Advanced topics in Finite Element Method 3D truss structures. Jerzy Podgórski

Transcription

1 Advanced topics in Finite Element Method 3D trss strctres Jerzy Podgórski

2 Introdction Althogh 3D trss strctres have been arond for a long time, they have been sed very rarely ntil now. They are difficlt to solve. Thogh a series method simplifying the calclation of internal forces has been devised for statically determined plane trsses, in case of space trsses, only the method of nodal eqilibrim has remained.

3 Introdction arge sets of eqations which are generated by this method for space trsses have discoraged engineers from designing this type of strctre. 3D strctres looking like trsses, in fact, are seldom trsses. For instance spport colmns of overhead power lines are most often space frames becase they keep their geometric stability with bent elements which don t exist in classical trsses.

4 Introdction Both the se of compters and new methods of statics analysis of a strctre making se of new technical possibilities (the finite element method is one of the main methods among them) have enabled considerable progress in designing space trsses. One of the most poplar ses of these strctres is in strctral roofs. Examples of space trsses are presented in the next slide.

5 Introdction

6 Introdction

7 Notation and basic relations The node of a space trss has three degrees of freedom becase in order to describe its movement, we have to give three components of a displacement vector. The displacement vector and forces acting on an element of the space trss are shown in next figre. As in previos presentation components of forces and displacements vector are collected in colmn matrices which will be called vectors.

8 Notation and basic relations nodal displacements vector of the first node i in the global and local coordinate system: i ix iy iz ' i vector of nodal forces acting at the first node i in the global and local coordinate system: f i F F F ix iy iz f ' i ix iy iz F F F ix iy iz

9 Notation and basic relations The above vectors form forces and displacements vectors of an element: vector of the nodal displacements of an element e with the node i and j is written in the global and local coordinate system: e i j ix iy iz jx jy jz ' e ' ' i j ix iy iz jx jy jz

10 Notation and basic relations vector of the nodal forces of an element e in the global and local system: f e f f i j F F F F F F ix iy iz jx jy jz f ' e f ' f ' i j F F F F F F ix iy iz jx jy jz

11 Notation and basic relations Global system

12 Notation and basic relations ocal system

13 The element stiffness matrix of a space trss The relationship between nodal forces and nodal displacements for a space trss is identical to that for a plane trss if we analyse it in the local coordinate system. Obviosly, the third force is F iz or F jz bt the eqilibrim eqation of moments with respect to the y axis reslts in the zero vale of this force:

14 The element stiffness matrix of a space trss F F F 0 F F x ix jx ix jx after considering eq. f Fy Fiy Fjy 0 Fiy 0 after considering eq. e Fz Fiz Fjz 0 Fiz 0 M x 0 M F 0 F 0 y jz jz M F 0 F 0 z jy jy

15 The element stiffness matrix of a space trss The relationship between an axial force and displacements which is identical to the relation presented in previos presentation (comp. N EA ) allows s to express the jx ix searched dependence as follows: K' e J' J' e e e f ' K' ' J' J' J' EA

16 The element stiffness matrix of a space trss The transformation from the local system to the global one will be done analogosly to the transformation performed in case of a 2D T trss (Eqn. f ' i R i f i, K e R e K e R e T ', f e K e e ). In order to complete the transformation to the global system, we need the rotation matrix of a node R i, and then we can determine components of J similar to J 2 EA c sc sc 2 s

17 The element stiffness matrix of a space trss The trss element arrangement with regard to the global coordinate system.

18 The element stiffness matrix of a space trss Since the location of the y and z axes of the local system is not essential for trss elements, we will choose the direction of the y axis in sch a way that it will always be parallel to the XY plane of the global system bt for bars parallel to the Z axis there will be an additional assmption that the y axis is parallel to the Y axis.

19 The element stiffness matrix of a space trss The rotation from the local coordinate system to the global one will be composed of two intermediate rotations. First, we rotate the system xyz to the intermediate system x''y''z'' selected so that the x'' axis is parallel to the XY plane and next we rotate the system x''y''z'' by an angle γ so that the x'' and X axes are parallel.

