"CESE 656 – Advanced Mechanicsfor Structural EngineersCivil Structural EngineeringUniversity of Alabama at BirminghamBeams on Elastic Foundations General Theory andBeams with Concentrated Loads2 Beams on Elastic Foundations CESE 656 Assumptions? Foundation is linear-Elastic – Winkler Foundation ? Reaction pressure, q, is proportional to beam displacement? ? = ??? q can be toward the beam (+) or away from the beam (-)? Beam is linear-elastic? Beam exhibits small deflections2 ? ? ? ? ? = - ?? ? 2 ? z 3 Beams on Elastic Foundations CESE 656 General Theory? ? ? = 0 = -? + ? + ? ? - ? ? ?? ? ? ? ? ? ? ? ? ? ? ? = ? lim ? ? =? ? ?? ? ? 0 2 ? ? ? ? ? = 0 = -? + ? + ? ? - ? - ? ? ?? ? ? ? 2 ? ? ? ? ? =? ?? 2 ? ? ? ? ? = -?? ? 2 ? z 4 Beams on Elastic Foundations CESE 656 General Theory?? ? = ??? 2 ? ? ? ? ? = -?? ? 2 ? z 3 ? ? ? ? ? = -?? ? 3 ? z 4 ? ? ? ? ? = -? = -??? 4 ? z 5 Beams on Elastic Foundations CESE 656 General Theory4 ? ? ? ? ? = -??? 4 ? z ? Foundation k is generally expressed using k0 with units of3 force / length? ? = ? ?0 ? b = width of beam6 Beams on Elastic Foundations CESE 656 General Theory4 4 ? ? ? ? ? ? ? = -?? ? ? ? + ?? = 0? ? 4 4 ? z ? z ? ? ? Sub.? = ?? 4 ? ? ? ? 4 ? ? ? ? ? + ? ? = 0 ? ? = -? ? ? ? ? Roots are4 ? ? ? = - ? = ? + 11 3 4? ? ? 4 ? ? ? = - ? = ? - 12 4 4? ? ? 7 Beams on Elastic Foundations CESE 656 General Theory4 ? ? ? ? ? + ?? = 0? 4 ? z ? General solution to this DE is? ? ? ? ? ? ? ? 1 2 3 4 ? ? = ? ? + ? ? + ? ? + ? ?1 2 3 4 ? Simplify using4 ? ? ? =4? ? ? ? ?? ? ? = cos ? ? + ? sin ? ? - ? ?? ? ? = cos ? ? - ? sin ? ?? With new constants? ? = ? + ? , ? = ? ? - ?1 1 4 2 1 4 ? ? = ? + ? , ? = ? ? - ?3 2 3 4 2 3 8 Beams on Elastic Foundations CESE 656 General Theory?? - ?? ? ? = ? ? sin ? ? + ? cos ? ? + ? ? sin ? ? + ? cos ? ?1 2 3 4 ? For large + values of z, y ? 0- ?? ? ? ? 0?? ? ? ? 8, so? = ? = 0 to make first term 01 2 ? Thus- ? ? ? ? = ? ? sin ? ? + ? cos ? ? , for ? = 03 4 ? Symmetry provides values for –z? y(z) = y(-z) ?(z) = -?(-z)? M(z) = M(-z) V(z) = -V(-z)9 Beams on Elastic Foundations CESE 656 Infinite Beam with Concentrated Load- ? ? ? ? = ? ? sin ? ? + ? cos ? ?3 4 -? ? ? ? = -? ? ? + ? sin ? ? - ? - ? cos ? ?3 4 3 4 ? Boundary Conditions? When? = 0,? = 0? 0 = - ? ? + ? 0 - ? - ? 1?? = ? = ?3 4 3 4 3 4 ? Area under q = P8 ? 2 ? ? ?? = ?? 0 ? ? ? ? 2 ? = ? ? ? =? 2? 10 Beams on Elastic Foundations CESE 656 "