The goal is to provide insight into the design of continuous footing under multiple columns, considering low and high soil and foundation stiffness. The study highlights their mutual interaction leading to different structural responses, analyzed using CSFM and verified using CDP.
Beam theory is overly conservative for continuous footings under concentrated column loads. Both nonlinear models show that soil stiffness governs load transfer and failure mechanisms, but:
Bottom line:
CSFM captures the real mechanics of footing–soil interaction with the right level of conservatism; CDP confirms the physics but exceeds what is defensible for design.
This study rigorously examines the structural performance of a continuous footing supporting multiple columns under varying soil and foundation stiffness parameters. The primary aim is to elucidate the mutual interaction between columns and the underlying soil, and to evaluate how this interplay influences load distribution and the overall structural behavior of the footing. Both low-stiffness (LSS) and high-stiffness (HSS) soil conditions are systematically analyzed to determine their impact on displacement, stress distribution, and load transfer mechanisms, particularly in scenarios involving concentrated column loads.
The analysis utilizes the Compatible Stress Field Method (CSFM) in three dimensions. The outcomes derived from CSFM are meticulously validated against simulations conducted using the Concrete Damage Plasticity (CDP) model as well as traditional verification methodologies, ensuring a high degree of reliability and precision in the 3D predictions.
The results of this investigation offer an enhanced comprehension of footing-soil-structure interaction, identify limitations inherent in conventional design assumptions, and underscore the efficacy and robustness of the CSFM for designing and verifying continuous footings under localized loading and variable soil conditions. This research contributes to advancing foundation design methodologies and provides valuable insights for developing more resilient structural solutions across diverse geotechnical scenarios.
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"value": "<p>The study investigates the <strong>structural response of continuous footings under concentrated loads</strong> resting on an <strong>elastic foundation</strong>. The analysis aims to verify the interaction between <strong>beam bending stiffness</strong> (foundation flexural rigidity) and <strong>subgrade stiffness</strong> (soil modulus), which together govern the deformation profile, bending moments, and shear force distribution along the footing.</p>\n<p>The analytical model follows the <strong>Euler–Bernoulli beam theory</strong> on a <strong>Winkler-type foundation</strong>, assuming an infinitely long beam subjected to a single concentrated load. This approach allows a direct comparison of deformation shapes and internal force gradients for different stiffness ratios between the foundation and the supporting soil.</p>\n<p>Let us discuss the four possible combinations:</p>\n<ol>\n <li><strong>Low beam bending stiffness + Low soil stiffness </strong></li>\n <li><strong>High beam bending stiffness + Low soil stiffness (next verification article)</strong></li>\n <li><strong>Low beam bending stiffness + High soil stiffness </strong></li>\n <li><strong>High beam bending stiffness + High soil stiffness (next verification article)</strong></li>\n</ol>\n<p>For the purpose of this verification, continuous footings with <strong>low bending stiffness were chosen for a study on numerical models</strong>.</p>\n<p>Fig. 1 shows the four combinations of footing systems. </p>\n<figure data-asset-id=\"34aa0996-5c6f-4325-ba9b-10c1ff90adc0\" data-image-id=\"34aa0996-5c6f-4325-ba9b-10c1ff90adc0\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/eb583615-5432-4411-abd7-e697581ad9ed/04.png\" data-asset-id=\"34aa0996-5c6f-4325-ba9b-10c1ff90adc0\" data-image-id=\"34aa0996-5c6f-4325-ba9b-10c1ff90adc0\" alt=\"\"></figure>\n<p><em>01) Continuous footing strip with multiple columns (use-case)</em></p>\n<h3>Material models</h3>\n<p>Material behavior and properties have been adopted from <strong>EN 1992-1-1</strong> [1]. The <strong>design</strong> properties of concrete grade <strong>C30/37</strong> and the corresponding reinforcement <strong>B500B</strong> with hardening have been specified (Fig.2).</p>\n<figure data-asset-id=\"2b4568bf-ae6f-4807-98e8-462231211bee\" data-image-id=\"2b4568bf-ae6f-4807-98e8-462231211bee\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/6b48c98e-0dd2-4185-afc1-b5c218bc2ef9/56.png\" data-asset-id=\"2b4568bf-ae6f-4807-98e8-462231211bee\" data-image-id=\"2b4568bf-ae6f-4807-98e8-462231211bee\" alt=\"\"></figure>\n<p><em>02) Material models</em></p>"
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"value": "<p>An <strong>infinite Euler–Bernoulli beam on a Winkler elastic foundation</strong> describes how a long (theoretically infinite) beam behaves when supported continuously by an elastic medium, such as soil or bedding. The Winkler model assumes that the foundation reacts proportionally to local deflection, like a bed of independent springs. The governing differential equation <strong>EIyw(z)^(4) + kw(z) = q(x)</strong> balances bending stiffness <strong>EI </strong>and foundation stiffness <strong>k </strong>under load <strong>q(x) </strong>that represent, in this case, the local force. The key parameter is characteristic length <strong>L = (EI/k)</strong><strong><sup>1/4</sup></strong>, defining how far deformations spread. For a concentrated load, deflection decays exponentially and oscillates slightly as it propagates along the beam. The solution enables the prediction of <strong>deflection, rotation, bending moment, and shear force</strong>, critical for <strong>designing foundations</strong>, pavements, rails, or pipelines resting on elastic supports.</p>\n<h3>Model assembly</h3>\n<figure data-asset-id=\"84663926-36ea-497a-b35f-e6217a96b9ce\" data-image-id=\"84663926-36ea-497a-b35f-e6217a96b9ce\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/687afa2b-83af-41cc-a849-1c822fef4304/05.png\" data-asset-id=\"84663926-36ea-497a-b35f-e6217a96b9ce\" data-image-id=\"84663926-36ea-497a-b35f-e6217a96b9ce\" alt=\"\"></figure>\n<p><em>03) Infinite beam on the elastic foundation </em></p>\n<h3>Solution for low-stiffness soils (LSS)</h3>\n<h4>Low beam bending stiffness + Low soil stiffness</h4>\n<ul>\n <li>Suitable for:\n <ul>\n <li>Better energy dissipation</li>\n <li>Moderate risk of punching failure</li>\n </ul>\n </li>\n <li>Be cautious:\n <ul>\n <li>Excessive deformations</li>\n <li>Sensitive to differential settlements</li>\n </ul>\n </li>\n</ul>\n<figure data-asset-id=\"932cd712-3d30-489c-91ce-5af765d17221\" data-image-id=\"932cd712-3d30-489c-91ce-5af765d17221\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/b9cbdea1-6ce6-4ba8-bf07-06bdb55e8abd/10%20LOW%20EI%20LOW%20K.png\" data-asset-id=\"932cd712-3d30-489c-91ce-5af765d17221\" data-image-id=\"932cd712-3d30-489c-91ce-5af765d17221\" alt=\"\"></figure>\n<p><em>04) Linear beam model, deformations, reactions, moments, shear forces </em></p>\n<h4>High beam bending stiffness + Low soil stiffness</h4>\n<ul>\n <li>Suitable for:\n <ul>\n <li>Improved global stiffness.</li>\n </ul>\n </li>\n <li>Be cautious:\n <ul>\n <li>Risk of cracking due to high bending stresses.</li>\n <li>Limited adaptability to uneven soil.</li>\n </ul>\n </li>\n</ul>\n<figure data-asset-id=\"c89c66e9-2c15-45a6-93d6-435ccb52860c\" data-image-id=\"c89c66e9-2c15-45a6-93d6-435ccb52860c\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/d1843ebe-d949-4365-b62c-98ffa0566c25/12%20HIGH%20EI%20LOW%20K.png\" data-asset-id=\"c89c66e9-2c15-45a6-93d6-435ccb52860c\" data-image-id=\"c89c66e9-2c15-45a6-93d6-435ccb52860c\" alt=\"\"></figure>\n<p><em>05) Linear beam model, deformations, reactions, moments, shear forces </em></p>\n<p><strong>Figure 06 illustrates the behavior for a relatively low-stiffness soil</strong> with a subgrade modulus of<strong> 16,000 kN/m³</strong> and <strong>varying heights of the footing strip</strong>.