General 1D Element
Within the application the three basic types tension, torsion and bending can occur in an arbitrary combination. This chapter serves to introduce how the stiffness relation for a general 1D element can be gained. The stiffness relation of the basic types
- PDF / 895,655 Bytes
- 17 Pages / 439.37 x 666.142 pts Page_size
- 69 Downloads / 250 Views
General 1D Element
Abstract Within the application the three basic types tension, torsion and bending can occur in an arbitrary combination. This chapter serves to introduce how the stiffness relation for a general 1D element can be gained. The stiffness relation of the basic types build the foundation. For ‘simple’ loadings the three basic types can be regarded separately and can easily be superposed. A mutual dependency is nonexistent. The generality of the 1D element also relates to the arbitrary orientation within space. Transformation rules from local to global coordinates are provided. As an example, structures in the plane as well as in three-dimensional space will be discussed. Furthermore there will be a short introduction in the subject of numerical integration.
6.1 Superposition to a General 1D Element A general 1D element can be derived from the basic types of tension, bending and torsion without mutual dependency. For an arbitrary point, the three forces and three moments can be represented as • normal force N (x), • respectively a shear force and a bending moment around an axis of the crosssection: Q z (x), M yb (x), Q y (x), Mzb (x) and • torsional moment Mt (x) around the body axis. The six kinematic parameters are described as follows: • the three displacements u x (x), u y (x) and u z (x). Usually the displacement in the body axis equals the displacement u x (x). • the three rotations ϕx (x), ϕ y (x), ϕz (x). Figure 6.1 shows the kinematic parameters, the forces and the moments. The arrangement of the single parameters in the vectors defines the structure of the total stiffness matrix. If the kinematic parameters are arranged in the order that follows A. Öchsner and M. Merkel, One-Dimensional Finite Elements, DOI: 10.1007/978-3-642-31797-2_6, © Springer-Verlag Berlin Heidelberg 2013
111
112
6 General 1D Element
Fig. 6.1 State variables for the general three-dimensional case
u = [u x , u y , u z , ϕx , ϕ y , ϕz ]T ,
(6.1)
the order of entries for the vector of generalizes forces in the stiffness relation results in: (6.2) F = [N x , Q y , Q z , Mx , M y , Mz ]T . An alternative order results, if the vector of generalized forces is established in the following order • normal force (in the direction of the x-axis), • bending (around the y-axis and around the z-axis) and • torsion (around the x-axis), meaning F = [N x , Q z , M y , Q y , Mz , Mx ]T .
(6.3)
For this order, the single stiffness relation in Eq. (6.4) is illustrated. Under the assumption of a two-node element the stiffness matrix consists of the 6 respective entries on both nodes. The dimension of the stiffness matrix results in 12 × 12. ⎤ ⎡ ⎤ ⎡ ⎤⎡ Z 0 0 0 0 0 Z 0 0 0 0 0 u 1x N1x ⎢ Q 1z ⎥ ⎢ 0 B y B y 0 0 0 0 B y B y 0 0 0 ⎥ ⎢ u 1z ⎥ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎢ M1y ⎥ ⎢ 0 B y B y 0 0 0 0 B y B y 0 0 0 ⎥ ⎢ ϕ1y ⎥ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎢ Q 1y ⎥ ⎢ 0 0 0 Bz Bz 0 0 0 0 Bz Bz 0 ⎥ ⎢ u 1y ⎥ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎢ M1z ⎥ ⎢ 0 0 0 Bz Bz 0 0 0 0 Bz Bz 0 ⎥ ⎢ ϕ1z ⎥ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎢ M1x ⎥ ⎢ 0 0 0 0 0 T 0 0 0 0 0 T ⎥ ⎢ ϕ1x ⎥ ⎥=⎢ ⎥ ⎢ ⎥⎢ (6.4) ⎢ N2x ⎥ ⎢ Z 0 0 0 0 0 Z 0 0 0 0 0 ⎥ ⎢ u 2x
Data Loading...
