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Research of maximum stresses zones in circular perforated plates made of S235JR steel

loaded with concentrated forceMateusz Konieczny*, Henryk Achtelik, Grzegorz Gasiak

Opole University of Technology, Department of Mechanics and Machine Design, Opole, Poland; *mateuszmarcinkonieczny@wp.pl

The paper proposes a numerical approach to determining the internal forces in a circular perforated plate loaded with concentrated force Pi applied in the geometric centre of the plate. The finite element method program Femap was used for numerical calculations. The test plate with diameter D = 300 mm has holes in ten circles. The plate made of S235JR steel has holes with diameter d1 = 3.5 mm on the first inner circle, and holes on the tenth outside circle have a diameter d10 = 20.5 mm. The plate of the above geometry was simply supported and loaded with different values of concentrated force. By means of numerical calculations using the finite element method, the coordinates of concentration zones of reduced stress in the perforated plate were determined. These zones were located on the plate bridges between perforation holes. The most hazardous place in the analysed perforated plate is associated with the 8 zone with the hole radius d3 = 5.0 mm at the circle radius R3 = 33.0 mm, where the highest stress concentration occurs. In this zone, the reduced stress is σred max = 217.20 MPa (point with the coordinates x, y, z [mm], i.e. P2[5.2; 30.6; 4.5]). The results of numerical calculations were verified with experimental results. The differences between the results of numerical calculations of the state of stress and those obtained experimentally did not exceed 10%.

Key words: circular perforated plate, S235JR steel, concentrated force, reduced stress, numerical calculations, experimental research.

Inżynieria Materiałowa 2 (228) (2019) 53÷60DOI 10.15199/28.2019.2.4

MATERIALS ENGINEERING

1. INTRODUCTION

Present engineering constructions are characterized by a decrease in terms of the safety factor, that is why calculations of strength, stiffness and stability of thin-walled construction elements are becoming some of the important aspects in all branches of tech-nology, especially such as chemical engineering. There are many applications of perforated plates, e.g. in pressure vessels, chemi-cal equipment (heat exchanger tube screens) [1÷3], oil refineries, aircraft and spacecraft parts, aircraft, robots, transport containers, box conveyors or as elements of loose material screens, which can be subjected to a load that varies over time. This type of girders (perforated plates) can also act as mounting boards in which holes are made for various operational reasons [4].

In practical problems of design, it is often necessary to determine stresses and deflections in such plates, which are loaded with static forces acting perpendicular to their surface. Due to the wide ap-plication of perforated elements in many branches of industry, it is advisable to undertake studies involving the analysis of the state of stress such elements subjected to a normal load. Knowledge of the state of stress will offer optimal design; and thus, safe operation of devices in which these types of components are used has been made more feasible [5÷7].

In the literature many attempts were made to analyse stress con-centration, deflections, strains, stress concentration factor and stress intensity factor in perforated plates weakened because of openings comprised in them. The analytical method [8÷10], the experimental method [11÷14] and the numerical method [15÷21] were used to evaluate the parameters of perforated plates. During the analysis of perforated plates, various design parameters were varied, which have a significant impact on the nature of the correct operation of these plates, i.e.: type of material (isotropic, orthotropic, composite and steel), material parameters, plate thickness, plate dimensions, among others diameters of the hole, the method of supporting the plate and applying a gradually increasing load on the plate. The value of stresses and strains as well as deflections for a variety of designs applied in plates, e.g. square, rectangular and circular, were

