{"id":11125,"date":"2023-11-30T14:06:20","date_gmt":"2023-11-30T14:06:20","guid":{"rendered":"https:\/\/support.buildsoft.eu\/?post_type=ht_kb&p=11125"},"modified":"2026-01-12T09:00:58","modified_gmt":"2026-01-12T09:00:58","slug":"tll-01-beam-under-temperature-load","status":"publish","type":"ht_kb","link":"https:\/\/support.buildsoft.eu\/fr\/knowledge-base\/tll-01-beam-under-temperature-load\/","title":{"rendered":"TLL 01: Beam under temperature load"},"content":{"rendered":"

Description<\/h2>\n

\"\"<\/p>\n

\n\n\n\n\n\n\n\n\n\n\n\n
Material<\/th>\nModulus of elasticity<\/td>\n\"E=210 000 N/mm^2\"<\/td>\n<\/tr>\n
Thermal dilation<\/td>\n\"\alpha =0.000 012/ ^{\circ}C\"<\/td>\n<\/tr>\n
Geometry<\/th>\nCross-section<\/td>\nIPE 500<\/td>\n<\/tr>\n
Boundary conditions<\/td>\nCase 1: cantilever<\/td>\n<\/tr>\n
<\/td>\nCase 2: fixed beam<\/td>\n<\/tr>\n
Loads<\/th>\nLoad 1:<\/td>\nGlobal temperature change \"\Delta T = 40 ^{\circ}C\"<\/td>\n<\/tr>\n
Load 2:<\/td>\nLinear gradient in local y’ direction \"\Delta T = 40 ^{\circ}C\"<\/td>\n<\/tr>\n
Load 3:<\/td>\nLinear gradient in local z’ direction \"\Delta T = 40 ^{\circ}C\"<\/td>\n<\/tr>\n
Mesh<\/th>\nNo. of divisions<\/td>\n8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

Independent reference results<\/h2>\n
<\/span>Open handcalculations<\/div>
\n

Load 1: global temperature change \"\Delta T = 40 ^{\circ}C\"<\/h4>\n
    \n
  • Case 1: cantilever beam
    \nSince the beam can lengthen\/ shorten, the temperature load will extend the beam<\/li>\n<\/ul>\n

      <\/span>   <\/span>\"\[ \delta_x=\alpha \cdot \Delta T\cdot L=0.000 012 / ^{\circ}C\cdot 40^{\circ}C\cdot 5m=2.4mm\]\"<\/p>\n

      \n
    • Case 2: fixed beam
      \nSince the beam cannot lengthen\/ shorten, the temperature load will result in axial force.
      \nThe increase in temperature will want to lengthen the beam. To counteract this deformation, a compressive force must be created. So the normal force in Diamonds will have a negative sign.<\/li>\n<\/ul>\n

        <\/span>   <\/span>\"\[N=\alpha \cdot \Delta T\cdot A\cdot E=0.000 012 / ^{\circ}C\cdot 40^{\circ}C\cdot 11553mm^2\cdot 210000N/mm^2=1164.46kN\]\"<\/p>\n

      Load 2: linear gradient in local y’ direction \"\Delta T = 40 ^{\circ}C\"<\/h4>\n
        \n
      • Case 1: cantilever beam
        \nSince the beam can lengthen\/ shorten, the gradient will cause an elongation of the upper fiber and a shortening of the non-fiber. The combination of both results in a downward deformation.<\/p>\n

          <\/span>   <\/span>\"\[ \varphi_z=\frac{\alpha \cdot \Delta T\cdot L}{h}=\frac{0.000 012 / ^{\circ}C\cdot 40^{\circ}C\cdot 11553mm^2\cdot 5m}{500mm}=0.2750^{\circ}\]\"<\/p>\n

