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

2. 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 Topics 2 GTSUG, 2011, Delray Beach,FLJune 22-25, 2011

3. Overview of Basic Concepts Equilibrium Formulation 3 GTSUG, 2011, Delray Beach,FLJune 22-25, 2011

4. Overview of Basic Concepts Equilibrium Formulation 4 GTSUG, 2011, Delray Beach,FLJune 22-25, 2011

5. Overview of Basic Concepts Element Nodal Forces 5 GTSUG, 2011, Delray Beach,FLJune 22-25, 2011

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 6 GTSUG, 2011, Delray Beach,FLJune 22-25, 2011 0 0

7. Overview of Basic Concepts Element Implementation Behavior Assumptions Summary of GTSTRUDL NLG Behavior Assumptions 1.Plane and Space Frame −Small strains; σ = Eε remains valid −Internal rotations and curvatures are small; θ ≈ sinθ −Member chord rotations are small −P and M are coupled −U axial and U Transverse are uncoupled −θ Torsion and U Transverse are uncoupled −Other member effects are not affected by member displacement −Member loads are not affected by member displacement 2.Plane and Space Truss −Small strains ; σ = Eε remains valid −No assumptions limiting magnitude of displacements 7 GTSUG, 2011, Delray Beach,FLJune 22-25, 2011

8. Overview of Basic Concepts Element Implementation Behavior Assumptions Summary of GTSTRUDL NLG Behavior Assumptions 3.SBHQ and SBHT Plate Elements −Small strains; σ = Dε remains valid −BPH + PSH + 2 nd order membrane effects Internal rotations and curvatures are small U in-plane and U Transverse are coupled in 2 nd order membrane effects BPH and 2 nd order membrane effects are uncoupled −Element loads are not affected by element displacements 4.The IPCABLE Element −Small strains ; σ = Eε remains valid −No assumptions limiting magnitude of displacements −Regarding NLG, 2-node version and the truss are the same 8 GTSUG, 2011, Delray Beach,FLJune 22-25, 2011

9. Overview of Basic Concepts The Tangent Stiffness Matrix 9 GTSUG, 2011, Delray Beach,FLJune 22-25, 2011

10. GTSUG, 2011, Delray Beach,FL 10 Overview of Basic Concepts The Tangent Stiffness Matrix u P PiPi P i+1 uiui u i+1 a 1 b 2 u1u1 u2u2 u 1 =u i +  u 1 u 2 =u 1 +  u 2 K T = [K σ + K u ]

11. June 22-25, 2011GTSUG, 2011, Delray Beach,FL 11 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! Simple Basic behavior Examples

12. 12 GTSUG, 2011, Delray Beach,FLJune 22-25, 2011 Simple Basic Behavior Examples Simply-supported beam under axial load, imperfect geometry 20 @ 1 ft Imperfection: Y imp = -0.01sin( π x/L) ft P E = 10,000 ksi Plane Frame: Ax = 55.68 in 2, Iz = 100.00 in 4

13. June 22-25, 2011GTSUG, 2011, Delray Beach,FL 13 Simple Basic Behavior Examples Simply-supported beam under axial load, imperfect geometry P e = 171.2 kips

14. June 22-25, 2011GTSUG, 2011, Delray Beach,FL 14 Simple Basic Behavior Examples Simply-supported beam under axial load, imperfect geometry Push-over Analysis Procedure 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 $ f 1 CONVERGENCE RATE 0.8 CONVERGENCE TOLERANCE COLLAPSE 0.0001 CONVERGENCE TOLERANCE DISPLACEMENT 0.001 END PERFORM PUSHOVER ANALYSIS f1Pf1P Displacement Load P 1

15. June 22-25, 2011GTSUG, 2011, Delray Beach,FL 15 Simple Basic Behavior Examples Simply-supported beam under axial load, imperfect geometry Push-over Analysis Procedure 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 $ f 1 CONVERGENCE RATE 0.8 CONVERGENCE TOLERANCE COLLAPSE 0.0001 CONVERGENCE TOLERANCE DISPLACEMENT 0.001 END PERFORM PUSHOVER ANALYSIS f1Pf1P (2f 1 )P Displacement Load P 1 2

16. June 22-25, 2011GTSUG, 2011, Delray Beach,FL 16 Simple Basic Behavior Examples Push-over Analysis Procedure 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 $ f 1 CONVERGENCE RATE 0.8 CONVERGENCE TOLERANCE COLLAPSE 0.0001 CONVERGENCE TOLERANCE DISPLACEMENT 0.001 END PERFORM PUSHOVER ANALYSIS Simply-supported beam under axial load, imperfect geometry f1Pf1P (2f 1 )P (3f 1 )P Displacement Load P 1 3 2

17. June 22-25, 2011GTSUG, 2011, Delray Beach,FL 17 Simple Basic Behavior Examples Push-over Analysis Procedure 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 $ f 1 CONVERGENCE RATE 0.8 $ r CONVERGENCE TOLERANCE COLLAPSE 0.0001 CONVERGENCE TOLERANCE DISPLACEMENT 0.001 END PERFORM PUSHOVER ANALYSIS Simply-supported beam under axial load, imperfect geometry f1Pf1P (2f 1 )P (3f 1 )P Displacement Load P (2f 1 + rf 1 )P 1 3 4 2

18. June 22-25, 2011GTSUG, 2011, Delray Beach,FL 18 Simple Basic Behavior Examples Shallow truss arch: snap-through behavior E = 29,000 ksi Plane Truss: Ax = 1.0 in 2

19. June 22-25, 2011GTSUG, 2011, Delray Beach,FL 19 Simple Basic Behavior Examples Shallow truss arch: snap-through behavior

20. June 22-25, 2011GTSUG, 2011, Delray Beach,FL 20 Simple Basic Behavior Examples Shallow arch toggle: SBHQ6 model, snap-through behavior SBHQ6 Arch Leg, 20 x 4 Θz = 0

21. June 22-25, 2011GTSUG, 2011, Delray Beach,FL 21 Simple Basic Behavior Examples Shallow arch toggle: SBHQ6 model, snap-through behavior Note: P buck = 152.4 lbs (linear buckling load)

22. June 22-25, 2011GTSUG, 2011, Delray Beach,FL 22 Slender cantilever shear wall under axial load -- in-plane SBHQ plate behavior Simple Basic Behavior Examples 0.01 kips P Mesh = 2X50 Material = concrete POISSON = 0.0 Thickness = 4 in 100 ft 2 ft

23. June 22-25, 2011GTSUG, 2011, Delray Beach,FL 23 Slender cantilever shear wall under axial load -- in-plane SBH plate behavior Simple Basic Behavior Examples P buck (FE) = 41.95 kips (P e (SF) = 28.42 kips)

24. June 22-25, 2011GTSUG, 2011, Delray Beach,FL 24 The P-δ Question Does GTSTRUDL Include P-δ? E = 10,000 ksi, Plane Frame: Ax = 55.68 in 2, Iz = 100.0 in 4 No Mid Span Nodes 1 Mid Span Node

25. June 22-25, 2011GTSUG, 2011, Delray Beach,FL 25 The P-δ Question

26. June 22-25, 2011GTSUG, 2011, Delray Beach,FL 26 The P-δ Question M tot = M 0 + Pδ mid