{"id":1407,"date":"2026-09-11T03:02:22","date_gmt":"2026-09-11T00:02:22","guid":{"rendered":"https:\/\/setaf2018.com\/?p=1407"},"modified":"2026-09-11T03:02:28","modified_gmt":"2026-09-11T00:02:28","slug":"modulus-of-subgrade-reaction","status":"publish","type":"post","link":"https:\/\/setaf2018.com\/en\/post\/modulus-of-subgrade-reaction\/","title":{"rendered":"Modulus of Subgrade Reaction: How to Determine kv and kh"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">The modulus of subgrade reaction is the ratio between the pressure applied to the ground at a given point and the displacement produced at that point. It is expressed in force per unit volume, typically kN\/m\u00b3, and it allows the ground to be represented as a system of independent elastic springs rather than as a continuous medium.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Engineers use this parameter whenever a structural element is analysed as a beam, plate, or wall supported by the ground rather than as part of a full soil continuum. Raft foundations on elastic support, laterally loaded piles, and embedded retaining walls are the most common applications.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Two forms are used in practice. The vertical modulus of subgrade reaction, kv, describes the ground response beneath a foundation. The horizontal modulus, kh, describes the lateral response of soil against a wall or a pile. They are calculated in different ways and are not interchangeable.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>What Is the Modulus of Subgrade Reaction?<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The modulus of subgrade reaction relates contact pressure to displacement at a point:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>k = q \/ s<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where q is the contact pressure at that point (kN\/m\u00b2) and s is the corresponding displacement (m). The result has units of kN\/m\u00b3.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The terminology surrounding this parameter is a frequent source of confusion, because several related quantities are all referred to as &#8220;k&#8221; in practice:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th><strong>Term<\/strong><\/th><th><strong>Symbol<\/strong><\/th><th><strong>Units<\/strong><\/th><th><strong>Meaning<\/strong><\/th><\/tr><\/thead><tbody><tr><td>Modulus of subgrade reaction<\/td><td>k, ks, kv, kh<\/td><td>kN\/m\u00b3<\/td><td>Pressure per unit displacement, distributed over an area<\/td><\/tr><tr><td>Spring stiffness at a node<\/td><td>K<\/td><td>kN\/m<\/td><td>The modulus multiplied by the tributary area of that node<\/td><\/tr><tr><td>Plate load test coefficient<\/td><td>k\u2083\u2080, k\u2080.\u2083<\/td><td>kN\/m\u00b3<\/td><td>The value measured on a 305 mm diameter standard plate<\/td><\/tr><tr><td>Soil elastic modulus<\/td><td>Es, E\u2032<\/td><td>kN\/m\u00b2<\/td><td>A material stiffness property, independent of geometry<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The distinction between the third and fourth rows matters more than it may appear. The soil elastic modulus is a material property. The modulus of subgrade reaction is not.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Is the Modulus of Subgrade Reaction a Soil Property?<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">No. Although k is often presented in tabulated form alongside soil descriptions, it depends on the loaded area, the geometry and rigidity of the structural element, the depth of the compressible zone, and the stress level at which it is evaluated.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Terzaghi noted this explicitly in 1955: the same soil will produce different values of k beneath a small footing and beneath a large raft, because the two foundations stress different volumes of ground. A 1 m wide footing mobilises soil to roughly one or two metres depth; a 30 m wide raft mobilises soil to a depth many times greater, engaging layers that the smaller footing never reaches.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This is why a value taken from a general table, or measured on a small plate and applied directly, can be wrong by an order of magnitude.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The modulus of subgrade reaction is a property of the soil\u2013foundation system, not of the soil.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The practical consequence is that k should generally be derived from the specific problem rather than looked up. That derivation is the subject of the remainder of this guide.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>How Is the Vertical Modulus of Subgrade Reaction (kv) Determined?<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Three approaches are commonly used for shallow foundations. They require different input data and are appropriate to different stages of design.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. From Calculated Foundation Settlement<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The most direct route follows the definition of the parameter itself:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Soil profile + foundation geometry + applied load \u2192 Settlement analysis \u2192 kv = q \/ s<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Settlement is calculated for the actual foundation on the actual soil profile, and the resulting displacement is divided into the contact pressure. Because the settlement analysis already accounts for layering, groundwater, stress distribution with depth, and the thickness of the compressible zone, the resulting kv is consistent with the problem being solved.