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

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

Strut and Tie Model

Fig. 1 Strut and Tie Model

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.

Equilibrium (Plasticity) Truss Model

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

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

Fig. 4 Equilibrium (Plasticity) Truss Model in Torsion

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

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.

Mohr Compatibility Truss Model

Fig. 5 Transformation of stresses

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

Fig. 6 Graphical expression of principle stresses

Fig. 7 Definition of Strains and Transformation Geometry

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

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

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

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

Softened Truss Model for Membrane Element

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

Fig. 9 Compressive stress-strain curve of concrete

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

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

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

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