【正文】 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 eL2xT2 t 1 eL2yT2 s t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 eL2xT2 t 1 eL t yT2 s t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif。 yT e8T For the NSA roof, f ex。zi 2f exi。ades, the largest vertical reactions are found at the courtyard corners. This clearly shows that the boundary zones of the shell itself acts as truss along the boundary walls. After the numerical formfinding process, the resulting, generated geometry of the shell is subjected to a nonlinear analysis. The real values of elastic and bending stiffness need to be used during the structural analysis of the grid shell, the results of which are verified against the Building Codes (European Committee for Standardization 1990, 1991, 1993). In the structural analysis, the shell is subjected to the loading binations of selfweight (glass kN/m2, aluminum profile kN/m, and steel profile in function of crosssectional area steel density kN/m3), live load ( kN/m2), maintenance load (1 kN/m2), impact load ( kN over Fig. 5. The shape of the shell is formfound to achieve membrane action 210 / an area of 10 3 10 cm), thermal load (), snow load (varying between and kN/m2), and wind load (varying between and kN/m2). Because the cupola should express a clarity of form resembling a fine line drawing against the sky, all 3368 elements are dimensioned as steel sections with widths of 40 or 60 mm and with variable height (100–180 mm). The total weight of the steel roof is 100,000 kg, and the ring beamweighs 40,000 kg. The largest ultimate limit state axial forces occur in the grid diagonals (pressive force 940 kN) and edge beam (tensile force 2,600 kN). A static analysis shows that all elements are loaded far below their critical buckling load by a factor of 2. The maximum shell deflection is 170 mm. The deflection values under wind loadings are relatively small because of the suction effect. A dynamic analysis finds an eigenfrequency value of . The different analyses show that the shell satisfies all structural criteria. The glass cladding has two layers: one bottom layer with two panes of 6mm half toughened glass and one top layer of 8mm toughened glass. The issue of facet planarity needed for glass panes imposes a slight modification of the form found geometry of the shell. For this project, a specific method based on origami folding was derived and will be discussed next. Sometimes, planarity of mesh might not be desired (., Foster and Partners’ design for the Smithsonian Institute). Because of steel digital fabrication techniques [pioneered in the design of the roof over the great courtyard of the British Museum (Barnes and Dickson 2021)], standardization of meshes and, thus, elements and nodes, is no longer considered crucial, but mesh planarity of nontriangular meshes is still a vital issue. Construction Constraints Adapt the Irregular Faceted Catenary Surface In this project, the plan geometry of the roof is based upon Fig. 6, in which 16 equally spaced points around a circle are all joined by a total of 120 straight lines. The square plan of the roof itself (Fig. 6) is the central square part of the circle with only the four corner points remaining from the original 16. Thus, one can calculate exi。 and _vt ix 5acceleration at node i in direction x at time t. Expressing the acceleration term in Eq. (1) in finite difference form and rearranging the equation gives the recurrence equation for updating the velocity ponents vt1Dt=2 ix Dt Mi Rt ix t vt2Dt=2 ix e2T Hence, the updated geometry projected to time t1Dt=2 xt1Dt i xti t Dtvt1Dt=2 ix e3T Eqs. (2) and (3) apply for all unconstrained nodes of the grid in each coordinate direction. These equations are nodally decoupled, in the sense that the updated velocity ponents are dependent only on previous velocity and residual force ponents at a node. They are not directly influenced by the current t1Dt=2 updates at other nodes. Having obtained the plete updated geometry, the new link forces can be determined and resolved together with the applied gravity load ponents Pix to give the updated residuals Rt1Dt ix Pix t P _ F L _t1Dt m _ xj 2xi _t1Dt e4T for all elementsm connecting to i, where Ft1Dt m 5force in member m connecting node i to an adjacent node j at time。 this load value differs for most nodes because of the plex 2D geometry of the initial pattern. An adapted version of the dynamic relaxation method with kiic damping, which takes into account the contextual boundary conditions, performs the formfinding. Summarized, the dynamic relaxation technique traces stepbystep for small time increments Dt the motion of each interconnected node of the grid until the structure es to rest in static equilibrium. The motion of the grid is caused by applying a negative fictitious gravity tributary load at all the grid nodes. The upwards load avoids having to turn the structure upside down to get the hanging tension form. During the formfinding process, the values of all numerical quantities (elastic stiffness EA, bending stiffness EI, and load) are arbitrary because it is only their ratios that effect the shape. The dynamic relaxation formulation for this project uses Newton’s second law governing the motion of any node i in direction x at time t Fig. 4. The square courtyard of the NSA is covered by a steel glass structure whose irregular mesh is based on a loxidrome map with 16 wind roses (top left photograph courtesy of Hella Bauma