MECHANICS. DIFFERENTIAL EQUATION SYSTEM OF ELASTICITY AND ITS BOUNDARY ELEMENTS PROBLEM
8. ANALYSIS OF THREE-DIMENSIONAL BENT BARS USING FI- FI-NITE ELEMENT METHOD BASED PROGRAM SYSTEM FI-NITE ELEMENT METHOD BASED PROGRAM SYSTEM
8.1. Three-dimensional beam structures
In case of two-dimensional bent bar structures discussed in chapter 6, the deflections may be generated in plane of structure. In engineering practice, using three-dimensional models are required many of the cases.
Such cases usually are:
Two dimensional construction, with asymmetrical cross section beams,
Two dimensional construction, with loads perpendicular to plane of structure,
The general three-dimensional beam structures.
This chapter deals with these structures. The chapter 9-10. deals with buckling of the compressed bars.
Because the buckling of the compression chords and the shear buckling of the web sheets require different calculations, so we do not deal with this.
Questions to be answered
The magnitude and direction of the reaction forces and moments generated in supports,
Magnitude and direction of the axial and shear forces, bending and torque moments in each bar,
The and τ stresses which characterized of the stressed state,
Displacements of each point of the structure, and deformation of each beam.
These structures may be testing for the stability of the structure and dynamic behavior (the critical forces of compressed bars and natural frequencies). We deal with these problems later.
The previous chapter has mentioned the externally and internally determination and inde-termination structures. We will see that it is irrelevant in this case too.
8.2. The used finite elements in modeling
The chapter 4 clarified that program system based on the finite element method use two types of element for modeling beam structures. The TRUSS element for modeling structure loaded axial forces only and BEAM element for modeling loaded axial and shear forces, bending and torque moments. Both TRUSS and BEAM elements can be two- or three-dimensional.
In all cases, the finite elements are characterized by a single straight line.
The properties of the TRUSS and BEAM2D elements already described in the previously chapters.
8.2.1. The properties of the BEAM3D elements
The properties of the BEAM2D elements have already written in chapter 6.
The BEAM3D characterized by three dimensional stressed state, and it is general three-dimensional bar structures or displaced perpendicular to the own plane under the loads.
The BEAM3D elements are two or three-node, uniaxial element, have six degrees of free-dom (three translations and three rotations) per each end node. The third node points towards
y-axis in the element local coordinate system. It or an orientation angle (as real constant) is required only for determine the element orientation.
The element coordinate system shown in figure 8.1. The coordinate system x-axis pointing from the first to the second node, the y-axis perpendicular to x axis and central principal axes of cross section, z axis perpendicular to x-y plane and create a right-handed Cartesian coordi-nate system.
Figure 8.1 BEAMD3D element local coordinate system
The linear static analysis requires some real constant (marking as shown Figure 8.2):
The cross-sectional area,
Moment of inertia about the element Y axis,
Moment of inertia about the element Z axis,
Depth of the beam,
Width of the beam,
Relationship between the ends of the connected elements (end release code, two sets of data),
Torsional constant J (see also 8.2.2),
Shear factor in the element y axis (see also 8.2.2),
Shear factor in the element z axis (see also 8.2.2),
Orientation angle of the cross section (only if the orientation does not define by the third node),
Constant for maximum shear stress calculation (see also 8.2.2),
x, y, z distance of the section centroid relative to the nodal point in each node of the beam (total six data),
y, z distance of the shear center relative to the section centroid at each node of the beam (total four data),
y, z distance of the point where stresses are to be calculated at each node of the beam (total four data),
Centroidal product if inertia of the element cross section
Usually there can be defined tapered beam properties and more real constant for thermal analysis also. These properties are not dealt in this chapter.
In usually, we can specify often used cross-sections in engineering practice, such as rec-tangular, a square hole, circle, ring, I, L, T sections, by geometrical dimensions. In this case the other sectional properties will be calculated by the program.
Figure 8.2 BEAM3D elements properties
We also need the material properties of the elements. In this case, is sufficient to specify the value of the modulus of elasticity, Poisson’s coefficient and density of the beam elements.
If necessary, we can define more material properties for the buckling or heat transfer analysis.
The interpretation of the bending moments and shear forces shown in Figure 8.3.
y
x
1
2
z
Vs1 Ms1
Tr1 Fr1
Vt1
Mt1
Vs2 Ms2
Tr2 Fr2
Vt2
Mt2
Figure 8.3 Forces and moments in BEAM3D elements
8.2.2. The special properties of BEAM3D elements
The shear deformation is usually neglected. The chapter 6.2.2 has shown that how this can be taken into account. Also in this chapter we have properties of several common used sections.
We have dealt with the simplified definition of the shear factor (see Figure 8.4). This con-cept will also used in this chapter.
The cross section fs The Shear factor
A/AWeb AWeb/A
Figure 8.4 The simplified definition of Shear factor
The calculations will be needed to determine a shape factor (Ctor) for calculate the maximum stress τ comes from torsion.
In case of circular and thin-walled ring section, the maximum stress τ generated on peri-meter of the circle, so:
I r T
P max
In case of non-circular cross section, the maximum stress τ depends on the section shape.
In such cases we can use only approximate procedures, such as Constantin Weber approx-imate method:
W tor W
max K
C T I
T
where: IW: Weber's centroidal product of inertia, KW: Weber's polar section modulus, Ctor: the shape factor.
Circular cross section, of course, IW=IP, KW=KP and Ctor= r.
An open cross-section (see figure 8.5), where h>> v, we can apply the splitting, and so:
3 ) h v (
I i
i 3 i W
,
max W
W v
K I
max
tor v
C
h1
h2
v2v1
h3
v3
Figure 8.5 Splitting of open cross-sections The η is a factor to correction error of splitting.
Figure 8.6 The η factor of some cross-section