【正文】 connecting to i, where Ft1Dt m 5force in member m connecting node i to an adjacent node j at time。 and Lt1Dt m 5 current length of member m at time, calculated using Pythagoras’s theorem in three dimensions. This process is continued, cycle by cycle, to trace the motion of the structure. So far, no damping has been introduced and, thus, the grid continues to oscillate. This phenomenon can be prevented by introducing ―kiic damping‖ in all the velocities that are set to zero when a kiic energy peak is detected. This process will never truly converge, but once the residual forces are measured in, for example, thousandths of Newtons, convergence has occurred for all practical purposes. At that point, a shape is found that is in static equilibrium and that holds the ―correct‖ spatial surface. This formfinding process yields a 3D cupola with a height of m, as shown in Fig. 5 (ratio height/span 5 ). The steel skeletal shell mainly works in pression under selfweight. As to be expected, large tensile forces arise in the ring beam framing the shell. The structural elements radiating out from the corners experience the largest pressive forces. Although all boundary nodes can transmit vertical forces onto the fa231。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。 yiT, the plan coordinates of the ith vertex at which two lines cross. Then, the heights of the nodes, zi, need to be calculated so that all of the glass facets are flat, although the shape of the structure is domelike, as shown in Fig. 5. Clearly, this is only Fig. 6. Plan geometry of the roof a problem for facets with four or more sides because a flat triangle can always be constructed with three arbitrary vertices. Formulation of the Problem It is supposed that the equation describing the jth flat facet is z ajx t bjy t cj e5T If the ith vertex is on the jth facet zi ajxi t bjyi t cj e6T To get the faceted surface to form the dome, it needs to be pulled toward the desired shape. Imagine that the dome was connected to vertical springs at each vertex, such that the tension in each spring is equal to si189。zi 2f exi。 yiT_ e7T Thespring stiffness, si, is chosen to be proportional to the plan area in the region of the ith vertex. This will have the effect of pulling the roof toward the form z f ex。 yT e8T For the NSA roof, f ex。 yT was chosen such that b z 188。 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 eL2xT2 t 1 eL2yT2 s t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 eL2xT2 t 1 eL t yT2 s t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi