1. Nonlinear Analysis of Reinforced Concrete Xuehui AN

2. Today’s Agenda Review of last week’s lecture How to calculate average yielding strength of steel Introduction of WCOMD, model version Today’s topic: size effect of RC

3. 1 Method of Modeling 2 Concrete Model Prior to Generation of Cracks (1) Elasto-Plastic and Fracture Model (2) Cracking Criteria 3 Modeling of Cracked Concrete (1) Modeling of Concrete under Tensile Stress (2) Modeling of Concrete under Compressive stress (3) Modeling of Concrete under Shear Stress (4) Inter-relation of Each Model 4 Modeling of Reinforcing Bar in Concrete

4. (1) Modeling of Concrete under Tensile Stress

5. Concrete Steel CrackingFEM Model One Element Fine Elements

6. Total Force Average Strain Steel Force Total Force – Average Steel Force = Average Concrete Force Average Concrete Force Average Steel Force Total Force Steel strain/stress at cracking location

7. (1) Modeling of Concrete under Tensile Stress

9. Total Force Steel Force Average Concrete Force Average Steel Force Average yielding Steel strain at cracking location Average Strain Average yielding strength Stress-strain for steel in RC Yielding strength

10. Total Force Steel strain at cracking location Stress-strain for steel in RC ? ? Strain

11. Total Force Steel strain at cracking location Stress-strain for steel in RC ? ? Strain

12. Total Force Steel strain at cracking location Stress-strain for steel in RC ? ? Strain

13. Total Force Stress-strain for steel in RC ? ? Strain

14. Introducing of Simple Version of WCOMD for Analyzing of RC Plate Elements Subject to Plane Stresses M. Soltani M. Concrete Lab., Dec. 2000

15. Reinforced concrete structures can be considered as an assemblage of simpler elements, if the behavior of the component elements is understood, then the behavior of the complete structure can be predicted more accurately. Introduction: Bridge Structure Beam Containment Vessel Shear Wall Tow dimensional, rectangular reinforced concrete element subject to plane stress.

16. Link Shear Key RC PANEL (SPECIMEN) Jack Experiment Methods: Test of panels Jack and Link assembly to apply shear,tension and compression to reinforced concrete elements Test of specimens under biaxial stress Direct tension Tension stiffening

17. Modeling method: The reinforced concrete plate model The constitutive law for concrete The constitutive law for reinforcing bars =+ Reinforced concrete Cracked concrete Reinforcing bars FEM analysis of RC Structures For checking the constitutive model at the element level under different loading condition and combination of lads (biaxial stress, Cyclic loading,…) 1. Using the equilibrium and compatibility equations. 2. Using FEM and analyzing of plate element by considering the adequate boundary conditions ~

18. Analyzing of plate elements subject to plane stresses: WCOMD-SJSimple version (PANEL) Capability of program: 1. 2D analysis of plate elements subject to different loading condition. 2. Analysis subject to reversed cyclic loading. 3. Load Control Analysis. Load Control Analysis: Loads (Tension, compression and shear) is incrementally applied on the panel You can get more information from the User Guide and solved examples

19. A Sample for Input Data: The panel (Collins and Vichico test 1982) f'c=195 ft=21 Rx=1.785% Fy=4580 Ry=1.524% Fy=4200 Loading= Pure Shear Output files: Input file name+.RCS (Strain Stress relationship of RC in X and Y dir.) Input file name+.STL (Strain-Stress relationship of Steel in Steel Dir.) Input file name+.CON Input file name+.RCS x y Analysis of Panel Subject to Pure Shear (PV22, Collins 1982) 00020 CONC 196 21.0 0.0020 0.15 1.0 1.0 STXY 2100000.0 4580.0 2100000.0 4200.0 ELEM 0.01785 0.01524 0.0000 0.2 0.2 LOAD STEP 1 0.00 0.10 1000 1 LOAD 1 1.0 END

20. Output results:

21. LOAD STEP 1 0.00 0.10 300 1 LOAD 1 1.0 STEP 1 0.00 0.10 300 -1 STEP 1 0.00 0.10 400 1 END Reversed Cyclic Loading:

22. Some other examples: Effect of Reinforcement Ratio on Average Strain-Stress Relationship of Steel F'c=200 Ft=20 Fy=4000 Output file name= Input file name +. stl Direct Tension

23. Effect Tension Stiffening Factor on Average Strain-Stress Relationship of Steel F'c=200 Ft=20 Fy=4000

24. Biaxial Stresses: Analysis of panel subject to biaxial Stresses, Reinforcement angel=45 Deg. P T Tensile Stress-Strain in X dir. Stress-Strain of concrete normal to crack

25. Today’s Topic Size effect

26. Shear strength=Vc+Vs Improving shear strength ： Larger concrete area More shear reinforcement Shear strength by experimental results: Main reinforcement Shear reinforcement Area Vc Effective depth (m)

27. Large Beam

28. Tension stiffening behavior of concrete in RC domain softening behavior of plain concrete

29. Shear failure and size effect of large RC beams without web reinforcement