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The refinement that this method requires is recognising that a foundation does not settle uniformly. Settlement is greatest near the centre and smallest at the edges, which means a single value derived from the centre will overestimate the stiffness of the edge regions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A more representative approach evaluates settlement at a series of points along two orthogonal lines passing through the centroid of the foundation, computes an average modulus in each direction, and then takes the mean or the minimum of the two directional values depending on how conservative the design needs to be. <a href=\"https:\/\/setaf2018.com\/en\/bearing-capacity-of-foundations\/\">SETAF2018<\/a> follows this procedure for rectangular, circular, and polygonal foundation geometries.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This method is appropriate when the soil profile is layered, when the compressible zone is deep relative to the foundation width, or when settlement has already been calculated as part of the design.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2. From the Vesi\u0107 Equation<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Vesi\u0107 (1961) derived a closed-form expression for a beam resting on an elastic half-space:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>ks = (0.65 \/ B) \u00b7 \u00b9\u00b2\u221a( Es\u00b7B\u2074 \/ (Eb\u00b7Ib) ) \u00b7 Es \/ (1 \u2212 \u03bd\u00b2)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where B is the foundation width, Es and \u03bd are the elastic modulus and Poisson&#8217;s ratio of the soil, and Eb and Ib are the elastic modulus and second moment of area of the foundation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Because the twelfth root of the bracketed term is close to unity for most practical combinations of soil and concrete stiffness, the expression is often simplified to:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>ks \u2248 Es \/ [ B \u00b7 (1 \u2212 \u03bd\u00b2) ]<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The equation is useful for preliminary design and for uniform profiles, and it makes the dependence on foundation width explicit. Its limitation is the assumption of a homogeneous elastic half-space, which becomes restrictive when a compressible layer is underlain by a much stiffer stratum, or when the profile changes significantly within the influence depth. Establishing a representative Es for a layered profile is itself an engineering judgement, and correlations between field test data and deformation moduli \u2014 such as those discussed in <a href=\"https:\/\/setaf2018.com\/en\/post\/spt-n-value-vs-friction-angle\/\">SPT N value and friction angle relationships<\/a> \u2014 carry their own scatter.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3. From a Plate Load Test<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A plate load test measures the pressure\u2013settlement response directly in the field, usually on a 305 mm diameter rigid plate, giving the coefficient k\u2083\u2080.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">That value cannot be used for the foundation without correction. A 305 mm plate stresses only the upper few hundred millimetres of ground, while the foundation may stress several metres. Terzaghi&#8217;s classical size corrections are:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For granular soils: <strong>ks = k\u2083\u2080 \u00b7 [ (B + 0.3) \/ (2B) ]\u00b2<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For cohesive soils: <strong>ks = k\u2083\u2080 \u00b7 (0.3 \/ B)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">with B in metres. A further shape correction is applied for rectangular foundations, since the plate is square or circular while the foundation may be elongated.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The plate load test is valuable because it is a direct measurement rather than a derived quantity. It is also the method most frequently misused, because the measured value is often carried into design without any size correction at all.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Comparing the Three Methods<\/h3>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th><strong>Method<\/strong><\/th><th><strong>Input required<\/strong><\/th><th><strong>Particularly suited to<\/strong><\/th><th><strong>Main limitation<\/strong><\/th><\/tr><\/thead><tbody><tr><td>From calculated settlement<\/td><td>Full soil profile, geometry, loads<\/td><td>Layered profiles, deep compressible zones, raft foundations<\/td><td>Requires a completed settlement analysis<\/td><\/tr><tr><td>Vesi\u0107 equation<\/td><td>B, Es, \u03bd, Eb, Ib<\/td><td>Preliminary design, relatively uniform profiles<\/td><td>Assumes a homogeneous elastic half-space<\/td><\/tr><tr><td>Plate load test<\/td><td>k\u2083\u2080 from field test, foundation dimensions<\/td><td>Verification, granular near-surface soils<\/td><td>Samples a very shallow zone; requires size correction<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Why Do the Three Methods Not Give the Same Answer?