CC BY
7doi: 10.18720/MCE.82.12
Analytical calculation of the frame with an arbitrary number of panels
Аналитический расчет рамы с произвольным числом панелей
M.N. Kirsanov,
National Research University "Moscow Power Engineering Institute", Moscow, Russia
Д-р физ.-мат. наук, профессор М.Н. Кирсанов,
Национальный исследовательский университет "МЭИ", Москва, Россия
Key words: core frame; deflection; Maple; Ключевые слова: стержневая рама; прогиб; analytical solution; induction; truss; kinematic Maple; аналитическое решение; индукция; degeneracy ферма; кинематическое вырождение
Abstract. A rod model of a statically determinate planar frame with four supports is proposed. The task is to obtain an analytical dependence of the deflection of the truss under the action of various loads on the number of panels in the crossbar and in vertical support trusses. To solve the problem, the computer mathematics system Maple is used. The system of equations of equilibrium is solved in symbolic form. It is shown that for a certain number of panels in the crossbar the determinant of the system of equations of equilibrium turns to zero, which indicates a kinematic changeability of the structure. The corresponding velocity distribution of the nodes of the truss is found. For cases of unchangeable construction, the deflection is found from the Maxwell-Mohr's formula. A series of solutions for various numbers of panels is generalized by the method of double induction. Using the operators of the Maple system, homogeneous linear recurrence equations are derived and solved for the terms of the coefficients of the desired formula. On the graphs of the dependence of the deflection on the number of panels, significant jumps and extreme points are revealed. To evaluate the strength and stability of the construction, expressions for the forces in the individual most stretched and compressed rods in the middle of the span are found. The cases of loading the truss along the upper belt, the lower belt and the force in the middle of the span are considered. The solutions found can be used to evaluate the operability of the designed design and to optimize it.
Аннотация. Предложена стержневая модель статически определимой плоской рамы с четырьмя опорами. Ставится задача получения аналитической зависимости прогиба фермы под действием различных нагрузок от числа панелей в ригеле и в вертикальных опорных фермах. Для решения задачи используется система компьютерной математики Maple. Система уравнений равновесия решается в символьном виде. Показано, что при некотором числе панелей в ригеле определитель системы уравнений равновесия обращается в ноль, что свидетельствует о кинематической изменяемости конструкции. Найдено соответствующее распределение скоростей узлов фермы. Для случаев неизменяемой конструкции прогиб находится по формуле Максвелла -Мора. Серия решений для различных чисел панелей обобщается методом двойной индукции. С помощью операторов системы Maple для членов последовательностей коэффициентов искомой формулы выводятся и решаются однородные линейные рекуррентные уравнения. На графиках зависимости прогиба от числа панелей выявлены значительные скачки и экстремальные точки. Для оценки прочности и устойчивости конструкции найдены выражения для усилий в отдельных наиболее растянутых и сжатых стержнях в середине пролета. Рассмотрен случай нагружения рамы по верхнему поясу, нижнему поясу и силой в середине пролета. Найденные решения могут быть использованы для оценки работоспособности проектируемой конструкции и для ее оптимизации.
1. Introduction
Most modern calculations of building structures are performed numerically using standard software packages based on the finite element method [1-4]. A very important tool in the evaluation of both the design itself and the accuracy and reliability of the numerical methods used are analytical formula solutions. In [5], an algorithm for the analytical solution of the problems of rod structures in combination with the finite element method is given. Analytical solutions in the form of finite formulas are extremely rare. In many cases the field of application of the computational form is narrow, but with the advent of computer mathematics (Maple, Mathematica, Maxima, etc.) the number of such solutions is growing. In [6, 7] reviews
are given of some papers that use the induction method to generalize particular solutions in the derivation of formulas suitable for a sufficiently wide class of problems. The number of schemes of such regular-type trusses with a periodic structure is relatively small. R.G. Hutchinson and N.A. Fleck [8, 9] announced the "hunt" for schemes of statically determinate periodic trusses. For some types of spatial regular trusses an attempt [10] was made to sort the periodic structures. Formulas for the dependence of deflection and effort on the number of panels in the rods of various planar [11-16] and spatial [17-20] statically determinate trusses are known. In [21], an algorithm is given for analyzing the limiting state of spatial rod frames using kinematic relations. Almost all formulas are designed to evaluate statically determinate structures with a periodic structure of trusses, to which the induction method with one parameter is applicable, for example, the number of panels in the span. Analytic solutions with two independent natural parameters of the structure are much more interesting. In this paper, we propose a two-parameter analytic calculation of the core frame with an arbitrary number of panels in the crossbar and the number of panels in the vertical support parts of the frame. The problems of the topology of the trusses and their optimization are considered in [22-24].
