Titel_SS06

If the number of design variables is then a design optimization problem can be
formulated:

N& 1

lu

ii i

()

.. ( , , , ) 0 , 1,...,

( , , , ) 0 , 1,...,

z z z , 1,...,

min

ic j e

ic j e

C

st c i m

cim

iN

6

6



*

 

z

xpz

xpz m

(11.12)

C(z) is the objective function and cii , 1,...,m are the constraints. The objective function

C(z) is often chosen as the weight of the structure. The me equality constraints in Equation

zi

(11.12) can be used to model design requirements (e.g. constraints on the geometrical
quantities) and to relate the load on the structure to the response (e.g. finite element
equations). Often equality constraints can be avoided because the structural analysis is
incorporated directly in the formulation of the inequality constraints. The inequality
constraints in Equation (11.12) ensure that response characteristics such as displacements and
stresses do not exceed codified critical values as expressed by the design Equation (11.11).
The inequality constraints may also include general design requirements for the design
variables. The lower and upper bounds, and zil ziu, to in Equation (11.12) are simple

bounds. Generally, the optimization problem Equation (11.11) is non-linear and non-convex.
Next, the reliability index j() 6 is estimated by FORM/SORM or simulation on the basis of

the limit state equations (Equation (11.6)) using the optimal design z* from Equation (11.11)
or Equation (11.12).

Ad 7. As discussed above a first guess of the partial safety factors is obtained by solving these
optimization problems. Next, the final partial safety factors are determined taking into account
current engineering judgment and tradition. Examples of reliability-based code calibration can
be found in Nowak (1989), Sørensen et al. (2001) and SAKO (1999).

Example 11.1 – Calibration of partial safety factors using the JCSS CodeCal software

The following simple, but representative limit state function is considered:

gzRX RQ (1 555 )G Q 1 (1 (^5) Q)Q 2  (11.13)
where:
R load bearing capacity
XR model uncertainty
z design variable
G permanent load
Q 1 variable load: type 1, e.g. wind load
Q 2 variable load: type 2, e.g. snow load
5 factor between 0 and 1, modelling the relative fraction of variable load