Dr.-Eng Ryszard Drozdowicz,
Dr.-Eng Wieslaw Paczkowski, Technical University of Szczecin, Poland

(Conference - Nothingam, 1998)


The paper presents the results of investigations performed on the parabolic conoidal shell designed as the roofing of the Summer Theatre in Szczecin. Wind load distributions for the designed roofing were determined in the wind tunnel on the specially prepared model at the Technical University of Szczecin. The numerical analysis was performed for two different construction's alternatives: the reinforced concrete shell and the steel lattice. Investigations have indicated the significant wind load influence on internal forces in the designed construction.


The paper presents the results of aerodynamical model tests and numerical analysis for the Szczecin Summer Theatre's shell cover concept. Two alternatives were designed: the conoidal reinforced concrete shell and the steel lattice with geometry complying with the middle shell surface. The shell dimensions are: the span - 60.68 m, the length - 43.30 m, and the height - 22.73 m (see Fig. 1). The equation for an internal cover surface is as follows:

The cover surface was created as the result of the shell extension below the plane passing through the leading line. It intersects the ground level at curvilinear supporting edges shown on fig.1 (model "A"). The final roofing shape takes into account the existing state of the Summer Theatre, the object specific functional requirements and the general architectonic concept as well. The Summer Theatre is located at very exposed and picturesque place in Szczecin. It was designed as the area for different open-air enterprises - with an amphiteathral auditorium, a stage and technical back up facilities as well. Later the parabolic reinforced concrete arch was constructed over the stage. It was intended as the specific architectonic accent and the supporting element for the steel back up platform over the stage. For the protection against atmospheric influences the canvas cover was hanged on the platform. The stage without that canvas cover is shown on Fig. 2.

Fig. 1

Fig. 2

2.Objectives and scope of tests

The activities concerning roofing type selection have started from a preliminary analysis related to the accepted geometry. Due to the construction type, its functional requirements and the lack of proper data in available literature it was decided to perform necessary laboratory tests and the numerical analysis of the roofing. The scope of tests comprised: - determination of the wind load distribution for inner and outer surfaces of the cover "A" shown on Fig. 1; - the numerical analysis of internal forces distribution produced by wind- and dead weight load (for both alternative constructions); - the rationalisation of tested roofing geometry and supports. The doubts have arised if the results of aerodynamical tests, preliminary numerical calculations and possible recalculations from the model to the real object can be treated as credible data. Therefore it was decided to start from the complete aerodynamical tests for the smaller cover "B" (Fig. 1). The equation for the middle surface of the roofing "B" is as follows:

There are available analytical solutions for the roofing "B" (Menyhárd1). They were applied for verification and complementation of experimental tests and numerical calculations. For the shell marked "A" wind load distribution measurements were performed and the internal forces were determined on the basis of "Cadro" software package results.

3. The description of tests

The subject of the tests was the roofing "A" model in the scale 1:100. The tests were performed in the wind tunnel of Szczecin Technical University described in (Paczkowski2). The model was made of epoxy resin with cement filling material (see fig.3.). For the purpose of pressure measurements it was provided with drain pipes of polyvinyl chloride laminated inside the shell.

Fig. 3

The battery of water manometers was used to determine the distribution of pressures. The measurements were registered on the microfilms. The exemplary wind load distributions on the upper surface of the cover "A" are shown on fig.4 (they concern the cross-section "1" on Fig. 1.; the wind direction was 270o). Dashed lines present pressure distribution in the case of the saw tooth disturbance barrier located before the tested model at various distances, and the solid line presents the pressure distribution without that barrier. The wind suction is marked over the reference line and the wind pressure - below. Pressure values are given in the form of dimensionless aerodynamic coefficients (marked C). These coefficients enable to determine wind load for the real scale construction (prototype Sachs3 and Zaks4 ). The measurements were performed for wind profiles typical for open flat areas (Sachs3 and Zuranski3). The wind load value according to the standard PN-77/B-02011 was assumed for numerical calculations (2nd wind load zone).
The coefficient b was calculated according to the standard mentioned above. The worst case was assumed, i.e. gust of wind in the case of the construction susceptible for dynamic wind influence. It was used in the preliminary analysis of the cover and calculated as follows:

b = 1+ Y *Ö r/Ce*(kb +kr)

Y - the coefficient of the peak load value (dependent on constructionís natural frequency). The value Y max = 4.0 was assumed;

r - the coefficient of the surface roughness. The value r = 0.10 was assumed;

Ce - the exposure coefficient (dependent on the area development type and configuration). The value Ce = 1.20 was assumed;

kb - the turbulent influence coefficient for over-resonance frequencies (dependent on constructionís height and width). The value kb = 0.4 was assumed;

kr - the turbulent influence coefficient for resonance frequency complying with constructionís natural frequency.

The value kr is calculated due to the following formula:

kr = 2p *KL*Ko/D


KL - the coefficient of the resonance wind blasts reduction (dependent on the constructionís size). The value KL = 0.001 was assumed;

Ko - the coefficient of the resonance wind blasts energy (dependent on the constructionís natural frequency). The value Ko = 0.045 was assumed;

D - the logarithmic damping decrement (dependent on constructionís material and type). The value D = 0.15 was assumed.