2.. Methods
2.1. The calculation of the stresses in the rods A frame with a cross-shaped grate and four supports, one of which is fixed, contains 2n1 panels of length a in the crossbar and panels in side parts of height h (Figure 1).
Figure 1. The scheme of the truss, n1 = 4, n2 = 3
Calculation of the forces in the rods of individual trusses with specific numbers of panels n1 and n2
is done by cutting out the nodes. A distinctive feature of the considered truss is its external static indeterminacy. This means that the forces in the supports can only be found from the general system of equilibrium equations for all nodes, together with the forces in the rods. The periodic structure of the truss allows you to write equations of node equilibrium in cycles using the Maple programming language. To do this (also using the loop operator), the node coordinates and the connection structure of the rods are defined by analogy with the task of graphs.
The elements of the matrix G of the system of equations of equilibrium of nodes contain the direction cosines of the forces. Here is a fragment of the program for specifying a matrix in the Maple language:
> M5:=M+5:
> for i to M5 do
> Lxy[1]:=x[N[i][2]]-x[N[i][1]]:
> Lxy[2]:=y[N[i][2]]-y[N[i][1]]:
> L[i]:=sqrt(Lxy[1]A2+Lxy[2]A2);
> for j to 2 do
> t:=2*N[i][2]-2 + j:
> if t<= M5 then G[t,i]:=-Lxy[j]/L[i]:fi;
> t:=2*N[i][1]-2 + j:
> if t<= M5 then G[t,i]:= Lxy[j]/L[i]:fi;
> od;
> od:
Here, M=8(n1+n2)+13 - the number of rods, Lxy - the projection of the oriented rods on the coordinate axis, x,y - the coordinates of the hinges. The vector N[i][1] indicates the hinge number in the conditional beginning of the rod i, in N[i][2] - the end. In the vector of the right-hand part B of the system of equations G.S=B where S is the force vector in the rods, information about the load is recorded. For the case of uniform load in the upper-belt nodes, we have the following entry in the program
for i from 3*n2+2*n1+4 to 3*n2+4*n1+7 do B[2*i]:=1: end:
To solve a system of linear algebraic equations in symbolic form, the inverse matrix method is used in a very simple form: 1/G. This way the solution is faster than with the help of the special operator LinearSolve.
The first solutions for a number of trusses with different numbers n1 and n2 have shown that for n1 = 2, 5, 8, 11, ... and for any n2 the determinant of the system of equilibrium equations vanishes. This indicates a kinematic variability of the structure with such a number of panels. In general, inadmissible numbers of panels with step 3 can be represented as the following formula n1 = 3k-1, k = 1,2,... . For
some other flat trusses, this effect was previously seen in [11, 15, 16, 20]. To confirm the found feature, you can specify a scheme of possible speeds of the hinges (Figure 2).
Figure 2. Scheme of possible speeds of an instantly variable truss
The joints 2, 3, 5, 6, 8, 10-12, 14-15, 17-19 remain stationary, rods 89 and 9-10 perform a rotation about the instantaneous joint 9, 12-13 and 13-14 are rotated about the joint 13, the rods 9-10 and 4-13 move instantaneously. The same movements make symmetrical rods on the right side of the truss. Obviously the ratio of speeds. To notice this feature is not easy. In calculations this is manifested in the degeneration of the system of equilibrium equations. In this case, numerical methods can skip this feature hidden behind the errors of the count. And when modeling a truss with finite elements with rigid nodes instead of hinged, this feature is not visible at all. A review of similar cases, where the values of panel numbers in regular trusses were also found, and the schemes of possible speeds of nodes of variable trusses are given in [6].
