Residual Stresses in Hot Rolled WideFlange Steel Members
Residual Stresses in Hot Rolled Wide-Flange Steel Members Yaze Chena, Thomas Hookerb, Ming Songc Civil Engineering Master of Engineering (Structural) c a yc 964@cornell. edu b tdh 47@cornell. edu ms 2832@cornell. edu
What is “Hot Rolling? ” Di re c tio n of r oll ing
What are residual stresses?
What are residual stresses? Rolling process Straightening procedures Nonuniform cooling Cross-sectional geometry Cooling conditions Steel material properties
What are residual stresses? Rolling process Straightening procedures Nonuniform cooling Cross-sectional geometry Cooling conditions Steel material properties }
What are residual stresses? Rolling process Straightening procedures Nonuniform cooling Cross-sectional geometry Cooling conditions Steel material properties
Why are we interested? Partial yielding of cross-section
Why are we interested? Partial yielding of cross-section
Why are we interested? Partial yielding of cross-section
Basic Equation of Transient heat Problems The temperature distribution inside the body is varies with time. The basic equation of transient thermal problems is Where [C] is the specific heat matrix [k] is thermal conductivity matrix
Heat Conduction Equation A basic law of heat conduction A heat flow is controlled by: Governing equation of temperature:
Generalized Finite-Element Method The boundary conditions: Green formula:
Finite-Element Formulation Temperature at any point: Temperature gradient at any point: Heat flux in each point:
The Expression of Matrix The basic equation of transient thermal problems is: The element matrices and external heat load vector:
Mode Superposition Step 1: Find the eigenvalues λn, and the associated eigenvectors from establish matrix [A], whose columns are the eigenvectors. Step 2: Calculate the elements Cnn in matrix [C], using
Mode Superposition Step 3: Solve differential equation as below, to obtain the vector {a}. Step 4: Use equation below to obtain the nodal temperature solution {T(t)}.
Time Integration θ-family of approximation
Time Integration θ-family of approximation Where,
Time Integration θ-family of approximation
Thermal Stresses σ =Eε = E α dt σ = stress due to temperature expansion E = Young’s Modulus ε = strain α = temperature expansion coefficient dt = temperature difference
ANSYS SIMULATION A 36 W 14 x 730 Dimensions in inches
Young’s modulus vs. Temperature 202 GPa 25 GPa
Yield stress & Tangent modulus Temperature Yield stress Tangent modulus C MPa 38 248 9929 149 219 8770 343 184 7364 454 161 6454 593 136 5443 900 78 3107
Thermal Properties 7832 kg/m 3 Density Isotropic Specific Film Thermal Conductivity 60 W/(m*°C) Heat 434 J/(kg*°C) Coefficient 193 W/m 2 *°C
Analysis Process Transient Heat Transfer Analysis Ø Initial Temperature: Uniform 900 °C Ø Ambient Convection: 20 °C Ø End Time: 5000 Sec. Substeps: 250 Thermal Stress Analysis (Static Structural) σ=Eε = E α dt
Mesh W 14 X 730 element size: 1’’ 0. 5’’ 0. 25’’ 0. 125’’
Temperature
Normal Stress
Verification Empirical (W 14 x 730) Maximum compression stress: Analysis 10. 3 ksi AISC Steel Construction Manual 0. 3 Fy=0. 3*36=10. 8 ksi Error: 4. 63% Mesh Convergence (W 14 x 730) 12 Max Normal Stress (ksi) 10. 1 10 9, 02 7, 98 8 10, 3 6 4 2 0 1 2 3 Trial of Element Size (1'', 0. 5'', 0. 25'', 0. 125'') 4
Max Stress vs Size Flange thickness/Web thickness=1. 6 Max Normal Stress vs Flange Thickness 12 10 8 6, 44 Normal Stress (ksi) 6 4 3 4, 07 7, 37 8, 43 10, 3 9, 28 max C (ksi) 4, 93 max T (ksi) 2 Flange thickness (in) 0 2, 11 2, 51 -2, 31 -4 -2, 39 -2 -6 2, 91 -2, 64 3, 31 3, 71 4, 11 4, 51 4, 91 (W 14 X 730) -3, 58 -4, 13 -4, 15 -4, 59 -4, 99
Thank You!
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