1 Photoelastic determination of stress con- centration factors · tension bending torsion a KK>...

1 Photoelastic determination of stress concentration factors

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1 Photoelastic determination of stress con-centration factors

1.1 Introduction More cost-effective, safer, more durable, lighter, ever faster and better. These are the requirements for development and structural engineers. But how can these requirements be met? In practice, analyses of mechanical loading states have currently become the ba-sis for predicting component behaviour by using a high standard. This state of affairs, together with knowledge of the load-related material’s / component's be-haviour, guarantees structural designs having suitable strengths. In the same way, optimising components and thereby improving the use of materials is pro-vided by means of savings on unnecessary weight. Thus, lighter components or machines can be manufactured. Lower consumption of materials saves re-sources, the manufacturer's material costs, the utilisation costs - more is saved, the higher the machine's degree of movement - as well as protecting the envi-ronment. Stress analyses, which can no longer be simply performed on components pos-sessing complex geometries, can be undertaken using various computer or ex-perimental methods. These methods correspondingly exhibit both advantages and disadvantages. Due to the computer's rapid development, which is expressed by the computing power, the capability of graphic's software and, last but not least, the reduction of hardware costs, the application of Finite Element Methods (FEM) is available to a broad domain of users. For instance, designing variants or studying parameters, FEM applications provide relatively quick solutions. Time consuming and expensive tests can be reduced. However, experimental investigations for observing real processes are mostly indispensible, particularly for new developments, for assessing and validating FEM models. Sufficiently accurate computations of specific stress states are fre-quently only determined with considerable effort or are not even possible. For example, this concerns very complex stress states, complex load applications, assembly stresses, residual stresses, thermal stresses and many others. In these cases, it is an advantage if experimental and computational analyses complement each other. Photoelastic methods have the virtue of allowing the stress distribution as well as the stresses' direction to be directly observed. They are therefore employed for quantitatively measuring and demonstrating complex stress states. The accu-racy is ±10%.


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Planar (PPE) just as spatial photoelastic (SPE) investigations are performed on models whose geometry correspond, or are similar, to the component. In contrast to this, the surface layer method (SLM) or PhotoStress-Method is carried out using the component itself. In industrial practice, the SLM and FEM analyses are frequently employed together. The planar photoelastic method always illustrates the complete stress field. This is a decisive advantage - compared, for instance, with strain gauges - (SG) - which makes the PPE eminently suitable for optimising components. SG's pro-vide only local measurements but with a significantly higher accuracy than ±10%. Photoelastic component optimisation should establish stress peaks and aim to reduce weights (lightweight structures). In this way, the component's de-sign strength will be increased. The procedure for employing PPE in, for in-stance, lightweight structures is briefly described in the following. One starts with the low stressed regions in the photoelastic model. These re-gions are removed by rigorously cutting them away. Following their removal, the stress state of the modified model is reconsidered using the PPE and reana-lysed. In the process of implementing this method, the stresses in the reduced residual cross-section stepwise increase such that the stress distributions become increasingly uniform. The optimisation is terminated when the stresses in the "remaining model" attain a predefined limiting value. The optimised geometry is then transferred to the component. Photoelastic methods are employed using static and vibrational loading. A few examples follow which are mainly taken from the field of automobile construc-tion: - PPE using a model • Static: Band brake component, automatic gearbox, threads, gear wheel meshing, „crackling“, chain connector, welded seams • vibrational: Valves - SPE using a model • Static: Cylinder head, crankshaft housing, pedestal of a mill - SLM using a component • Static: Motorblock, wheel rim, pipe clamp, spot-welded joints • vibrational: Noise absorber-operating vibrations and strength analysis, rear axle


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1.2 Notch effect - elastic loading Dimensional changes and deviations in cross-sections cause notch effects in me-chanically loaded components. This leads to local inhomogeneities and multiax-ial stress states with stress peaks in the notch root. These stresses can no longer be simply computed using the theory of elasticity. One speaks of stress concen-trations. For example, holes, openings, keyways, shoulders, bifurcations, threads and gear teeth. 1.2.1 Notched tensile bar Fig. 1.1 depicts the stress distributions for a smooth and a notched flat bar sub-ject to the same tensile force F and the same cross-sectional area A = AK. Fig. 1.1: Stress distribution across the cross-section of a smooth and a notched flat bar (schematic) The double edge notched flat bar has notch radii r, the notched ligament dK, notch depth t and the notch flank angle w. The bar's thickness is b for both spec-imens (smooth and notched). The base cross-section A0 = (dK+2t) b of the

r

A

s

sn

x

F

F

d

AK

sn smax

x

s

t

F

F

dK

A0

D0

s1(x)

w


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notched bar is reduced to the notched ligament area (nominal cross-section) AK = dK b. As a consequence of the tensile force F, a homogeneous longitudinal stress

distribution, , only exists in the base cross-section A0, and then only at

a sufficient distance from the notch. The longitudinal stress distribution in the ligament's cross-section AK is inhomogeneous. The maximum stress which occurs at the notch root is extremely significant. 1.2.2 Stress concentration factor To assess the maximum stress smax at the notch root, it is compared with a theo-retical nominal stress

, (1.1)

where sn would exist in a smooth bar with the same cross-section AK = A and for the same tensile force F (cf. Fig. 1.1). One defines the stress concentration factor as

(1.2)