Degeneration of the system of node equilibrium (or kinematic changeability of the truss) at can also be noticed without cutting out all the nodes. It suffices to consider the nodes of the chain 1-9-13-4-16-207, recording only one equation for each node. If we write the equations of equilibrium of nodes 13, 14 and 16 on the vertical axis, and for the remaining nodes on the horizontal axis, then the inconsistency of the resulting system of equations becomes immediately clear. For example, if the entire truss is free of loads and only the support node 7 is loaded with horizontal force, then the forces in the rods 1-9, 9-13, 13-4, 416, 16-20 and 20-7 must be zero, that contradicts the equation of equilibrium of the loaded node 7 in the projection to the horizontal, from which it follows that the force in rod 20-7 is not zero.
2.2. Deflection
We use the formula of Maxwell-Mohr for the computation of deflection:
m S s f
1=1
EF
(1)
where E and F - the modulus of elasticity and the cross-sectional area of the rods, lj and Sj - length
and stress in v-th core under the action of a given load, Sj - forces under a single vertical force applied to
the central node C. In order to construct a sequence of solutions, the generalization of which gives the required dependence of the deflection on the number of panels, while avoiding degeneracy, we give the
following relationship n1 = (6k-1 + (-1)k)/4 . The obtained sequence, when changing k = 1,2,3...
passes the unacceptable values and gives for the deflection of the structure under the action of the load distributed over the upper belt, a solution of the form
A = P(Chn^ + C^C + C3n2kkh3)/(2h2EF) . (2)
The coefficients in this expression depend only on k and n2 form sequences for which common
3
terms can be found. For the coefficient at a the operator rgf_findrecur returns an equation of the ninth order
C1,n2,k = C1,n2,k-1 + 4C1,n2,k-2 -4C1,n2,k-3 - 6C1,n2,k-4 + 6C1,n2,k-5 + 4C1,n2,k-6 - 4C1,n2,k-7 -C1,n2,k-8 +C1,n2,k-9-
The initial conditions obtained from the solutions of the problems of deflection of trusses with k = 1, ...,9 and arbitrary numbers n2 have the form
C„,1 = 8,Cn2,2 = 72,Cn2,3 = 268, ...,Q,n2,9 = 15016. The operator rsolve gives the following solution of the recurrence equation C1,n,k = (30k4 + 4((- 1)k + 15)k3 + (6(- 1)k + 34)k2 + 20(1 - (- 1)k )k +13(- 1)k -13) /16. (3) Similarly, we obtain other coefficients:
C2,n2,k = 3(2 k2 + 2(1 - (-1)k )k - (-1)k +1)/8, (4)
C3,n2,k = (2(n2 + 2 - 2(n2 + 1)(-1)k)k - (11n2 + 8)(-1)k + 7n2 + 8)/2. (5)
3 3
The coefficients of a and c do not depend on the number of panels n2. And to obtain a coefficient C3,n ,k , which also depends on the number n2 of panels along the vertical, it was necessary to perform induction on this parameter. Solving the problem sequentially for n2 = 1,2,...,6, and performing each time by induction on n1, we obtain a series of solutions
C31 k = (6 - 8(-1)k )k -19(-1)k +15)/2, C3,2,k = (8 -12(-1)k )k - 30(-1)k + 22)/2, C33k = (10 -16(- 1)k )k - 41(-1)k + 29) / 2, C34k = (12 - 20(- 1)k )k - 52(- 1)k + 36) / 2,
The coefficients in these formulas form the obvious arithmetic sequences in steps of 2, 4, 11 and 7. This allows us to obtain the general formula (5) for C3 k . Thus, formula (2) with coefficients (3-5) gives
the required dependence of the deflection of the structure on its dimensions, the magnitude of the load and the number of panels.
A similar solution is obtained for the case of loading the truss by the lower belt of the crossbar (Figure 3). For this, the program needs to write the vector of loads in the following form
for i from n2+2 to n2+2*n1 do B[2*i]:=1: end:
The form of the solution (2) does not change. By induction for the coefficients, we obtain expressions
Q,n,k = (30k4 + 4((-1)k + 15)k3 - 2(5(-1)k + 1)k2 - 9(-1)k + 9)/16, C^2,k = 3(4k2 + 4(1 - (- 1)k )k - 2(- 1)k + 2)/16,
Q,„2,k = (2(«2 + 2-4(^2 + 1)(-1)k)k -3^(-1)k + «2)/2.