<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Because they do not measure the same thing. Each method samples a different volume of ground, at a different stress level, under a different assumption about how the load is distributed.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A settlement-based value reflects the entire compressible zone beneath the foundation. A Vesi\u0107 value reflects an idealised uniform half-space characterised by a single Es. A corrected plate test value reflects a near-surface measurement extrapolated to a much larger loaded area.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Divergence between them is informative rather than alarming. A large discrepancy usually indicates that the profile is strongly layered, that Es was selected from a depth interval that does not govern the response, or that the compressible zone extends well beyond the influence depth of the plate.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Three methods that disagree are not evidence of a calculation error. They are evidence that k depends on how it was obtained.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Where the difference is significant, the practical response is to check the sensitivity of the structural result to the range of values rather than to select one and proceed. For a flexible raft, bending moments can be more sensitive to the variation of k across the foundation than to its absolute magnitude.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>How Is kv Determined for Piles and Rigid Columns?<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A pile or rigid inclusion does not transfer load to the ground in a single mechanism. Part of the load is carried by shaft resistance along its length, and the remainder by base resistance at the toe. Representing this with a single vertical spring loses the distinction between the two.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A more representative model uses two separate vertical moduli:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>kvs<\/strong>, the vertical modulus associated with shaft resistance<\/li>\n\n\n\n<li><strong>kvp<\/strong>, the vertical modulus associated with base resistance<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The load split is established from the ratio of shaft resistance to total capacity, Qs\/Qtotal. The portion of the service load carried by the shaft is applied to a single column as if transferred entirely by shaft friction, and the resulting elastic settlement Hs gives:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>kvs = \u03c3s \/ Hs<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The base portion is treated in the same way, applied entirely at the toe, giving kvp = \u03c3p \/ Hp. The service load used in this calculation is normally the maximum axial load within the column group rather than the average.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This distinction matters in structural models where the pile cap or raft is analysed on springs, because shaft and base springs mobilise at very different displacements. Full shaft resistance typically develops within 2.5 to 10 mm of relative movement, while full base resistance requires displacement of the order of 8 to 10 per cent of the column diameter. Applications and modelling considerations for these systems are discussed further in <a href=\"https:\/\/setaf2018.com\/en\/post\/pile-foundation-fem\/\">pile foundation modelling<\/a>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The same approach applies to rigid inclusions formed by ground improvement \u2014 deep soil mixing columns, jet grout columns, and stone columns \u2014 where the column stiffness differs from the surrounding soil but the load transfer mechanism is comparable.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>How Is the Horizontal Modulus of Subgrade Reaction (kh) Determined?<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The horizontal modulus governs how much lateral soil resistance develops for a given wall or pile displacement. It has a direct influence on calculated deflections, bending moments, and support forces in <a href=\"https:\/\/setaf2018.com\/en\/excavation-support-structures\/\">excavation support design<\/a>, which makes its determination one of the more consequential inputs in a beam-on-spring analysis.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Two established methods are used for embedded retaining walls.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">The Schmitt Method<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Schmitt (1995) relates kh to the oedometer modulus of the soil and the flexural rigidity of the wall:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>kh = 2.1 \u00b7 Eoed^(4\/3) \/ (EI)^(1\/3)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where Eoed is the oedometer (constrained) modulus of the soil and EI is the flexural rigidity of the wall per metre run.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The oedometer modulus can be derived from the shear modulus and effective Poisson&#8217;s ratio:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Eoed = 2G \u00b7 (1 \u2212 \u03bd\u2032) \/ (1 \u2212 2\u03bd\u2032)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The form of the Schmitt expression contains a result that surprises many engineers on first encounter: kh decreases as the wall becomes stiffer. This is not an error. A stiffer wall deflects over a longer wavelength and therefore mobilises a larger volume of soil for the same displacement at a point, which reduces the pressure developed per unit of local movement.