20 The element stiffness matrix of a space trss The first rotation arond the y axis gives the following reslt: c s s c x y z x y z '' '' '' R or c cos s Z sin i j X X X i j Y Y Y i j Z Z Z 2 2 X Y 2 2 Z

21 The element stiffness matrix of a space trss The second rotation arond the z axis leads the eqations to the global system: when ''=0 we assme γ=0, hence and c s s c X Y Z x y z '' '' '' or R c X cos s Y sin 1 c 0 s

22 The element stiffness matrix of a space trss The composition of both rotations which means J' J' ptting K' e into R, gives the J' J' searched rotation matrix of a node R R ' i i i i where R R i R i i

23 The element stiffness matrix of a space trss After mltiplying matrices, we obtain the final form of the rotation matrix R i : c c s c s R i s c c s s s 0 0 We calclate the transformation of the block J of the element stiffness matrix of the space trss from the local coordinate system to the global one as in previos presentation

24 The element stiffness matrix of a space trss J R i J' R i Inserting relations T J' EA and into the above eqation we obtain: J EA 2 2 c c c s c c c s 2 2 c s c s c s c c cc s s cs s 2 R R ' i i i i

25 The element stiffness matrix of a space trss After the introdction of a convenient notation: C X X C Y Y which are called direction cosines of an element, we obtain a very simple form of the block J of the stiffness matrix: J EA C 2 C C C C C 2 C C C C C 2 CX CZ CY CZ CZ X X Y X Z X Y Y Y Z Z Z K e J J J J

26 The temperatre loads for 3D trss Forming a loads vector of a trss for concentrated forces is identical to forming it for a 2D trss. We will also not discss the vector p. We will discss the vector of nodal forces de to a temperatre load. This vector in the local coordinate system is similar to the components for a plane trss f ' et EA t to 1 0 0

27 The temperatre loads for 3D trss The transformation to the global system proceeds in agreement with f R f following way: et e et f R f ' e e e ' in the R e R i 0 0 R j Since a trss element is straight, R i =R j, where the matrix R i is defined by R R R. i i i

28 The temperatre loads for 3D trss After inserting Ri Ri Ri into f et and mltiplying them, we obtain: f et EA t to c c CX s c C Y s or C f c c et Z EA t to C s c C s C The remaining procedre is identical to the one employed in case of a plane trss. X Y Z

29 The bondary element In previos presentation, we explained widely different types of bondary conditions and also elastic bondary elements. Since they are very sefl elements for modelling many different bondary conditions, we will pay more attention to them in this chapter concentrating on differences between plane and space trsses.

30 The bondary element We will discss the most general elastic element with stiffness k b dropping with respect to axes of the global system with the angles α X, α Y, α Z. c X cos X c Y Y cos c Z cos Z The stiffness matrix of this element in the local system is analogos to the matrix stiffness of an ordinary trss element bt this element has three degrees of freedom, so the stiffness matrix contains only one block J.

31 The bondary element K' b kb Transforming this element to the global coordinate system we obtain a matrix which is similar to the one obtained for a plane trss: K b k b c c c c c c c c c c 2 cx cx cy cx cz 2 X Y Y Y Z 2 X Z Y Z Z

32 The bondary element Bondary elements can form for example, an element with three different types of stiffness k x, k y, k z parallel to axes of the local system xyz: K' b k x 0 k 0 y k z The transformation of this matrix to the global system is analogos to the transformation of the T block J : J R i J' R i

33 Stresses and Internal forces We present here eqations to calclate stresses and internal forces in an element: x E t E jx ix t o t or E E t x t jx ix t o The transformation of the vector to the global system gives the relationship: x E e e R T E t t o

34 Stresses and Internal x forces E e e R T E t t After mltiplication it gives components of direct stress in an element as follows: o x e e E c c R T T T 1 t t o where c is the vector of element direction cosines: c T c X cy cz

35 Stresses and Internal forces Calclating the normal force consists of integrating stresses on the srface of a cross section with an assmption of homogeneity of the stress field 1 N A EA T T e T e x c c R t t The remaining spport reactions are calclated with the help of r = K - p. We can do it exactly in the same way as it has been done for the 2D trss. o