The recurrence equations for the coefficients remain the same as for loading the nodes of the upper belt. Only the initial data differ.
The method of double induction by k and n2 is used for obtaining C3n k .
The coefficients in the deflection formula (2) have a somewhat simpler form, in the case of the action on the structure of one force at the point C:
C,n,k = (4k3 + 6(1 - (-1)k)k2 + 2(4 - 3(-1)k)k + (-1)k -1) / 4,
= 3(2 к +1 - ( -1)к )/4, Cw = 3 П2 + 2 - 2( П2 +1)( -1)к.
Figure 3. The scheme of the truss, n1 = 6, n2 = 2
The formulas obtained require verification. This can be done in the derivation with an inverse induction sequence. First, we perform an induction on the parameter n2, then on the parameter n1. In
some cases, changing the sequence of inductions significantly speeds up the solution. The final check can be a count on the found formulas in the numerical mode of the program, the speed of which is much larger than the symbolic transformations. For example, with a dozen panels along the vertical of the truss and 4050 in the crossbar, the time for symbolic obtaining of the formula is unrealistically large and is calculated in days. The numerical solution for the same program is executed in fractions of seconds. This fact is one of the reasons for applying the induction method. Otherwise, you could get an analytical solution each time for a truss with a specific number of panels and a given load. A similar feature is possessed by other systems of symbolic mathematics (Mathematica, Maxima, etc.).
2.3. Reaction of supports and forces in critical rods
Simultaneously with the derivation of formulas for deflection by induction, formulas were obtained for the reactions of supports and forces in certain rods (Figure 1). On the basis of these formulas it is possible in the future to evaluate the stability and strength of the rods. Consider the rods from the middle of the span. When loading the upper belt, we have the following expressions
Ya = P( k ((-1)k + 3) + 5 + (-1)k )/2,
YB = - P(2( -1) kk + (-1)k + 3) / 4,
01 = -Pa(6k2 + 2((-1)k + 3)k + (-1)k -1) / (8h),
02 = -U = - Pa(6 k2 + 2(5( -1)k + 3) k + 5( -1)k + 3) / (8 h),
U2 = Pa(6 k2 + 2(( -1)k + 3) k + (-1)k + 7) / (8 h).
The case of loading the lower belt:
Ya = Pk ((-1)k + 3)/2,
YB = - P(2( -1)k k - (-1)k + 3) / 4,
01 = -Pa(6k2 + 2((-1)k + 3)k + (-1)k -1) / (8h),
02 = - Pa(6 k2 + 2(5( -1)k + 3) k + 5( -1)k + 3) / (8 h), U = Pa(6k2 + 2(5(-1)k + 3)k - 3(-1)k - 9)/(8h), U2 = Pa(6k2 + 2((-1)k + 3)k + (-1)k - 5)/(8h).
The case of force loading in the middle of the span:
Ya = P(1 -(-1)k)/2, Yb = P(-1)k/2, O1 = O2 = - Pa(2k +1 - (- 1)k )/ (4h), Uj = U2 = Pa(2k -1 - (- 1)k )/ (4h).
3. Result and Discussion
In the formulas obtained, some features of the solution are hidden, which the graphical representation of them for specific cases will help to see.
In order to clarify the qualitative picture of the deformation of the construction that the solution obtained, we construct the curves of the dependence (2) of the deflection on the number of panels in the crossbar in the case of loading the lower belt of the truss at n2 = 1. We assume that the span length is constant and does not depend on the number of panels L = 40m , hence a = L / (2n1). Let us also fix the total load on the structure Psum = (2n1 - 1)P. We introduce the dimensionless deflection
A' = AEF / (PsumL) . With an increase in the number of panels, the deflection in this formulation of the
problem decreases first, then increases (Figure 4). In this case, the curve has very significant jumps, but the minimum is weakly expressed.
Figure 4. The dependence of the deflection of the panels. 1 - h = 4m, 2 - h = 6m
Despite this, this fact can be used to optimize stiffness. Analytical expressions for the minimum value cannot be obtained.