also measured. In [22], the authors propose a mathematical ap-proach to determining internal forces in a circular perforated plate of a heat exchanger in a chemical reactor. The plate was subjected to a load that was symmetrical about the centre axis of the plate and perpendicular to the centre surface of the plate. The assumed math-ematical algorithm provides a manner in which the state of stress in the perforated plate can be evaluated by means of analytical re-lations. Paper [23] contains the development of experimental re-search methodology, which can be used to determine the real stress values in perforated plates loaded centrally by concentrated force and verification of the mathematical model. The axially symmet-rical perforated plate was subjected to bending with concentrated force applied centrally. In addition, the bending of the plate was monitored by means of a dial indicator during strain measurements. On the basis of the measured strain values, radial, peripheral and reduced stresses in the plate were determined. An experimental ap-proach to the analysis of perforated plates was also elaborated in [24], in which they assessed the state of stresses and deflections of a perpendicularly supported perimeter, perforated axiomymet-ric plate loaded with loaded with a concentrated force applied in the geometric centre of the plate. For this purpose, a special setup was designed and a methodology was developed for the purposes of testing the perforated plate. Tensometric rosettes were used in the measurements of plate deformations. Radial and circumferen-tial stresses were measured in the tests, which were measured de-pending on the radius of the plate’s perforation and load. The most common approach to solving problems related to perforated plates applies the use of numerical methods by finite element methods (FEM). Examples of the application of the finite element method to determine the state of stress and deformation of perforated plates are given in [25, 26]. Using the FEM, the study defined effective elasticity constants, which made it possible to determine the state of stress for any load state of a perforated plate. However, in work [27], by application of various hole patterns and using a gradually increasing load to the plate, the authors determined the value of shear stress for the perforated plates. The subject of research in [28÷30] involved a numerical analysis of the state of stress and

54 INŻYNIERIA MATERIAŁOWA MATERIALS ENGINEERING ROK XXXX

deflection in restrained and freely supported, perforated rectangular and circular plates, loaded centrally with focused force. In the cases analysed, stress distributions over the entire surface of the perfo-rated plate were obtained.

The aim of the study reported in the present paper is to locate the zones in which maximum stresses occur in a circular axisymmetric perforated plate made of S235JR steel, simply supported and loaded with concentrated force Pi applied in the geometric center of the plate. Tests of stress concentration zones were performed numeri-cally using the finite element method.

2. MODEL OF PERFORATED PLATE

The calculation was based on a circular axisymmetric perforated plate with dimensions: diameter D = 300 mm, thickness h = 5 mm. 200 holes with different radii placed on the plate. These holes were arranged in 10 circles with 20 holes in each circle, as shown in Fig-ure 1. The plate was made of S235JR steel sheet with the following material parameters: Young’s modulus E = 200 GPa, Poisson’s ratio ν = 0.3 (Tab. 1).

The perforated plate was simply supported on the entire perime-ter and loaded with a concentrated force Pi applied in the geometric centre of the plate (Fig. 2). The following values of the load were assumed Pi: P1 = 410 N; P2 = 875 N; P3 = 1365 N; P4 = 1925 N; P5 = 2510 N. Eight measuring zones, i.e. from Z1 to Z8 (Fig. 1), were placed on the axisymmetric perforated plate.

3. CALCULATE OF THE STATE OF STRESS

The subject of numerical calculations of the state of stress applied a circular axisymmetric perforated plate, whose geometry, i.e. the

dimension, the method of restraint and the load method, was out-lined in section 2 of this paper. Numerical calculations were carried out using the finite element method (FEM) by application of the Femap program [31]. The work uses a spatial tetrahedral finite ele-ment with four nodes in the corners, where the walls have the shape of a triangle, i.e. W4 (Fig. 3).The location of stress concentration zones in a circular axisym-metrical perforated plate, simply supported and loaded with con-centrated force Pi applied in the geometric centre of the plate with the value P5 = 2510 N is shown in Figure 4. The figure illustrates the distribution of reduced stresses given in MPa in eight measur-ing zones (Fig. 1). Point PT1 define the values of reduced stresses obtained numerically at the spots where strain gauges were attached in the circumferential and radial directions, and P1, P2 and P3 are points of identification of the maximum numerical values of stress.

Fig. 1. Model of perforated plate with eight measuring zones, i.e. from Z1 to Z8 [23]Rys. 1. Model płyty perforowanej z ośmioma strefami pomiarowymi, tj. od Z1 do Z8 [23]

Table 1. The chemical composition of S235JR steel, wt %Tabela 1. Skład chemiczny stali S235JR, % mas

C Mn S N P

0.2 1.4 0.040 0.009 0.040

Fig. 2. The method of supporting the entire perimeter and the load of concentrated force Pi on the perforated plateRys. 2. Sposób podparcia na całym obwodzie i obciążenia siłą skupioną Pi płyty perforowanej

Fig. 3. Spatial tetrahedral finite element with four nodes in the corners, i.e. from W1 to W4 (a) and the division of the plate into finite elements (distribution of finite elements around the holes in zone Z1 Fig. 1) (b)Rys. 3. Element skończony przestrzenny (3D) czworościenny z czterema węzłami w narożach, tj. od W1 do W4 (a) oraz podział płyty na elemen-ty skończone (rozmieszczenie elementów skończonych wokół otworów w strefie Z1 rys. 1) (b)