</p>\n<figure data-asset-id=\"bb64b849-a702-4dfb-a2f8-202b558af3c2\" data-image-id=\"bb64b849-a702-4dfb-a2f8-202b558af3c2\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/4f605057-ab31-4b7d-8129-177c72f19ff8/06-Loose%20sand.png\" data-asset-id=\"bb64b849-a702-4dfb-a2f8-202b558af3c2\" data-image-id=\"bb64b849-a702-4dfb-a2f8-202b558af3c2\" alt=\"\"></figure>\n<p><em>06) Interaction of relatively low-stiffness soil with varying stiffness of the beam (closed-form solution)</em></p>\n<h3>Solution for high-stiffness soils (HSS)</h3>\n<h4>Low beam bending stiffness + High soil stiffness</h4>\n<ul>\n <li>Suitable for:\n <ul>\n <li>Efficient stress transfer to the stiff soil</li>\n <li>Lower moment demand</li>\n </ul>\n </li>\n <li>Be cautious:\n <ul>\n <li>High local shear forces</li>\n <li><strong>The most significant chance of punching shear failure</strong></li>\n </ul>\n </li>\n</ul>\n<figure data-asset-id=\"0f7f92a8-b63f-4850-a77c-75506c5f3106\" data-image-id=\"0f7f92a8-b63f-4850-a77c-75506c5f3106\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/5f084214-5e3d-4ef8-8200-6455d39005dd/11%20LOW%20EI%20HIGH%20K.png\" data-asset-id=\"0f7f92a8-b63f-4850-a77c-75506c5f3106\" data-image-id=\"0f7f92a8-b63f-4850-a77c-75506c5f3106\" alt=\"\"></figure>\n<p><em>07) Linear beam model, deformations, reactions, moments, shear forces </em></p>\n<h4>High beam bending stiffness + High soil stiffness</h4>\n<ul>\n <li>Suitable for:\n <ul>\n <li>Stable system, minimal deflections</li>\n <li>Predictable linear response</li>\n </ul>\n </li>\n <li>Be cautious:\n <ul>\n <li>Higher construction cost</li>\n </ul>\n </li>\n</ul>\n<figure data-asset-id=\"584ad683-e4d6-4587-9f2f-d45f81e1cefc\" data-image-id=\"584ad683-e4d6-4587-9f2f-d45f81e1cefc\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/ca70cf12-42f5-442e-b81b-91c3b2ada235/13%29HIGH%20EI%20HIGH%20K.png\" data-asset-id=\"584ad683-e4d6-4587-9f2f-d45f81e1cefc\" data-image-id=\"584ad683-e4d6-4587-9f2f-d45f81e1cefc\" alt=\"\"></figure>\n<p><em>08) Linear beam model, deformations, reactions, moments, shear forces </em></p>\n<figure data-asset-id=\"58deb677-5a2d-4651-8750-0c66465e729d\" data-image-id=\"58deb677-5a2d-4651-8750-0c66465e729d\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/baeb2825-9cac-44bb-9b9d-e833be9d3f42/07%20-%20DENSE%20SAND.png\" data-asset-id=\"58deb677-5a2d-4651-8750-0c66465e729d\" data-image-id=\"58deb677-5a2d-4651-8750-0c66465e729d\" alt=\"\"></figure>\n<p><em>09) Interaction of high-stiffness soil with varying stiffness of the beam (closed-form solution)</em></p>\n<h3>Response of a beam for low/high stiffness soils</h3>\n<figure data-asset-id=\"a034528c-d865-4d6c-b744-60c6cd29bad9\" data-image-id=\"a034528c-d865-4d6c-b744-60c6cd29bad9\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/cd1b748e-8bfb-459d-8ba2-ba94063e9de1/08%29ElasticFoundation_LooseDense.png\" data-asset-id=\"a034528c-d865-4d6c-b744-60c6cd29bad9\" data-image-id=\"a034528c-d865-4d6c-b744-60c6cd29bad9\" alt=\"\"></figure>\n<p><em>10) Interaction of low and high-stiffness soil with varying stiffness of the beam </em></p>"
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"value": "<p>The most frequently employed solution adopted by structural engineers for the current model is a beam model integrated with code compliance checks in accordance with applicable standards. The setup of the testing model remains consistent across all levels of model complexity and represents a <strong>column</strong> with a square cross-section measuring <strong>500 x 500 mm and a length of 1,000 mm</strong>, a <strong>footing strip</strong> with a unit width of <strong>1,000 mm</strong>, and a <strong>length of 6,000 mm</strong>. The height of the footing strip is a variable parameter. For the current verification, a <strong>height of 250 mm is utilized</strong>.</p>\n<p>The bottom face of the footing strip is supported by compression-only springs with <strong>either </strong><a href=\"https://www.researchgate.net/figure/Values-of-modulus-of-subgrade-reaction-for-different-soils_tbl3_338402472\"><strong>low soil stiffness of 16,000 kN/m³ or the high soil stiffness of 128,000 kN/m³</strong></a>. Symmetry boundary conditions constrain the left and right ends of the footing strip. </p>\n<p>It is essential to note that <strong>all models are design models</strong>. <strong>For simulation and code-check verification, the partial factors for materials have been applied.</strong></p>\n<figure data-asset-id=\"004cda5c-2a80-4dca-9f39-57d5aca82fa0\" data-image-id=\"004cda5c-2a80-4dca-9f39-57d5aca82fa0\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/071ca304-f4e6-4364-abed-bfab3331ff53/09.png\" data-asset-id=\"004cda5c-2a80-4dca-9f39-57d5aca82fa0\" data-image-id=\"004cda5c-2a80-4dca-9f39-57d5aca82fa0\" alt=\"\"></figure>\n<p><em>11) Dimensions and analytical model</em></p>\n<h3>Linear beam model – Low-Stiffness Soil (LSS)</h3>\n<p>Once the simulation is conducted on the beam model, the standard code checks can be employed. The designed reinforcement adheres to the minimum detailing requirements specified by EN 1992-1-1 [1]. A minimum reinforcement ratio is applied to both longitudinal bars and stirrups. <strong>The simulation is executed using a modulus of elasticity of 10 GPa, representing the secant modulus of the designated concrete material</strong>. Due to the hyperstatic nature of the structure, the modulus influences the redistribution of internal forces. </p>\n<figure data-asset-id=\"02133f65-3602-4046-b5a7-292ed9ce7f60\" data-image-id=\"02133f65-3602-4046-b5a7-292ed9ce7f60\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/c2ba2d5a-c8a6-43e1-9983-fee1224c2375/14.png\" data-asset-id=\"02133f65-3602-4046-b5a7-292ed9ce7f60\" data-image-id=\"02133f65-3602-4046-b5a7-292ed9ce7f60\" alt=\"\"></figure>\n<p><em>12) Linear beam model – ultimate load for passing ULS checks</em></p>\n<p>The bending moment directly beneath the column reaches the <strong>ultimate value of 60.1 kNm</strong> under an <strong>axial force in the column of -245 kN</strong>. The second critical point is situated within the zone of maximum shear, where the <strong>interaction of a shear force of -86.4 kN and a corresponding bending moment of 44.8 kNm</strong> results in an interaction check, which also remains within acceptable limits with a utilization ratio of <strong>96.6%</strong>. The most critical location on the structure is directly beneath the column, and the <strong>failure mode involves the concrete in compression and the longitudinal reinforcement bars in tension. The shear capacity indicates that it is not critical for this case.</strong></p>\n<figure data-asset-id=\"99a4f0cd-9bca-41f7-9d41-e5449f14df36\" data-image-id=\"99a4f0cd-9bca-41f7-9d41-e5449f14df36\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/6bb1085f-0551-4118-a77e-c5c73d89f432/15.png\" data-asset-id=\"99a4f0cd-9bca-41f7-9d41-e5449f14df36\" data-image-id=\"99a4f0cd-9bca-41f7-9d41-e5449f14df36\" alt=\"\"></figure>\n<p><em>13) Linear beam model – code-check for low-stiffness soil</em></p>\n<h3>Linear beam model – High-Stiffness Soil (HSS)</h3>\n<p>The <strong>high-stiffness soil</strong> in this scenario, <strong>dense sand with a subgrade modulus of 128,000 kN/m³</strong>, significantly alters the behavior of the structure. The load is concentrated directly beneath the column area. The contact area exhibits a higher stress contact gradient and magnitude. <strong>The ultimate resistance in the column of -540 kN</strong> has increased by a <strong>factor of 2.2 compared to low-stiffness soil</strong>. The shear force profile is steeper, and the bending moment is more localized. It leads to a more-prone structure to punching shear failure.