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# Michael H. Swanger Georgia Tech CASE Center June, 2011

GTStrudl Training … Nonlinear Geometric Analysis of Structures … Some Practical Fundamentals and Insights. Michael H. Swanger Georgia Tech CASE Center June, 2011. Topics. Lite Overview of Basic Concepts - Equilibrium Formulation - Element Nodal Forces

## Michael H. Swanger Georgia Tech CASE Center June, 2011

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1. GTStrudl Training…Nonlinear Geometric Analysis of Structures…Some Practical Fundamentals and Insights Michael H. Swanger Georgia Tech CASE Center June, 2011

2. Topics • Lite Overview of Basic Concepts • - Equilibrium Formulation • - Element Nodal Forces • Element Implementation Behavior Assumptions • Tangent Stiffness • Simple Basic behavior Examples • Simply-supported beam under axial load, imperfect geometry • Shallow truss arch: snap-through behavior • Shallow arch toggle: SBHQ6 model, snap-through behavior • Slender cantilever shear wall under axial load -- in-plane • SBHQ plate behavior • The P-δ Question! • Additional Examples GTSUG, 2011, Delray Beach,FL

3. Overview of Basic Concepts Equilibrium Formulation GTSUG, 2011, Delray Beach,FL

4. Overview of Basic Concepts Equilibrium Formulation GTSUG, 2011, Delray Beach,FL

5. Overview of Basic Concepts Element Nodal Forces GTSUG, 2011, Delray Beach,FL

6. Overview of Basic Concepts Element Implementation Behavior Assumptions Assumptions related to the scope of nonlinear geometric behavior are introduced into the definition of strain and the equilibrium equation: Example: Frame Member Strain and Equilibrium 0 0 GTSUG, 2011, Delray Beach,FL

7. Overview of Basic Concepts Element Implementation Behavior Assumptions Summary of GTSTRUDL NLG Behavior Assumptions • Plane and Space Frame • Small strains; σ = Eεremains valid • Internal rotations and curvatures are small; θ ≈ sinθ • Member chord rotations are small • P and M are coupled • Uaxial and UTransverse are uncoupled • θTorsion and UTransverse are uncoupled • Other member effects are not affected by member displacement • Member loads are not affected by member displacement • Plane and Space Truss • Small strains; σ = Eεremains valid • No assumptions limiting magnitude of displacements GTSUG, 2011, Delray Beach,FL

8. Overview of Basic Concepts Element Implementation Behavior Assumptions Summary of GTSTRUDL NLG Behavior Assumptions • SBHQ and SBHT Plate Elements • Small strains; σ = Dεremains valid • BPH + PSH + 2nd order membrane effects • Internal rotations and curvatures are small • Uin-plane and UTransverse are coupled in 2nd order membrane effects • BPH and 2nd order membrane effects are uncoupled • Element loads are not affected by element displacements • The IPCABLE Element • Small strains; σ = Eεremains valid • No assumptions limiting magnitude of displacements • Regarding NLG, 2-node version and the truss are the same GTSUG, 2011, Delray Beach,FL

9. Overview of Basic Concepts The Tangent Stiffness Matrix GTSUG, 2011, Delray Beach,FL

10. Overview of Basic Concepts The Tangent Stiffness Matrix P KT = [Kσ+ Ku] b a Pi+1 2 1 u2=u1+u2 Pi u1 u2 u1=ui+u1 ui ui+1 u GTSUG, 2011, Delray Beach,FL

11. Simple Basic behavior Examples • Simply-supported beam under axial load, imperfect geometry • Shallow truss arch: snap-through behavior • Shallow arch toggle: SBHQ6 model, snap-through behavior • Slender cantilever shear wall under axial load -- in-plane SBHQ plate behavior • The P-δ Question! GTSUG, 2011, Delray Beach,FL

12. Simple Basic Behavior Examples • Simply-supported beam under axial load, imperfect geometry P 20 @ 1 ft Imperfection: Yimp = -0.01sin(πx/L) ft E = 10,000 ksi Plane Frame: Ax = 55.68 in2, Iz = 100.00 in4 GTSUG, 2011, Delray Beach,FL

13. Simple Basic Behavior Examples • Simply-supported beam under axial load, imperfect geometry Pe = 171.2 kips GTSUG, 2011, Delray Beach,FL

14. Simple Basic Behavior Examples • Simply-supported beam under axial load, imperfect geometry Push-over Analysis Procedure Load P UNITS KIPS LOAD 1 JOINT LOADS 21 FORCE X -1000.0 \$ Load P NONLINEAR EFFECTS GEOMETRY MEMBERS EXISTING PUSHOVER ANALYSIS DATA INCREMENTAL LOAD 1 MAX NUMBER OF LOAD INCR 200 MAX NUMBER OF TRIALS 20 MAX NUMBER OF CYCLES 100 LOADING RATE 0.005 \$ f1 CONVERGENCE RATE 0.8 CONVERGENCE TOLERANCE COLLAPSE 0.0001 CONVERGENCE TOLERANCE DISPLACEMENT 0.001 END PERFORM PUSHOVER ANALYSIS 1 f1P Displacement GTSUG, 2011, Delray Beach,FL

