### Checking calculations

Calculation support Designing Аэродинамические расчеты Calculation of joints Equipment designing

## Calculation and investigation of causes of Transvaal Water Park cover collapse

A range of expert’s studies devoted to the analysis of structural strength of Transvaal Water Park building have been conducted so far. However, none of these studies definitely indicated the cause of collapse of this building.

 Water Park after collapse

As the main tool to determine stressedly-deformed state and dynamic properties of the structure under different loads, a finite element numeric method was used in these studies, realized in various program systems: LIRA, SCAD, ANSYS and STADIO. Finite element models formed from beam and shell elements were used to determine stressedly-deformed state of the cover structure with a system of support pillars. Strength characteristics of reinforced concrete were specified to model a reinforced concrete shell of the cover. These finite element models included from several dozens to a hundred thousand elements.

Real cover was made as reinforced concrete shell of variable thickness having inhomogeneous reinforcement and a system of cross reinforced ribs. In modeling of such complex structure, the above mentioned degree of discretization does not make it possible to take into consideration any local features of stressedly-deformed state of the structure conditioned by nonlinear behavior of concrete and reinforcement system realized in the structure.

 Finite element model of cover structure

For more accurate determination of local features it was decided: to build a finite element model of cover with modeling of support contour and adjacent to it shell areas by solid elements; define all reinforcement installed in the concrete by beam elements in accordance with the drawings; define a “thin” part of the shell (thickness=70 mm) and supporting ribs by two-dimensional shell elements; define rib reinforcement by beam elements; take account of nonlinear behavior of material with different compression and tension characteristics for concrete; model the structure of support pillars by shell elements with a detailed elaboration of connections and support joints. As a result, a model was developed, consisting of about 2 million elements, 20 times above a detailed elaboration of the structure in the models presented earlier in the calculations of expert organizations. Number of joints in the model =1,851,000, number of elements =1 894 000.

LLC Hexa was the first to use this combined method of concrete and reinforcement modeling in the study of the water park structure. A simplified roof model was used in all previous calculations and examinations.

The model of pillars with connections is also different from the model used in the analysis of expert organizations. Simplified framing models used in the calculations of expert organizations lead to 450% inaccuracy in the results of force analysis. Such inaccuracy resulted from the fact that compliance of pillar shell was not taken into account in the model.

Calculation results.

1. with permanent load including weight of structural elements + roof weight – “weight” load

The calculations and analyses performed in this study include:

• comparison of design data for linear-elastic and nonlinear models of materials;
• effect of concrete properties on stressedly-deformed state of the structure;
• determination of breaking loads in welded joints of top pillar points with embedded items of support contour of the shell;
• comparison of stressedly-deformed state of the structure determined with different variants of connections between top pillar points and support contour of the shell;
• Determination of snow load value resulting in plastic (critical) deformation in the reinforcement of reinforced concrete cover.

Initially, it was considered whether it was reasonable to solve the problem of large-span sloping shell loading in the linear-elastic setting. For this purpose, calculations for loading with permanent load (“weight” load) were performed for the models:
1) without regard to geometric and physical nonlinearity;
2) with regard to geometric nonlinearity;
3) with regard to geometric and physical nonlinearity.

Nonlinear behavior of concrete under tension and compression was taken into consideration in the calculations by defining an elasto-plastic material model. Link connections of top pillar points with the support contour of the shell were defined.

Comparison of design data shows significant difference in maximum displacements of the cover determined for models with linear and elasto-plastic properties of concretel.

 Calculation type Maximum displacement U, мм Stresses in concrete σmax / σmin, MPa Stresses in reinforcement σmax / σmin, MPa Without regard to geometric and physical nonlinearity 60 15.1 / -12.2 86.0 / -54.7 With regard to geometric nonlinearity 65 15.4 / -12.8 89.2 / -55.2 With regard to geometric and physical nonlinearity 144 2.4 / -21.9 302.5 /-101.2

When geometric nonlinearity is taken into account in the calculation for the linear-elastic concrete model it is possible to determine the areas of large local deflections of the shell, but deflection values are understated: 65 mm for the linear-elastic model, 144 mm for the linear-plastic model. In the calculations performed in the design of the construction under study, nonlinear behavior of concrete was not taken into account thus resulting, as shown above, in significantly understated estimate of shell deflections.
Stresses in the structural elements determined in the calculations for linear and elasto-plastic model of concrete are different in their values (See the Table) and nature of distribution. Assignment of nonlinear behavior of concrete in the calculation leads to 3.4 increase of tensile stresses in the reinforcement and 1.8 increase of compression stresses.
It follows from the above given comparison that determination of stressedly-deformed state of the structure under study with no account taken of geometric and physical nonlinearities results in significant understating of maximum shell deflections and maximum stresses both in concrete and reinforcement.

