{"id":213,"date":"2025-01-29T02:22:38","date_gmt":"2025-01-29T02:22:38","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/3-9-additional-problems\/"},"modified":"2026-03-16T13:58:29","modified_gmt":"2026-03-16T13:58:29","slug":"3-9-additional-problems","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/3-9-additional-problems\/","title":{"raw":"3.9 Additional problems","rendered":"3.9 Additional problems"},"content":{"raw":"

3.9.1<\/h2>\nYou are called to verify Saint-Venant\u2019s principle on the difference between the effects of two dissimilar yet statically equivalent loadings becoming negligible at sufficiently large distances from the loading. For that, you should calculate the vertical stress increment \u0394\u03c3z<\/sub><\/em> below the axis of an infinite strip pressure of magnitude qext <\/sub><\/em>= 100 kPa and width B <\/em>= 2 m, and compare it with the corresponding value of \u0394\u03c3z<\/sub><\/em> due to a statically equivalent<\/em> line load, Qext<\/sub><\/em>. Try the comparison at depths z <\/em>= 1 m, z <\/em>= 3 m and z= 9 m from the ground surface.\n\n[caption id=\"attachment_212\" align=\"aligncenter\" width=\"400\"]\"Diagram Additional problem 3.9.1. Problem description and input parameters.[\/caption]\n

Answer:<\/h2>\n\n\n\n\n\n
Additional problem 3.9.1 Answers<\/caption>\n
z (m)<\/em><\/th>\n\u0394\u03c3z<\/sub> (strip pressure, kPa)<\/em><\/th>\n\u0394\u03c3z<\/sub> (line load, kPa)<\/em><\/th>\nDifference (\u0394\u03c3z,line<\/sub>-\u0394\u03c3z,strip<\/sub>)\/\u0394\u03c3z,line<\/sub> (%)<\/em><\/th>\n<\/tr>\n
1.00<\/em><\/td>\n81.83<\/em><\/td>\n127.32<\/em><\/td>\n55.59<\/em><\/td>\n<\/tr>\n
3.00<\/em><\/td>\n39.58<\/em><\/td>\n42.44<\/em><\/td>\n7.22<\/em><\/td>\n<\/tr>\n
9.00<\/em><\/td>\n14.03<\/em><\/td>\n14.15<\/em><\/td>\n0.82<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n
\n\n

3.9.2<\/h2>\nA line load of magnitude Qext<\/sub> <\/em>= 100 kN\/m is applied at 2 m from the crest of a 4 m-deep excavation, which is supported by a flexible<\/em> retaining wall e.g., a sheet pile wall. Calculate the horizontal stress \u0394\u03c3x<\/sub><\/em> due to the external load acting on the wall at z <\/em>= 1 m and z <\/em>= 3 m from the top of the wall. Calculate the stresses again, considering this time a rigid<\/em> retaining wall e.g., a basement reinforced concrete wall.\n\n[caption id=\"attachment_212\" align=\"aligncenter\" width=\"800\"]\"The Additional problem 3.9.2. Problem description and input parameters.[\/caption]\n

Answer:<\/h2>\n\n\n\n\n
Additional problem 3.9.2 Answers<\/caption>\n
z(m)<\/em><\/th>\n\u0394\u03c3x<\/sub> (flexible wall, kPa)<\/em><\/th>\n\u0394\u03c3x<\/sub> (rigid wall, kPa)<\/em><\/th>\n<\/tr>\n
1.00<\/em><\/td>\n10.19<\/em><\/td>\n20.38<\/em><\/td>\n<\/tr>\n
3.00<\/em><\/td>\n4.52<\/em><\/td>\n9.04<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n
\n\n

3.9.3<\/h2>\nTwo alternative routes are examined for a sewage pipeline, passing below locations A and B. You are the designers of the pipeline. Which one would you prefer, and why? Justify your answer.\n\n[caption id=\"attachment_212\" align=\"aligncenter\" width=\"800\"]\"Schematic Additional problem 3.9.3. Problem description and input parameters.[\/caption]\n

Answer:<\/h2>\nLocation A.\n\n
\n\n

3.9.4<\/h2>\nCalculate the vertical stress below the axis of the embankment shown below, at a depth z <\/em>= 10 m.\n\n[caption id=\"attachment_212\" align=\"aligncenter\" width=\"650\"]\"Schematic Additional problem 3.9.4. Problem description and input parameters.[\/caption]\n

