Design of the bridges structures using MathCAD.

Tierean, Mircea ; Baltes, Liana ; Eftimie, Lucian 等

1. INTRODUCTION

There are thousand of papers that describe F.E.A. applications, one of there realised with individual soft, others with commercial programs. We consider useful to present two simple F.E.A. applications, realised with MathCAD, for truss and frame bridge structures.

Because there are simple structures realised from straight bars, the meshing is easy, the element boundaries being the successive junctions.

2. APLICATION FOR TRUSS STRUCTURES

We considered as application the parabolic truss structure of a bridge (fig. 1). The forces are [Q.sub.1] = 2x[10.sup.5]N and [Q.sub.2] = [10.sup.5]N. The application starts with the drawing of the structure. The input data are the number of bars, the horizontal dimension "b" and height of first column "c". The structure drawing is generating by successive movement of the position vector across the nodes (fig. 2) (Tierean et al., 1999, 2006).

Next, a nodal junction matrix will be built, which defines the elements of the structure. In this matrix the column number is the same with the element number, the first line shows the start node and the second line the arrival node. Thus the column 0 defines the 0 bar, the start node being 0 and the arrival node 1. The last column defines the 24 bar the start node being 7 and the arrival node 10.

In the next step is generate the expansive incidence matrix, which has the same number for the lines and for the structure nodes; the number of columns is the same with the number of its elements.

Additional, with these definitions (nodal junction and expansive incidence matrix) the elements are toggled in plan by polar co-ordinates. In order to assembly the stiffness matrix (K), the computing of the rotated matrix of incidence (SER) is necessary,

[SER.sub.2xi + u,2xz + v] := [SE.sub.i,z] x ROT[(z).sub.u,v] (1)

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

trough the multiplication of the incidence matrix (SE) and the rotation matrix (ROT), and for the local stiffness matrix (KL) also.

[KL.sub.2xz + u,2xz + v] := [KLOC.sub.u,v] x (Ex[A.sub.z]/[1.sub.z]) K := SER x KL x [SER.sup.T] (2)

After this step, it is necessary to explain the boundaries of structure; also the loadings are introduced in nodes. Every node has two values: the first refers on horizontal force and the second refers on vertical force. In the next step there are computed the nodal displacements (two for each node) by multiplying the inverse of stiffness matrix with the loadings vector.

These nodal displacements draw the distorted structure (fig. 3), amplified by 100.

The forces on the bar direction, in the starting node, are obtained by projecting the nodal distortions in the bar local base, amplified by their stiffness.

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After the nodal equilibrium and the calculation error verification, (in the first occurrence the error is less than [10.sup.-6], and in the second one the error is less than [10.sup.-7]), the structure bars are checking at traction and buckling. The results were checked with ANSYS.

3. APLICATION FOR FRAME STRUCTURES

As application is considered the frame structure (fig. 4), loaded by the forces [Q.sub.2] = [Q.sub.6] = 25x[10.sup.3]N, [Q.sub.3] = [Q.sub.4] = [10.sup.4]N and the bending moments [M.sub.3] = [M.sub.4] = [10.sup.4]Nm. As in the previous chapter, the calculation program starts with the drawing of the structure, the input data being the number of bars, the horizontal "b" and the vertical dimension "c".

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Because in this case are three displacements the rotation matrix is:

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and by multiplying with the translation matrix (because of bending) result the roto-translation matrix.

RT(z) := ROT(z)xTd[(z).sup.T] (6)

The components of rotated matrix of incidence (SER) are the same as the elements of rotation matrix, for the starting node and same as roto-translation matrix, in the case of arrival node.

[SER.sub.3xi+u, 3xz+v] := if([SE.sub.i,z] > 0, [SE.sub.i,z]xROT[(z).sub.u,v], [SE.sub.i,z]xRT[(z).sub.u,v]) (7)

The stiffness matrix results by projecting the free elements in the incidence space of the structure (transformation of the transformation).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

After the fixing of the structure, the loadings are introduced in nodes. Every node corresponds with three values: the first refers on elongation force; the second refers on shearing force and the third to the bending moment.

After that step the nodal displacements are calculated (three for each node) by multiplication of the inverse stiffness matrix with the loadings vector. These nodal displacements draw the distorted structure, using cubic Hermite interpolation.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

In the fig. 5 is presented the deformed frame structure considering the nodes stiffness.

To obtain the forces is necessary the displacements imagine in the global reference system.

TD(z) := ROT (z)xTd(z)xROT [(z).sup.T] (10)

[FIGURE 5 OMITTED]

It is also necessary to calculate the strain matrix in the start ([delta]EF) and in the arrival node ([DELTA]EF). With these matrix are calculated the force matrix in the local (FL) and in the global reference system (F), in the start (i) and in the arrival node (j) (Tofan et al., 1995).

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After the bars' equilibrium checking (less than [10.sup.-6]), nodal equilibrium (less than [10.sup.-8]) and the computing of error (less than [10.sup.-9]), the structure bars are checking at traction, bending and buckling, considering the nodes stiffness. The results were checked with ANSYS.

4. CONCLUSION

The presented programs give the deformed structures, the displacement values, and the bars stresses, in the local and in the global reference system. Using a medium equipped Windows XP PC configuration, the computing time is 4s for the first program and 9s in the second case.

Due to the simplicity and the higher velocity, the program is easily to implement in any type of calculation for the structure bars avoiding the laboriously methodology of the drawing, introducing of the data and calculation, claimed by commercial programs for F.E.A.

5. REFERENCES

Mathsoft, Inc. (2001)MathCAD User's Guide, Cambridge, MA "Tierean, M., Tofan, M., Goia, I., F.E.A. of bars structures using MathCAD, NAFEMS World Congress '99 on "Effective Engineering Analisys", Newport, USA, 25-28, April, 1999, pag. 947-952.

Tierean, M.H., Baltes, L.S., Mirza Rosca, J., Santana Lopez, A., Applications for finite element analysis of strength structures using Mathcad, Third international conference "Mechanics and Machine Elements", 2-4 Nov. 2006, Sofia, Bulgaria, 2006, ISBN 10: 954-438-587-8, pag. 240-245.

Tofan, M.C.; Goia, I.; Tierean, M.H. & Ulea, M. (1995). Deformatele structurilor (Structure's strains), Editura Lux Libris, ISBN 973-96854-2-0, Brasov, Romania http://www.ptc.com/go/mathsoft/mathcad/