</p>\n<figure data-asset-id=\"36ec34a0-63ed-452b-a152-210792ac4163\" data-image-id=\"36ec34a0-63ed-452b-a152-210792ac4163\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/823bf064-f7a6-4abf-bb42-c49ce6b78af5/16.png\" data-asset-id=\"36ec34a0-63ed-452b-a152-210792ac4163\" data-image-id=\"36ec34a0-63ed-452b-a152-210792ac4163\" alt=\"\"></figure>\n<p><em>14) Linear beam model – ultimate load for passing ULS checks</em></p>\n<p>The <strong>maximum bending moment</strong> concentrated beneath the column<strong> is 60.7 kNm</strong>, <strong>attributable to the maximum bearing capacity of the section in bending</strong>. The extreme shear force is displaced proximally to the column area and reaches a magnitude of -132 kN, with the corresponding moment being 38.1 kNm. In the interaction code check, <strong>the theta angle for the concrete strut </strong>has been adjusted from 21.5 degrees to 23 degrees. The Eurocode permits the adjustment of the strut angle within the range of 21.5 to 45 degrees. It has been observed that an angle of 21.5 degrees results in overutilization of capacity, primarily attributed to bending. By accommodating the variability prescribed by code requirements, the failing check has been successfully addressed through the application of an alternative strut angle.</p>\n<p><strong>The critical mode of failure involves the concrete in compression and the longitudinal reinforcement bars in tension</strong>. </p>\n<figure data-asset-id=\"5ae18730-c254-462d-b15e-136db425a08c\" data-image-id=\"5ae18730-c254-462d-b15e-136db425a08c\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/e3262261-9493-432b-b5ab-0b0f6f0e4d45/17.png\" data-asset-id=\"5ae18730-c254-462d-b15e-136db425a08c\" data-image-id=\"5ae18730-c254-462d-b15e-136db425a08c\" alt=\"\"></figure>\n<p><em>15) Linear beam model – code-check for high-stiffness soils</em></p>"
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}Name: Nonlinear solution - CSFM (plane stress)
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"value": "<h3>Assumptions and model assembly</h3>\n<p>The theory employed in the nonlinear solution is called CSFM (Compatible Stress Field Method) and is traced in the <a data-item-id=\"0000c94c-b603-48c4-8d31-bc56d7c95886\" href=\"\">theoretical background</a>[2].</p>\n<p><strong>Assumptions and attributes of the model: </strong></p>\n<ul>\n <li>Materially Nonlinear Analysis (MNA)</li>\n <li>Plane stress model. </li>\n <li>Compression-only line supports (low/high stiffness).</li>\n <li>Symmetry constraints are positioned on the left and right edges of the footing strip.</li>\n <li>A thick plate 100 mm on the top of the column to stem local stress concentration underneath the point force load.</li>\n <li>All the material properties for concrete C30/37 and reinforcement bars B500B are engaged as design values with partial factors according to EN 1992-1-1 [1]. </li>\n <li>Mesh factor 1 - a minimum of four elements over the shortest edge.</li>\n</ul>\n<figure data-asset-id=\"dacf243d-0927-4906-9c15-f985d5b516a8\" data-image-id=\"dacf243d-0927-4906-9c15-f985d5b516a8\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/96ff2219-ce1d-48a4-9ed5-1458db4db5ad/18.png\" data-asset-id=\"dacf243d-0927-4906-9c15-f985d5b516a8\" data-image-id=\"dacf243d-0927-4906-9c15-f985d5b516a8\" alt=\"\"></figure>\n<p><em>16) 2D model + reinforcement bars layout</em></p>\n<h3>2D CSFM – Low-Stiffness-Soil (LSS)</h3>\n<p>The <strong>maximum applied force</strong> capable of addressing the failure modes <strong>has reached -1,340 kN</strong>. The vertical force has resulted in a<strong> contact stress of 0.59 MPa</strong>. The observed trend in <strong>contact stress indicates nonlinearity in tension</strong>, attributable to the uplift of the left and right sections near the symmetry constraints. <strong>The failure modes occurred in compression at the interface between the column edge and the face in contact with the footing, concurrently, by the tensile rupture of longitudinal reinforcement</strong>.</p>\n<figure data-asset-id=\"11572d08-089b-4a6f-ac2d-3f2b28e50870\" data-image-id=\"11572d08-089b-4a6f-ac2d-3f2b28e50870\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/5c6642db-5389-47ea-8fd5-3f70de6b1ba7/19.png\" data-asset-id=\"11572d08-089b-4a6f-ac2d-3f2b28e50870\" data-image-id=\"11572d08-089b-4a6f-ac2d-3f2b28e50870\" alt=\"\"></figure>\n<p><em>17) Maximum applied force, contact stress, and failure modes</em></p>\n<figure data-asset-id=\"af2f3d14-3c25-41e5-a24b-bbab599caebd\" data-image-id=\"af2f3d14-3c25-41e5-a24b-bbab599caebd\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/5b1bff8d-6f36-427f-bf19-b5b9520c82f9/20.png\" data-asset-id=\"af2f3d14-3c25-41e5-a24b-bbab599caebd\" data-image-id=\"af2f3d14-3c25-41e5-a24b-bbab599caebd\" alt=\"\"></figure>\n<p><em>18) Principal stress in compression, compression plastic strain, stress in reinforcements</em></p>\n<p><strong>The stress in the stirrups has reached a maximum of 201 MPa</strong>, leading to the consensus that this stress level is significantly below the ultimate limit of utilization. <strong>The failure mode in shear does not pose a threat in this context. </strong></p>\n<figure data-asset-id=\"dfba1c83-83aa-48be-855b-eb822aa476c9\" data-image-id=\"dfba1c83-83aa-48be-855b-eb822aa476c9\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/88595738-da33-4606-a0ff-1adf87d8170b/21.png\" data-asset-id=\"dfba1c83-83aa-48be-855b-eb822aa476c9\" data-image-id=\"dfba1c83-83aa-48be-855b-eb822aa476c9\" alt=\"\"></figure>\n<p><em>19) Nonlinear deflections, stress in stirrups, and detailed insight into the failure modes of longitudinal bars</em></p>\n<h3>2D CSFM – High-Stiffness-Soil (HSS)</h3>\n<p>The maximum load at which all governing failure mechanisms can still be resisted is <strong>–2,652 kN</strong>. The corresponding vertical reaction induces a <strong>contact stress of 1.99 MPa</strong> at the footing–soil interface. The evolution of contact stress exhibits marked nonlinearity in tension, resulting from the <strong>uplift of the footing edges</strong>. This loss of contact occurs primarily along the left and right ends of the model.</p>\n<p>The dominant failure mechanism is <strong>compressive crushing</strong> at the interface between the column edge and the loaded face of the footing. <strong>Simultaneously,</strong> <strong>tensile rupture of the bottom-layer longitudinal reinforcement within the footing occurs.</strong></p>\n<figure data-asset-id=\"7fecdd4e-6adb-4fbd-b83b-75957f9e94de\" data-image-id=\"7fecdd4e-6adb-4fbd-b83b-75957f9e94de\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/a42dc56d-2921-4d73-b5ad-acd8ce31cd85/22.png\" data-asset-id=\"7fecdd4e-6adb-4fbd-b83b-75957f9e94de\" data-image-id=\"7fecdd4e-6adb-4fbd-b83b-75957f9e94de\" alt=\"\"></figure>\n<p><em>20) Maximum applied force, contact stress, and failure modes</em></p>\n<figure data-asset-id=\"a9f8d875-01ff-4b59-9b6f-699edfcd69d8\" data-image-id=\"a9f8d875-01ff-4b59-9b6f-699edfcd69d8\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/65bc11c5-c3e9-48c5-9982-29045448a648/23.png\" data-asset-id=\"a9f8d875-01ff-4b59-9b6f-699edfcd69d8\" data-image-id=\"a9f8d875-01ff-4b59-9b6f-699edfcd69d8\" alt=\"\"></figure>\n<p><em>21) Principal stress in compression, compression plastic strain, stress in reinforcements</em></p>\n<p>The <strong>nonlinear deflections demonstrate substantially smaller displacements under higher loads compared to the LSS variants</strong>. Stress is predominantly concentrated beneath the column area, with <strong>stirrups being underutilized at approximately 186 MPa</strong>. However, the model exhibits evidence of <strong>local softening</strong> on the bottom face of the footing strip due to high tensile stress in reinforcement bars.</p>\n<figure data-asset-id=\"dd1fbbc5-74e5-4727-adda-608626c3ef90\" data-image-id=\"dd1fbbc5-74e5-4727-adda-608626c3ef90\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/bdbeea55-80d6-48e4-ab45-bbcd0229d67c/24.png\" data-asset-id=\"dd1fbbc5-74e5-4727-adda-608626c3ef90\" data-image-id=\"dd1fbbc5-74e5-4727-adda-608626c3ef90\" alt=\"\"></figure>\n<p><em>22) Nonlinear deflections, stress in stirrups, and localized compression softening</em></p>"