Answer:<\/h2>\n\u0394\u03c3z <\/sub><\/em>= 136 kPa","rendered":"

3.9.1<\/h2>\n

You are called to verify Saint-Venant\u2019s principle on the difference between the effects of two dissimilar yet statically equivalent loadings becoming negligible at sufficiently large distances from the loading. For that, you should calculate the vertical stress increment \u0394\u03c3z<\/sub><\/em> below the axis of an infinite strip pressure of magnitude qext <\/sub><\/em>= 100 kPa and width B <\/em>= 2 m, and compare it with the corresponding value of \u0394\u03c3z<\/sub><\/em> due to a statically equivalent<\/em> line load, Qext<\/sub><\/em>. Try the comparison at depths z <\/em>= 1 m, z <\/em>= 3 m and z= 9 m from the ground surface.<\/p>\n

\"Diagram
Additional problem 3.9.1. Problem description and input parameters.<\/figcaption><\/figure>\n

Answer:<\/h2>\n\n\n\n\n\n\n
Additional problem 3.9.1 Answers<\/caption>\n
z (m)<\/em><\/th>\n\u0394\u03c3z<\/sub> (strip pressure, kPa)<\/em><\/th>\n\u0394\u03c3z<\/sub> (line load, kPa)<\/em><\/th>\nDifference (\u0394\u03c3z,line<\/sub>-\u0394\u03c3z,strip<\/sub>)\/\u0394\u03c3z,line<\/sub> (%)<\/em><\/th>\n<\/tr>\n
1.00<\/em><\/td>\n81.83<\/em><\/td>\n127.32<\/em><\/td>\n55.59<\/em><\/td>\n<\/tr>\n
3.00<\/em><\/td>\n39.58<\/em><\/td>\n42.44<\/em><\/td>\n7.22<\/em><\/td>\n<\/tr>\n
9.00<\/em><\/td>\n14.03<\/em><\/td>\n14.15<\/em><\/td>\n0.82<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n

3.9.2<\/h2>\n

A line load of magnitude Qext<\/sub> <\/em>= 100 kN\/m is applied at 2 m from the crest of a 4 m-deep excavation, which is supported by a flexible<\/em> retaining wall e.g., a sheet pile wall. Calculate the horizontal stress \u0394\u03c3x<\/sub><\/em> due to the external load acting on the wall at z <\/em>= 1 m and z <\/em>= 3 m from the top of the wall. Calculate the stresses again, considering this time a rigid<\/em> retaining wall e.g., a basement reinforced concrete wall.<\/p>\n

\"The
Additional problem 3.9.2. Problem description and input parameters.<\/figcaption><\/figure>\n

Answer:<\/h2>\n\n\n\n\n\n
Additional problem 3.9.2 Answers<\/caption>\n
z(m)<\/em><\/th>\n\u0394\u03c3x<\/sub> (flexible wall, kPa)<\/em><\/th>\n\u0394\u03c3x<\/sub> (rigid wall, kPa)<\/em><\/th>\n<\/tr>\n
1.00<\/em><\/td>\n10.19<\/em><\/td>\n20.38<\/em><\/td>\n<\/tr>\n
3.00<\/em><\/td>\n4.52<\/em><\/td>\n9.04<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n

3.9.3<\/h2>\n

Two alternative routes are examined for a sewage pipeline, passing below locations A and B. You are the designers of the pipeline. Which one would you prefer, and why? Justify your answer.<\/p>\n

\"Schematic
Additional problem 3.9.3. Problem description and input parameters.<\/figcaption><\/figure>\n

Answer:<\/h2>\n

Location A.<\/p>\n

\n

3.9.4<\/h2>\n

Calculate the vertical stress below the axis of the embankment shown below, at a depth z <\/em>= 10 m.<\/p>\n

\"Schematic
Additional problem 3.9.4. Problem description and input parameters.<\/figcaption><\/figure>\n

Answer:<\/h2>\n

\u0394\u03c3z <\/sub><\/em>= 136 kPa<\/p>\n","protected":false},"author":1,"menu_order":11,"template":"","meta":{"pb_show_title":"","pb_short_title":"3.9 Additional problems","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-213","chapter","type-chapter","status-publish","hentry"],"part":165,"_links":{"self":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/213","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":1,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/213\/revisions"}],"predecessor-version":[{"id":214,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/213\/revisions\/214"}],"part":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/parts\/165"}],"metadata":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/213\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/media?parent=213"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapter-type?post=213"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/contributor?post=213"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/license?post=213"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}