The dependence of the deflection on the number of panels n2 along the vertical reveals a more
pronounced minimum (Figure 5, k = 5 , L = 40m). In addition, the curves have an intersection point, after which the effect of the height of the truss on the deflection is reversed. Here the height of the truss H is adopted, which is independent of the number of panels in the vertical direction: h = H / n2. For small
values of n2, the deflection increases with the greater the height of the truss. After the intersection point (in Figure 5 this is n2 = 7), the dependence is inverse.
A'
2.4 2.32.2-
2
1.9 1.8 1.7-
/
/Л
7 — ///
//
1 /А / 3
//У
у
¿г
¿г
[V
tu
Figure 5. The dependence of the deflection of the number n2. 1 - H = 50m ; 2 - H = 52m ;
3 - H = 54m
The obtained dependences of the forces in the rods on the number of panels also have some peculiarities. Help them to help with graphics. Consider the case of loading the lower belt at n2 = 1,
h = 4m , L = 40m (Figure 6). The force values depend essentially on the parity of the number of panels
n1 and do not depend on the number of panels in height. The same effect can be observed under other
loads. In this case, the maximally stretched rod can be not a rod U2 in the middle of the span, but a rod
U1 adjacent to it. The same alternation is observed for compressed elements O1 and O2. With increasing
k , the amplitude of the oscillations of the graph decreases.
S/P
0.
-0.4 -0.6
\ у -/ V / N1 \ / \
\ / - —\ / - \ / - \ / —ч ч Х2 ч
1 0 1 1 1 г 1 5 1 1 5 1 6 1 1 1
3 ✓ Ч ч
/ V / \ у N и 7" 4Ч у Ч
/ \ !
Figure 6. Dependence of the forces in the rods on the number of panels. 1—U1; 2— U2;
3 - O1 ; 4 - O2
4. Conclusion
1. The analytical dependences of the deflection of a planar truss structure on its dimensions, load and number of panels have been found and analyzed. To generalize particular solutions, we carry out induction on two parameters.
2. It is revealed that the proposed scheme of the truss has some hidden feature, which can be
considered a defect. Externally, a fully functional design with certain values of the panel numbers turns out
to be kinematically variable.
3. Another feature of the proposed scheme is its external static uncertainty. This creates difficulties in determining the reaction of supports, but allows modeling of structures with many supports with a statically determinate scheme. Of course, in design practice, such schemes can be used as basic ones for more complex statically indeterminate structures with additional constraints, rigid knots and rods of different cross-sections and elastic properties. However, the resulting formulas, due to their simplicity, accuracy and ease of use, can always be useful both for estimating the structural deformity and for estimating the accuracy and reliability of numerical solutions.
4. The solutions essentially depend on the parity of the number of panels in the crossbar. This means that changing the number of panels in a project by just one can significantly reduce or increase the rigidity of the structure or the force in the rod. Judging from Figure 6, the force in the middle rod of the lower belt with a small number of panels varies in this setting to 30 %. Interesting is also the effect of changing the alternation of curves depending on the height of the truss when the number of panels is changed vertically.
References
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21. Ferrari R., Cocchetti G., Rizzi E. Effective iterative algorithm for the Limit Analysis of truss-frame structures by a kinematic approach // Computers & Structures. 2018. Vol. 197. Pp. 28-41. https://doi.org/10.1016/j.compstruc.2017.11.018
22. Topping B.H.V., Khan A.I., Leite J.P.D.B. Topological design of truss structures using simulated annealing // Structural engineering review. 1996. Vol. 2. № 8. Pp. 301-314.
23. Topping B.H.V. Shape optimization of skeletal structures: a review // Journal of Structural Engineering. 1983. Vol. 109. № 8. Pp. 1933-1951.
24. Kaveh A., Laknejadi K. A hybrid evolutionary graph-based multi-objective algorithm for layout optimization of truss structures //Acta Mechanica. 2013. Vol. 224. No. 2. Pp. 343-364.
Mikhail Kirsanov,
+7(495)362-73-14; [email protected]
Михаил Николаевич Кирсанов,
+7(495)362-73-14;
эл. почта: [email protected]
© Kirsanov, M.N., 2018