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Fig. 4. Reduction stress zones σred given in MPa for an axisymmetric perforated plate simply supported and loaded with concentrated force Pi ap-plied in the geometric centre of the plate with the value P5 = 2510 N: 1) zone 1, where σred max = 88.7 MPa, 2) zone 2, where σred max = 99.5 MPa, 3) zone 3, where σred max = 100.1 MPa, 4) zone 4, where σred max = 141.5 MPa, 5) zone 5, where σred max = 160.2 MPa, 6) zone 6, where σred max = 183.6 MPa, 7) zone 7, where σred max = 204.3 MPa, 8) zone 8, where σred max = 217.2 MPa Rys. 4. Strefy spiętrzenia naprężeń redukowanych podanych w MPa dla osiowosymetrycznej płyty perforowanej swobodnie podpartej i obciążo-nej siłą skupioną Pi przyłożoną w środku geometrycznym płyty o wartości P5 = 2510 N: 1) strefa 1, gdzie σred max = 88,7 MPa, 2) strefa 2, gdzie σred max = 99,5 MPa, 3) strefa 3, gdzie σred max = 100,1 MPa, 4) strefa 4, gdzie σred max = 141,5 MPa, 5) strefa 5, gdzie σred max = 160,2 MPa, 6) strefa 6, gdzie σred max = 183,6 MPa, 7) strefa 7, gdzie σred max = 204,3 MPa, 8) strefa 8, gdzie σred max = 217,2 MPa

56 INŻYNIERIA MATERIAŁOWA MATERIALS ENGINEERING ROK XXXX

The research concerned with the location of stress concentra-tion zones (Fig. 4) shows that the diameter of the perforation has a significant effect on the distribution of stresses on the en-tire surface of the plate, and in particular in the area of perfora-tion holes. In the first zone, maximum reduced stress σred amounted to σred max = 88.70 MPa, in the second zone, maximum reduced stress σred amounted to σred max = 99.50 MPa, in the third zone, maximum reduced stress σred amounted to σred max = 100.1 MPa, in the fourth zone, maximum reduced stress σred amounted to σred max = 141.50 MPa, in the fifth zone, maximum reduced stress σred amounted to σred max = 160.20 MPa, in the sixth zone, maxi-mum reduced stress σred amounted to σred max = 183.60 MPa, in the seventh zone, maximum reduced stress σred amounted to σred max = 204.30 MPa. In contrast, in the eighth zone (the nearest central part of the perforated plate) the maximum reduced stress σred was σred max = 217.20 MPa. The most dangerous place in the circular axisymmetric perforated plate, where the highest stress concentra-tion occurs, exists in 8 zone with the hole radius d3 = 5.0 mm and the radius of the circle on which the opening R3 = 33.0 mm is lo-cated. In this zone, the reduced stress is σred max = 217.20 MPa (point with the coordinates x, y, z [mm], i.e. P2[5.2; 30.6; 4.5]).

4. EXPERIMENTAL RESEARCH

The aim of experimental research was to verify, i.e. assess the ac-curacy associated with solving a given design with respect to its strength. The basis for such an evaluation involves the study of the effect exerted on the material defined by stress and strain states in the considered structural elements. Both real objects and their models made of different materials can be used to test these states. As a result of these tests, it is possible to verify the results derived numerically.

The subject of experimental research of the state of stress was a circular axisymmetric perforated plate, whose geometry, i.e. the dimension, the method of restraint and the load method, was de-scribed in section 2. The strain gauges were stuck on the previously prepared plate surface at the points shown in Figure 5a. Foil strain gauges type TF – 1/120, TFW – 2/120, TFP – 2/120 were used for the measurement of strains in the perforated plate. For practical reasons, the strain gauges were placed in eight circles, that is ones with radii R3, R4, ..., R10. Since the tested perforated plate was axi-ally symmetric, the strain gauges were placed in the first and fourth quarter as shown in Figure 5b [2].

