Comparative study of the stress state in plates assembled with screws.

Rusu-Casandra, Aurelia Liliana ; Iliescu, Nicolae ; Baciu, Florin 等

Abstract: For the calculation of the screw joint with initial preload, the most widespread joint type in machine building industry, it is necessary to know the rigidity of the assembled parts. In order to determine the latter, this paper presents a comparative study carried out using the finite element method and the photoelasticity technique of the stress state in the tightened plates of a screw assembly. The results of the analysis led to a more accurate calculation of the rigidity of the screw joint.

Key words: stress analysis, FEM, photoelasticity, screw joint

1. INTRODUCTION

The screw joints represent the most widespread joint type in machine building industry and the most important of all separable assemblings, over 60% of the parts having threads. These include fasteners (screws, bolts and nuts), housing-type parts (fastened together by screws and bolts or other parts are fastened to them by screws), shafts (with treads for securing bearings, gears), pulleys and gears (with threaded holes to receive set screws), etc. The extensive use of the screw joints is justified by many advantages:

(a) High axial forces can be developed as a result of the wedging effect of the thread and the large wrench length to thread radius ratio.

(b) A screw-type clamping facility can be fixed in any position because the self locking effect of the thread.

(c) Screw fasteners are of convenient shape and small size.

(d) Simple manufacture with the possibility of maintaining high accuracy.

According to their use, screw threads are classified as: fastening threads, intended for fastening parts together, fastening and sealing threads, intended both for fastening parts together and for preventing the leakage of fluids, power threads for transmitting motion, etc.

Fastening screws are among the most highly stressed machine parts. Screw joints subjected to separating forces and moments are designated with initial preload. This preload is necessary to prevent a relative sliding motion between the parts due to chance forces and shocks in variable loads and to ensure the rigidity and tightness of the joint. The initial condition in the calculation is to retain the given pressure at the surfaces of the contact after the external forces are applied. For the calculation of the screw joints with initial preload subjected to steady or variable loads, it is necessary to find out the relationships between the forces that occur in the assembly. For this reason, the operating diagram i.e. the force-deformation diagram for the screw and assembled parts respectively must be plotted. The rigidity of the screw [c.sub.s] and the rigidity of the joined parts [c.sub.p] occur in this diagram. In order to calculate the latter using the next formula [Reshetov, 1978]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represent the thicknesses of the parts and [E.sub.i] are the modules of elasticity of the materials of the joined parts, difficulties arise in the estimation of the sections [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that should be considered in the calculation. Most authors (Lingaiah, 2003; Shigley et al., 2004) assume that the volume of the material of the fastened parts that is elastically deformed is extended over the so-called pressure cones (Fig. 1), being limited at the exterior by two truncated cones having the generators inclined with an angle [alpha] and at the interior by the hole for the screw made in the joined parts. The notation d is for the diameter of the bearing circle of the nut. For these cones is advisable to take tan [alpha] = 0.4 to 0.5 (Reshetov, 1978).

[FIGURE 1 OMITTED]

A comparative study, numerical and experimental, of the stress state in the tightened plates of an assembly was developed in order to perform a more accurate calculation of the rigidity of the screw joint. A good agreement occurred between the results obtained and those given in references.

2. FINITE ELEMENT ANALYSIS

A finite element study was performed using SOLIDWORKS software (***2010). The finite element mesh was generated for the model using tetrahedral elements. The model has the Poisson's ratio value applicable to photoelastic materials. Both the applied load and boundary conditions used for the finite element model were chosen to be similar to those of the photoelastic model. The contour plot of the principal stress [[sigma].sub.1] obtained with the finite element method is presented in Fig. 2. Fig. 3 shows the volume of the elastically deformed material of the two plates loaded with an initial preload, corresponding to the photoelastic experiment for the fringe order N = 1.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

3. PHOTOELASTIC INVESTIGATION

The experimental investigation was conducted using the photoelastic technique (Iliescu & Atanasiu, 2006; Paipetis, 1990) The model made of an epoxy resin represents the cross section through two plates of equal thicknesses (24 mm) and equal lengths (120mm) assembled with a metric thread screw M12. A disc made of the same material as the model, diametrical compressed, was used to calibrate the material resulting the stress photoelastic constant of the model ([[sigma].sub.0] = 2,71 MPa/fringe). The model loaded by tightening a nut on the M12 screw through two washers made of Plexiglas was examined in a circular polariscope. Fig. 4 shows the isochromatic patterns. In Fig. 5 are plotted the curves of the principal stress [[sigma].sub.1] along the hole for the screw made in the assembled plates. The volume of the material of the joined plates that is elastically deformed is represented for the isochromatic pattern having the fringe order N = 1 in Fig. 6. The diameter of the hole [d.sub.h] is equal to 12.5mm.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

4. CONCLUSIONS

Comparison of the results of the finite element analysis with the photoelastic ones regarding the distribution of the stresses around the hole made in two plates assembled by a thread screw with initial preload led to the following conclusions:

(a) The principal stress [[sigma].sub.1] (Fig. 2) and the fringe order of the isochromatic patterns (Fig. 4) decrease with the increasing distance from the axis of the hole, this suggesting the magnitude of the stresses in different areas of the joined plates.

(b) The stress distribution on the contour of the hole shows also that the stresses increase towards the upper and lower edges of the hole and are minimal at the contact area of the two plates (Fig. 2 and Fig. 5).

(c) The volume of the material of the fastened plates that is elastically deformed is approximately extended over two identical truncated cones and a cylinder (Fig.3 and Fig. 6). Each of the truncated cones has the height equal to a third of the thickness of a plate (i.e. 8mm) and the generators inclined with an angle of [alpha] = 24[degrees], starting from the diameter of the bearing circle of the nut (d = 18.05mm).The above scheme made for the stressed area consisting of a cylinder with two truncated cones at the top and bottom is closer to the references (Fig. 1).

(d) The calculated results are in good agreement with the measured ones, the maximum stress being predicted in both analyses with reasonable accuracy.

(e) To consolidate the above conclusions on which the calculation of the rigidity of the joined parts is based, as future research, studies for the stress state in assembled plates of different thicknesses with screws of different diameters can be conducted.

5. REFERENCES

Iliescu, N.; Atanasiu, C. (2006). Metode tensometrice in inginerie (Stress Analysis Techniques in Engineering), Editura AGIR, ISBN 973-720-078-0, Bucuresti

Lingaiah, K. (2003). Machine Design Databook, McGraw-Hill, ISBN 0-07-136707-1, New York

Paipetis, S. (1990). Photoelasticity in Engineering Practice, Routledge, ISBN 978-0853343639, United Kingdom

Reshetov, D. (1978). Machine Design, Mir Publishers, Moscow

Shigley, J.; Mischke, C. & Brown, T. (2004). Standard Handbook of Machine Design, McGraw-Hill, ISBN 0-07-144164-6, New York

*** (2010) Solidworks User Manual, Dassault Systemes SolidWorks Corp, Concord, MA, USA