C. Machelski/R. Toczkiewicz · Effects of connection flexibility in bridge girders under moving loads
Mb a·Nb
Table 1. Geometrical and physical characteristics of girder analysed
[MNm]
Element
Ab/Ap [m2]
Ib/Ip [m4]
vg/yg [m]
vd/yd [m]
Eb/Ep [GPa]
Steel beam
0.0448
0.0191
0.990
0.640
205
Concrete plate
0.5820
0.0020
0.095
0.145
32.6
μ =2.036
1.0
0
0.8
Mb 0.6
0.2
μ 0 0
0.4
0.8
1.2
1.6
2.0
2.4
Fig. 2. Internal forces in steel beam versus value of index R
bending moment M " 1 MNm is assumed. The diagram shows that the increase in R value is associated with an increase in axial force Nb (multiplied in the example by distance a) and with a decrease in bending moment Mb. These changes are significant at low values of R and stabilize as R increases. The index R has two characteristic values [15]: – in the case of full beam-plate interaction, then μ = μ0 =
a · ad · A b Ib
(12)
– in the case of no interaction (when Nb " 0), then R " 0 according to Eq. (11). Values of R | R0 mean that there is partial beam-slab interaction and that shear slip occurs at the steel-concrete interface. Examples of functions R(x) created for different cases of connection stiffness function k(x) are given and commented on later in this paper. In the case of partial interaction, index R depends linearly on the position of the girder’s neutral axis described by distance aed (see Fig. 1), according to the relationship a′d =
μ · Ib μ = ad μ0 a · Ab
(13)
Eq. (10), providing the solution for the partially composite girder subjected to bending, is solved in the analysis using the finite difference method [19]. This method allows the function R(x) to be analysed for arbitrary functions of variables included in Eq. (10). Values of Eb, Ipn and Apn are usually constant in bridge structures, whereas beam characteristics Ib and Ab (and hence distance a) and connection stiffness k are most often constant only along sections of a girder. All the analyses described below were conducted for a typical steel-concrete girder of a beam bridge with a span L " 28 m. The geometrical and material characteristics of the girder are given in Table 1.
4.1 Local effect of concentrated load For the girder analysed here (simply supported beam loaded with single force at mid-span and constant connection stiffness), the problem was solved several times, each time changing the connection stiffness constant along the girder k(x) " k. The resulting diagrams of the function R(k, x) are shown in Fig. 3. It can be seen that along the whole length of the girder, index values fulfil the relationship R(x) f R0. Both the decrease in index R and the range of decrease along the girder depend on the connection stiffness k(x). The lowest values of R are obtained in the section where force P is applied. Function R(k, x) results directly from the functions of internal forces Nb(x) and Mb(x) determined from the solution of Eq. (10) and is connected with changes in slip increments at the beam-plate interface [19]. In the case of constant stiffness k(x) " k, the result is R(x) f R0.
4.2 Change in connection stiffness along beam In real composite girders, connection stiffness results from the distribution and type of connector (e.g. headed shear studs). The connectors are arranged according to the enve-
If there is no interaction (R " 0), the neutral axis coincides with the axis of inertia of the beam Ob (aed " 0), and in the case of full interaction (R " R0), the relation aed " ad is obtained. After transformation, Eq. (13) allows the value of R to be calculated on the basis of the position of the neutral axis that can be determined in loading tests [15].
2.1 2.0
μ = μ0 k [MN/m 2]
1.9
μ [-]
4 Function of partial interaction R(x)
2.2
x=L /2
a·Nb
0.4
1000
P
k
1.8
2000
1.7
The analytical model of a girder with partial interaction described in section 2 was used in the numerical analyses given below. The main aim of the analysis was to illustrate the influence of moving loads, changing their position on the structure, on the beam-plate interaction characterized by the proposed index R.
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Steel Construction 9 (2016), No. 1
5000
1.6
L/2
10000
L/2
20000
1.5
x [m]
1.4 0
2
4
6
8
Fig. 3. Effect of a single point load
10
12
14