2.2.5 Limitation of Deformations

Limitation of Deformations

EN 1992-1-1, clause 7.4.3 allows for a simplified design of the limitation of deformations via direct calculation. The deflections must be determined realistically: The calculation method has to match the real structural material performance with an accuracy that corresponds to the design purpose.

The deflection is determined by double integration from the differential equation of the bending line. However, as the stiffness of a reinforced concrete cross-section changes in parts due to cracking, the moment-curvature diagram is nonlinear. There are big differences in curvature, and thus in deflection, for uncracked (state I) and cracked sections (state II).

Therefore, the deflection is determined with the principle of virtual work for the location of the maximum deformation. An approximation line is used for the curvature, connecting the extreme values of the curvature with a line that is affine to the moment distribution.

When calculating manually, three values of the deflection are determined according to [3]:

Minimum deflection is achieved when the calculation is performed for a completely uncracked cross-section (state I). This type of deflection is described as f_I.

Maximum deflection is achieved when the calculation is performed for a completely cracked cross-section (state II). This type of deflection is referred to as f_II.

It is fair to assume that some parts of the cross-section are uncracked, and other, highly stressed parts are cracked. The moment-curvature relation runs up to the first crack after state I, after which it shows some cracks. This assumption results in the probable value of the deflection f, which lies between the lower and upper calculated value. According to EN 1992-1-1, clause 7.4.3 (3), Eq. (7.18), the value can be derived from the following relation:

$\alpha = \zeta · {\alpha }_{II} + \left(1 - \zeta \right) · {\alpha }_1$

Equation 2.16 EN 1992-1-1, Eq. (7.18)

The values α_I and α_II represent general deflection parameters (e.g. f_I or f_II). This can be a strain, curvature, deflection, or rotation. ζ is the distribution coefficient between state I and state II, and lies between 0 ≤ ζ < 1, as shown in EN 1992-1-1, Eq. (7.19). Generally, the deformation calculation is to be performed with a quasi-permanent combination (see EN 1992-1-1, clause 7.4.3 (4)).

Chapter 9.1 describes an example where the manually performed calculation of a deformation analysis is compared with the results of the program.