1. Practical Application of Finite Element Analysis to the Design of Post-Tensioned and Reinforced Concrete Floors Jonathan Hirsch, P.E.

2. Computer Assisted Design of Concrete Floors Types of programs available Advantages of each Why specialized finite element software is necessary for PT design

3. Computer Assisted Design of Concrete Floors The design process using 3-D finite element analysis Project examples

4. Types of Programs Available 2-D strip method 3-D finite element method Linear elastic Non-linear

5. 2-D Strip Method Structure analyzed with one model per beam, one-way slab, or two-way slab bay Equivalent frame method used for two- way slabs Easy to understand behavior Good for highly repetitive structures

6. Flat Plate Example

7. Transverse direction

8. Longitudinal direction

9. 3-D finite element method Visual modeling / input Accuracy continuity effects (elastic reactions) load path complicated loads (including lateral) restraint effects torsion

10. 3-D finite element method Graphical presentation of results Less cumbersome – work with one model instead of numerous Easier to incorporate changes Loadings Concrete geometry Construction Issues »Low Concrete Strength »Broken Strands

11. Las Olas River Condominiums 43 Story Fort Lauderdale, FL Suncoast Post- Tension

12. Visual modeling / input Speed CAD like interface Reduce chances for input error Automatic mesh generation

13. River City Apartments, Brisbane 1650 mm Transfer Slab

14. River City Apartments – Tendons Robert Bird and Partners

15. Accuracy of 3-D FE Analysis Continuity Effects Load Path Complicated Loads Generally leads to more optimal design

16. Accuracy of 3-D FE Analysis Restraining Effects Torsion

17. Continuity Effects

19. Beam and Slab: Relatively straightforward load path

20. Beam and Slab: More difficult load path

21. Prestress tendon profile variations

22. Bending moments …

23. Loads ….. Self weight is automatically calculated Superimposed loadings easily input

24. Straightforward line load

25. Complicated point and line loads

26. Restraining Effects Normally ignored by 2-D programs Can be calculated and accounted for by 3-D finite element programs Important for serviceability of structure Important for strength of structure (hyperstatic effects)

27. Torsion Normally ignored by 2-D programs (potentially creating a conservative design) Can exist in 3-D finite element model and therefore should be designed for

28. Torsion

30. Graphical Presentation of Results

32. Finite Element Basics Using shell elements to model concrete floors In plane forces Out of plane forces Related in irregular slabs (change of centroid)

33. In Plane Forces

34. Out of Plane Forces

35. Plate Considerations Resolution of Txy Integrated forces in equilibrium with nodal loads

36. Interaction of In Plane/Out of Plane Fx’ = Fx Vxy’ = Vxy Vxz’ = Vxz My’ = My - Fx d Mxy’ = Mxy - Vxy d

37. Using Shell Elements to Model Beams Deep beam behavior Torsion stiffness of beams using shell elements Transfer of moment through large step

38. Deep Beam Behavior

40. Torsion Stresses

41. Moment Transfer Through Step Beam

42. Orthotropic Element Properties

43. Hyperstatic (Secondary) effects …..

44. Hyperstatic effects …

45. Hyperstatic effects …..

46. “Complete Secondary (Hyperstatic) Effects” Allan Bommer PTI Journal - January 2004

47. Post-Tensioning Loadings Balance Loading Hyperstatic Loading

48. The 3-D Finite Element Design Process Model the structure Apply the loads Lay out the tendons (if PT) Draw design strips (define cross-sections) Perform the design Process results

49. Model the Structure

52. Apply the Loads (Dead Loads)

53. Apply the Loads (Live Loads)

54. Lay Out Tendons (Banded)

55. Lay Out Tendons (Distributed)

56. Lay Out Tendons

57. Deflection With Initial Tendon Layout

58. Draw the Design Strips

64. Perform the Design

68. Process Results

83. Special Considerations Lateral Analysis / Design Punching Shear Analysis / Design Restraining Effects Pour Strips, etc. Mat Foundations

84. Punching Shear

86. SR=1.25 Punching Shear …….. stress ratio exceeds unity

87. Punching Shear …….. without penetrations, stress ratio < 1

88. Mats

90. 48” Mat: DL + LL + WL Bearing pressure Max = 2560 psf Min = 690 psf

91. 24” Mat: DL + LL + WL Bearing pressure Max = 3450 psf Min = 0 psf (10 iterations)

92. 24” Mat: DL + LL + WL Bearing pressure Max = 3450 psf Min = 0 psf (10 iterations)

93. Bridgewater Place, Leeds Mixed Use: Office / Residential Connell Mott McDonald Matthew Consultants

94. Minneapolis Institute of Arts …….Soffit view Top view………

95. Sheraton Keauhou Bay Resort Keauhou Kona, Hawaii

96. Structural Systems (UK) Ltd

97. Westbridge Wharf Leicester Strongforce / Laing O’Rourke

98. 3 residential buildings 9 levels each Westbridge Wharf

99. St. Lucia Luxury Condominiums Destin, Florida Suncoast Post-Tension

101. Royal Palm Plaza Boca Raton, FL Tendon Systems, Inc.

103. Opus Architects and Engineers

104. Jonathan Hirsch, P.E. Jonathan.Hirsch@bentley.com