The perforated plate with the collocated strain gauges was sub-sequently mounted on a suitable annular support, concentric in rela-tion to the loading mandrel of the strength test stand. A schematic of the plate mounting and the applied concentric load application P are shown in Figure 6. In order to control the plate deflection, a dial gauge was used with an accuracy of 0.01 mm. The gauge was in-stalled so that the axis of the gauge plunger coincided with the axis of the loading mandrel.

During the first stage of the study, a preload P0 was applied to the perforated plate. This caused a preliminary deflection in the plate centre, f0 = 1 mm. This deflection corresponded to the force P0 = 570 N. Having obtained the preliminary loading and deflec-tion, the readings of the strain gauges were taken in each measuring region in radial and circumferential directions. Next, the plate was loaded with a force Pi, which caused the deflection fi = 1.5 mm. The readings of the strain gauge were taken again and they were marked with the index “k” standing for the “final readings”. The value of force Pn and the ensuing deflection fn were calculated from the rela-tions:

Pn = Pi – P0 (1)

fn = fi – f0 (2)

where: n – successive natural numbers, i.e. 1, 2, 3, 4, 5; i = I, II, III, IV, V.

The successive loads applied by forces: PI, PII, PIII, PIV, PV, were used so as to obtain deflections equal to fII = 2 mm, fIII = 2.5 mm, fIV = 3 mm, fV = 3.5 mm respectively. In this case, loading val-ues were PI = 410 N, PII = 875 N, PIII = 1365 N, PIV = 1925 N, PV = 2510 N.

The deformations of the perforated plate subjected to the above loadings were measured in the radial and circumferential directions, at points along the x- and y-axes. Finally, the strain gauge readings and the calculated radial and circumferential strains, εr and εϴ, were used for the determination of the radial stresses σr and circumferen-tial stresses σϴ in the investigated perforated plate.

It was assumed that in the tested perforated plate, a plane stress state occurs. Having determined empirically the strains εr and εϴ the principal stress components in the radial and circumferential direc-tions, σr and σϴ, were defined on the basis of the following relations:

r r

E

1 2 ( )

(3)

Fig. 5. The arrangement of extensometers on a circular axisym-metric perforated plate: a) schematically (numbered 1÷8), b) in the experiment [23]Rys. 5. Sposób rozmieszczenia tensometrów na okrągłej osiowosyme-trycznej płycie perforowanej: a) schematycznie (oznaczono numerami 1÷8), b ) podczas badań [23]

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Er1 2 ( )

(4)

The reduced stress was determined using the relations:

red r r2 2

(5)

4. COMPARATIVE ANALYSIS OF NUMERICAL RESULTS WITH EXPERIMENTAL ONESBased on numerical calculations using the finite element method (FEM) and experimental research, the results that were obtained could be compared. Table 2 and Figures 7÷9 presents the values of reduced stresses σred in the zones of accumulated stresses for a cir-cular axisymmetric perforated plate, simply supported and loaded with concentrated force Pi applied to the geometric centre of the plate, numerically and experimentally determined in eight measur-ing points shown in Figure 6a and the difference between the results obtained methods δσred.

The reduced stresses for a circular axisymmetric perforated plate, simply supported and loaded with concentrated force Pi applied to the geometric centre of the plate of monotonically are reduced start-ing from the center towards the edge (supporting the plate) as il-lustrated in Figure 10.

The results of calculations indicate that the highest value of re-duced stress σred occurred when the perforated plate was loaded with the concentrated force Pi applied in the geometric centre of the plate with the value P5 = 2510 N, both in the case of experi-mental calculations and numerical calculations, which was equal to σred

e = 160.03 MPa and σredn = 158.40 MPa (Fig. 4, zone 8). The

relative difference in this case was 1.02%.The diameter of the perforation hole in the perforated plate has

a significant impact on the stress distribution in the plate in which

Fig. 6. A scheme of support of a circular axisymmetric perforated plate loaded with concentrated force Pi in the geometric centre of the plate, where Rp – radius of supports distribution, h – plate thickness [23]Rys. 6. Schemat podparcia okrągłej osiowosymetrycznej płyty perforo-wanej obciążonej siłą skupioną Pi w środku geometrycznym płyty, gdzie Rp – promień rozmieszczenia podpór, h – grubość płyty [23]

