5.9 Supports

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Standard support conditions are

hinged: w = 0,mn = 0

clamped: w = 0, wn = 0

free: mn = vn = 0

Hinged or clamped supports are often idealized as being completely rigid.

But the correct assessment of the stiffness of a load-bearing wall or an edge

beam is important, because the distribution of the support reactions strongly

depends on the stiffness of the supports; see Fig. 5.23. In Fig. 5.24 a point

load is applied at the end of the load-bearing wall. If the support were really

rigid, the applied load would cause no stresses in the slab.

The more flexible the supports, the more “beautiful”the results, because

the slab has a chance to circumvent constraints that might otherwise lead to

singularities; see Fig. 5.25.

It seems that intermittent supports which typically occur at doors and

window openings (see Fig. 5.26) can be modeled as continuous supports as

long as l/h ≤ 7, where l = length of the opening, h = slab thickness. The effect

of a sleeping beam on the structural behavior is often overrated. The increase

in stiffness due to additional reinforcement is too little to be noticeable.

The vertical stiffness of a load-bearing wall with modulus of elasticity

E is

442 5 Slabs

270.1

263.2 kN/m

7.3

1.9

3.6

-532.2

1035.1

85.0

83.2

7.9

-0.7

6.2

104.9

338.0

Fig. 5.23. Support reactions of a slab (8m×8 m) under gravity load g = 9 kN/m2

having a free edge on the right-hand side. a) Rigid supports, b) soft support (brickwork)

k = E d

h

(kN/m2) , (5.68)

where d is the thickness of the wall and h is its height. This coefficient k times

the displacement w of the wall yields the support reaction (kN/m). In the

same sense,

k = EA

h

(kN/m) (5.69)

is the stiffness of a column with cross-sectional area A, height h, and modulus

of elasticity E.

The rotational stiffness cϕ of a wall is the bending moment (kN m/m) that

effects a rotation of 45◦ of the upper edge. The rotational stiffness of the head

of a column depends on the support conditions at the bottom of the column:

kϕ =

3 EI

h

hinged support (5.70)

kϕ =

4 EI

h

clamped support . (5.71)

It is obvious that if a column forms a rigid joint with the slab, the support

reaction will increase, because the influence function for the support reaction

will widen.

5.10 Columns 443

Fig. 5.24. Slab on a system of brickwork walls. a) System and single force applied

at the end of an interior wall; b) deflection surface of the slab; c) principal moments;

d) support reactions and assumed punching shear

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