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A Mathematical Solution to Predict the Shape of a Catenary Arch Or Shell Subjected to Non-Uniform Loads

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A Mathematical Solution to Predict the Shape of a Catenary Arch Or Shell Subjected to Non-Uniform Loads A Mathematical Solution to Predict the Shape of a Catenary Arch or Shell Subjected to Non-uniform Loads Mitchell Gohnert School of Civil and Environmental Engineering, University of the Witwatersrand, P.O. Wits, 2050, South Africa [email protected] Keywords: Shells, arches, catenary, finite element, form finding, low-cost housing Abstract. Thin shell vaults have been used extensively in antiquity. These structures were constructed primarily of masonry and are therefore only capable of resisting a compressive force. In recent times, these types of structures are being applied to arches and shells, especially in the design of low-cost housing using earth bricks. Similarly, these types of structures are only capable of resisting compressive forces. However, the problem is defining the shape. The mathematical equation for a catenary curve is well known, but the equation is only valid for a uniformly distributed load (i.e., self-weight). The design of catenary shell structures are therefore severely limited, since reality requires shells and arches to withstand point and varying uniform loads— maintenance, snow and wind loads are examples of non-uniform distributed loads that must be accounted for in the design. The proposed method requires the catenary shell or arch to be divided into elements, similar to finite elements. However, the first element at a support is solved using equilibrium equations, which determines the magnitude and direction of the compressive thrust- lines in successive elements. This approach allows the curve to naturally grow and form into a pure compression structure, without prior knowledge of the required shape. This method differs radically from conventional finite element methods, which requires the shape to be known and defined at the start of the analysis. The proposed theory was applied to a couple of loading configurations and the results were compared to a finite element analysis, which confirmed that the predicted shape is in pure compression. Introduction Arches and shells are highly efficient structures, in terms of their capacity to carry loads [1]. The source of their efficiency is that the majority of the load is carried along its axis, or in-plane (referred to as membrane action in shells)[2]. If bending moments are present, the stresses may be 12 to 24 times higher than if the load is carried by membrane, or axial forces [3]—thus, bending and shears are uneconomical and cause wastage of building materials [4]. The word catenary, meaning chain, are unique types of arches, in which the stresses are carried in pure compression, without bending and shears. A chain that is suspended between two points and allowed to drape will form this unique curve, which is an extremely important shape in structural applications, especially in masonry. A hanging chain is in pure tension, and incapable of resisting bending and shear forces. If the chain links are locked and the shape is flipped upright, the forces are reversed and in pure compression. This form is commonly applied to arches, and sometime to shells (which is not entirely correct since the theory does not include Poisson’s effects). Common usage of the word catenary typically refers to a shape where the load is continuous and uniform, and the supports are at the same level. However, in reality, structures may not be level and the structure may be exposed to a host of different loadings, such as point loads or non- uniform continuous loads (e.g., wind loads). If these loads are applied to a catenary shape, the arch will no longer be in pure compression, but may have tensile or bending stresses. It is therefore essential to be able to define a new catenary shape that is resistant to a set of point, or non-uniform loads. The new shape will be referred to as a funicular form—a generic word to describe an infinite number of catenary shapes, depending on the magnitude, direction and type of applied loads. Currently, only physical models are used to define funicular shapes [5]. For this reason, a numerical algorithm is needed to determine the shape of a pure compressive structure, especially if the loads are complex and non-uniform. Other works are applicable to cable and tensile structures, which utilize closed-form mathematical solutions or the adaptation of finite elements to define a catenary shape [6,7,8]. Numerical Algorithm A combination of loading will result in a complex shape that is, in many cases, extremely difficult to predict. The solution