According to formula (1) and assumed above values the coefficient b = 2.82. That value was multiplied by the basic wind load value and the result of multiplication was used as the static equivalent wind load for numerical calculations.
A critical Reynolds number was determined experimentally according to the general rules given in the literature (Sachs3 and (Żurański5).
The tests were performed for 5 basic (0 o, 45 o, 90 o, 135 o and 180 o) and 3 additional (225 o, 270 o and 315 o) wind directions. Pressure values were determined at 7 parallel cross-sections (120 measuring points - see fig.1). Tests results were applied in the numerical analysis of tested roofing.

Fig. 4

4.Numerical analysis

Numerical analysis was performed with "Cadro" software package. The subject of analysis was the conoidal reinforced concrete shell with constant shell thickness t = 0.2 m. Material parameters for B35 concrete were assumed (resulting dead weight density = 5.0 kN/m2).
A continuum analogy was applied for preliminary calculations concerning the lattice steel structure Bródka6). Calculations based on PoradnikÖ7). The results enabled to replace the basic construction (net mesh size approx. 2.0x2.0 m, 2.0 m of structural height, made of pipes f101.6/5.0)by an equivalent continuous configuration (thickness t = 1.0 m, equivalent modulus of elasticity E = 372 kN/m2 and Poisson ratio n = 0.0).The value 0.63 kN/m2 was assumed for the dead weight of the cover.
For C = 1 the assumed wind load was pk = 0.85 kN/m2.
The snow, temperature and foundation subsidence loads were neglected at this stage of the analysis. The dynamic wind influence was taken into account according to the clause 3. Firstly the numerical analysis was performed for the roofing with the free edge x = 0 (as on Fig. 1). The values for wind load were taken due to the results of wind tunnel tests for the model "A". Calculations indicated significant values for the following quantities : the free edge displacement, internal forces in the middle area of the roofing and reactions. Therefore the support of the roofing was modified - the free edge was supported on existing pillars (see - Fig. 2) with the help of the special girder. The exemplary internal forces distributions after modifications are shown on Fig. 5 and 6.
Figure 5 presents bending moments (My) distribution related to the strip 1.0 m wide (a longitudinal shell expansion at cross-section "1" on Fig. 1). There are shown bending moments caused by the dead weight of the concrete shell , the dead weight of the steel lattice structure and the wind from the selected direction (90o). Figure 6 presents normal forces (Ny) distribution related to the strip 1.0 m wide (the same cross-section "1" on Fig. 1). Presented normal forces are generated by the concrete shell's dead weight, the steel lattice structure's dead weight and the wind load from the selected direction (90o).

Fig. 5

Fig. 6

5.Verification of results

It was necessary to verify the correctness of the experimental wind load internal forces determination. Therefore appropriate investigations were performed. During these investigations the internal forces caused by dead weight for the shell model "B" were determined numerically and experimentally as well. The prepared shell model "B" (Fig. 1) was loaded with the simulated concrete shell's dead weight. Obtained strains were recalculated to proper internal forces and then these forces were determined numerically. The exemplary bending moments (MY) distributions are presented on Fig. 7. The accordance of results confirms the correctness of both applied methods (numerical and experimental).

Fig. 7

6. Results analysis and conclusions

The significant values of internal forces occurring in wide areas of the shell (particularly bending moments produced by wind load and dead weight) for both construction's alternatives have forced the change of the construction support. The shell supported in its flat part by the transom and the row of existing back-stage pillars was assumed for the second phase of numerical analysis. It should be mentioned that wind load distributions assumed in that phase of analysis did not take into consideration the existence of new supports. It was a priori assumed that their influence on perturbations of wind load distribution could be neglected.

Wind load distributions obtained for the tested shell are essentially ununiform and asymmetrical (see Fig. 4). The only exceptions are wind directions 0o and 180 o. Asymmetrical and locally concentrated wind load produces complex distributions of normal forces and bending moments, especially disadvantageous for great concrete shells. Those forces, added to the forces caused by the dead weight and snow in many cross-sections, significantly load the construction. When an average of wind load is about 19 % of the total load (including concrete shell's dead weight and snow) - bending moments produced by wind load can reach in local areas up to 300 % of dead weight moments. In the case of light roofing (e.g. steel lattice structure) the wind load is the main criterion for their dimensioning.

Estimation of extreme values for wind load internal forces is possible in the wind tunnel. These values can be important for the construction's safety. During the computer registration of extensometer read-outs the tested model should be rotated fluently in the wind tunnel's test section.

The tests performed on the small conoidal shell "B" were the substantial part of the research program and confirmed the correctness of analysis.

7. References

1. Menyhárd I.: Konstrukcje powlokowe, Arkady, Warszawa, 1971 2. Paczkowski W., Drozdowicz R., Mielczarek M.: Badania modelowe powloki walcowej w tunelu aerodynamicznym, Inzynieria i Budownictwo Nr 5/96, s.307-309
3. Sachs P.: Wind forces in engineering. Pergamon Press, 1978
4. Zaks N.: Podstawy aerodynamiki doswiadczalnej. MON, 1957
5. Zuranski J.A.: Obciazenia wiatrem budowli i konstrukcji, Arkady, Warszawa 1978
6. Bródka J. : Przekrycia strukturalne, Arkady, Warszawa, 1985
7. Poradnik: projektanta konstrukcji metalowych, Cz.2. Arkady, Warszawa, 1982