          <\/span>   <\/span>\"\[ \delta _y=\frac{\alpha \cdot \Delta T\cdot L^2}{2 \cdot h}=\frac{0.000 012 / ^{\circ}C\cdot 40^{\circ}C\cdot (5m)^2}{2 \cdot 500mm}=12.0000mm\]\"<\/p>\n<\/li>\n

      • Case 2: Fixed beam
        \nSince the beam cannot lengthen\/ shorten, the gradient will cause an elongation of the upper fiber and a shortening of the non-fiber. To counteract this deformation, a compressive force must be created in the upper fiber and a tensile force in the lower. So the bending moment in Diamonds will be drawn on the lower side of the beam.<\/p>\n

          <\/span>   <\/span>\"\[M_{y'}=\frac{\alpha \cdot \Delta T\cdot E\cdot I_y}{h}=\frac{0.000 012 / ^{\circ}C\cdot 40^{\circ}C \cdot 4.82\cdot 10^8 mm^4}{500mm}=97.17kNm\]\"<\/p>\n<\/li>\n<\/ul>\n

        Load 3: linear gradient in local z’ direction \"\Delta T = 40 ^{\circ}C\"<\/h4>\n
          \n
        • Case 1: cantilever beam
          \nSince the beam can lengthen\/ shorten, the gradient will cause an elongation of the upper fiber and a shortening of the non-fiber. The combination of both results in a downward deformation.<\/p>\n

            <\/span>   <\/span>\"\[ \varphi_z=\frac{\alpha \cdot \Delta T\cdot L}{b}=\frac{0.000 012 / ^{\circ}C\cdot 40^{\circ}C\cdot 11553mm^2\cdot 5m}{200mm}=0.6875^{\circ}\]\"<\/p>\n

            <\/span>   <\/span>\"\[ \delta _y=\frac{\alpha \cdot \Delta T\cdot L^2}{2 \cdot b}=\frac{0.000 012 / ^{\circ}C\cdot 40^{\circ}C\cdot (5m)^2}{2 \cdot 200mm}=30.0000mm\]\"<\/p>\n<\/li>\n

        • Case 2: Fixed beam
          \nSince the beam cannot lengthen\/ shorten, the gradient will cause an elongation of the upper fiber and a shortening of the non-fiber. To counteract this deformation, a compressive force must be created in the upper fiber and a tensile force in the lower. So the bending moment in Diamonds will be drawn on the lower side of the beam.<\/p>\n

            <\/span>   <\/span>\"\[M_{y'}=\frac{\alpha \cdot \Delta T\cdot E\cdot I_y}{b}=\frac{0.000 012 / ^{\circ}C\cdot 40^{\circ}C \cdot 2.14\cdot 10^7 mm^4}{200mm}=10.79kNm\]\"<\/p>\n<\/li>\n<\/ul>\n<\/div><\/div><\/div>\n

          Diamonds results and comparison<\/h2>\n

          Load 1: global temperature change \"\Delta T = 40 ^{\circ}C\"<\/h4>\n

          \"\"
          \n

          \n\n\n\n\n\n
          <\/th>\n<\/th>\nIndependent reference<\/th>\nDiamonds<\/th>\nDifference<\/th>\n<\/tr>\n
          Case 1<\/td>\nHorizontal deformation \"\delta_x\"<\/td>\n2.4mm<\/td>\n2.4mm<\/td>\n0,00%<\/td>\n<\/tr>\n
          Case 2<\/td>\nAxial force \"N\"<\/td>\n-1164.46kN<\/td>\n-1164.46kN<\/td>\n0,00%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

          Load 2: linear gradient in local y’ direction \"\Delta T = 40 ^{\circ}C\"<\/h4>\n

