5.12 Variable thickness

Back

If a slab has a smooth surface but the thickness varies the midsurfaces of the

single panels will lie at differing levels; see Fig. 5.38 a. To accurately model

such a plate would require elements for which such a shift of the midsurface

5.12 Variable thickness 453

0.6

0.9

0.8

0.6

0.8

0.9

0.6

0.9

0.8

d = 22.0 cm

x

y

23.2

17.3

Fig. 5.34. Slab: a) system, b) shear stresses at the supports usually remain below

the threshold values for shear reinforcement, here 0.5 MN/m2

is possible. Conventional plate elements model such a slab with a uniform

midsurface; see Fig. 5.38 e.

Variations in the thickness of the slab will produce jumps in the internal

actions; see Fig. 5.38. At the interface between two such zones the bending

moment mxx and the curvature κyy = w,yy must be the same,

mL

xx = −KL(wL,xx +ν w,yy ) = −KR(wR,xx +ν w,yy ) = mR

xx , (5.81)

while the bending moment myy will be discontinuous.

454 5 Slabs

x

qy in a horizontal and

supports, EA = ∞;

b) 3-D view of qx

If Poisson’s ratio is assumed to be zero, ν = 0, then the ratio of these two

bending moments becomes approximately

mL

yy

mR

yy

= KL

KR

(w,yy +ν wL,xx )

(w,yy +ν wR,xx )

_ KL

KR = h3

L

h3

R

=

0.23

0.43 =

1

8 . (5.82)

Hence if the thickness h doubles, then because of the h3 the bending moment

increases by a factor of eight.

At column capitals or drop panels, the bending moments peak at an earlier

stage, and they stay at that level for a longer time; see Fig. 5.39.

In the slab in Fig. 5.40 the singularity in the support reactions is very

pronounced and mainly due to the rather large change in the thickness of

the slab from 0.25 m to 0.60 m. Such situations are not uncommon in the

analysis of slabs and then elaborate mathematical theories will not help very

much—rather a sound engineering judgement must cope with such problematic

results.

Fig. 5.35. Slab on

masonry walls: a)

shear forces q and

respectively. The values

in brackets are

the results for rigid

a vertical section,

3.2 3.1

x

y

8.0

8.0

X

Y

Z

5.12 Variable thickness 455

5797.3 kN/m

- 2628 kN/m

668 kN/m

668 kN/m

125 kN/m

93 kN/m

0.4 m

5797.3 kN/m

- 2628 kN/m

890 kN/m

0.4 m

R =101 kN

0.4

R = 50.5 kN

0.2

0.2 m x0.2 m

R =143 kN

0.4

R = 71.5 kN

0.2

0.2 m x0.2 m

0.2 m 0.2 m

Fig. 5.36. Support reactions and equivalent punching shear for assumed columns

0.4 × 0.4 and 0.2 × 0.2 respectively at the end of a wall

node 2

QR= 13.5 kN

A-SS= 0 cm2

node 19

QR= 14.9 kN

A-SS= 0 cm2

node 33

QR= 42.6 kN

A-SS= 0 cm2

node 68

QR= 11.4 kN

A-SS= 0 cm2

node 76

QR= 77.6 kN

A-SS= 0 cm2

node 85

QR= 68.8 kN

A-SS= 0 cm2

node 121

QR= 12.8 kN

A-SS= 0 cm2

node 124

QR= 8.0 kN

A-SS= 0 cm2

node 25

QR= 42.6 kN

A-SS= 0 cm2

Fig. 5.37. Nowadays punching shear checks are done routinely by FE programs at

the end points of load-bearing walls

456 5 Slabs

Fig. 5.38. Hinged slab: a) cross section; b) system; c) principal moments;

d) bending moments mxx and myy; e) 3D-view

5.12 Variable thickness 457

xx,

b) bending moments myy

In a vertical section myy

mxx would be discontinuous

Fig. 5.39. Interior cola

a)

bending m

in a horizontal section.

would be continuous and

umn of a

moments

hinged slab

with drop panel:

-26.5

x

y

6.0

6.0

-26.6

x

y

6.0

6.0

458 5 Slabs

Fig. 5.40. The edge load rotates the slab downward but the slab is stabilized by the

torque built up by the support reactions: a) slab and loading b) support reactions of

the continuous support c) support reactions with an intermission in the supporting

wall

5.13 Beam models 459

Fig. 5.41. Slab with an attached balcony load case g + p (balcony): a) system,

bending moments mxx in various sections, c) 3-D view of the deflection surface,

bending moment mxx

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