General calculation method

Right from the design or planning state, one of the problems of construction engineering to be resolved is the cambering of the individual bridge components. The design planes show the geometric outline of the bridge, i.e. its shape under the designated load condition. This is commonly the full dead load including the additional dead load due to pavement etc. at normal temperature. The contractor, however, fabricates the bridge segments under the no-load condition, and at “shop temperature”. The difference between the shape of a member under full dead load and normal temperature, and its shape at the no-load condition and shop temperature, is the segment camber. Figure 4-25 illustrates the camber control by a camber and the deformation sketch.

Figure 4-25: Camber and deformation

The general procedure of calculating the camber data is demonstrated in the following. The figure below shows the cantilever, which is assumed to be installed in three steps. In this case, not only the actual deformation under its direct loading, but also the additional deflection due to the weight of following segments must be considered in the camber calculation. Usually, the subsequent segments are connected with a prefabricated angle to balance the deformations.

Figure 4-26: Cantilever

Figure 4-27 a) describes the current deflection of installing a new segment and Figure 4-27 b) the fabrication camber to realize a zero displacement condition in the last stage of the construction.

δ¹¹ δ²¹ δ³¹

δ¹²

δ²² δ³²

δ¹³

δ²³

δ³³

δ³² δ³³ δ³¹

δ¹¹ δ¹² δ¹³

δ²¹ δ²² δ²³

δ³¹ δ³² δ³³

1 2 3

Figure 4-27: a) Current displacement b) Construction camber

The current displacement is only that of the new installed segment without considering any other existing displacements of previous installations. The real displacement of each segment can be calculated on the basis of the current displacement.

Chapter 4: Example of a Cable-Stayed Bridge including temporary supports 122

After all segments are put in place, the real displacement can be calculated for each node as:

Node 1 =

δ

₁₁+

δ

₁₂+

δ

₁₃

Node 2 =

δ

₂₁+

δ

₂₂+

δ

₂₃ ⇒ Real displ., fabrication camber Node 3 =

δ

₃₁+

δ

₃₂+

δ

₃₃

where δ expresses the deflection, the first indices describes the location of the deformation (node number) and the second indices describes due to which segment the deformation occurs.

The real displacement has the meaning of the general fabrication camber.

The total net displacement, which is the meaning of the construction camber, can be determined as:

Node 1 =

δ

₁₁+

δ

₁₂+

δ

₁₃

Node 2 =

δ

₂₂ +

δ

₂₃ ⇒ Total net displ., construction camber Node 3 =

δ

₃₃

In case of the given example, the first segment must be installed with a specific angle, which produces an upward deformation of

δ

₁₁+

δ

₁₂ +

δ

₁₃at the tip of the segment. Due to its self-weight, the segment will be subject to the downward movement of δ₁₁. The next segment must be installed with an angle again, producing a deformation at the tip of

δ

₂₂+

δ

₂₃, which will be pulled downward by -

δ

₂₂because of its self-weight. The remaining value of

δ

₂₃^{will be}

eliminated with the installation of the last segment, which has to be fixed to the previous segment with an angle generating a deformation of

δ

₃₃ . The total structural upward displacement produced by the angle-connections, will be balanced by the structural self-weight.

The described method is illustrated in Figure 4-27 b).

A simple example is calculated to illustrate the procedure as explained before. Figure 4-28 shows a cantilever which is erected in three stages. A distributed load of 25 tonf/m, a stiffness I of 0.92 m⁴, an elasticity E = 2.1*10⁵ N/mm and cross section area A = 4.38 m² is assumed in the calculation.

Figure 4-28: Erection of a cantilever CS 1

10 tonf/m 10 tonf/m

10 tonf/m

CS 2

CS 3

The figure below shows the current displacement for each erection step. All elements are activated in the first step in order to prove the tangential activation of the elements in the MiDAS programme. The loads are applied sequentially.

Current step displacement [mm]

CS 1

CS 2

CS 3

Figure 4-29: Erection of a cantilever, current displacement value

The current displacement for construction stage CS 1 is controlled by a simple calculation. With the displacement u1 (δ11) at Node 1

q 25tonf

:= m E 2.1 10⋅ ⁵ N mm²

:= I:=0.92 m⋅ ⁴ L:=10 m⋅

u1 q L⋅ ⁴ 8 E⋅ ⋅I

:= u1 1.586mm=

and the angle ψ, the displacement for Node 2 (δ12) and 3 (δ13) can be calculated as:

ψ q L⋅ ³ 6E I⋅

:= ψ =2.115×10⁻⁴

∆u2ψ:=ψ⋅L u2:=∆u2ψ+u1 ∆u3ψ:=ψ⋅2L u3:=∆u3ψ+u1

∆u2ψ=2.115mm u2 3.701mm= ∆u3ψ=4.23mm u3 5.816mm=

From the current displacement, the real and the net displacement can be calculated as already explained. The real displacement values are calculated in Table 4-12. The real displacements are also calculated with MiDAS by using a tangential erection for the new activated elements. The computed results, given in the table, are identical with the calculated values.

Real displacement [mm] Real displacement MiDAS [mm]

CS 1 uR1 = δ11 = 1.586

CS 2 uR1 = δ11+δ12 = 1.586+7.402 = 8.988 uR2 = δ21+δ22 = 3.701+21.678 = 25.379

CS 3

uR1 = δ11+δ12+δ13 = 1.586+7.402+13.747 = 22.735 uR2 = δ21+δ22+δ23 = 3.701+21.678+46.529 = 71.908 uR3 = δ31+δ32+δ33 = 5.816+36.483+86.185 = 128.484

Table 4-12: Real displacement table

Chapter 4: Example of a Cable-Stayed Bridge including temporary supports

Table 4-13: Total net displacement and construction camber data

The required angle, which must be considered between two segments to achieve a final structure with no deformation, can be evaluated from the real displacement data. Figure 4-30 indicates the angles which have to be calculated for the fabrication of the individual segments.

Figure 4-30: Fabrication camber, real displacement [mm]

With the real displacement and the current displacement uR1 22.735mm:= ⋅ δ12:=3.701 mm⋅ uR2 71.908mm:= ⋅ δ13:=5.816 mm⋅ uR3 128.484mm:= ⋅ δ23:=36.483mm⋅

the angles can be calculated as the following:

ψ1 atan uR1

∆uSeg2:=uR2 2− ⋅∆uSeg1 ∆uSeg2 26.438mm=

ψ2 atan

∆uSeg2 lSeg









:= ψ2=0.151deg

∆uSeg3:=uR3 3 uR1− ⋅ −2⋅∆uSeg2 ∆uSeg3 7.403mm=

ψ3 atan

∆uSeg3 lSeg







:=  ψ3=0.042deg

To prove that the required girder elevation at the time of installing the segment is achieved by the calculated angles, the flowing calculation shall control the condition:

uCS1 lSeg tan^{:= ⋅}

( )

ψ1 uCS1 22.735mm=

uCS2 2^:= ⋅∆uSeg1−δ12⁺lSeg tan^⋅

( )

ψ2 uCS2 68.207mm= uCS3^:=

(

3 tan^⋅

( )

ψ1 ^{+2 tan⋅}

( )

^ψ2

)

^{⋅lSeg−δ13−δ23+ lSeg tan⋅}

( )

^ψ3 uCS3 86.185mm= The girder elevation is the same as given in the construction camber data in Table 4-13.

In document cable stayed (Page 142-147)