          \"\"<\/p>\n

          \n\n\n\n\n\n\n
          <\/th>\n<\/th>\nIndependent reference<\/th>\nDiamonds<\/th>\nDifference<\/th>\n<\/tr>\n
          Case 1<\/td>\nAngular rotation \"\varphi_z\"<\/td>\n0.2750\u00b0<\/td>\n0.2750\u00b0<\/td>\n0,00%<\/td>\n<\/tr>\n
          <\/td>\nHorizontal deformation \"\delta_y\"<\/td>\n12.0000mm<\/td>\n12.0000mm<\/td>\n0,00%<\/td>\n<\/tr>\n
          Case 2<\/td>\nBending moment \"M_y\"<\/td>\n97,17kNm<\/td>\n97,17kNm<\/td>\n0,00%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

          Load 3: linear gradient in local z’ direction \"\Delta T = 40 ^{\circ}C\"<\/h4>\n

          \"\"<\/p>\n

          \n\n\n\n\n\n\n
          <\/th>\n<\/th>\nIndependent reference<\/th>\nDiamonds<\/th>\nDifference<\/th>\n<\/tr>\n
          Case 1<\/td>\nAngular rotation \"\varphi_y\"<\/td>\n0.6875\u00b0<\/td>\n0.6875\u00b0<\/td>\n0,00%<\/td>\n<\/tr>\n
          <\/td>\nHorizontal deformation \"\delta_z\"<\/td>\n30.0000mm<\/td>\n30.0000mm<\/td>\n0,00%<\/td>\n<\/tr>\n
          Case 2<\/td>\nBending moment \"M_z\"<\/td>\n10.79kNm<\/td>\n10.79kNm<\/td>\n0,00%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

          References<\/h2>\n
            \n
          • Hibbeler, R. (2006). Sterkteleer, 2\/e<\/i>. Pearson Education.<\/li>\n
          • Weaver, W., & Gere, J. M. (1990). Matrix analysis of framed structures Table B-2. In Springer eBooks<\/i>.<\/li>\n<\/ul>\n
              \n
            • Tested in Diamonds 2026.<\/li>\n<\/ul>\n\n","protected":false},"excerpt":{"rendered":"

              Description Independent reference results Diamonds results and comparison Load 1: global temperature change Load 2: linear gradient in local y’ direction Load 3: linear gradient in local z’ direction References Hibbeler, R. (2006). Sterkteleer, 2\/e. Pearson Education. Weaver, W., & Gere, J. M. (1990). Matrix analysis of framed structures Table…<\/p>\n","protected":false},"author":3,"comment_status":"closed","ping_status":"closed","template":"","format":"standard","meta":[],"ht-kb-category":[1206],"ht-kb-tag":[],"_links":{"self":[{"href":"https:\/\/support.buildsoft.eu\/fr\/wp-json\/wp\/v2\/ht-kb\/11125"}],"collection":[{"href":"https:\/\/support.buildsoft.eu\/fr\/wp-json\/wp\/v2\/ht-kb"}],"about":[{"href":"https:\/\/support.buildsoft.eu\/fr\/wp-json\/wp\/v2\/types\/ht_kb"}],"author":[{"embeddable":true,"href":"https:\/\/support.buildsoft.eu\/fr\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/support.buildsoft.eu\/fr\/wp-json\/wp\/v2\/comments?post=11125"}],"version-history":[{"count":3,"href":"https:\/\/support.buildsoft.eu\/fr\/wp-json\/wp\/v2\/ht-kb\/11125\/revisions"}],"predecessor-version":[{"id":23729,"href":"https:\/\/support.buildsoft.eu\/fr\/wp-json\/wp\/v2\/ht-kb\/11125\/revisions\/23729"}],"wp:attachment":[{"href":"https:\/\/support.buildsoft.eu\/fr\/wp-json\/wp\/v2\/media?parent=11125"}],"wp:term":[{"taxonomy":"ht_kb_category","embeddable":true,"href":"https:\/\/support.buildsoft.eu\/fr\/wp-json\/wp\/v2\/ht-kb-category?post=11125"},{"taxonomy":"ht_kb_tag","embeddable":true,"href":"https:\/\/support.buildsoft.eu\/fr\/wp-json\/wp\/v2\/ht-kb-tag?post=11125"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}