Rotation mechanisms – cantilever action

Consider a pure rotation system like the one illustrated in Fig. A-4. For a small rotation ϕ of the pure rotation mechanism, the displacement of a tie connection i can be expressed approximately as

ϕ

⋅

≈ _i

i l

w (A-6)

where li = radial distance between the rotation axis and tie connection i and the vertical displacement of the driving force as

ϕ

⋅

≈ _qx,0

qz l

a (A-7)

where lqx,0 = horizontal projection of the radial distance between the rotation axis and the driving force Q in the undeflected system

Fig. A-4: Rotation mechanism for a multi-storey wall cantilever above a local damage

As follows from eqs. (A-6) and (A-7) there is a simple geometric relationship between the vertical displacement of the driving force and the displacement of each tie connection.

qz

For plane cantilevers with a depth to length ratio of h/l ≤ 2 this expression gives satisfactory accuracy for rotations ϕ ≤ 0,3 rad. For a ratio h/l = 10 the rotation should be limited to about ϕ ≤ 0,03.

These conditions are normally satisfied for applications in ordinary building structures and also in the case of multi-storey cantilevers.

For a certain small rotation ϕ of the multi-storey cantilever the condition of static equilibrium yields

i

where lqx = actual horizontal distance between the rotation axis and the driving force Q wi is determined by (A-6)

The static resistance of the cantilever system expresses the ability of the system to balance a driving force in the centre of gravity. Since the tie forces develop with the rotation, the static resistance varies with the rotation and can be expressed as a function of the vertical displacement of the driving force as

For estimation of the dynamic resistance it is convenient to introduce a formal value of the maximum static resistance in the undeflected state. It is obtained from eq. (A-9) by replacing the actual tensile forces Ni with the tensile force capacity Ni,u of each tie connections independently of the actual displacements, and lqx by lqx,0.

This value is ‘formal’ since the tie connections are contributing with maximum capacities simultaneously and without any displacement of the cantilever system. In the real system the tie connections may rupture one after the other depending on their locations and deformation capacities.

For a certain tolerable value of the maximum displacement aqz,max of the cantilever system the corresponding dynamic resistance Rdyn(aqz,max) can be determined by means of the energy equilibrium in eq. (A-4).

Eq. (A-11) expresses the maximum load Q that can be bridged by cantilever action in case of a sudden support removal. This is the dynamic resistance of the system. Hence, the dynamic resistance is a function of the vertical displacement aqz,max that is used to slow down the motion

u

Hence, the dynamic resistance can be regarded as a reduced value of the maximum static resistance according to eq. (A-10), where the contributions from the respective tie connections have been reduced by factors ξi(wi,max). The relative strain energy will always, according to the definition in eq. (A-3), be less than or equal to 1. The actual value depends on to what extent the elongation capacity wi,u of the respective tie connections is used.

It is important to note that the displacements wi,max of the respective tie connections are related by the geometry of the system and should be determined for the same maximum rotation of the bridging system. It is convenient to carry out the analysis of a rotation mechanism according to the following steps:

1) Choice of maximum vertical displacement aqz,max of the driving force with due regard to free space for displacements and elongation capacities of the tie connections.

2) Calculation of the corresponding maximum displacements wi,max for the connections by means of eq.(A-8).

3) Calculations of the corresponding relative strain energy ξi(wi,max) for the tie connections according to the relevant load-displacement relationships, see Section 7.2.

4) Calculation of the maximum static resistance Rmax, formal value in the undeflected state, according to eq. (A-10)

5) Calculation of the dynamic resistance Rdyn by means of the condition of energy equilibrium in eq. (A-13).

6) Check of static equilibrium in the state of maximum displacement according to eq. (A-9).

This approach to analyse collapse mechanisms where tie connections are strained in the plastic range was confirmed by tests on precast floor specimens subjected to sudden support removal, see Fig. A-5, [Engström (1992)].

Fig. A-5: Tests on a collapse mechanism in a precast floor with various types of tie connections, according to Engström (1992)

Example A-1, two-wall system – cantilever action

A load-bearing precast wall consists of 6 wall elements arranged in three stories, see Fig. A-6. The possibility that one of the bottom elements is totally damaged by accidental action should be considered in the design against progressive collapse. Check if it is possible to bridge over the damaged area by cantilever action of the wall elements above the damaged area, either as separate wall cantilevers, or as a combined two-storey cantilever. Rotation mechanisms can be assumed. Each wall elements is loaded by the dead weight G = 80 kN and a uniformly distributed load from the floor q = 40 kN/m (gravity load). The horizontal joints are provided with two tie bars φ16 B500 anchored in concrete of strength class C20/25, ‘good’ bond conditions (the same connection as in Examples 7-1 and 7-5).

2120

1610 3000 265 Q

3,0

3,0

3,0

Fig. A-6: Analysis of alternative load-bearing system in a precast wall in Example A-1, a) location of damaged area, b) collapse mechanism of separate wall cantilevers

Separate wall cantilevers:

The resultant Q, considered as a gravity load, and its position (gravity centre) in the undeflected system is determined as, see Fig. A-7.

2,5 2,5

Fig. A-7: Resultant driving force Q and its location on single-storey cantilever 280

The tie connection across the vertical joint consists of two tie bars φ16 and has a total yield capacity of Ny = 202 kN and an ultimate capacity of Nu = 218 kN. The schematic load-displacement relationship of the tie connections has been determined according to Sections 7.2.3 and 7.4.1 (Example 7-5) and is shown in Fig. A-8. On basis of the schematic relationship the relative strain energy has been determined to ξ(wu) = 0,835 for the maximum displacement, and ξ(0,5wu) = 0,670 for half the maximum displacement, see Example 7-5.