Determination of the Spring Constant for the Horizontal Movement of Rigid Rectangular Foundations Resting on Elastic Subsoil
In this paper, a formula for the spring constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow>&l...
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| description | In this paper, a formula for the spring constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> for the horizontal movement of rigid rectangular foundations resting on elastic subsoil and spring coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> in this formula was derived, which demonstrates that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> depends on Poisson’s ratio <i>ν</i>. It was also shown that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula>, which has already been presented in the literature, was determined for a constant value <i>ν</i> = 0.3. It was shown that the values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula>, obtained from the formulas derived in this paper and the calculations based on the formulas and nomogram for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> given in the literature, may differ by 8–11%. For the adopted parameters, among others <i>ν</i> = 0.5, and for the side ratio α = 10, the value of spring constant is in the first case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>45.9 MN/m and in the second case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> = 158.3 MN/m (differ by 8%), while for <i>ν</i> = 0.1, is, respectively, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> = 517.8 N/m and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> = 464.3 MN/m (differ by 11%). Numerical FEM 3D analysis verified analytical solutions for the concrete foundation footing and the soil layer beneath it. This paper also provides useful nomograms that can be used to easily read the values <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula>. The use of the proposed formulas in this paper, refined formulas for determining <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub><mtext> </mtext></mrow></semantics></math></inline-formula>and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> in engineering calculations, can improve the accuracy of the analyses related to the influence of soil stiffness in horizontal movement. |
| format | Article |
| id | doaj-art-bde12862e283474b883a42a1b0cae7f3 |
| institution | Kabale University |
| issn | 2076-3417 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Applied Sciences |
| spelling | doaj-art-bde12862e283474b883a42a1b0cae7f32025-08-20T03:49:22ZengMDPI AGApplied Sciences2076-34172025-04-01159490610.3390/app15094906Determination of the Spring Constant for the Horizontal Movement of Rigid Rectangular Foundations Resting on Elastic SubsoilWojciech Pakos0Faculty of Civil Engineering, Wroclaw University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, PolandIn this paper, a formula for the spring constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> for the horizontal movement of rigid rectangular foundations resting on elastic subsoil and spring coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> in this formula was derived, which demonstrates that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> depends on Poisson’s ratio <i>ν</i>. It was also shown that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula>, which has already been presented in the literature, was determined for a constant value <i>ν</i> = 0.3. It was shown that the values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula>, obtained from the formulas derived in this paper and the calculations based on the formulas and nomogram for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> given in the literature, may differ by 8–11%. For the adopted parameters, among others <i>ν</i> = 0.5, and for the side ratio α = 10, the value of spring constant is in the first case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>45.9 MN/m and in the second case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> = 158.3 MN/m (differ by 8%), while for <i>ν</i> = 0.1, is, respectively, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> = 517.8 N/m and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> = 464.3 MN/m (differ by 11%). Numerical FEM 3D analysis verified analytical solutions for the concrete foundation footing and the soil layer beneath it. This paper also provides useful nomograms that can be used to easily read the values <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula>. The use of the proposed formulas in this paper, refined formulas for determining <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>x</mi></mrow></msub><mtext> </mtext></mrow></semantics></math></inline-formula>and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> in engineering calculations, can improve the accuracy of the analyses related to the influence of soil stiffness in horizontal movement.https://www.mdpi.com/2076-3417/15/9/4906foundationspring constantfoundation stiffnesselastic subsoil |
| spellingShingle | Wojciech Pakos Determination of the Spring Constant for the Horizontal Movement of Rigid Rectangular Foundations Resting on Elastic Subsoil Applied Sciences foundation spring constant foundation stiffness elastic subsoil |
| title | Determination of the Spring Constant for the Horizontal Movement of Rigid Rectangular Foundations Resting on Elastic Subsoil |
| title_full | Determination of the Spring Constant for the Horizontal Movement of Rigid Rectangular Foundations Resting on Elastic Subsoil |
| title_fullStr | Determination of the Spring Constant for the Horizontal Movement of Rigid Rectangular Foundations Resting on Elastic Subsoil |
| title_full_unstemmed | Determination of the Spring Constant for the Horizontal Movement of Rigid Rectangular Foundations Resting on Elastic Subsoil |
| title_short | Determination of the Spring Constant for the Horizontal Movement of Rigid Rectangular Foundations Resting on Elastic Subsoil |
| title_sort | determination of the spring constant for the horizontal movement of rigid rectangular foundations resting on elastic subsoil |
| topic | foundation spring constant foundation stiffness elastic subsoil |
| url | https://www.mdpi.com/2076-3417/15/9/4906 |
| work_keys_str_mv | AT wojciechpakos determinationofthespringconstantforthehorizontalmovementofrigidrectangularfoundationsrestingonelasticsubsoil |