Deformation Capacity of Beams, Columns and Walls (Eurocodes)

The value for the chord rotation capacity of rectangular beams and columns is calculated through different expressions, depending on the state of damage in the structure.

Limit State of Near Collapse (NC)
The value for the chord rotation capacity for the limit state of near collapse (NC) is the value of the total chord rotation capacity (elastic plus inelastic part) at ultimate of concrete members under cyclic loading, which is calculated from the following equations (A.1) and (A.3) of EC8: Part 3 (CEN, 2005b):

Where γ_el is equal to 1,5 for primary seismic elements and to 1,0 for secondary seismic ones and L_V is the ratio between bending moment, M, and shear force, V. The remaining relevant parameters are defined in section A.3.2.2 of EC8: Part 3.

In walls the value given by the equation (A.1) is multiplied by 0,58.

The total chord rotation capacity at ultimate of concrete members under cyclic loading may be also calculated as the sum of the chord rotation at yielding and the plastic part of the chord rotation capacity calculated from the following expression:

Where γ_el is equal to 1,8 for primary seismic elements and to 1,0 for secondary seismic ones; the chord rotation at yielding, θ_y, is calculated in accordance with the section A.3.2.4 of EC8: Part 3 and the remaining relevant parameters are defined in section A.3.2.2 of EC8: Part 3.

In walls the value given by the equation (A.3) is multiplied by 0,6.

Limit State of Significant Damage (SD)
The chord rotation capacity corresponding to the limit state of significant damage (SD) is assumed to be 3/4 of the ultimate chord rotation, calculated from the equations above.

Limit State of Damage Limitation (DL)
The chord rotation capacity that corresponds to the limit state of damage limitation (DL) is given by the chord rotation at yielding, evaluated as:

For rectangular beams and columns:

For walls or rectangular T- or barbelled section:

Or from alternative and equivalent expression for rectangular beams and columns:

For walls or rectangular T- or barbelled section from expression:

Where α_V is equal to zero if the yielding bending moment is lower than L_V multiplied by the concrete shear resistance – V_R,c - and 1.0 otherwise. V_R,c is calculated according to EN1992-1-1:2004 provisions for concrete elements without shear reinforcement. The remaining relevant parameters are defined in section A.3.2.4 of EC8: Part 3.

The yield curvature of the end section is calculated according to the following expressions. The lower from the two values below, (a) for the case that the section's yielding is due to steel yielding and (b) for the case that the section yields due to the deformation non-linearities of the concrete in compression, is used for the calculation of the chord rotation capacity.

                        (a)

                       (b)

According to Annex A of EN1998-3 the chord rotation capacity is highly influenced by a number of different factors such as the type of the longitudinal bars. If cold-worked brittle steel is used the plastic part of chord rotation is divided by 2, whereas if smooth (plain) longitudinal bars are applied, section A.3.2.2(5) of Annex A is employed, taking, also, into consideration whether the longitudinal bars are well lapped or not. In case of members with lack of appropriate seismic resistant detailing the plastic part of chord rotation capacity is multiplied by 0,825. Moreover, if the deformed longitudinal bars have straight ends lapped starting at the end section of the member, the plastic part of chord rotation is calculated with the value of the compression reinforcement ratio, ω’, doubled over the value applying outside the lap splice. In addition, in sections where the reinforcement lap length lo is less than the minimum lap length for ultimate deformation lou,min, the plastic part of the chord rotation capacity is multiplied by the ratio lo/lou,min, for more information about the calculation of lou,min you may refer to A.3.2.2(4) of Annex A, while the value for chord rotation at yielding, θy accounts for the effect of the lapping in accordance with A.3.2.4(3) of Annex A.