"CESE 656 – Advanced Mechanicsfor Structural EngineersCivil Structural EngineeringUniversity of Alabama at BirminghamBeams on Elastic Foundations Beams with Uniform Patch Loads2 Beams on Elastic Foundations CESE 656 Infinite Beam with Distributed Patch Load? The solution for a beam with a concentrated load can beused to solve a beam with a uniform load? For the portion of the beam underneath w? Examine the effect of ?P = w?z? Since?P is a concentrated load? ? ? ? - ? ? ? ? ? = ? cos + sin? 2? ? ? ? = ? ? as lim?z ? 0? ? ? ?? ? ?? - ? ? - ? ? ? ? = ? cos ?? + sin ?? ? ? + ? cos ?? + sin ?? ??? ? ? 0 0 2? 2? 3 Beams on Elastic Foundations CESE 656?? ??Infinite Beam with Distributed Patch Load? -? ? -? ? ? ? = 2 - ? cos ?? - ? cos ? ?? 2? - ? ? ? ? = ? cos? ? ? ? ? = 2 - ? - ?? ? ? ? ? These are valid as2? ? ? long as point H is? ? = ? - ?? ? ? ? ? 2? located under the? ? ? = ? + ?? ? ? ? ? distributed load2 4 ? ? ? ? = ? - ?? ? ? ? ? 4 ? 4 Beams on Elastic Foundations CESE 656??Infinite Beam with Distributed Patch Load? For point H outside the loaded portion? ? ? = ? - ?? ? ? ? ? 2? These are valid as? ? long as point H is? ? = ? - ?? ? ? ? ? 2? NOT located? ? ? = - ? - ?? ? ? ? ? 2 under the4 ? ? distributed load? ? = ? - ?? ? ? ? ? 4 ? 5 Beams on Elastic Foundations CESE 656 Location of Maximum y and M? The maximum deflection occurs at the center of theloaded segment? Maximum Moment' ? ? ? = ? ? M occurs at the center of the loaded segmentmax ? ' ? ? ? = 8 ? ? ? 0, ? ? 0, ? ? 0, ? ?? ' ? ? = ? ? = 8 ? M may be outside of L' and can be foundmax by trial and error? However, is it reasonably accurate to take M as the momentmax ? within L' a distancefrom the end of L' 4? 6 Beams on Elastic Foundations CESE 656 Example – Beam with Patch Load? The piece of heavy equipment from the previous example is reconfigured so that itloads the beam with a 5 ft long uniform load instead of 2 point loads. ? The beam on which it rests is timber (E = 1,800 ksi) and is 8 in. wide and 12 in. tall.3 ? The elastic foundation has a spring constant of k = 110 psi/in (lb/in ). 0 a. Determine the maximum deflection of the beamb. Determine the moment at the center of the distributed load. Is this themaximum moment?? From previous problem? ? = 880 ? - 1 ? ? = 0.0180 ? ?7 Beams on Elastic Foundations CESE 656??a) Maximum Deflection? The maximum deflection occurs at the center of the beam.? For point H in the center of the beam? ? = 30 ?? .? ? = 30 ?? .- 1 ? ? ? = ? ? = 0.0180 ? ? 30 ?? = 0.54? ? ? = 2 - ? - ?? ? ? ? ? 2? - ? ? ? ? = ? cos ??? ? - 0. 54 ? ? = ? = ? cos 0.54 = 0.50? ? ? ? 1 ?? ? /? ? ? ? = 2 - 0.5 - 0.5? 2( 0. 880? ) ? ? = 0.57 ?? (16% larger than for two 30 kip loads, 5 ft apart)? 8 Beams on Elastic Foundations CESE 656??Deflection Along the Beam60 kip spread over 5 ft60 kip spread over 20 ftq = 110 psi/in (0.57 in) = 63 psi = 9,100 psf q = 110 psi/in (0.30 in) = 33 psi = 4,800 psfmax max 9 Beams on Elastic Foundations CESE 656 b) Moment at Center? For point H in the center of the beam? ? = 30 ?? .? ? = 30 ?? .- 1 ? ? ? = ? ? = 0.0180 ? ? 30 ?? = 0.54? ? ? = ? + ?? ? ? ? ? 2 4 ? - ? ? ? ? = ? sin ??? ? - 0. 54 ? ? = ? = ? sin 0.54 = 0.30? ? ? ? 1 ?? ? /? ? ? ? = 0.3 + 0.3? - 1 2 4 0. 01 80? ? ? ? = 463 ?? · ? ? ? (29% larger than for two 30 kip loads, 5 ft apart)? 10 Beams on Elastic Foundations CESE 656 "