Reliability Analysis 305 26） s6,)o36,) (6.32) ∂bab 3(g) -ELg1-3Ag1o(g川。g Inserting the limit state function g from(6.29)into (6.30)-(6.32)and assuming that the random variable of allowable stresses and the random field of actual stresses are uncorrelated it is obtained that (6.33) e-6-+06-腰 (6.34) S6)o6,）-E[o-o】 and,finally sg)=aa-o月 oi") (6.35) s6,)6) a-o:loo-o) Comparing the second order second moment(SOSM)approach with the second order third moment (SOTM)approach,it is seen that the expected values are described by exactly the same equation,while standard deviations(or variances) have some extra components connected with the skewness of analysed PDF;the third order parameter of the output PDF is taken into account in the SOTM-based analysis [282].Reliability Analysis 305 ( ) ( ) ( ) ( ) () () } ( ) 1 [ ] 3 [ ] ( ) ( ) 1 3 2 2 3 3 3 2 3 1 3 2 2 0 3 1 2 2 2 2 0 2 0 3 0 g E g E g g g S b b b g b g g b g b b g g b g S g g g n i i i i i i n i i i i σ σ σ σ σ − − ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = + ∑ ∑ = = (6.32) Inserting the limit state function g from (6.29) into (6.30)-(6.32) and assuming that the random variable of allowable stresses and the random field of actual stresses are uncorrelated it is obtained that ∑ ( ) = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = − − n i i i z all z b b E g 1 2 2 2 2 0 0 1 [ ] σ σ σ σ (6.33) ( ) ( ) ( ) ( ) () () [ ] all z n i i i i z i z n i i i z all z i z all z S b b E b b b b b g σ σ σ σ σ σ σ σ σ σ σ σ σ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = − + ∑ ∑ = = 2 1 3 2 2 1 2 2 2 0 0 2 2 2 0 0 (6.34) and, finally ( ) ( ) ∑ ( ) ( ) ( ) = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ + − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + − ⎩ ⎨ ⎧ = − n i i i z all z i z all z all z b b b S g 1 2 2 2 2 0 0 2 0 0 3 0 0 2 2 3 σ σ σ σ σ σ σ σ σ ∑ () () = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + n i i i i i i S b b b g b g g b g 1 3 2 2 0 3 3 σ [ ][ ] ( )} ( ) 3 0 0 3 0 0 0 0 2 0 0 1 3 all z E all z E all z all z σ σ σ σ σ σ σ σ σ σ − − − − − − (6.35) Comparing the second order second moment (SOSM) approach with the second order third moment (SOTM) approach, it is seen that the expected values are described by exactly the same equation, while standard deviations (or variances) have some extra components connected with the skewness of analysed PDF; the third order parameter of the output PDF is taken into account in the SOTM-based analysis [282]