Spatial structure form finding analysis (2)—force density method form finding
Spatial structure form finding analysis (2)—force density method form finding
Official Account: Non-Deconstructive · Author: Zhou Wenqi (Venchy)
- The significance of structural form-finding
What is the significance of form-finding analysis of structures?Let’s look at two cases:
Are you amazed?How can a long-span and special-shaped structure be realized when the shear bearing capacity between the joints of the blocks is so small?
Let’s compare two simple mechanics diagrams:
The above is a simplified diagram of the internal force analysis at any position of a simple simply supported beam.There is a vertical force V at any position on the beam.M=V×L (Equation 1) at any position of the beam section, the shear force borne by the beam support and the section together form a couple.In order for the structural forces to reach equilibrium, there must be a bending moment to balance the force couple.And this couple can only be balanced by the bending moment borne by the beam section.
Let’s look at the simplified internal force diagram of an arch, M+N×H=V×L (Equation 2).The space arch structure has both a beam cross-section bending moment and a force couple formed by a pair of horizontal forces that balance the vertical force.When N×H=V×L (Equation 3), there is no need for bending resistance of the section at all.At this time, the structure uses axial force to resist the bending moment caused by external forces. Let us compare the above internal force diagrams of simply supported beams and arch structures from two dimensions: (1) Comparing the above simply supported beams and arches, simply supported beams use the bending moment borne by the beam section itself, while arches use the axial force and the moment caused by the structural shape to resist the bending moment caused by external forces.In terms of the effectiveness of the structure in bearing moments, the bending moment formed by the internal tension and compression of the section itself is far less than the bending moment caused by the axial force of the full section and the relatively large force square.
From this point of view, the arch structure can resist large bending moments through axial force.At the same time, comparing Equations 2 and 3, the left-hand term T×H>>M means that the arch structure can bear a larger span. (2) Compare Formula 2 and Formula 3 again. The terms on the left are N×H+M and T×H respectively.In Formula 2, the existence of M causes unevenly distributed stress inside the section, which may also cause local tension and compression in the section.Since the tensile capacity of concrete or masonry structures is extremely weak, this uneven distribution should be avoided as much as possible. Therefore, the internal force state of Formula 3 is our final ideal choice.However, in Formula 3, for the structure, H and L are determined by the shape of the structure. Only when H and L meet the relevant dimensional requirements can the structure satisfy the state of fully resisting the bending moment caused by external loads through axial force. Let’s sort out Formula 3 again, V/N=H/L, “H/L”. Do you think of a concept in space structure? Yes, this is the span ratio. The smaller the span ratio, the smaller the bending moment borne by the horizontal force, and the greater the requirement for cross-section bending resistance. Therefore, the specification puts forward relevant limit requirements for the span ratio of the lattice shell structure. The goal of form finding, in the above example, is to find the relationship between H and L.For spatial structures, the goal of form-finding is to put each component of the structure in a state of pure tension and compression under certain specific boundary conditions, so as to maximize the structural efficiency.This state of pure tension and compression only depends on the geometry of the structure and external load distribution, and has nothing to do with other factors.
- Traditional architectural form-finding analysis cases
In order to find the ideal structural form, engineers started trying very early.Taking advantage of the fact that the rope does not have bending resistance and can only bear tension, when a specific load is applied to the rope, the system formed by the rope and the load can automatically deform and put the rope in a pure tension state.This is what we often call the catenary method.
As early as the 16th century, Simon Stevin (1548-1620) made a mathematical demonstration of the catenary method.
In the 17th century, when Robert Hook (1635-1703) and Christopher Wren (1632-1723) were designing Paul’s Cathedral in London, they used the catenary method to design the masonry structure dome of the church.
Paul’s Cathedral The Spanish architect Antoni Gaudí (1852–1926) also used form-finding methods when designing the famous Sagrada Familia (Colònia Güell).
Sagrada Familia In addition to using ropes, structural engineers also use two-dimensional continuous materials to find the shape of the spatial structure. Heinz Isler (1926–2009), an expert in thin concrete shells, pioneered the use of textiles for structural form-finding.For example, in the cold winter, leave a piece of damp cloth outdoors to let it freeze.Ice on the cloth is equivalent to a uniformly distributed load.Finally the corresponding geometric shape is found.By using textile form-finding methods, he designed famous thin concrete shell buildings such as the Grötzingen outdoor theater and Restaurant Los Manantiales.
