2.4.7.2 Safety Design

Safety Design

According to EN 1992-1-1, clause 5.7, we have to design the safety of nonlinear calculations by means of a global safety factor γ_R. We can do this with a "trick", though it is disputed: modifying the mean stiffnesses of structural components (f_cR, f_yR, etc.). The calculational steel stress has been increased and the calculational concrete stress has been reduced, which allows for a return to the global safety factor γ_R = 1.3 (or 1.1 for extraordinary action combinations).

To ensure sufficient bearing capacity, the following conditions are required:

$E_d \le R_d = \frac{R}{{\gamma }_R} \left(f_{cR }, f_{yR }, f_{tR} ...\right)$

where

• E_d : design value of governing action combination
• R_d : design value of load-bearing capacity
• γ_R : uniform partial safety factor on side of ultimate load

RF-CONCRETE Members calculates with a γ_R-fold action. It can be applied in load steps, corresponding to an incremental calculation of the ultimate load.

The design is fulfilled when the γ_R-fold action is higher than the ultimate load. This corresponds to a conversion of the equation above.

${\gamma }_R · E_d \le R_d = R \left(f_{cR }, f_{yR }, f_{tR} ...\right)$

This also takes account of the aspect for determining the reduction of imposed internal forces.

The most important advantage of this approach is obvious: Only one material rule is used for the entire calculation. This leads to easier handling as well as economy of time when calculating, because the determination of internal forces and the design are performed in one go.

The disadvantage is only explicitly visible when we assume that the terms

$\frac{R}{{\gamma }_R} \left(f_{cR }, f_{yR }, f_{tR }, ...\right) = R \left(\frac{f_{cR}}{{\gamma }_R}, \frac{f_{yR}}{{\gamma }_R}, \frac{f_{tR}}{{\gamma }_R}, ...\right)$

are compatible. In nonlinear calculations, of course, this compatibility is not given without restrictions. An example, which shows that such an approach can be very much on the unsafe side, is the consideration of imposed internal forces. The use of material properties divided by γ_R leads to strongly reduced stiffnesses resulting in a strong reduction of the imposed internal forces. However, this representation is quite useful to illustrate the problem of the reduced elastic modulus for steel.

The direct reduction of the stiffnesses is described in detail by Quast [10] and is evaluated critically with regard to slender compression elements.

To clarify the correlations, we simplify the conditions and assume a horizontal branch of the characteristic curve for reinforcing steel (f_yd = f_td). This results in the reduced design resistance R_d for:

$R_d=\frac{R}{{\gamma }_R}=\frac{1}{{\gamma }_R}\int a·{\sigma }_R\left[\epsilon \left(y,z\right)\right]dA \text{where} a=\left\{\begin{array}{c}1\\ z\\ -y\end{array}\right\}$

$$\begin{gathered} R_d = \frac{1}{{\gamma }_R} \int a \left[-f_{cR} \le {\sigma }_{cR} \left(\epsilon , f_{cR}\right) \le 0; -f_{yR} \le {\sigma }_{sR} \left(\epsilon \right) \le f_{yR}\right] dA \\ {R}_d = \int a \left[\frac{-f_{cR}}{{\gamma }_R} \le \frac{{\sigma }_{cR} \left(\epsilon , f_{cR}\right)}{{\gamma }_R} \le 0; \frac{-f_{yR}}{{\gamma }_R} \le \frac{{\sigma }_{sR} \left(\epsilon \right)}{{\gamma }_R} \le \frac{f_{yR}}{{\gamma }_R}\right] dA \end{gathered}$$

If we set σ_sR = E_s ⋅ ε, the result is the following:

$R_d = \int a \left[\frac{-f_{cR}}{{\gamma }_R} \le \frac{{\sigma }_{cR} \left(\epsilon , f_{cR}\right)}{{\gamma }_R} \le 0; \frac{-f_{yR}}{{\gamma }_R} \le \frac{E_s }{{\gamma }_R} \epsilon \le \frac{f_{yR}}{{\gamma }_R}\right] dA$

For a practical determination of internal forces according to the linear static analysis without imposed internal forces, it is absolutely legitimate to calculate with the reduced stiffnesses. In this case, the diagram of internal forces is affected by the relation of the stiffnesses from different areas to each other anyway.

However, this concept proves to be problematic when designing slender compression elements according to the second-order analysis. The deformations are overestimated because of the reduced stiffness in the system. This results in an overestimation of internal forces for calculations according to the second-order analysis.

Slender compression elements generally fail when the yield strain in the reinforcement is reached. Hence, it becomes obvious that deformations are overestimated due to the reduced modulus of elasticity and the resulting larger curvatures when the yielding starts. This leads to a smaller allowable column load, or the reinforcement must be increased accordingly. Quast [10] sees no reason for that.

According to EN 1992-1-1, clause 5.8.6 (3), it is possible to directly perform the design for sufficient structural safety on the basis of design values (f_cd, f_yd, ...) of the material properties. In accordance with clause (3), the stress-strain curves defined on the basis of design values must also be used for the determination of internal forces and deformations. The modulus of elasticity E_cd to be applied must be calculated with the safety factor γ_CE (E_cd = E_cm / γ_CE).

According to the National Annex for Germany EN 1992-1-1, clause 5.8.6 (NDP 5.8.6 (3)), the internal forces and deformations may be determined by means of average material properties (f_cm, f_ctm, ...). However, the design for the ultimate load capacity in the governing sections must be performed with the design values (f_cd, f_yd, ...) of the material properties.

The problem with this approach is that it is impossible for some parts in statically indeterminate systems to reach a convergence of results: The internal forces calculated with the mean values of the material properties cannot be taken over in the design with the design values to be applied. Increasing the reinforcement results in an increase of stiffness of the respective parts and areas, which again requires an increase of reinforcement in the subsequent iteration step. It is also important to note that a utilization of the plastic resources in the ultimate limit state is hardly possible, as the calculational design moment M_Ed (design values for strengths of materials) will not reach the value of the yield moment M_y (mean material properties).

RF-CONCRETE Members performs the safety design according to the standard by contrasting the provided reinforcement with the required reinforcement that is determined for the design values of the material properties. This must always be observed when manually correcting the reinforcement (keyword: "increase in stiffness").