Table 2. The course of reduced stress changes σred in of accumulation stress zones for a circular axisymmetric perforated plate, simply sup-ported and loaded with concentrated force Pi applied in the geometric centre of the plate, numerically determined σred

n and experimentally σred

e in 8 measuring points and the percentage difference between the results obtained by these methods δσTabela 2. Przebieg zmian naprężeń redukowanych σred w strefach spiętrzenia naprężeń dla okrągłej osiowosymetrycznej płyty perforowanej, swobodnie podpartej i obciążonej siłą skupioną Pi przyłożoną w środku geometrycznym płyty, wyznaczonych numerycznie σred

n i eksperymental-nie σred

e w 8 punktach pomiarowych oraz różnice procentowe pomiędzy rezultatami uzyskanymi tymi metodam δσred

Rn mm

σred MPa

Pi, N

410 875 1365 1925 2510

33.0σred

n 35.70 72.40 102.76 132.30 158.40σred

e 34.17 72.07 102.25 131.18 160.03δσred, % 4.48 0.46 0.50 0.85 1.02

40.5σred

n 28.50 60.70 86.20 115.10 138.60σred

e 28.84 60.31 88.09 113.22 138.62δσred, % 1.18 0.65 2.15 1.66 0.01

49.0σred

n 24.60 50.70 72.40 94.50 116.60σred

e 23.92 50.71 71.96 93.26 114.62δσred, % 2.84 0.02 0.61 1.33 1.73

60.0σred

n 19.20 39.70 62.10 75.30 95.70σred

e 18.68 39.25 59.22 75.45 93.18δσred, % 2.78 0.89 4.86 0.20 2.70

72.5σred

n 17.20 34.20 44.90 57.80 79.30σred

e 17.59 33.71 47.27 58.42 73.17δσred, % 2.22 1.45 5.01 1.06 8.38

88.0σred

n 11.80 24.50 37.50 47.30 62.50σred

e 12.07 25.37 37.02 47.79 59.52δσred, % 2.24 3.43 1.30 1.03 5.01

107.5σred

n 8.25 17.60 27.40 37.40 57.60σred

e 8.63 17.97 27.73 38.79 55.62δσred, % 4.40 2.06 1.19 3.58 3.56

130.5σred

n 3.48 9.90 18.70 27.70 38.50σred

e 3.32 9.83 18.65 27.40 36.14δσred, % 4.81 0.71 0.27 1.09 6.53

such geometric discontinuity occurs. Significant values of reduced stress σred form the common reason for the development of micro-cracks in the area of the perforation hole. In the case of variable (fatigue) loads, microcracks start to develop and, as a consequence, they may cause the failure of a machine element or in a part in which the perforated element is located. In connection with the above, it is very important to determine the value and location of stress concentration in machine components representing the rea-sons responsible for their occurrence.

6. CONCLUSIONS

The following conclusions can be drawn from the above analysis:1. The present study has demonstrated that the way in a circular

axisymmetric perforated plate is loaded and the diameter of per-foration has a significant impact on the distribution of stresses on the entire surface of the plate, in particular in the area of holes and on the bridges between the holes.

2. The most hazardous place in a circular axisymmetric perforated plate with the highest stress concentration is located in zone 8 with the hole radius d3 = 5.0 mm and the radius of the circle on which the hole R3 = 33.0 mm is located, the reduced stress is σred

max = 217.20 MPa (point with the coordinates x, y, z [mm], tj. P2[5.2; 30.6; 4.5] (Fig. 4, zone 8).

3. Maximum relative disaggregations expressed in terms of the val-ues of reduced stresses σred obtained by the numerical method

58 INŻYNIERIA MATERIAŁOWA MATERIALS ENGINEERING ROK XXXX

Fig. 7. The course of reduced stresses changes σrede using the von Mises hypotesis for a circular axisymmetric perforated plate, simply supported

and loaded with concentrated force Pi applied in the geometric centre of the plate, experimentally determinedRys. 7. Przebieg zmian naprężeń redukowanych σred

e wg hipotezy Hubera-Misesa dla okrągłej osiowosymetrycznej płyty perforowanej, swobodnie podpartej i obciążonej siłą skupioną Pi przyłożoną w środku geometrycznym płyty, wyznaczonych eksperymentalnie