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"value": "<p>The theory used in the nonlinear solution is called 3D CSFM and is outlined in the <a data-item-id=\"66c6fbb8-b380-43c7-8b4f-9d41d29a42f2\" href=\"\">theoretical background</a> [3]. All the presumptions for the designed calculation procedure are explained in detail there.</p>\n<p><strong>Assumptions and attributes of the model: </strong></p>\n<ul>\n <li>Materially Nonlinear Analysis (MNA)</li>\n <li>3D solution – volume elements.</li>\n <li>Mohr-Coulomb plasticity theory - zero angle of internal friction for concrete behaviour.</li>\n <li>Compression-only surface supports (low/high stiffness).</li>\n <li>Symmetry constraints are positioned on the left and right edges of the footing strip.</li>\n <li>A thick plate 100 mm on the top of the column to stem local stress concentration underneath the point force load.</li>\n <li>Bond model and tension stiffening are considered.</li>\n <li>Stress triaxiality and confinement effect.</li>\n <li>Compression softening is not a part of the implemented solution.</li>\n <li>Mesh factor 1 - recommended computational settings.</li>\n</ul>\n<figure data-asset-id=\"32e584d7-3ca8-433a-a83f-39697e5bb222\" data-image-id=\"32e584d7-3ca8-433a-a83f-39697e5bb222\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/8ece7aac-dbb0-44be-98d2-7e6b0ac6432d/25.png\" data-asset-id=\"32e584d7-3ca8-433a-a83f-39697e5bb222\" data-image-id=\"32e584d7-3ca8-433a-a83f-39697e5bb222\" alt=\"\"></figure>\n<p><em>23) 3D model + reinforcement bars layout</em></p>\n<h3>3D CSFM – Low-Stiffness-Soil (LSS)</h3>\n<p>The <strong>maximum axial force designated in the model reached -980 kN</strong> due to failure modes involving tensile rupture of the longitudinal reinforcement in the encircling area of the column. Transverse compression forces are restrained by the stirrups, which in the column zone are utilized during yielding and contribute to <strong>additional failure mode of the horizontal stirrup legs</strong> <strong>caused by transverse tensile stress evolutions that cannot be captured in the plane stress solution</strong>. Over-compression and crushing of the concrete occur at the interface area between the column and footing. The confinement effect is localized in this region, based on the reinforcement effect and the stiffness of the footing strip. <strong>The failure mechanism involves concrete crushing, tensile rupture of the longitudinal reinforcement, and the horizontal legs of the stirrups in tension</strong>.</p>\n<figure data-asset-id=\"56f0c91a-6aa6-498c-a304-4a4d3381b5da\" data-image-id=\"56f0c91a-6aa6-498c-a304-4a4d3381b5da\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/bf293d64-4a41-4259-a57e-58b4da9e4490/26.png\" data-asset-id=\"56f0c91a-6aa6-498c-a304-4a4d3381b5da\" data-image-id=\"56f0c91a-6aa6-498c-a304-4a4d3381b5da\" alt=\"\"></figure>\n<p><em>24) Maximum applied force, failure modes, and transverse stress distribution</em></p>\n<figure data-asset-id=\"aaa29b8a-13f3-4a48-a83f-2fabb5fa090f\" data-image-id=\"aaa29b8a-13f3-4a48-a83f-2fabb5fa090f\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/6ec29758-6357-4db5-a5b6-402159054543/27.png\" data-asset-id=\"aaa29b8a-13f3-4a48-a83f-2fabb5fa090f\" data-image-id=\"aaa29b8a-13f3-4a48-a83f-2fabb5fa090f\" alt=\"\"></figure>\n<p><em>25) Minimum principal stress Sigma 3, confinement effect – ratio between triaxial vs uniaxial stress</em></p>\n<figure data-asset-id=\"f5cbabe3-07cc-44c6-bf5b-4f3387a02bca\" data-image-id=\"f5cbabe3-07cc-44c6-bf5b-4f3387a02bca\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/2d53ac43-976f-4e2a-af79-510979b664b5/28.png\" data-asset-id=\"f5cbabe3-07cc-44c6-bf5b-4f3387a02bca\" data-image-id=\"f5cbabe3-07cc-44c6-bf5b-4f3387a02bca\" alt=\"\"></figure>\n<p><em>26) Compressive plastic strain and stress in reinforcements</em></p>\n<figure data-asset-id=\"9080d1b3-fdef-4024-97bd-66518280464e\" data-image-id=\"9080d1b3-fdef-4024-97bd-66518280464e\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/a0cbe418-a7e6-41ca-82cd-143ac9fa41ea/29.png\" data-asset-id=\"9080d1b3-fdef-4024-97bd-66518280464e\" data-image-id=\"9080d1b3-fdef-4024-97bd-66518280464e\" alt=\"\"></figure>\n<p><em>27) Detailed detection of critical stress on the longitudinal bars and stirrups </em></p>\n<figure data-asset-id=\"5e0c37c2-77ec-4c53-9953-4431eb8fe9dd\" data-image-id=\"5e0c37c2-77ec-4c53-9953-4431eb8fe9dd\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/cc255426-0143-44cc-ba0e-b9a1225c739e/30.png\" data-asset-id=\"5e0c37c2-77ec-4c53-9953-4431eb8fe9dd\" data-image-id=\"5e0c37c2-77ec-4c53-9953-4431eb8fe9dd\" alt=\"\"></figure>\n<p><em>28) Nonlinear deflections</em></p>\n<h3>3D CSFM – High-Stiffness-Soil (HSS)</h3>\n<p><strong>The force absorbed by the footing strip reached -2,116 kN</strong>, which is approximately 215% higher bearing capacity than in LSS. <strong>The failure mode involves concrete crushing, tensile rupture of the longitudinal reinforcement, and the horizontal legs of the stirrups in tension</strong>.</p>\n<figure data-asset-id=\"b513e904-ffce-4b69-8fce-c497f3f8231b\" data-image-id=\"b513e904-ffce-4b69-8fce-c497f3f8231b\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/73fe5bde-c925-4a8f-a558-9c7e36b3ff1b/32.png\" data-asset-id=\"b513e904-ffce-4b69-8fce-c497f3f8231b\" data-image-id=\"b513e904-ffce-4b69-8fce-c497f3f8231b\" alt=\"\"></figure>\n<p><em>29) Maximum applied force, failure modes, and transverse stress distribution</em></p>\n<figure data-asset-id=\"ad4dbdae-d05a-4a2b-8cc5-2fdb76c33b83\" data-image-id=\"ad4dbdae-d05a-4a2b-8cc5-2fdb76c33b83\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/d2579180-ce0d-4230-b8c2-7efd38cd9068/33.png\" data-asset-id=\"ad4dbdae-d05a-4a2b-8cc5-2fdb76c33b83\" data-image-id=\"ad4dbdae-d05a-4a2b-8cc5-2fdb76c33b83\" alt=\"\"></figure>\n<p><em>30) Minimum principal stress Sigma 3, confinement effect – ratio between triaxial vs uniaxial stress</em></p>\n<figure data-asset-id=\"32387b59-aea2-48de-b2ec-17a0d10d6733\" data-image-id=\"32387b59-aea2-48de-b2ec-17a0d10d6733\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/b40080e6-7843-4073-a7a0-8fa658e49f3e/34.png\" data-asset-id=\"32387b59-aea2-48de-b2ec-17a0d10d6733\" data-image-id=\"32387b59-aea2-48de-b2ec-17a0d10d6733\" alt=\"\"></figure>\n<p><em>31) Compressive plastic strain in concrete and stress in reinforcements</em></p>\n<p><strong>The maximum shear stress exerted on the inner closed stirrups has reached a value of 298 MPa</strong>, which remains within the elastic range as defined by the material. This observation leads to the conclusion that <strong>punching shear failure was not the predominant failure </strong>mode in this particular instance.</p>\n<figure data-asset-id=\"57249239-205d-4f94-8f8e-1b04c5bebd9b\" data-image-id=\"57249239-205d-4f94-8f8e-1b04c5bebd9b\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/e418ffb5-b3d3-4954-a919-2f38cb61f5b7/35.png\" data-asset-id=\"57249239-205d-4f94-8f8e-1b04c5bebd9b\" data-image-id=\"57249239-205d-4f94-8f8e-1b04c5bebd9b\" alt=\"\"></figure>\n<p><em>32) Detailed detection of critical stress on the longitudinal bars and stirrups </em></p>\n<figure data-asset-id=\"9836b5bc-365c-44f3-bc0d-d66621ff70e3\" data-image-id=\"9836b5bc-365c-44f3-bc0d-d66621ff70e3\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/3109ce36-5ef3-41e4-a497-693ffa6b2e5a/36.png\" data-asset-id=\"9836b5bc-365c-44f3-bc0d-d66621ff70e3\" data-image-id=\"9836b5bc-365c-44f3-bc0d-d66621ff70e3\" alt=\"\"></figure>\n<p><em>33) Nonlinear deflections </em></p>"
