w, https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-content\/uploads\/sites\/9\/2026\/03\/5.41-hr.png 1288w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><figcaption id=\"caption-attachment-385\" class=\"wp-caption-text\">Figure 5.41. Modified bearing capacity factor <em>N<sub>\u03b3<\/sub>*<\/em> for different values of <em>\u03c6\u2032<sub>1<\/sub><\/em>, <em>\u03c6\u2032<sub>2<\/sub><\/em>\u00a0and <em>\u0397\/\u0392<\/em> (Salimi Eshkevari et al. 2019b) and upper bound bearing capacity factor <em>N<sub>\u03b3<\/sub><\/em> for generalised failure according to Meyerhoff.<\/figcaption><\/figure>\n<hr \/>\n<h2>5.5.5 Footings on multi-layered soil profile<\/h2>\n<p>A lower-bound estimate of the bearing capacity of a footing resting on a multi-layered soil profile can be found by simply considering a homogeneous soil with the properties of the weakest layer, an approach that may lead to perhaps over-conservative results. Numerical methods are recommended for the treatment of such problems.<\/p>\n","protected":false},"author":1,"menu_order":8,"template":"","meta":{"pb_show_title":"","pb_short_title":"5.5 Bearing capacity of footings on layered soils","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-386","chapter","type-chapter","status-publish","hentry"],"part":325,"_links":{"self":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/386","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\/386\/revisions"}],"predecessor-version":[{"id":387,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/386\/revisions\/387"}],"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\/386\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/media?parent=386"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapter-type?post=386"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/contributor?post=386"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/license?post=386"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}