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A stiffer wall does not attract a stiffer spring.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">The Chadeisson Method<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The Chadeisson approach originates from a design chart developed from observed wall behaviour, and it introduces soil strength rather than stiffness alone. It evaluates kh from the passive earth pressure coefficient Kp, the at-rest coefficient K\u2080, the effective cohesion c\u2032, the soil unit weight \u03b3, the flexural rigidity EI of the wall, and a cohesion influence coefficient Ap that typically ranges between 1 and 15.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Its practical advantage over Schmitt is the explicit treatment of cohesion, which makes it better suited to stiff clays and to profiles where c\u2032 contributes meaningfully to lateral resistance. <a href=\"https:\/\/setaf2018.com\/en\/excavation-support-structures\/\">SETAF2018<\/a> evaluates both methods from the defined soil profile and wall section, allowing the engineer to select the approach appropriate to the ground conditions.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">For Piles and Rigid Columns<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">For laterally loaded piles and rigid inclusions, kh is evaluated from the oedometer moduli of the individual soil layers and the flexural rigidity EpIp of the column. The result varies with depth as the layers change, which is why a single horizontal spring value applied along the full length of a pile rarely represents the response of a layered profile.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>How kh Is Used: The Dependent Pressures Method<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The horizontal modulus does not act alone. In an embedded wall analysis, kh determines the response only within an elastic range that is bounded by the limiting active and passive states.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The assumption underlying the dependent pressures method is that the ground behaves as an ideal elasto-plastic Winkler material. Pressure acting on an undeformed wall is the at-rest pressure. As the wall deflects, pressure migrates away from the at-rest condition in proportion to displacement and to kh, but it cannot pass beyond the active or passive limits:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">At-rest state (K\u2080) \u2192 wall deflects \u2192 pressure migrates toward Ka on the retained side, Kp on the excavation side \u2192 bounded at those limits<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Computationally, the wall is divided into elements with nodes placed at all topological points \u2014 the wall head and toe, anchor and strut levels, excavation levels, and section changes \u2014 with intermediate nodes distributed so that elements remain approximately equal in size. Each element is assigned a value of kh, the structure is loaded with at-rest pressure, and the analysis is performed by the matrix\u2013displacement method. Where the calculated pressure violates an active or passive limit, kh is set to zero at that location and the wall is loaded with the limiting pressure instead. The iteration continues until horizontal equilibrium is satisfied.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Because the analysis proceeds stage by stage, plastic deformation accumulated in earlier construction stages is carried forward, and prestressed anchors are introduced as forces at the stage in which they are stressed and as springs thereafter. This is why construction stages must reflect the actual execution sequence rather than a simplified final condition.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The same framework governs the analysis of <a href=\"https:\/\/setaf2018.com\/en\/post\/soldier-pile-wall-design\/\">soldier pile walls<\/a>, <a href=\"https:\/\/setaf2018.com\/en\/post\/sheet-pile-wall-design\/\">sheet pile walls<\/a>, and bored pile and diaphragm wall systems described in the <a href=\"https:\/\/setaf2018.com\/en\/post\/shoring-system-design-and-analysis\/\">shoring system design guide<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>What the Winkler Model Does Not Represent<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The spring idealisation is efficient and well established, but its central simplification should be understood before the results are interpreted.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Winkler springs are independent. Each spring responds only to the displacement at its own location, with no shear transfer to its neighbours. Real ground is continuous, and a load applied at one point produces displacement at adjacent points.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The most visible consequence appears beneath a uniformly loaded flexible raft on a uniform value of k. The spring model returns uniform settlement across the entire foundation, whereas the real settlement profile is dish-shaped, with the centre settling more than the edges. Bending moments derived from a uniform k can therefore be underestimated in the central region of a large raft.