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"value": "<p>The theory used in the nonlinear solution is called CDP and is outlined in the <a href=\"https://classes.engineering.wustl.edu/2009/spring/mase5513/abaqus/docs/v6.6/books/usb/default.htm?startat=pt05ch18s05abm36.html\">theoretical background</a> [4]. The material model is a part of the ABAQUS library for concrete simulation.</p>\n<p>The simulation was terminated when the model reached its maximum bearing capacity, subsequently transitioning to the plastic state and the post-critical state, as observed on the load-deformation curve. No predefined stop criteria were applied in this case, as in CSFM.</p>\n<p> <strong>Assumptions and attributes of the model: </strong></p>\n<ul>\n <li>Utilizes concepts of isotropic damaged elasticity in conjunction with isotropic tensile and compressive plasticity to characterize the inelastic behavior of concrete.</li>\n <li>It is designed for applications in which concrete is subjected to monotonic, cyclic, and/or dynamic loading under low confining pressures.</li>\n <li>Consists of the combination of non-associated multi-hardening plasticity and scalar (isotropic) damaged elasticity to accurately describe the irreversible damage that occur during the fracturing process.</li>\n <li>Compression softening and tension stiffening are employed under assumptions of perfect bonding for reinforcement bars modeled independently. </li>\n <li>Total number of nodes 46,003</li>\n <li>Total number of elements 37,892\n <ul>\n <li>27,600 linear hexahedral element C3D8 - full integration, element deletion-on</li>\n <li>10,192 linear line elements T3D2</li>\n <li>Mesh size - 50 mm on the concrete and reinforcements</li>\n </ul>\n </li>\n <li>The interlayer between compression-only constraints representing soil and concrete footing strip provides information about the contact status and contact stress.</li>\n <li>A thin layer of 10 mm with modulus elasticity 1,000 MPa to emulate an interlayer for the results outputs from soil pressure.</li>\n</ul>\n<figure data-asset-id=\"73d3cf14-158d-4730-970d-dab692bcbe55\" data-image-id=\"73d3cf14-158d-4730-970d-dab692bcbe55\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/ce5d3295-ac83-44c6-b88a-818b0d104543/40.png\" data-asset-id=\"73d3cf14-158d-4730-970d-dab692bcbe55\" data-image-id=\"73d3cf14-158d-4730-970d-dab692bcbe55\" alt=\"\"></figure>\n<p><em>34) Mode + reinforcements, mesh</em></p>\n<h3>Material models for Concrete-Damage-Plasticity</h3>\n<p>The evolution of the material model under <strong>compression exhibits softening after reaching 20 MPa</strong>, while in tension, it exhibits a value of 0.2 MPa, which approximately simulates zero tensile strength. This precise zero value results in the model diverging. </p>\n<figure data-asset-id=\"50a71029-9570-4c9b-bf26-cd689bdc0354\" data-image-id=\"50a71029-9570-4c9b-bf26-cd689bdc0354\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/d65eb1d5-df9a-4483-9a18-8f4d62d82d86/37.png\" data-asset-id=\"50a71029-9570-4c9b-bf26-cd689bdc0354\" data-image-id=\"50a71029-9570-4c9b-bf26-cd689bdc0354\" alt=\"\"></figure>\n<p><em>35) Material models for concrete in compression, tension, and reinforcement</em></p>\n<h3>Concrete-Damage-Plasticity - Low-Stiffness-Soil (LSS)(GMNA)</h3>\n<p>The ultimate loading force imparted to the model is -2,029 kN. <strong>The minimum (compressive) strain observed is -0.04</strong>, located at the intersection of the column and footing. Conversely, <strong>the maximum (tensile) strain is identified on the bottom face of the footing, measuring 0.105</strong>. <strong>Excessive compressive strains </strong>have been assessed as the primary failure mechanism, characterized by <strong>concrete crushing</strong>.</p>\n<figure data-asset-id=\"ac396d36-fc3c-4f91-a637-ea762da60e49\" data-image-id=\"ac396d36-fc3c-4f91-a637-ea762da60e49\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/2b86cada-607d-4a5c-bdec-9fccab53fb20/41.png\" data-asset-id=\"ac396d36-fc3c-4f91-a637-ea762da60e49\" data-image-id=\"ac396d36-fc3c-4f91-a637-ea762da60e49\" alt=\"\"></figure>\n<p><em>36) Maximum applied force, Minimum principal stress</em></p>\n<figure data-asset-id=\"aa5536bc-b750-4205-8023-31924068e961\" data-image-id=\"aa5536bc-b750-4205-8023-31924068e961\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/442c7985-6454-4fbb-9d44-279bb84e8a7c/47.png\" data-asset-id=\"aa5536bc-b750-4205-8023-31924068e961\" data-image-id=\"aa5536bc-b750-4205-8023-31924068e961\" alt=\"\"></figure>\n<p><em>37) Minimum plastic strain, Maximum plastic strain</em></p>\n<figure data-asset-id=\"b1cb32c0-190e-44ee-9e21-dd6df150b309\" data-image-id=\"b1cb32c0-190e-44ee-9e21-dd6df150b309\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/c2cabeea-9ba3-43a5-93bc-79a12cf06042/42.png\" data-asset-id=\"b1cb32c0-190e-44ee-9e21-dd6df150b309\" data-image-id=\"b1cb32c0-190e-44ee-9e21-dd6df150b309\" alt=\"\"></figure>\n<p><em>38) Damage in tension, Damage in compression</em></p>\n<p>Regarding the reinforcement capacity, the analysis has been terminated at a <strong>plastic strain of 6% on the rebars, corresponding to a Von-Mises stress of 439 MPa</strong>. The longitudinal bars, transverse horizontal stirrups, and shear legs of the stirrups are utilized within the hardening plastic branch of the diagram. A simultaneous failure of both longitudinal and shear reinforcement is observed. <strong>This interaction results in a combined failure mechanism, where the longitudinal bars experience bending, the stirrups undergo tension due to transverse bending, and the vertical legs of the stirrups, subjected to shear forces within the concrete, experience axial tensile rupture.</strong></p>\n<figure data-asset-id=\"dd4a36ce-2d66-4f02-be2f-dd4389924a9c\" data-image-id=\"dd4a36ce-2d66-4f02-be2f-dd4389924a9c\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/26ccedc1-8bcd-40f5-b759-6a7fed6549fb/43.png\" data-asset-id=\"dd4a36ce-2d66-4f02-be2f-dd4389924a9c\" data-image-id=\"dd4a36ce-2d66-4f02-be2f-dd4389924a9c\" alt=\"\"></figure>\n<p><em>39)Stress in reinforcements</em></p>\n<figure data-asset-id=\"a71f090f-b25b-49bb-a2d7-489ea1e8afd4\" data-image-id=\"a71f090f-b25b-49bb-a2d7-489ea1e8afd4\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/0b431d1a-592a-4fee-a204-2098a1f19499/44.png\" data-asset-id=\"a71f090f-b25b-49bb-a2d7-489ea1e8afd4\" data-image-id=\"a71f090f-b25b-49bb-a2d7-489ea1e8afd4\" alt=\"\"></figure>\n<p><em>40)Nonlinear deflections</em></p>\n<figure data-asset-id=\"9a46bb87-ffcf-4b79-92b6-59638be773b9\" data-image-id=\"9a46bb87-ffcf-4b79-92b6-59638be773b9\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/f296848f-47d7-4bc6-b078-2e7a1db818eb/45.png\" data-asset-id=\"9a46bb87-ffcf-4b79-92b6-59638be773b9\" data-image-id=\"9a46bb87-ffcf-4b79-92b6-59638be773b9\" alt=\"\"></figure>\n<p><em>41)Contact area and contact stress</em></p>\n<h3>Concrete-Damage-Plasticity – High-Stiffness-Soil (HSS)(GMNA)</h3>\n<p>The ultimate loading force exerted on the model has been documented at -4,181 kN. <strong>The minimum (compressive) strain observed is -0.0175</strong>, which represents approximately a 56% reduction compared to the values recorded in LSS. A noteworthy change is identified in the location of this strain, shifting to the bottom face of the footing rather than the interface between the column and footing. This shift is primarily attributed to the predominance of vertical stress, which resulted in the peak strain relocating. Concurrently, <strong>the maximum (tensile) strain is observed on the bottom face of the footing, measuring 0.0451</strong>.