Dec 30, 20194 min read

The Complexity of the Copenhagen Opera House roof |Finite Element Analysis using LUSAS

Updated: Jul 2, 2020

Quick FACTS 🧾

The roof of the new Copenhagen Opera House is one of the largest canopy roof structures in the world.
With a plan dimensions of 158m by 90 m, it equates to the size of three football fields.
LUSAS finite element analysis was used for the design of the structure in order to ensure the necessary strength, stability and dynamic response was achieved.
Static, dynamic and thermal assessment of the roof for in-service loadings were made.

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Architectural rough sketches (side view)

What is Finite Element Analysis (FEA)?

Finite Element Analysis or FEA is the simulation of a physical phenomenon using a numerical mathematic technique referred to as the Finite Element Method, or FEM. Engineers can use these FEM to reduce the number of physical prototypes and run virtual experiments to optimize their designs.

Consider a concrete beam with support at both ends, facing a concentrated load on its centre span. The deflection at the centre span can be determined mathematically in a relatively simple way, as the initial and boundary conditions are finite and in control.

However, once you transport the same beam into a practical application, such as within a bridge, the forces at play become much more difficult to analyze with simple mathematics.

FINITE ELEMENT ANALYSIS OF A TRUSS BRIDGE

Roof design and construction

To design the roof a number of technical challenges had to be overcome:

It had to be shown that the structure possessed the necessary strength and stiffness as preliminary calculations had shown that it was almost impossible to design the entire structure using common truss girders in two directions
It was important that in any final roof design the first mode shapes involved not only localised deformations of the outer corners but included deformations of the whole roof structure to ensure a dynamic response of the roof within acceptable limits.
It had to be able to safely resist the large temperature differences in winter between the cantilevered part and the internal roof over the foyer.

Ramboll chose to construct the cantilever roof as a closed steel box because a significantly higher flexural as well as torsional rigidity is obtained compared to that for a traditional lattice roof structure.

The roof structure could not be designed as a closed box over the total area because of differential temperatures but was divided into a section made as a box and the remaining section made of a number of girders.

The outer ring beam forms the inner closure of the box, and the radial beams are designed so that the flaring of the beams can absorb the horizontal axial stresses from the box structure...

To analyze stability problems in the slender plates, Ramboll developed comprehensive new formulas for biaxial stress combinations, which included post-critical stresses and not only initial buckling stresses. These formulas led to significantly lower plate thicknesses in the cantilevered roof.

Plan of the cantilevered West roof showing area of box construction (light grey) and roof over foyer area (dark grey)

Static Analysis

The roof was designed for wind, snow and dead loads, as well as for stresses caused by temperature and for any settlement of the foyer columns.

The wind load was based on results from wind tunnel tests. A 3D LUSAS model of the closed box and the girders of the roof structure was used for both static and dynamic analyses and these calculations determined all the normal stresses in the plates, parallel and perpendicular to the troughs, as well as the shear stresses and all internal forces in each beam in the girders.

The static analysis proved the box structure to be an optimal solution, due to the use of stresses in both directions of the plates in combination with the shear stresses from the large torsional moments in the structure.

Dynamic Behaviour

Preliminary studies for a truss girder roof indicated an unacceptable response. By constructing the roof as a closed box the dynamic wind load was reduced to an acceptable level, and damping devices were not required. Eigenvalue analyses with LUSAS showed that the first mode shape for the closed box roof involved not only local deflections of the outer corners but also a global deflection of the entire front of the roof.

A wind tunnel test was carried out to determine the time-averaged wind load on the structures and the fluctuating wind load, which were combined with mode shapes, natural frequencies and modal masses of the structures to determine the dynamic response.

Differential Temperatures

The external cantilevered part of the roof forms a horseshoe around the foyer area, and contracts in winter compressing the structure over the foyer. Roller bearings sit between the cantilevered box and the foyer girder portion of the roof and release the differential horizontal deformations in the north-south direction, and can transfer compression/tension forces in the vertical direction.

Carrying out a differential temperature analysis with LUSAS showed how the structure over the foyer contracts in the north-south direction, the girders will deflect horizontally and the entire foyer structure moves towards the east.

Hans Exner, Senior Chief Engineer at Ramboll said: "All of us at Ramboll are really proud of this building. It was a very valuable project for our client, for Copenhagen and for ourselves.

The Opera House opened on 15 January 2005. It received the 2008 International Association for Bridge & Structural Engineering's Outstanding Structure Award, principally in recognition of the innovative design of its roof.