15. Simple Basic Behavior Examples • Simply-supported beam under axial load, imperfect geometry Push-over Analysis Procedure Load P UNITS KIPS LOAD 1 JOINT LOADS 21 FORCE X -1000.0 NONLINEAR EFFECTS GEOMETRY MEMBERS EXISTING PUSHOVER ANALYSIS DATA INCREMENTAL LOAD 1 MAX NUMBER OF LOAD INCR 200 MAX NUMBER OF TRIALS 20 MAX NUMBER OF CYCLES 100 LOADING RATE 0.005 \$ f1 CONVERGENCE RATE 0.8 CONVERGENCE TOLERANCE COLLAPSE 0.0001 CONVERGENCE TOLERANCE DISPLACEMENT 0.001 END PERFORM PUSHOVER ANALYSIS 2 1 (2f1)P f1P Displacement GTSUG, 2011, Delray Beach,FL

16. Simple Basic Behavior Examples • Simply-supported beam under axial load, imperfect geometry Push-over Analysis Procedure Load P UNITS KIPS LOAD 1 JOINT LOADS 21 FORCE X -1000.0 NONLINEAR EFFECTS GEOMETRY MEMBERS EXISTING PUSHOVER ANALYSIS DATA INCREMENTAL LOAD 1 MAX NUMBER OF LOAD INCR 200 MAX NUMBER OF TRIALS 20 MAX NUMBER OF CYCLES 100 LOADING RATE 0.005 \$ f1 CONVERGENCE RATE 0.8 CONVERGENCE TOLERANCE COLLAPSE 0.0001 CONVERGENCE TOLERANCE DISPLACEMENT 0.001 END PERFORM PUSHOVER ANALYSIS 3 2 (3f1)P 1 (2f1)P f1P Displacement GTSUG, 2011, Delray Beach,FL

17. Simple Basic Behavior Examples • Simply-supported beam under axial load, imperfect geometry Push-over Analysis Procedure Load P UNITS KIPS LOAD 1 JOINT LOADS 21 FORCE X -1000.0 NONLINEAR EFFECTS GEOMETRY MEMBERS EXISTING PUSHOVER ANALYSIS DATA INCREMENTAL LOAD 1 MAX NUMBER OF LOAD INCR 200 MAX NUMBER OF TRIALS 20 MAX NUMBER OF CYCLES 100 LOADING RATE 0.005 \$ f1 CONVERGENCE RATE 0.8 \$ r CONVERGENCE TOLERANCE COLLAPSE 0.0001 CONVERGENCE TOLERANCE DISPLACEMENT 0.001 END PERFORM PUSHOVER ANALYSIS 3 4 2 (2f1 + rf1)P (3f1)P 1 (2f1)P f1P Displacement GTSUG, 2011, Delray Beach,FL

18. Simple Basic Behavior Examples Shallow truss arch: snap-through behavior E = 29,000 ksi Plane Truss: Ax = 1.0 in2 GTSUG, 2011, Delray Beach,FL

19. Simple Basic Behavior Examples Shallow truss arch: snap-through behavior GTSUG, 2011, Delray Beach,FL

20. Simple Basic Behavior Examples Shallow arch toggle: SBHQ6 model, snap-through behavior Θz = 0 SBHQ6 Arch Leg, 20 x 4 GTSUG, 2011, Delray Beach,FL

21. Simple Basic Behavior Examples Shallow arch toggle: SBHQ6 model, snap-through behavior Note: Pbuck= 152.4 lbs (linear buckling load) GTSUG, 2011, Delray Beach,FL

22. Simple Basic Behavior Examples Slender cantilever shear wall under axial load -- in-plane SBHQ plate behavior P 0.01 kips Mesh = 2X50 Material = concrete POISSON = 0.0 Thickness = 4 in 2 ft 100 ft GTSUG, 2011, Delray Beach,FL

23. Simple Basic Behavior Examples Slender cantilever shear wall under axial load -- in-plane SBH plate behavior Pbuck(FE) = 41.95 kips (Pe (SF) = 28.42 kips) GTSUG, 2011, Delray Beach,FL

24. The P-δ Question Does GTSTRUDL Include P-δ? E = 10,000 ksi, Plane Frame: Ax = 55.68 in2, Iz = 100.0 in4 No Mid Span Nodes 1 Mid Span Node GTSUG, 2011, Delray Beach,FL

25. The P-δ Question GTSUG, 2011, Delray Beach,FL

26. The P-δ Question Mtot = M0 + Pδmid GTSUG, 2011, Delray Beach,FL

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