</p>\n<p>The reduction in strain values can be attributed to the increased stiffness of the soil, confinement phenomena, and reduced deformation relative to LSS. Furthermore, the confined stress within the <strong>concrete reaches a value of -166 MPa</strong>. <strong>The confined strain highlights the post-critical behavior of concrete, including compression softening and concrete crushing</strong>.</p>\n<figure data-asset-id=\"e57976ff-60d2-4653-99b8-7b2ce3286501\" data-image-id=\"e57976ff-60d2-4653-99b8-7b2ce3286501\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/82042850-c4a4-4c23-8a3b-1f90e72fb0c1/50.png\" data-asset-id=\"e57976ff-60d2-4653-99b8-7b2ce3286501\" data-image-id=\"e57976ff-60d2-4653-99b8-7b2ce3286501\" alt=\"\"></figure>\n<p><em>42) Maximum applied force, Minimum principal stress</em></p>\n<figure data-asset-id=\"6b368948-5b6c-4240-8fd9-f05f1f668edb\" data-image-id=\"6b368948-5b6c-4240-8fd9-f05f1f668edb\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/0e214c24-e4dc-40f7-a6a7-708059187fbc/51.png\" data-asset-id=\"6b368948-5b6c-4240-8fd9-f05f1f668edb\" data-image-id=\"6b368948-5b6c-4240-8fd9-f05f1f668edb\" alt=\"\"></figure>\n<p><em>43) Minimum plastic strain, Maximum plastic strain</em></p>\n<figure data-asset-id=\"ba6700ab-c1ce-407e-8387-df1cb76e9d51\" data-image-id=\"ba6700ab-c1ce-407e-8387-df1cb76e9d51\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/bc2df111-fa00-4ff6-89c3-17b18e242629/52.png\" data-asset-id=\"ba6700ab-c1ce-407e-8387-df1cb76e9d51\" data-image-id=\"ba6700ab-c1ce-407e-8387-df1cb76e9d51\" alt=\"\"></figure>\n<p><em>44) Damage in tension, Damage in compression</em></p>\n<p><strong>The concentration of stress is predominantly centralized beneath the column area</strong>, resulting in elevated <strong>contact stress 3.41 MPa</strong> and a <strong>significant gradient of shear</strong>. This condition increases the likelihood of punching shear failure. <strong>The longitudinal reinforcement bars and stirrups play a pivotal role in accommodating plastic behavior</strong>. The localized stress induces yielding in the immediate vicinity of the column area on the footing strip. <strong>The tensile forces in the reinforcement bars, arising from the bending of the footing in both directions, combined with the shear force traction captured by the vertical legs of the stirrups, contribute to the manifestation of plasticity</strong>. The primary mode of failure is characterized by tension-induced stress along the reinforcement bars.</p>\n<figure data-asset-id=\"f2824406-5362-4b73-81a9-7eb5f5434a11\" data-image-id=\"f2824406-5362-4b73-81a9-7eb5f5434a11\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/da0a2a19-f16c-442a-839c-fbd53b5c97b6/53.png\" data-asset-id=\"f2824406-5362-4b73-81a9-7eb5f5434a11\" data-image-id=\"f2824406-5362-4b73-81a9-7eb5f5434a11\" alt=\"\"></figure>\n<p><em>45)Stress in reinforcements</em></p>\n<figure data-asset-id=\"4c76b980-ef91-4342-bfd8-9a3a436d8067\" data-image-id=\"4c76b980-ef91-4342-bfd8-9a3a436d8067\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/6def03a1-bef4-481e-b3d0-a20037f599b2/54.png\" data-asset-id=\"4c76b980-ef91-4342-bfd8-9a3a436d8067\" data-image-id=\"4c76b980-ef91-4342-bfd8-9a3a436d8067\" alt=\"\"></figure>\n<p><em>46) Nonlinear deflections</em></p>\n<figure data-asset-id=\"7ccde9be-e903-4cf7-8333-655852612ce9\" data-image-id=\"7ccde9be-e903-4cf7-8333-655852612ce9\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/65c01b80-ed77-4c5d-87a5-9bfab464ef34/55.png\" data-asset-id=\"7ccde9be-e903-4cf7-8333-655852612ce9\" data-image-id=\"7ccde9be-e903-4cf7-8333-655852612ce9\" alt=\"\"></figure>\n<p><em>47) Contact area and contact stress</em></p>"
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"value": "<p>The evidence that the model exhibits the same behavior becomes apparent when examining the phenomena under identical load levels. The maximum bearing capacity of the 3D CSFM will be compared with that of the CDP model.</p>\n<h3>Low-Stiffness-Soil (LSS)</h3>\n<p>The maximum bearing capacity of the <strong>3D CSFM</strong> model has reached <strong>-980 kN </strong>of axial force acting on the column. <strong>The forces have been used as the benchmark level for comparison</strong>. </p>\n<p>As observed, the minimum principal stress varies between output steps. This discrepancy arises from the nonlinear evolution of stress under compression, which depends on the constitutive behavior of the material. Due to triaxiality at the interface between the column and footing, the principal stress levels are higher than those in uniaxial compression.</p>\n<p>In the 3D CSFM model, the deviatoric stress remains constant. The deviatoric stress is insensitive to the level of mean stress, same as for Tresca theory. Conversely, the CDP model employs a dilation angle of 30°, which generates volumetric expansion in compression and causes the deviatoric stress to evolve along the stress path, particularly under higher triaxiality. The peak compressive stress of −94.6 MPa in CDP corresponds to a local maximum associated with the sharp corner in the stress path, reflecting combined effects of triaxiality and dilatancy.</p>\n<figure data-asset-id=\"2011f40e-0dd9-4d51-8c58-82a0ed17a04d\" data-image-id=\"2011f40e-0dd9-4d51-8c58-82a0ed17a04d\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/8f19815f-391c-4e9a-a386-67ebf2023c7c/57.png\" data-asset-id=\"2011f40e-0dd9-4d51-8c58-82a0ed17a04d\" data-image-id=\"2011f40e-0dd9-4d51-8c58-82a0ed17a04d\" alt=\"\"></figure>\n<p><em>48) Minimum principal stress by level of load -980 kN</em></p>\n<p><strong>The difference between the stress at critical places of 3D CSFM compared to CDP. </strong></p>\n<ul>\n <li>CDP approximately -70 MPa along the side of the column edge</li>\n <li>3D CSFM - 60 MPa along the side</li>\n</ul>\n<figure data-asset-id=\"1b215eb5-737f-4067-af40-f6be34b82dcf\" data-image-id=\"1b215eb5-737f-4067-af40-f6be34b82dcf\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/425fede7-a0c1-41d7-af81-2afc31140c1d/58.png\" data-asset-id=\"1b215eb5-737f-4067-af40-f6be34b82dcf\" data-image-id=\"1b215eb5-737f-4067-af40-f6be34b82dcf\" alt=\"\"></figure>\n<p><em>49) Detailed filtered stresses along the edge for CDP</em></p>\n<p>The variation in stress observed in the reinforcements has been quantified at approximately 8% for rebars under tension and 28% for those under compression. The reduced stress in compression and the 28% discrepancy can be attributed to the concrete material model utilized for compression and dilation angle and the exclusion of the bond interaction between the rebars and concrete (perfect bond) within the CDP model. The 3D CSFM demonstrates a tendency towards conservative results, indicating elevated stress levels in both compression and tension.</p>\n<figure data-asset-id=\"9cd68036-b465-4516-bdf6-3fcff8d4fbac\" data-image-id=\"9cd68036-b465-4516-bdf6-3fcff8d4fbac\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/95d56fe2-6364-4d4a-86c1-961832f3ab99/59.png\" data-asset-id=\"9cd68036-b465-4516-bdf6-3fcff8d4fbac\" data-image-id=\"9cd68036-b465-4516-bdf6-3fcff8d4fbac\" alt=\"\"></figure>\n<p><em>50) Stress in reinforcements by the same level of load </em></p>\n<p>Deformation level matches on 93%. </p>\n<figure data-asset-id=\"e7291422-3ea0-4936-a8bb-2c5d8b6cfdb9\" data-image-id=\"e7291422-3ea0-4936-a8bb-2c5d8b6cfdb9\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/94e4f5b0-a24a-494b-a8bc-1a0e27ed3380/60.png\" data-asset-id=\"e7291422-3ea0-4936-a8bb-2c5d8b6cfdb9\" data-image-id=\"e7291422-3ea0-4936-a8bb-2c5d8b6cfdb9\" alt=\"\"></figure>\n<p><em>51) Total deformation for the same level load</em></p>\n<h3>High-Stiffness-Soil (HSS)</h3>\n<p>The maximum bearing capacity of the <strong>3D CSFM</strong> model has reached <strong>-2,073 kN </strong>of loading force acting on the column. <strong>The forces have been used as the benchmark level for comparison</strong>. </p>\n<p><strong>The minimum principal stress for the CDP model reaches −127 MPa at peak.