Fig. 8. The course of reduced stresses changes σredn using the von Mises hypotesis for a circular axisymmetric perforated plate, simply supported

and loaded with concentrated force Pi applied in the geometric centre of the plate, numerically determinedRys. 8. Przebieg zmian naprężeń redukowanych σred

n wg hipotezy Hubera-Misesa dla okrągłej osiowosymetrycznej płyty perforowanej, swobodnie podpartej i obciążonej siłą skupioną Pi przyłożoną w środku geometrycznym płyty, wyznaczonych numerycznie

Fig. 9. Percentage difference in reduced stress variations δσred according to von Mises hypotesis for a circular axisymmetric perforated plate, simply supported and loaded with concentrated force Pi applied in the geometric centre of the plate, numerically and experimentally determined δσredRys. 9. Procentowa różnica zmian naprężeń redukowanych wg hipotezy Hubera-Misesa dla okrągłej osiowosymetrycznej płyty perforowanej, swo-bodnie podpartej i obciążonej siłą skupioną Pi przyłożoną w środku geometrycznym płyty, wyznaczonych numerycznie oraz eksperymentalnie δσred

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Fig. 10. The course of reduced stresses changes σred using the von Mises hypotesis for a circular axisymmetric perforated plate, simply supported and loaded with concentrated force Pi applied in the geometric centre of the plate, numerically and experimentally determinedRys. 10. Przebieg zmian naprężeń redukowanych σred wg hipotezy Hubera-Misesa dla okrągłej osiowosymetrycznej płyty perforowanej, swobodnie podpartej i obciążonej siłą skupioną Pi o wartości Pi = 2510 N przyłożoną w środku geometrycznym płyty, wyznaczonych numerycznie oraz ekspery-mentalnie

using the finite element method and the experimental method, in a round axisymmetric perforated plate, with free edge support, did not exceed 10% in the present research.

4. The reduced stresses of monotonically are reduced starting from the centre towards the edge (supporting the plate);

5. Numerical programs (FEM) such as Femap provide the tools to determine stresses induced in the machine element or any place in the structure;

6. The use of numerical methods (FEM) leads to the development of a topography of stress distribution over the entire surface of a perforated plate, in particular in the perforation holes area so as to offer details regarding the location of stress concentra-tion zones.

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60 INŻYNIERIA MATERIAŁOWA MATERIALS ENGINEERING ROK XXXX

Badanie obszarów maksymalnych naprężeń w okrągłych płytach perforowanych ze stali S235JR

obciążonych siłą skupionąMateusz Konieczny*, Henryk Achtelik, Grzegorz Gasiak

Politechnika Opolska, Katedra Mechaniki i Podstaw Konstrukcji Maszyn, Opole; *mateuszmarcinkonieczny@wp.pl

Inżynieria Materiałowa 2 (228) (2019) 53÷60DOI 10.15199/28.2019.2.4

MATERIALS ENGINEERING

Słowa kluczowe: okrągła płyta perforowana, stal S235JR, siła skupiona, naprężenie redukowane, obliczenia numeryczne, badania eksperymentalne.

1. CEL PRACY

Współczesne konstrukcje inżynierskie charakteryzuje obniżenie współczynnika bezpieczeństwa, dlatego obliczenia wytrzymałości, sztywności i stateczności cienkościennych elementów konstrukcji nabierają coraz większego znaczenia we wszystkich gałęziach tech-niki, a szczególnie takich jak inżynieria chemiczna. Istnieje wiele zastosowań płyt perforowanych, np. w zbiornikach ciśnieniowych, aparaturze chemicznej (ściany sitowe wymienników ciepła), rafine-riach ropy naftowej, budowie samolotów i statków kosmicznych, jednostkach powietrznych, robotach, kontenerach transportowych, taśmociągach skrzynkowych lub jako elementy przesiewaczy mate-riałów sypkich, które mogą być poddane obciążeniu zmieniającemu się w czasie. Tego typu dźwigary (płyty perforowane) mogą rów-nież pełnić rolę płyt montażowych, w których wykonano otwory z różnych względów eksploatacyjnych.

Celem pracy jest lokalizacja stref, w których występują maksy-malne naprężenia w okrągłej osiowosymetrycznej płycie perforo-wanej swobodnie podpartej i obciążonej siłą skupioną Pi przyłożo-ną w środku geometrycznym płyty. Badania obszarów koncentracji naprężeń wykonano numerycznie metodą elementów skończonych.

