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CURVATURE DUCTILITY
Structural Members like Beam undergoes bending due to external bending moments. The curvature is the measure of change in rotation with in small incremental length of beam. There is direct relationship between curvature and the internal bending moment at a section of a beam. As Internal moment at a section increases, the curvature at that section increases proportionally. This relationship was first given by Euler–Bernoulli and theory is known as Euler-Bernoulli Beam theory This theory is also known as Flexural beam theory. As per this theory,

$$EI\frac{d^2y}{dx^2}=M$$ Where, E = Modulus of elasticity of beam materials I = Moment of Inertia of the section y being deflection at section with distance x from reference point in beam. M is the moment at a section

The above equation can be rearranged considering slope of the beam at section x, $$dy/dx = \theta$$ $$EI\frac{d\theta}{dx} = M$$ The expression $$\frac{d\theta}{dx}$$ is known as curvature = $$\rho$$ of beam at a section x, In conclusion, $$\rho = M/EI$$ For a constant sectional dimensions, $$I$$ will be constant and for same material with in elastic range $$E$$ will be constant, Curvature at a section is directly related to the moment at a section only in this case.

The plot of Moment vs curvature gives useful informations and it is known as moment curvature diagram.

Practically all the materials are nonlinear, the value of E is never constant for the all range of loading. So there wont be linear relationship between sectional moment and sectional curvature. Considering Reinforced concrete beam as an example, With the increase of moment from Zero there will be some moment where the first crack in concrete would appear. With further increase of moment there will be first reinforcement bar yielding in beam. The curvature corresponding to the first reinforcement bar yield is know as yield curvature$$\rho_y$$ and corresponding rotation in beam will be the yield rotation$$d\theta_y$$. After rebar yielding, with slight increase in moment, the change in curvature or rotation is high. The section undergoes plastic deformations above yield point. There will be some curvature of beam for which either rebar fractures or concrete crushes. Corresponding curvature is known as ultimate curvature $$\rho_u$$ and corresponding rotation $$d\theta_u$$ is the yield rotation. The curvature above this $$\rho_u$$ causes drastic loss of strength with instability of beam leading to complete collapse.

The ratio of the ultimate curvature $$d\theta_u$$ and yield curvature $$d\theta_y$$ is defined as curvature ductility. Thus curvature ductility gives information regarding the extent of deformation that the beam can resist beyond yield limit.

Thus the curvature ductility of beam depends up on the ductility of confined concrete as well as ductility of reinforcement bar at element level. Curvature ductility, $$\mu = \frac{\rho_u}{\rho_y}$$