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Several practical responses exist. Distributing k across the foundation \u2014 higher at the edges, lower at the centre \u2014 reproduces the settlement trough more realistically, which is one reason the settlement-based derivation described earlier evaluates multiple points rather than a single one. Where the interaction itself governs the design, a continuum model that transfers shear between adjacent points may be appropriate; the distinction between spring-based and continuum approaches is examined in more detail in the guide to <a href=\"https:\/\/setaf2018.com\/en\/post\/finite-element-analysis-fea-geotechnical-guide\/\">finite element analysis in geotechnical engineering<\/a>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This does not make the Winkler model unsuitable. For most foundation and excavation support problems the spring representation, with a properly derived k, produces results consistent with observed behaviour at a fraction of the modelling effort. The judgement lies in recognising when the interaction between adjacent points is central to the engineering question and when it is not.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Common Errors in Determining k<\/strong><\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th><strong>Error<\/strong><\/th><th><strong>Consequence<\/strong><\/th><\/tr><\/thead><tbody><tr><td>Confusing kN\/m\u00b3 with kN\/m\u00b2 or kN\/m<\/td><td>Order-of-magnitude errors in structural forces; the most frequent single mistake<\/td><\/tr><tr><td>Using k\u2083\u2080 from a plate test without size correction<\/td><td>Foundation stiffness substantially overestimated<\/td><\/tr><tr><td>Applying one value of k across an entire raft<\/td><td>Settlement trough and central bending moments not captured<\/td><\/tr><tr><td>Taking k from a general table as a soil property<\/td><td>Foundation width, rigidity, and compressible depth ignored<\/td><\/tr><tr><td>Evaluating k at a stress level different from the design condition<\/td><td>Non-linear pressure\u2013settlement response misrepresented<\/td><\/tr><tr><td>Deriving kh from a vertical modulus<\/td><td>Lateral response governed by different mobilisation and different limits<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The recurring theme is that k carries information about the problem it was derived from. A value transferred between problems carries assumptions with it that may no longer apply.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Where SETAF2018 Fits<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">SETAF2018 derives subgrade reaction moduli directly from the project model rather than requiring them to be entered as independent assumptions.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For shallow foundations, the vertical modulus is obtained from the calculated settlement of the actual foundation on the defined <a href=\"https:\/\/setaf2018.com\/en\/soil-profile\/\">soil profile<\/a>, evaluated along two orthogonal axes through the centroid, with the Vesi\u0107 equation and plate load test correction available as alternative methods. Stress increments beneath the foundation are calculated using Boussinesq integration over the loaded area, so the settlement on which kv is based reflects the layering, groundwater conditions, and compressible depth defined in the borehole data.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For piles and rigid inclusions \u2014 bored piles, micropiles, deep soil mixing columns, and jet grout columns \u2014 shaft and base moduli kvs and kvp are derived separately from the load transfer ratio, using Mindlin\u2013Geddes stress distribution for the single column. Horizontal moduli follow from the layer oedometer moduli and the column flexural rigidity.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For embedded retaining walls, kh is evaluated by the Schmitt or Chadeisson method and applied within the dependent pressures framework across all defined construction stages. The resulting values, together with the equations and code references used to obtain them, appear in the calculation <a href=\"https:\/\/setaf2018.com\/en\/reports\/\">reports<\/a> rather than only as final numbers, which allows a reviewer to trace how a given spring stiffness was established.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The intent is not to remove the engineering judgement involved in selecting a method. It is to ensure that whichever method is selected remains consistent with the soil profile, geometry, and loading already defined in the project model, rather than being derived separately and transferred by hand. A broader description of how these analyses connect to design, drawings, and documentation is available in the <a href=\"https:\/\/setaf2018.com\/en\/features\/\">platform features overview<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Frequently Asked Questions<\/strong><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">What is the modulus of subgrade reaction in simple terms?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">It is the pressure required to produce one unit of displacement in the ground at a given point, expressed in kN\/m\u00b3. It allows the soil beneath a foundation or behind a wall to be represented as a set of elastic springs, so that structural elements can be analysed on elastic support rather than within a full soil continuum.