2. MATERIAŁ I METODYKA BADAŃ

Do obliczeń przyjęto okrągłą osiowosymetryczną płytę perforo-waną o wymiarach: średnica D = 300 mm, grubość h = 5 mm. Na płycie rozmieszczono 200 otworów o różnych promieniach. Otwory te rozmieszczono na 10 okręgach po 20 otworów na każ-dym okręgu, jak pokazano na rysunku 1. Płytę wykonano z blachy stalowej S235JR o następujących właściwościach: moduł Younga E = 200 GPa, liczba Poissona v = 0,3 (tab. 1). Płytę perforowaną swobodnie podparto na całym obwodzie i obciążono siłą skupio-ną Pi przyłożoną w środku geometrycznym płyty (rys. 2). Przy-jęto następujące wartości obciążenia Pi: P1 = 410 N, P2 = 875 N, P3 = 1365 N, P4 = 1925 N, P5 = 2510 N. Na osiowosymetrycznej płycie perforowanej naniesiono osiem stref pomiarowych tj. od Z1 do Z8 (rys. 1). Obliczenia numeryczne realizowano metodą ele-mentów skończonych (MES) za pomocą programu Femap.

3. WYNIKI I ICH DYSKUSJA

Uzyskane wyniki obliczeń wskazują, że największa wartość na-prężenia redukowanego σred wystąpiła, gdy płyta perforowana była obciążona siłą skupioną Pi przyłożoną w środku geometrycznym

płyty o wartości P5 = 2510 N zarówno w przypadku badań eks-perymentalnych, jak i obliczeń numerycznych, które wynosiło σred

e = 160,03 MPa oraz σredn = 158,40 MPa (rys. 4, strefa 8). Róż-

nica względna w tym przypadku wyniosła 1,02% .Średnica otworu perforacji w płycie perforowanej ma istotny

wpływ na rozkład naprężeń w płycie, w której występuje ten rodzaj nieciągłości geometrycznej. Znaczne wartości naprężenia redu-kowanego σred są z reguły przyczyną tworzenia się mikropęknię-cia w strefie otworu perforacji. W przypadku obciążeń zmiennych (zmęczeniowych) mikropęknięcia zaczynają się rozwijać i w kon-sekwencji mogą doprowadzić do zniszczenia elementu maszyny lub elementu konstrukcji, w której występuje element z perforacją. W związku z tym bardzo istotne staje się wyznaczenie wartości i miejsca występowania koncentracji naprężenia w elementach ma-szyn zawierających przyczyny ich występowania.

4. PODSUMOWANIE

Na podstawie przeprowadzonych obliczeń numerycznych oraz eks-perymentalnych w płycie perforowanej można sformować następu-jące wnioski:1. Wykazano, że sposób obciążenia okrągłej osiowosymetrycznej

płyty perforowanej oraz średnica otworów perforacji ma istotny wpływ na rozkład naprężeń na całej powierzchni płyty, a zwłasz-cza w strefie otworów i na mostkach pomiędzy otworami.

2. Najbardziej niebezpieczne miejsce w okrągłej osiowosyme-trycznej płycie perforowanej, gdzie występuje największa koncentracja naprężenia istnieje w strefie 8 przy promieniu otworu d3 = 5,0 mm i promieniu okręgu, na którym znajdu-je się otwór R3 = 33,0 mm, naprężenie redukowane wynosi σred max = 217,20 MPa (punkt o współrzędnych x, y, z [mm], tj. P2[5,2; 30,6; 4,5] (rys. 4, strefa 8).

3. Maksymalne rozbieżności względne w wartościach naprężeń redukowanych σred uzyskanych metodą numeryczną z zastoso-waniem metody elementów skończonych i metodą eksperymen-talną, w okrągłej osiowosymetrycznej płycie perforowanej, przy swobodnym podparciu brzegów, nie przekraczają 10%.

4. Przebiegi naprężeń redukowanych ulegają obniżeniu mono-tonicznie, poczynając od centrum (środka) w kierunku brzegu (podparcia płyty).

5. Wykorzystanie metod numerycznych (MES) umożliwia uzyska-nie rozkładu naprężeń na całej powierzchni płyty perforowanej, a w szczególności w strefie otworów perforacji oraz pozwala na zlokalizowanie stref spiętrzenia naprężeń.