Residual Stresses in Hot Rolled Wide-Flange Steel Members

1. Residual Stresses in Hot Rolled Wide-Flange Steel Members Yaze Chena, Thomas Hookerb, Ming Songc Civil Engineering Master of Engineering (Structural) a yc964@cornell.edu btdh47@cornell.edu cms2832@cornell.edu

2. What is “Hot Rolling?” Direction of rolling

3. What are residual stresses?

4. What are residual stresses? • Rolling process • Straightening procedures • Nonuniform cooling • Cross-sectional geometry • Cooling conditions • Steel material properties

5. What are residual stresses? • Rolling process • Straightening procedures • Nonuniform cooling • Cross-sectional geometry • Cooling conditions • Steel material properties }

6. What are residual stresses? • Rolling process • Straightening procedures • Nonuniform cooling • Cross-sectional geometry • Cooling conditions • Steel material properties

7. Why are we interested? • Partial yielding of cross-section

8. Why are we interested? • Partial yielding of cross-section

9. Why are we interested? • Partial yielding of cross-section

10. 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 the thermal conductivity matrix

11. Heat Conduction Equation • A basic law of heat conduction • A heat flow is controlled by: • Governing equation of temperature:

12. Generalized Finite-Element Method • The boundary conditions: • Green formula:

13. Finite-Element Formulation • Temperature at any point: • Temperature gradient at any point: • Heat flux in each point:

14. The Expression of Matrix • The basic equation of transient thermal problems is: • The element matrices and external heat load vector:

15. 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

16. 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)}.

17. Time Integration θ-family of approximation • A weighted average of the time derivatives at two consecutive time step is approximated by linear interpolation of the values of the variable at the two steps. • From

18. Time Integration θ-family of approximation • Plugging and in, we arrive at, • Rearranging terms as form of Where,

19. Time Integration θ-family of approximation • Different time integration schemes: • Θ= 0, the forward difference scheme (conditionally stable); , the Crank-Nicolson scheme (unconditionally stable); , the Galerkin method (unconditionally stable); 1, the back ward difference scheme (unconditionally stable)

20. Thermal Stresses • σ = E ε = E αdt • σ = stress due to temperature expansion • E = Young’s Modulus • ε = strain • α = temperature expansion coefficient • dt = temperature difference

21. ANSYS SIMULATION A36 W14x730 Dimensions in inches

22. Young’s modulus vs. Temperature 202GPa 25GPa

23. Yield stress & Tangent modulus

24. Thermal Properties • Density 7832kg/m • Isotropic Thermal Conductivity 60W/(m*°C) • Specific Heat 434J/(kg*°C) • Film Coefficient 193W/m *°C 3 2

25. Analysis Process Transient Heat Transfer Analysis • Initial Temperature: Uniform 900 °C • Ambient Convection: 20 °C • End Time: 5000 Sec. Substeps: 250 Thermal Stress Analysis (Static Structural) σ = E ε     = E α dt

26. Mesh • W14X730 element size: 1’’ 0.5’’ 0.25’’ 0.125’’

27. Temperature

28. Normal Stress

29. Verification • Empirical (W14x730) Maximum compression stress: Analysis 10.3 ksi AISC Steel Construction Manual 0.3Fy=0.3*36=10.8ksi Error: 4.63% • Mesh Convergence (W14x730)

30. Max Stress vs Size Flange thickness/Web thickness=1.6 (W14X730)

31. Thank You!