Calculation studies of effect produced on stressedly-deformed state of the structure by changes in concrete properties were conducted.
In these calculations, concrete characteristics were defined according to concrete strength to compression В35, В40, В50, В60. The results of calculation for loading with permanent load (“weight” load) are given in the Table below:

 Concrete strength to compression Maximum displacement U, mm Stresses in concrete σmax / σmin, MPa Stresses in reinforcement σmax / σmin, MPa В35 Еb = 34500 MPa 173 2.1 / -22.2 329.4/ -113.5 B40 Еb = 36000 MPa 160 2.2/ -22.1 318.4 / -108.5 B50 Еb = 39000 MPa 144 2.4 / -21.9 302.5 /-101.2 B60 Еb = 40000 MPa 133 2.6 / -21.1 288.7 / -96.7

The use of concrete with better elasticity and strength properties in the cover structure reduces deflections of the shell and stresses in the reinforcement.
The comparison of design data for B35 concrete (used in the structure of cover of this water park) and B60 shows 1.3 decrease in maximum deflection value and 1.14 decrease in maximum tensile stress in the reinforcement.
.

Results of calculations under snow load

Initially, the properties of В35 concrete strength (initial modulus of elasticity Еb = 34,500 MPa) were defined for the elements modeling concrete. With uniformly distributed snow load 90 kg/m² (design value of snow load is 180 kg/m²), maximum tensile stresses in the reinforcement exceeded yield point Rsn= 500 MPa (standard resistance for A500C reinforcement). It should be noted that creep of concrete is not taken into account in the calculation.
According to Building Rules SP 52-101-2003 Concrete and reinforced concrete structures without prestressing, under long-term load, a value of initial concrete modulus of deformation is determined according to the formula: Еb' = Еb / (1+цb,cr), where цb,cr = 2.1 – creep coefficient. According to the formula, the initial modulus of elasticity should be reduced 3 times. It was decided not to define significant decrease of the initial modulus of elasticity in the calculation (to give the structure a “chance to survive”).
The calculation was performed for concrete modulus of elasticity reduced 1.5 times Еb'=Еb/1.5=23,000 MPa. According to the results of calculation for “weight + snow” load it was found that maximum tensile stresses in the reinforcement exceeded the yield point under snow load of 30 kg/m².

Maximum deflections under the above given load achieve 300 mm.

 Combined displacement in the shell.

Stresses in concrete exceed yield points (standard resistances) to compression and tension. Principal stresses in concrete achieve 27.5 MPa (standard value is 25.5 MPa). The zone of plastic deformations in concrete covers a significant area of the shell.

 Plastic deformations in concrete of the shell

The entire red zone is exposed to cracking in concrete.

“Weight + snow 30 kg/m²” load. Concrete modulus of elasticity Еb = 23000 MPa

 Equivalent stresses in pillars with connections, мах=300 MPa (standard value is 370 MPa)

Stresses are right on size thus refuting experts’ assertion of loss in bearing capacity of pillars and side connections. We found that even if one pillar is excluded from the use, no significant (catastrophic) changes occur in the structure.

Conclusions

The design data given above show that determination of stressedly-deformed state of the structure under study with no account taken of geometric and physical nonlinearities results in significant understating of maximum shell deflections and maximum stresses both in concrete and reinforcement.
The calculations with account taken of geometric and physical nonlinearities for loading with weight + snow load showed:
- according to data from the calculation with concrete modulus of elasticity Еb = 34,500 MPa, plastic deformation occurs in the reinforcement with 50% of design snow load;
- according to data from the calculation with concrete modulus of elasticity Еb = 23,000 MPa (with account taken of creep), plastic deformation occurs in the reinforcement with 17% of design snow load.
A water park building is a unique and complex structure. The use of standard procedures for strength and bearing capacity assessment gives rather approximate and superficial result.
At the design stage of such unique constructions, it is insufficient to check bearing capacity of the structure only by traditional methods. It is necessary to use state-of-the-art program systems in combination with powerful computer systems
Expert organizations, in particular, Moscow Commission of Experts, shall approach to the assessment of unique projects in a more responsible manner. When a translucent light roof structure was replaced by rather heavy reinforced concrete structure, it was necessary to thoroughly check a new design, make detailed checking calculations, but not superficial estimations of strength which have been in use up till now.

If the analysis of structural strength similar to that described above was performed at the stage of design and final decision-making, it would be more reasonable to assert that all design errors were detected.
And the main thing is that it would be possible to avoid the tragedy that took lives of innocent people.

Hexa Company to order of KURORTPROJECT CJSC, 2007.