</strong> This large compressive value is primarily the result of an increased level of deviatoric stress combined with strong dilatancy in compression (high dilation angle), which drives the stress path toward larger compressive principal stresses. Compared to the LSS case, the applied load was increased by approximately 211%, which explains the higher principal compressive stress in the CDP model.</p>\n<p>In the case of 3D CSFM, the minimum principal stress reached about −60 MPa (≈3× the uniaxial compressive strength), i.e., substantially lower compression than in the CDP. The stress differences between models will rise further if the mean (hydrostatic) stress becomes higher.</p>\n<figure data-asset-id=\"2c2f006b-2b0f-44b7-ae3b-c6dc70b2dba7\" data-image-id=\"2c2f006b-2b0f-44b7-ae3b-c6dc70b2dba7\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/31ad0a16-5f79-4885-bab3-8b053bc9b747/61.png\" data-asset-id=\"2c2f006b-2b0f-44b7-ae3b-c6dc70b2dba7\" data-image-id=\"2c2f006b-2b0f-44b7-ae3b-c6dc70b2dba7\" alt=\"\"></figure>\n<p><em>52) Minimum principal stress by level of load -2070 kN</em></p>\n<p>The filtered stress distribution along the edge, with improved visualization and a properly scaled legend, indicates that the maximum compressive stress reaches approximately −70 MPa for the CDP model, compared with −60 MPa for the 3D CSFM model.</p>\n<figure data-asset-id=\"7e08dded-a19f-431b-a4aa-701488b0d201\" data-image-id=\"7e08dded-a19f-431b-a4aa-701488b0d201\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/bada283c-ca4d-4a00-9e72-8847777fe6af/62.png\" data-asset-id=\"7e08dded-a19f-431b-a4aa-701488b0d201\" data-image-id=\"7e08dded-a19f-431b-a4aa-701488b0d201\" alt=\"\"></figure>\n<p><em>53) Detailed filtered stress along the edge for CDP</em></p>\n<p>The variation in stress observed in the reinforcements has been quantified at approximately 8% for rebars under tension. The critical spot under tension has been identified in the exact location on the bottom longitudinal bars.</p>\n<figure data-asset-id=\"92d3e286-361e-4f2e-8025-8d22ee621a48\" data-image-id=\"92d3e286-361e-4f2e-8025-8d22ee621a48\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/4aa0b954-2b6b-4c4e-8190-85b67159db86/63.png\" data-asset-id=\"92d3e286-361e-4f2e-8025-8d22ee621a48\" data-image-id=\"92d3e286-361e-4f2e-8025-8d22ee621a48\" alt=\"\"></figure>\n<p><em>54) Stress in reinforcements by the same level of load</em></p>\n<p>The evidence pertaining to the level of deformation corresponds to an 85% match. <em> </em></p>\n<figure data-asset-id=\"ccdc40d9-fd40-4d03-9ef2-923ed6055255\" data-image-id=\"ccdc40d9-fd40-4d03-9ef2-923ed6055255\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/264de026-8e11-4e9b-8abe-e94523beca5f/64.png\" data-asset-id=\"ccdc40d9-fd40-4d03-9ef2-923ed6055255\" data-image-id=\"ccdc40d9-fd40-4d03-9ef2-923ed6055255\" alt=\"\"></figure>\n<p><em>55) Total deformation for the same level load</em></p>"
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"value": "<p>This verification study presents a thorough comparative analysis of analytical solutions of an infinite beam on an elastic medium, a standard beam solution and code-checks according to EN, as well as sophisticated nonlinear simulations utilizing CSFM in 2D/3D and CDP in 3D. The findings consistently illustrate the critical interaction between the model and soil stiffness in determining the structural behavior of continuous footings subjected to concentrated loading.</p>\n<h3>Overview of results:</h3>\n<p>The results indicate that the CSFM method occupies a distinctive position between analytical and conventional approaches and advanced numerical solutions as models. While standard methods tend to yield overly conservative outcomes, this can be attributed to the use of an inappropriate approach for analyzing areas subjected to concentrated loads, which are likely discontinuity regions where beam solution assumptions do not apply and should be replaced with the Strut-and-Tie method.</p>\n<p>Conversely, the higher bearing capacity observed in plasticity models arises due to the absence of internal criteria for terminating simulations, as implemented in the CSFM methods. The difference, which can play a key role in the results discrepancy, is geometrical nonlinearity, a dilation angle of 30 degrees, a minor contribution of tension in concrete, and a perfect bond considered for CDP.<strong> </strong>CSFM supports material nonlinearity, considering the bond between rebars and concrete, with zero strength in tension. Those effects evidently lead to a more conservative solution than CDP. </p>\n<p>Another aspect to note is that the current model is highly dependent on the soil's stiffness, and a very small increment of deformation leads to significant changes in the transferable load.</p>\n<p>In general, the contact stress in soil typically adheres to the standard recommendations. For loose sand used in this experiment, the maximum designed contact stress is 200 kPa, and for dense sand, 500 kPa. The calculated stress from simulations falls within the ranges of 0.59-1.56 MPa (loose sand) and 1.99-3.41 MPa (dense sand), which exceeds the standard criteria; however, this is irrelevant to the study's aim.</p>\n<p><strong>The CSFM method offers a balanced compromise between state-of-the-art numerical models, such as CDP, and beam theory models integrated into the codes. Notably, its advantages surpass those of conventional solutions</strong>.</p>\n<figure data-asset-id=\"e377ae4e-3677-491d-a2d1-2f3ca99f4160\" data-image-id=\"e377ae4e-3677-491d-a2d1-2f3ca99f4160\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/ebb90970-86dd-4a0d-82e6-7d6634f8497f/RESULTS.png\" data-asset-id=\"e377ae4e-3677-491d-a2d1-2f3ca99f4160\" data-image-id=\"e377ae4e-3677-491d-a2d1-2f3ca99f4160\" alt=\"\"></figure>\n<p><em>56) Results summary</em></p>\n<figure data-asset-id=\"8f63eeb3-9d2d-4863-ab87-357e36fb2567\" data-image-id=\"8f63eeb3-9d2d-4863-ab87-357e36fb2567\"><img src=\"https://assets-us-01.kc-usercontent.com:443/28eac049-c8ed-00e2-220c-12142a968dff/0e5465db-8d3b-4e53-b02b-83dafd27d27b/65.PNG\" data-asset-id=\"8f63eeb3-9d2d-4863-ab87-357e36fb2567\" data-image-id=\"8f63eeb3-9d2d-4863-ab87-357e36fb2567\" alt=\"\"></figure>\n<p><em>57) Graph representation of results split for LSS and HSS</em></p>\n<h3>Key takeaways</h3>\n<h4>Linear Beam Model (EN 1992-1-1 checks)</h4>\n<ul>\n <li>High ground stiffness significantly augments the bearing capacity of the model. <strong>The subgrade modulus of 128,000 kN/m³ in comparison with 16,000 kN/m³, resulting in a 2.2 times increased magnitude of applied force</strong>.</li>\n <li><strong>Failure modes occur in the bending region directly beneath the concrete column</strong>, where the concrete is subjected to compression at the interface with the column, as well as tension in the <strong>bottom layer of longitudinal reinforcement bars</strong>. </li>\n</ul>\n<h4>2D CSFM Solution</h4>\n<ul>\n <li><strong>The model accurately predicts identical failure modes</strong> as observed in the <strong>beam solution</strong>. Furthermore, the <strong>bearing capacity has been substantially enhanced for both LSS and HSS in comparison to the beam solution</strong>. This finding leads to the conclusion that <strong>beam theory is notably conservative</strong> when compared to a materially nonlinear solution using the 2D CSFM methodology.</li>\n <li>The concentrated load region is identified as the discontinuity region, so beam theory is not valid for this solution in this case due to the overly conservative approach.