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The Influence of Shear-Flexibility on the Elastic Critical Load for Frames of Pultruded Fibre Reinforced Plastic Section A. E. Abbaker and J. T. Mottram School of Engineering Advanced Polymer Composites for Structural Applications in Construction ACIC 2004 PFRP Frames Objectives Analysis of PFRP plane frames with moment resistant joints Analysis Approach Non-linear numerical frame analysis Non-linear Numerical Frame Analysis Matrix stiffness method Points to be considered • Shear flexible element • Semi-rigid connections • Second-order effects • Shear flexible stability functions • Stability analysis Shear Flexibility Deformation components • Bending controlled by EI • Shear controlled by GA Moduli ratio (E/G) • Steel is about 2.5 M A = k22θ A • FRP from 6 to 80 Modified slope–deflection equation Shear flexible stiffness k k22 = 1 3β ⎞ ⎛ l + ⎜ ⎟ ⎝ 4 EI 2 AG ⎠ Connection Behaviour and Modelling Moment-rotation curves Modelling connections • Rigid • Semi-rigid • Pinned Semi-Rigid Connections • PFRP sections have low Young’s modulus at 20-25 GPa and are relatively expensive • Design loading limited by deflection; not resistance • Practical moment-resistant connections will half deflection (or double loading) Second-order Effects Assumptions • Linear elastic material • Small deformation Axial load effect • Beam differential equation • Modified slope- deflection equation • Stiffness and carry-over factors (s and c) 2 d y 1 ( M ( x ) + PA y ) =− 2 EI dx QA ⎞ ⎛ MA = k22⎜ sθA + scθB − s(1+ c) ⎟ L⎠ ⎝ Shear Effect on φ Functions φ1 = α cot α α = 0.5π ρ ρ ρ = (1 − μρ ) Increasing shear flexibility μ= β Pe GA φ functions decreases with increase of shear-flexible parameter μ Stability Analysis • Due to non-linearity the stiffness matrix becomes function of load and can be used as stability criterion • Joint equilibrium equations • Positive determinant implies stable state • Negative determinant implies unstable state • Zero determinant equilibrium • Non-trivial solution of state of neutral equilibrium equation implies state of neutral Validation of Overall Stability Analysis PB and FB are Pinned Base and Fixed Base boundary conditions Comparison of elastic critical load of isotropic portal frame Previous Frame problem Author ρcr Abbaker Difference ρ cr % Sway with PB Timoshenko 0.184 0.183 -0.10 Sway with FB Chajes 0.74 4 0.745 0.10 No sway with PB Horne 1.285 1.283 -0.15 No sway with FB Coates 2.550 2.548 -0.10 Influence of Shear Flexible Parameter on Elastic Critical Load ρ ρsh 0.6 0.5 0.4 0.3 0.2 0.1 0 Increasing shear flexibility μ= 0 β Pe GA 20 = βπ 2 E 2 ⎛l⎞ G ⎜ ⎟ ⎝r⎠ 40 E/G Rigid joints 60 80 100 • As (E/G) increases from 2.5 to 80 elastic critical load decreases from 0.504 to 0.325 • Reduction in critical load is up to 35% Influence of Joint Action on Elastic Critical Load 0.8 0.7 0.6 0.5 ρρ cr 0.4 0.3 0.2 0.1 0 pinned joint Rigid joint Semi-rigid joint 0 5 10 15 S jL /EI 20 25 30 Sj is connection rotational stiffness Joint classification • As the relative stiffness factor (SjL/EI) is increased from 0.29 to 25 the elastic critical load for overall isotropic frame instability increases from 0.302 to 0.671 • Increase in elastic critical load by up to 122% Influence of Shear Flexibility and Semi-rigid Action 0.7 ρ crsh 0.6 0.5 ρ crsh 0.4 0.3 0.2 E/G = 6 E/G = 20 E/G = 80 Increasing shear flexibility 0 10 20 S jL /EI 30 Sj is connection rotational stiffness • High fixity achieved at low relative joint stiffness for very shear flexible members • Use Sj/ k rather than SjL/EI for the relative joint stiffness, where k is the shear flexible member stiffness Conclusions • Elastic critical load reduces with increase in member shear-flexibility. • It is necessarily to include the effect of shearflexibility when analysing the overall stability of frames of PFRP section. • There is no practical benefit to be gained in term of the load carrying capacity of frames when the connection relative stiffness is increase beyond 25, as specified in Annex J of Eurocode 3. Thank you for your attention Any questions? Email: [email protected]