Grötzingen outdoor theatre
Restaurant Los Manantiales
Frei Otto has also made many highly innovative attempts in structural form-finding since the 1950s.For example, he creatively searches for different structures based on different material properties.For example, when a soap bubble is stretched, the tensile stress at each position is constant, so a structure with uniform stress distribution can be found based on the shape of the soap bubble.The deformation of plastic sheeting usually depends on the strain of the material.The shape of the lattice shell structure mainly depends on the axial deformation of each rod and partly depends on the shear deformation.
Use soap bubbles to find shapes
Use plastic sheeting to find shapes
Form-finding using rope woven nets
- Principle of force density form finding
The above attempts were the main methods of early shell structure design.However, with the development of computer technology, the combination of structural design theory and computers is more and more widely used in engineering practice.Structural form-finding analysis can also break away from the complicated form-finding process through experiments and obtain the results we want through numerical analysis. In the previous article “Non-Deconstruction”, “A Brief Introduction to Form-Finding Analysis of Reticulated Shell Structures and Case Appreciation”, we have introduced the method of using the finite element method to perform structural form-finding.Today, we will discuss another numerical analysis method: Force Density Method (FDM).It was proposed by Schek H J in 1974 and improved accordingly by L.Grundi.The characteristic of the force density method is that it only needs to solve a system of linear equations, and the calculation accuracy can meet engineering requirements. The German membrane structure design software EASY uses the force density method to find shapes. Let us understand the basic principles of the force density method through a simple case.The following are nodes P0, P1, P2, P3, P4 and the four members a, b, c, d connected to them.There is load pz acting on P0.Among them, the nodes P1, P2, P3, and P4 are known, and the external load pz is known.
When load pz acts on this system, according to our mechanical foundation, the structure satisfies the following three criteria: (1) According to Hooke’s law, the axial force of each member on the structure is generated due to the deformation of the structure and is proportional to it.
Among them, F represents the axial force, E represents the elastic modulus, A represents the area, l0 represents the original length of the component, e represents the deformation, and i represents the number of the member. (2) The length of the component is equal to the distance between the two nodes of the component.
Deformation e is the length difference of the rod before and after deformation.
(3) At each node, the resultant force of all members acting on the node is balanced with the external force.Decompose the internal force of any rod into three forces in the x, y, and z directions.Taking the component in the x direction as an example
Among them, αi represents the angle between each rod and the x-axis, Fi, x represents the component force in the x direction.The resultant structural force at each node position is 0, and the following equation is obtained:
Converting each cosine term to a coordinate value yields the following equation:
Substituting the Hooke’s law formula into the above equation, we get the following equation:
It can be found that x0 and ld in the formula are unknown terms. Although x0 can be solved by simultaneously combining equations in the three directions of x, y, and z, solving such a nonlinear equation system is extremely complicated.Therefore, we introduce the concept of force density to simplify the process of solving equations. Then convert the above formula:
Define the following equation as the force density qi:
Introducing qi into the original equation, we get:
Finally, the coordinates of point p0 are obtained through transformation:
In this way, we get the coordinates of the unknown point P0. The above derivation process is the calculation process of using the force density method to find the shape of a single node.By extending this process to any number of nodes, we can obtain the node coordinates of each coordinate point of the spatial structure after shape-finding.According to the topological relationship between the original structures, the mesh of the form-finding structure can be obtained.
- Force density form finding case
Xiaonong used Python to write a form-finding program using the force density method. The following are several simple structural form-finding cases:
Case 1 before form finding
Case 1 After shape finding
Case 2 before form finding
Case 2 After shape finding
Case 3 before form finding
Case 3 After form finding (1)
Case 3 After form finding (2)
Case 4 before form finding
Case 4 After shape finding
- Summary
The purpose of structural form-finding analysis is to find the optimal structural form.In the traditional design process, architects often give full play to their imagination in the concept or plan stage, but ignore whether complex architectural shapes are structurally feasible or optimal.Especially for spatial special-shaped surfaces, the structure is involved in the architectural design at the conceptual stage. The architectural and structural majors work together to find the optimal spatial surface form, which can maximize the effectiveness of this major from both the architectural and structural levels.From an architectural point of view, the shape found in the structure not only brings unexpected beauty, but also maximizes the utilization efficiency of structural materials, meeting the architect’s requirements for larger space and thinner sections.From a structural perspective, it can also optimize the mechanical properties of the structure.
References: shell structure for architecture
Past selections:
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Collection of cases of layout of 200300m super high-rise structures
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The combination of form and force—Zhang Xianliang’s study notes
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