"CESE 656 – Advanced Mechanicsfor Structural EngineersCivil Structural EngineeringUniversity of Alabama at BirminghamBeams on Elastic Foundations Beams with Multiple DiscreteSupports2 Beams on Elastic Foundations CESE 656 Beam with Multiple Discrete Supports? Reaction under each spring is ? = ??? If this reaction is distributed over each tributary length, l,? then the equivalent k for a continuous support is ? =? 3 Beams on Elastic Foundations CESE 656 Limitations? Using the continuous support equations providesreasonable results is4 ? ? ? ? = , where? =4? 4? ? ? 4 Beams on Elastic Foundations CESE 656 Modeling Continuous Supports? The previous analysis obviously allows us to use thecontinuous support equations to analyze beam with“closely” spaced discrete supports? The analysis also gives us a way to model continuoussupports using discrete springs.? ? ? = - discrete springs must be no more than ? apart? ?? 4? ? ? = ? ? ?0 ? K = Stiffness of equivalent spring? b = width of the beam? ? = modeled spring spacing (smaller ? gives more accurate results)3 ? k = stiffness of continuous support in psi/in (or lb/in )0 5 Beams on Elastic Foundations CESE 656 Example – “Smeared” Springs? A long timber beam (E = 12.4 GPa) is 60 mm wide and a 200 mm tall.The beam issupported by elastic supports spaced at ? = 600 mm.The supports are 100 mm3 cubes with a spring stiffness of k = 0.330 MPa/mm (0.330 N/mm ).A load P is0 placed at the center directly over a support.a. Does the support spacing satisfy the limit for which support can be treated ascontinuous?b. If the timber has an allowable stress of 16 MPa, what is the largest P that can besupported?c. What is the maximum contact pressure between the beam and a support?6 Beams on Elastic Foundations CESE 656 a) Support Spacing Requirement? For the analysis of a continuous foundation to give “reasonable” results, the supportspacing must satisfy? ? ? = , where 4? 4 ? ? ? =4? ? ? ? × ? ? ? ? ? ?? 0 ? ? =? ? 0. 330 60 1003 ? ? ? ? ? = = 3.302 600 ? ? 3. 3? / ?? 4 - 1 ? ? = = 0.001136 ? ?3 60 ?? 2 00?? 4 12, 400 ? 12 ? ? ? = = 692 ?? > 600 ?? (springs can be “smeared”)?? ? - 1 4 ( 0. 00 11 36? ? ) 7 Beams on Elastic Foundations CESE 656?? ?? ?? ?? ?? ??b) Maximum P? The maximum flexural stress will occur where M is maximum.? M occurs at centermax ? ? ? = ? , with z=0?? ? 4 ? ? ? ? = = 220 ?? ( ? )?? ? - 1 4 0. 00 11 36? ? ? ? ? ? ? ? =? ? 220( ? ) ? 16 ? ?? =2 60200 ?? ? 6 ? ? = 29,100 ? = 29.1 ? ?8 Beams on Elastic Foundations CESE 656?? ?? ?? ??c) Maximum Contact Pressure? The maximum contact pressure occurs where y is maximum.? y occurs at centermax ? ? ? ? = ? , with z=0?? ? 2? - 1 29, 100? 0. 001 13 6? ? ? ? = = 5.01 ???? ? ? 2 ( 3. 30 ) ?? ? ? ? = ? ? = 0.330 5.01 ??? ?? 0 ? 3 ? ? ? ? = 1.65 ? ??? ?? ? ? = ? ? = ? ? ??? ? ?? ? ?? ? ? ? ? = 3.30 600 ?? 5.01 ???? ? 2 ? ? ? ? = 9,920 ? = 9.92 ? ??? ? 9 Beams on Elastic Foundations CESE 656?? ??Example – Discretization of ContinuousElastic Support? A piece of heavy equipment is moved along a timber (E = 1,800 ksi) beam? The beam is 8 in. wide and 12 in. thick.? The elastic foundation under the beam has a spring constant of k = 110 psi/in. 0 ? The machine produces two point loads on the timber beam of 30 kip each , spaced 5 ftapart. a. Determine the maximum distance discrete springs should be placed to model thecontinuous elastic foundation under the beamb. Determine the spring stiffness, K, for each discrete spring if the spring spacing is l =3 ft10 Beams on Elastic Foundations CESE 656 "