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

2. What is a Column Cap?

3. Our Problem
10 ”
1 ”

4. Simplification
8 ”

5. 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: > t >
Kirchhoff
Thin Plates
Reissner-Mindlin
Thick Plates
Accounts for shear deformations.
Normal to mid-surface does not remain normal after deformation.
Moderately thick plates: < t

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

7. WINT
WEXT

8. Linear Case Standard Displacement

9. 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

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

11. Element Performance

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

13. 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

14. 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

15. Final Results %

16. Thank you !

17. 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