Plate Bending of Steel Column Caps MAE 5700 Final Project Daniel Margolin MEng Structural Engineering Abdul Al-Mishwat MEng Structural Engineering

What is a Column Cap?

Our Problem 10 ” 1 ”

Simplification 8 ”

Plate Theory Kirchhoff Reissner-Mindlin Thin Plates Thick Plates Assumes out-of-plane components are negligible. Normal to mid-surface remains normal after deformation. Thin plates: 1 10 > t > 1 100 Kirchhoff Thin Plates Reissner-Mindlin Thick Plates Accounts for shear deformations. Normal to mid-surface does not remain normal after deformation. Moderately thick plates: 1 10 < t

Bi-Harmonic Equation of Plate Flexure: Thin Plate Approximation:

WINT WEXT

Linear Case Standard Displacement

Parameters PLATE PROPERTIES: DIMENSIONS: 8”- 8”- 0.1” MODULUS OF ELASTICITY (E): 29,000 KSI Poisson’s Ratio (ν): 0.3 Applied load: 1 psi SUPPORT CONDITIONS: SIMPLY SUPPORTED

ANSYS Deflection: 6.337x10-3 in. Stress: 1854 psi

Element Performance

MATLAB Maximum Deflection at Mid-Span: 1.56x10-4 m 6.14x10-3 in

Closed Form Solution Rectangular Kirchhoff Plate Subjected to uniform loading 𝐷= 𝑡 3 𝐸 12(1− ν 2 ) Maximum Deflection at Mid-Point: 𝑤 𝑘 = 6.266x10-3 in

Closed Form Solution 𝑀 𝑘 =−𝐷 𝛻 2 𝑤 𝐾 𝑤= 𝑤 𝑘 + 𝑀 𝑘 κ𝐺𝑡 , κ= 5 6 𝑤~ 𝑤 𝑘 𝑀 𝑘 =−𝐷 𝛻 2 𝑤 𝐾 𝑤= 𝑤 𝑘 + 𝑀 𝑘 κ𝐺𝑡 , κ= 5 6 𝑤~ 𝑤 𝑘 𝜎 𝑥𝑥 = 6 𝑡 2 𝑀 𝑥𝑥 𝜎 𝑦𝑦 = 6 𝑡 2 𝑀 𝑦𝑦 Maximum In-Plane Stress: σxx = 1839 psi

Final Results %

Thank you !

References M. Suneel Kumar “ULTIMATE STRENGTH OF SQUARE PLATE RECTANGULAR OPENING UNDER AXIAL XCOMPRESSION” Journal of Naval Architecture and Marine Engineering, June 2007 Alexander G. Losilevich “AN ANALYSIS OF FINITE ELEMENTS FOR PLATE BENDING PROBLEMS” Massachusetts Institute of Technology, 1996 A.J.M Ferreira “MATLAB CODES FOR FINITE ELEMENT ANALYSIS” Universidade do Porto, Portugal, 2008 Niels Ottosen & Hans Peterson “INTRODUCTION TO THE FINITE ELEMENT METHOD” University of Lund, Sweden, 1992 Thomas J.R.Hughes “THE FINITE ELEMENT METHOD-LINEAR STATIC AND DYNAMIC FINITE ELEMENT ANALYSIS” Stanford University, 1987