therefore, must be able to define a pure compressive structure, regardless of the number and type of loads that are applied. The proposed algorithm is for point loads, but a non-uniform load (e.g. wind loading) may also be solved by modelling the load as a series of point loads. The algorithm begins by defining the reactions of a catenary arch subjected to a uniform load (i.e., self-weight). These reactions are based on established theory. The next step is to define the reactions of individual point loads. The reactions are then collected and resolved into a single resultant. Subsequently, the forces in the first element of the curve are defined, at the base of the arch. Equilibrium equations for the first element are then formulated, and solved to determine the magnitude of the thrust and its orientation in the second element. This sequence is repeated and multiple elements are added until the funicular shape of a pure compression arch is determined. Reactions for a catenary curve subjected to self-weight. To begin, the catenary equation is defined as a hyperbolic expression, originally formulated by Gregory in 1706 [9]. 푦 = a cosh(푥/푎) (1) Eq. (1) is illustrated as the lower curve in Fig. 1. The catenary curve, however, is expressed in terms of an arch, as illustrated as the upper curve of Fig. 1. The equation of the arch is defined by Eq. (2), where the term “a” is determined by iterating Eq. 1 with predetermined values of H and L. 푦 = 퐻 − 푎 푐표푠ℎ(푥/푎) − 푎 (2) The length of the catenary curve (S) is solved by integrating Eq. (1) from 0 to L/2, and multiplying by 2. y H x H a x L Fig. 1: Catenary curve for self-weight 푆 = 2푎 푠푛ℎ(퐿/2푎) (3) The arch is then discretised into elements. The total weight of the arch (Wt) is equal to the weight per element length (w) multiplied by the length of the curve. The element height may be the height of individual masonry units, or any other height. However, the number of elements is related to the accuracy of the solution. 푊푡 = 푤푆 (4) The vertical reactions at each base of the arch are equal to the total weight divided by 2. 푅푐푣 = 푊푡 /2 (5) The angle of the diagonal reaction (Rc) is determined by first taking the derivative of Eq. (1). 푑푦 = sinh(푥⁄푎) (6) 푑푥 Since, 푑푦 = tan 휃 (7) 푑푥 Eqs. (6) and (7) are equated to determine the reaction angle at x = L/2. 휃 = tan−1[sinh 퐿⁄2푎] (8) The diagonal and the horizontal reactions are then solved (see Fig. 2). 푅푐 = 푅푐푣 / cos 휃 (9) 푅푐ℎ = 푅푐 sin 휃 (10) Wt Rch Rch Rc Rcv Rcv Rc Ɵ Fig. 2: Reactions of a catenary arch subjected to self-weight Point load reactions. If a point load is applied to a catenary arch, the structure will be subjected to bending. The objective, however, is to define the shape of a pure compression structure. If a chain of length S is weightless, supported at a distance L and if the direction of the load is reversed, the chain will form a triangle and the legs of the triangle will be in pure tension. However, if the chain links are locked, the triangle flipped upright and the load is reversed back to its original direction, the legs of the triangle will be in pure compression. The reactions will also be at the same angle of the legs of the triangle, as illustrated in Fig. 3. Furthermore, if the point load is located low in the arch, a triangular shape is unable to form, since S” is too long. For this case, the resistance to a low point load in compression is only achieved if the load flows directly into one leg of the triangle (S’), and if the leg is orientated at the same angle as the point load (see Fig. 4). The natural shape is simply one leg of the triangle (assuming the length S remains constant). Pn Pn γ S” S’ S’ S” β α β α RpL pR L R Fig. 3: The shape of a weightless structure that will resist a point load in pure compression Pn S” Pn S” S’ γ S’ β RpL Fig. 4: Load path of point loads located low in the catenary arch The boundary between high point loads and low point loads is determined by the scale underneath the arch of Fig. 5. When the length S’ becomes smaller and the triangle of Fig. 3 is flat, the scale in Fig. 5 defines the lengths S’ and S” at the boundary between the high and low point loads. Since S = S’ + S”, 푆′ = 푆⁄2 − 푎 sinh(푥′⁄푎) = 푆" − 퐿 (11) 푆" = 푆 − 푆′ (12) Substituting Eq. (12) into (11), the boundary is determined. 푆′ = (푆 − 퐿)/2 (13) Reactions for high point loads, 퐒′ > (퐒 − 퐋)/ퟐ. Referring to Fig. 3 and using the law of cosines for oblique triangles, the angles of the reactions are determined [10]. 푆"2+퐿2−푆′2 훼 = 푐표푠−1 [ ] S (14) P 2푆"퐿 푆′2+퐿2−푆"2 ” ’ 훽 = 푐표푠−1 [ ] (15) P 2푆′퐿 ’ Formulating equilibrium equations in the x and y directions, ∑ 퐹푥: 푃푛 cos 훾 + 푅푝퐿 cos 훽 − 푅푝푅 cos 훼 = 0 (16) S β ∑ 퐹푦: 푃푛 sin 훾 − 푅푝퐿 sin 훽 − 푅푝푅 sin 훼 = 0 (17) ’ Substituting Eq.
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