1. Unified Theory of Reinforced Concrete
Assist. Prof. Dr. /Abbas Abdul Majeed Allawi
University of Baghdad
February 2013

2. The Six Component Models of the Unified Theory:
Struts-and-ties Model
Bernoulli Compatibility Truss Model
Equilibrium (Plasticity) Truss Model
Mohr Compatibility Truss Model
Softened Truss Model

3. Strut and Tie Model

4. Fig. 1 Strut and Tie Model

5. Bernoulli Compatibility Truss Model
Principles: Equilibrium conditions, Bernoulli compatibility condition, and the unaxial constitutive laws of concrete and reinforcement. The constitutive laws may be linear or nonlinear.
Applications: Analysis and design of M and N in the main regions at both serviceability and the ultimate load stages.

6. Equilibrium (Plasticity) Truss Model

7. Fig. 2 Equilibrium in element shear Fig
Fig. 2 Equilibrium in element shear Fig. 2 Equilibrium (Plasticity) Truss Model in Element Shear

8. Fig. 3 Equilibrium (Plasticity) Truss Model in Beam Shear

9. Fig. 4 Equilibrium (Plasticity) Truss Model in Torsion

10. Table 1 : Summary of Basic Equilibrium Equations Bernoulli Compatibility Truss Model

11. Advantages Deficiencies
Satisfies completely the equilibrium conditions. It provides three equilibrium equations that are conceptionally identical in element shear, beam shear and torsion.
From a design point of view, the three equilibrium equations can be used directly to design the three components of the truss model, namely, the transverse steel, the longitudinal steel and the diagonal concrete struts.
The model provides a very clear concept of the interaction of bending, shear and axial load.
Deficiencies
The equilibrium truss model does not take into account the strain compatibility condition. As a result, it can not predict the shear or torsion deformation of a member.
The model can not predict the strains in the steel or concrete. Consequently, the yielding of steel or crushing of concrete can not be rationally determined, and the modes of failure cannot be discerned.

12. Mohr Compatibility Truss Model

13. Fig. 5 Transformation of stresses

14. Equilibrium equations:
(1)
(2)
(3)
Or, in a matrix form:
(4)

15. Fig. 6 Graphical expression of principle stresses

16. Fig. 7 Definition of Strains and Transformation Geometry

17. Compatibility equations:
(5)
(6)
(7)
Or, in a matrix form:
(8)

18. Stresses in terms of concrete and steel:
(9)
(10)
(11)

19. Fig. 8 Reinforced concrete membrane elements subjected to in-plane stress

20. Constitutive Laws:
(12)
(13)
(14)

21. Softened Truss Model for Membrane Element

22. Summery of Equations: Equilibrium Equations: Compatibility Equations:
[1]
[2]
[3]
Compatibility Equations:
[4]
[5]
[6]

23. Fig. 9 Compressive stress-strain curve of concrete

24. Ascending branch Descending branch Softening Parameter:
Constitutive law of concrete in compression:
Ascending branch
[7 a]
Descending branch
[7 b]
Softening Parameter:
[8]

25. Constitutive law of mild steel:
[9b]
[10a]
[10b]
[13]
[14]
[15]
[16]

26. Fig. 10 Flow chart showing the solution
algorithm (Constant Normal Stresses)

27. THANK YOU
Questions ?