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Is ks the same as the soil elastic modulus?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">No. The soil elastic modulus Es is a material property with units of kN\/m\u00b2 and does not depend on the size of the loaded area. The modulus of subgrade reaction has units of kN\/m\u00b3 and depends on foundation width, shape, rigidity, and the depth of the compressible zone. The two are related but not interchangeable.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Can I take k from a published table?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Tabulated values are useful for preliminary estimates and for sense-checking a calculated result, but they cannot account for the geometry of the specific foundation or the layering of the specific profile. For design, k should be derived from the settlement analysis, the Vesi\u0107 equation, or a corrected plate load test.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Why does k decrease as the foundation gets larger?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">A larger foundation stresses a greater depth of soil, so more compressible material contributes to the total settlement. Since k is pressure divided by settlement, a larger settlement for the same contact pressure produces a smaller value of k.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">What is the difference between kv and kh?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">kv describes the vertical response of the ground beneath a foundation or at the shaft and base of a pile. kh describes the lateral response of soil against a wall or a laterally loaded pile. They are derived by different methods, mobilise at different displacements, and are bounded by different limiting conditions.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Which is better, the Schmitt or the Chadeisson method?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Neither is universally preferable. Schmitt relates kh to the oedometer modulus and wall rigidity and is straightforward to apply where stiffness data is reliable. Chadeisson introduces cohesion explicitly through c\u2032 and Kp, which makes it better suited to cohesive profiles where cohesion contributes meaningfully to lateral resistance. Where the two differ significantly, examining the sensitivity of wall deflection to both values is more informative than selecting one in advance.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Does a spring model always require the same k along the whole wall or foundation?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">No, and a single value is often the source of unrealistic results. Layered profiles produce different moduli at different depths, and a foundation settling in a dish shape is better represented by a distribution of k across its area than by one value applied uniformly.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">When is a spring model not sufficient?<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">When shear transfer between adjacent points in the ground governs the engineering question \u2014 for example in strongly interacting foundation systems, tunnels, or deformation-sensitive excavations adjacent to existing structures \u2014 a continuum representation may be required. For routine foundation and excavation support design, a properly derived subgrade reaction model remains an efficient and widely accepted approach.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The modulus of subgrade reaction is the ratio between the pressure applied to the ground at a given point and the displacement produced at that point. It is expressed in force per unit volume, typically kN\/m\u00b3, and it allows the ground to be represented as a system of independent elastic springs rather than as a [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":1409,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","rank_math_title":"Modulus of Subgrade Reaction: How to Determine kv and kh ","rank_math_description":"How the vertical (kv) and horizontal (kh) modulus of subgrade reaction are determined \u2014 from settlement, the Vesi\u0107 equation, plate load tests, Schmitt and Chadeisson \u2014 and why k is not a soil property.","rank_math_focus_keyword":"Modulus of Subgrade Reaction"},"categories":[22],"tags":[],"class_list":["post-1407","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/setaf2018.com\/en\/wp-json\/wp\/v2\/posts\/1407","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/setaf2018.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/setaf2018.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/setaf2018.com\/en\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/setaf2018.com\/en\/wp-json\/wp\/v2\/comments?post=1407"}],"version-history":[{"count":1,"href":"https:\/\/setaf2018.com\/en\/wp-json\/wp\/v2\/posts\/1407\/revisions"}],"predecessor-version":[{"id":1410,"href":"https:\/\/setaf2018.com\/en\/wp-json\/wp\/v2\/posts\/1407\/revisions\/1410"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/setaf2018.com\/en\/wp-json\/wp\/v2\/media\/1409"}],"wp:attachment":[{"href":"https:\/\/setaf2018.com\/en\/wp-json\/wp\/v2\/media?parent=1407"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/setaf2018.com\/en\/wp-json\/wp\/v2\/categories?post=1407"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/setaf2018.com\/en\/wp-json\/wp\/v2\/tags?post=1407"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}