Transcript Document

Plate Bending of Steel
Column Caps
MAE 5700 Final Project
Daniel Margolin
Abdul Al-Mishwat
MEng Structural Engineering
MEng Structural Engineering
What is a Column Cap?
Our Problem
10 ”
1”
Simplification
8”
Plate Theory
• Kirchhoff
• Thin Plates
• Reissner-Mindlin
• Thick Plates
- Assumes out-of-plane components are
negligible.
- Normal to mid-surface remains normal after
deformation.
1
1
- Thin plates:
>
t
>
10
100
- Accounts for shear deformations.
- Normal to mid-surface does not remain normal
after deformation.
1
- Moderately thick plates:
<t
10
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
𝑡 3𝐸
𝐷=
12(1 − ν2 )
Maximum Deflection at Mid-Point:
𝑤 𝑘 = 6.266x10-3 in
Closed Form Solution
𝑀𝑘 = −𝐷𝛻 2 𝑤 𝐾
𝑘
𝑤= 𝑤 +
𝑀𝑘
κ𝐺𝑡
,
κ=5
6
𝑤~𝑤 𝑘
𝜎𝑥𝑥 =
6
𝑀
𝑡 2 𝑥𝑥
Maximum In-Plane Stress:
𝜎𝑦𝑦 =
6
𝑀
𝑡 2 𝑦𝑦
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