5.21 Sensitivity analysis

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The ideas which we developed in Section 1.27 can also be applied to slabs. If

the stiffness ks of a line support Γ—typically a wall—changes, ks → ks+Δks,

then the increment in the Dirac energy is the integral

− d(Gci

, w) = −

_

Γ

Δks Gci

(y, x)w(y) dsy

_ −

_

Γ

Δks Gi(y, x)w(y) dsy (5.113)

or if we use a one-point quadrature rule

− d(Gci

, w) _ −Δks · RG

ks

· Rp

ks

· 1

lΓ

(5.114)

that is if we replace the distributed forces by their resultants, RG = (Gi, ks)

and Rp = (w, ks) along the wall [0, lΓ ].

We applied this idea to the slab in Fig. 5.66 where each single value next

to a wall signals by how much the bending moment myy(x) at the mid point

of the central slab will change if the stiffness of this wall drops by 50 %.

The changes are very small probably because with ks → 0.5 ks the load is

transferred to the neighboring walls which—in this model—are assumed to

retain their original stiffness.

5.21 Sensitivity analysis 481

Fig. 5.61. Slab: a) system, b) FE SOFiSTiK mesh

482 5 Slabs

90.9 / 90.8 / 92.3 kN

60.7 / 59.7 / 59.5 kN

53.4 / 53.4 / 53.5

100 / 190 / 127 kN/m SOFiSTiK / Nemetschek / BE

185 / 247 / 36 kN/m

112 / 123 / 115 kN/m

7.3 / 7.3 / 7.7 kN/m

16.1 / 15.9 / 15.8 kN/m

7.4 / 6.9 / 5.3 kN/m

9.3 / 9.3 / 9.2 kN/m

30.6 / 29.5 /28.2 kN/m

9.1 / 9.7 / 8.6 kN/m

15.5/15.2/14.8kN/m

24.6 / 24.4 / 24.1 kN/m

T beam

free edge

Fig. 5.62. Gravity load, support reactions

60.1 / 60.1 / 61.5 kN

28.1 / 27.5 / 28.0 kN

- 4.0 / - 4.3 / - 5.4 kN

7.9 / 7.8 / 7.9 kN/m SOFiSTiK / Nemetschek / BE

- 2.7 / - 2.7 / - 3.1 kN/m

58.3 / 52.6 / 56.3 kN/m

- 3.4 / - 3.2 / - 2.6 kN/m

single force 100 kN

Fig. 5.63. Support reactions when a single force P = 100 kN is applied

5.21 Sensitivity analysis 483

- 22.6 / - 16.2 / -21.3 kNm/m SOFiSTiK / Nemetschek / BE

6.2 /6.0 / 6.1 kNm/m - 6.1 / - 4.6 / - 5.9 kNm/m

18.4/ 16.2/ 17.4 kNm/m

- 6.8/ - 5.0 / -6.4 kNm/m

5.0 / 4.8 / 4.9 kNm/m

- 12.9 / -11.5 / -10.4 kNm/m

6.6 / 7.1 / 6.8 kNm/m

m

xx

m

yy

Fig. 5.64. Gravity load: bending moments mxx and myy in two sections

9.9 / 10.0 / 9.9 kN/m

25.1 / 53.0 / 49.2 kN/m

24.6 / 53.0 / 46.3 kN/m

- 7.6 / -10.0 / -6.9 kN/m

- 5.2 / -5.5 / -6.1 kN/m

q

x

q

x

- 13.6 / -13.8 / -14.1 kN/m

17.6 /17.2 /17.4 kN/m

- 14.2 / - 13.8 / - 14.0 kN/m

- 10.1 / - 11.0 / - 11.0 kN/m

15.1 /15.8 /15.0 kN/m

- 6.3 / - 7.3 /- 1 kN/m SOFiSTiK / Nemetschek / BE

q

y

q

y

Fig. 5.65. Gravity load: shear forces qx and qy in two sections

484 5 Slabs

Fig. 5.66. Sensitivity analysis: the single numbers indicate how much myy will

change if the stiffness of the walls change by 50 %

6. Shells

Shell elements are the most sophisticated elements because they must represent

membrane and bending stresses equally well, and they must also model

analysis can be discussed in this chapter. Instead we concentrate on the typical

features.

Fig. 6.1. Shell roof

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