</li>\n</ul>\n<h4>3D CSFM Solution</h4>\n<ul>\n <li><strong>Captures confinement, triaxial stress effects, and transverse reinforcement involvement – none of which are accessible in 2D.</strong></li>\n <li>Failure modes are aligned with the two-dimensional plane stress solution. An additional failure mode arises due to the behavior in the transverse direction – stirrups are loaded up to the yield point, but this loading is limited to the horizontal bottom branches.</li>\n <li>Confirms that punching shear is not necessarily the governing mode even at high soil stiffness, provided adequate reinforcement is present.</li>\n</ul>\n<h4>3D CDP Solution</h4>\n<ul>\n <li><strong>Provides full volumetric concrete behaviour, including compression softening, tension stiffening, and progressive damage.</strong></li>\n <li>The geometrically nonlinear effect is the main reason for higher bearing capacity. This effect is the primary source of discrepancy between the models.</li>\n</ul>\n<h3>Engineering wisdom from the study</h3>\n<ul>\n <li><strong>The reinforcement layout depends on the stiff soil.</strong> Even heavily reinforced footings may fail prematurely due to soil-induced stress localization.</li>\n <li><strong>Linear beam models are useful for predesign but insufficient for capturing true behaviour</strong> when compression softening, uplift, or confinement occur.</li>\n <li><strong>Nonlinear models provide essential insight into failure mechanisms</strong>, especially when designing near capacity or verifying critical details.</li>\n <li><strong>3D effects matter.</strong> Transverse reinforcement and confinement significantly influence strength, ductility, and load redistribution.</li>\n <li><strong>Punching shear is not automatically governed.</strong> Many footings reach failure due to combined bending and tension in longitudinal bars – even under high soil stiffness.</li>\n</ul>\n<h3>Recommendations for IDEA StatiCa users</h3>\n<h4> 2D CSFM solution</h4>\n<ul>\n <li>Provides clear and physically meaningful failure modes.</li>\n <li>Ideal for quick yet accurate verification of simple strip footing or wall–base scenarios.</li>\n <li><strong>Highly efficient for comparing soil stiffness variants due to its low computational cost</strong>.</li>\n</ul>\n<h4>3D CSFM solution</h4>\n<ul>\n <li>Very strong at representing <strong>triaxial stress</strong>, <strong>confinement</strong>, <strong>transverse reinforcement action</strong>, and <strong>local crushing</strong>.</li>\n <li>Enables engineers to understand <strong>true spatial behaviour of complex details</strong> such as column–footing connections.</li>\n <li>Provides a realistic assessment of the contribution of stirrups and reinforcement legs in all directions.</li>\n</ul>\n<h4>3D CDP solution</h4>\n<ul>\n <li>Offers the most comprehensive representation of material softening, damage evolution, and collapse mechanisms.</li>\n <li>Ideal for <strong>research, advanced verification, and forensic analysis</strong>.</li>\n <li>Captures both progressive failure and redistribution, providing insight that cannot be obtained from code formulas.</li>\n</ul>\n<h3>Final recommendations for practice</h3>\n<p>These are my personal observations and recommendations based on the actual study.</p>\n<ul>\n <li>Use <strong>linear beam models</strong> for early-stage sizing and code-check verification.</li>\n <li>Use <strong>2D CSFM</strong> when uplift, nonlinear tension behaviour, or soil–structure interaction effects are critical.</li>\n <li>Use <strong>3D CSFM</strong> for evaluating complex stress fields, confinement, or the influence of transverse reinforcement.</li>\n <li>Use <strong>3D CDP</strong> for full verification of ultimate states, especially where material degradation or punching-like mechanisms are expected.</li>\n <li>Always evaluate <strong>soil stiffness</strong> in parallel with structural stiffness, this study confirms that it is a decisive parameter.</li>\n <li>For safety-critical components, prefer <strong>nonlinear analysis to supplement code checks</strong>.</li>\n</ul>\n<h2>References</h2>\n<p><strong>[1]</strong> <strong>EN 1992-1-1:2004+A1:2014</strong> – <em>Eurocode 2: Design of concrete structures – Part 1-1: General rules and rules for buildings.</em><br>\nEuropean Committee for Standardization (CEN), Brussels, 2014</p>\n<p><strong>[2] IDEA StatiCa, “Theoretical background for IDEA StatiCa Detail – Structural design of concrete discontinuities,” </strong>IDEA StatiCa Support Center. [Online]. Available: <a href=\"https://www.ideastatica.com/support-center/theoretical-background-for-idea-statica-detail?utm_source=chatgpt.com\">https://www.ideastatica.com/support-center/theoretical-background-for-idea-statica-detail</a> </p>\n<p><strong>[3] IDEA StatiCa, “IDEA StatiCa Detail – Structural design of concrete 3D discontinuities</strong>,” IDEA StatiCa Support Center. [Online]. Available: <a href=\"https://www.ideastatica.com/support-center/idea-statica-detail-structural-design-of-concrete-3d-discontinuities\">https://www.ideastatica.com/support-center/idea-statica-detail-structural-design-of-concrete-3d-discontinuities</a></p>\n<p><strong>[4] Dassault Systèmes, “ABAQUS Version 6.6 Documentation – Theory Manual</strong>,” [Online]. Available: <a href=\"https://classes.engineering.wustl.edu/2009/spring/mase5513/abaqus/docs/v6.6/books/usb/default.htm?startat=pt05ch18s05abm36.html\">https://classes.engineering.wustl.edu/2009/spring/mase5513/abaqus/docs/v6.6/books/usb/default.htm?startat=pt05ch18s05abm36.html</a></p>"
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"value": null
},
"options": {
"name": "Options",
"type": "multiple_choice",
"value": []
},
"url_slug": {
"name": "Url slug",
"type": "url_slug",
"value": "8-summary-and-key-takeaways"
},
"unique_url_slug": {
"name": "Unique URL slug",
"type": "custom",
"value": "[\"8-summary-and-key-takeaways\",\"[autogenerated]\"]"
},
"content_settings__sitemap": {
"name": "Show in sitemap",
"type": "multiple_choice",
"value": []
},
"content_settings__robots": {
"name": "Search engine indexing",
"type": "multiple_choice",
"value": []
},
"content_settings__is_hidden": {
"name": "Hidden nested content",
"type": "multiple_choice",
"value": [
{
"name": "yes",
"codename": "yes"
}
]
},
"metadata__page_title": {
"name": "Page title",
"type": "text",
"value": ""
},
"metadata__page_description": {
"name": "Page description",
"type": "text",
"value": ""
},
"metadata__page_keywords": {
"name": "Page keywords",
"type": "text",
"value": ""
},
"metadata__canonical_url": {
"name": "Canonical URL",
"type": "text",
"value": ""
},
"metadata__og_title": {
"name": "OG:title",
"type": "text",
"value": ""
},
"metadata__og_description": {
"name": "OG:description",
"type": "text",
"value": ""
},
"metadata__og_image": {
"name": "OG:image",
"type": "asset",
"value": []
},
"translation__translation_connector": {
"name": "Translation Connector",
"type": "taxonomy",
"value": [],
"taxonomyGroup": "languages"
},
"translation__force_translation": {
"name": "Force translation",
"type": "multiple_choice",
"value": []
},
"translation__last_translation": {
"images": [],
"linkedItemCodenames": [],
"linkedItems": [],
"links": [],
"name": "Last translation",
"type": "rich_text",
"value": "<p><br></p>"
},
"translation__ai_translated": {
"name": "AI translated",
"type": "multiple_choice",
"value": []
},
"page_tree_settings__page_label": {
"name": "Page label",
"type": "text",
"value": ""
},
"page_tree_settings__path_segment": {
"name": "Path segment",
"type": "text",
"value": ""
},
"page_tree_settings__breadcrumb_style": {
"name": "Breadcrumb style",
"type": "multiple_choice",
"value": []
},
"page_tree_settings__hide_in_breadcrumbs": {
"name": "Hide in breadcrumbs",
"type": "multiple_choice",
"value": []
}
}