The factor is influenced by various parameters. The stress concentration factor 1) depends on the components geometry. In particular, aK strongly increases

with decreasing notch radius r ("sharp" notches). 2) depends on the type of loading. For the same notch geometry, the following

is valid (1.3)

3) is independent of the component size if geometric similarities exist. 4) is independent of the absolute magnitude of the nominal stress if linear-

elastic material behaviour exists. 5) is independent of the material for isotropic, linear-elastic material behaviour. The stress concentration factor is determined analytically by elasticity theory (stress functions, not always possible), numerically (FEM) or experimentally (e.g. by photoelasticity).

l,00

FA

s =

maxs

K

K

d2

n 1 KK K d

2

F 1 (x)dAA A

-

s = = sò

maxK

n1s

a = ³s

tension bending torsionK KKa > a > a


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1.2.3 Multiaxial stress states Constraints on the transverse strain exert another important influence on notch effects. This constraint generates transverse stresses in addition to the longitudi-nal stresses. These multiaxial stress states are, in turn, strongly dependent (cf. stress concentration factor) on the component geometry. As an example, Fig. 1.2 compares the stress states in tensile loaded flat and round notched bars. Here, the same form of notch (e.g. notch radius r) and equal notched ligament areas ( ) are presumed. The stress state at the most highly loaded region, at the notch root, remains uniaxial (s1, flat bar) or biaxial (s1 and s2, round bar) because no stress can exist perpendicular to the load-free surface of the notch root. Within the bar, the uniaxial tensile loading produces a biaxial (s1 and s2, flat bar) or triaxial (s1,s2 and s3, round bar) notch stress state. For the above prerequisites, the following notch shape factors are valid (1.4) The reason this is the flat bars relatively small and relatively highly loaded sur-face region in comparison to the round bars relatively large and not so highly loaded surface region. Figure 1.2: Tensile loaded flat and round bars possessing multiaxial stress states

(schematic)

flat bar round barK KA A=

flat bar round barK Ka > a

F

s1(r)

s2(r) s3(r)

round barKA

F

r x

flat barKA

s1(x)

F

F

s2(x)

s

=flat bar round barK KA A


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F F

F F

Notch 1

Notch 2

1.2.4 Influence of notch stresses The stress elevation at the notch root can either be lowered by designing the component with a suitable line of force transmission, or intensified by an unfa-vourable design.

Multi notch effects exist when several notches are arranged close together. For instance, intersecting notches (cf. Fig. 1.3) intensify the notch effect. The resulting stress concentration factor can then be estimated from the product of each individual stress concentration factor.

(1.5) Figure 1.3: Intersecting notches for tensile or bending loads

Relief notches reduce the maximum force transmission concentration. Practical-ly, this is realised by means of multistep recessing or by more uniform cross-sectional transitions and by designing the notch region as remaining elastic (cf. Fig. 1.4).

Figure 1.4: Example for transmit-ting forces or mo-ments in a hub seating (Source: Dubbel)

a) Sharp edged (very unfavourable) b) Rounded with reduced edge pressure c) Gentle run-out of the hub d) Relief notch

K K,1 K,2a = a × a

a) b) c) d)


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By observing biomechanical structures in nature (trees, tiger's claws, chicken bones etc.), C. Mattheck developed simulation methods (among others, Comput-er Aided Optimisation, CAO) to specifically improve components. He remarked "that there is just one single and very general valid design rule in nature, which is the principle of constant stress." W. Nachtigall wrote, "bionically exploiting biomechanical self-optimisation in nature only then begins on applying it to op-timise the shapes of technical components". Thus "a machine's component which is dimensioned by means of 'growth' into a shape possessing a constant stress distribution" has "neither a predetermined point of fracture (locally elevat-ed stresses) nor wastes material (regions which are not loaded to capacity). It is, in a true sense, a biological design - ultra-light and high strength" /Mattheck/. The transmission of force is uniformly distributed and the notch stresses are eliminated. Fig. 1.5 shows an orthopaedic screw whose non-optimised shape frequently leads to fracture at the first thread /Mattheck/. Following its optimisation using the CAO-method, it was possible to fully eliminate the notch stresses /D. Erb/. Subsequent to this optimisation, no crack formation at all occurred even after load reversals with loads 20 times higher.

Figure 1.5: Orthopaedic screw, not optimised and optimised (Source: Mattheck)


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1.3 Test description The task of the laboratory test is to analyse notched model bars which are sub-ject to tensile loading and discuss the notch effect by means of the PPE. The models are cut from a 9 mm thick PC (polycarbonat) plate. 1.3.1 Testing equipment Fig. 1.6 depicts the set-up for the large surface stress testing machine, made by the company Schneider, Typ GS 2C/450 (year 1989), which is made available for the photoelastic tests. Using the equipment, planar and spatial models can be investigated.

Figure 1.6: Large surface stress testing machine and the investigated model schematic (Source: Fa. Schneider) White or monchromatic light leaves the light source and is homogenised by means of a diffuser. Light consists of eletromagnetic waves of different (white light) or constant (monochromatic) frequency having periodically vibrating elec-tric and magnetic fields. The vectors of the electric and magnetic fields mutually span a right-angle. The fields vibrate perpendicular to the wave's direction of propagation (transverse vibration). After the diffuser, the light enters a polariser with all its frequencies and vibrat-ing directions (cf. Fig. 1.6 and 1.7).

Light source

Diffuser Protec-tive discs

Model

l/4-Plates Polariser’s sheet


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Polarisers are optically anisotropic due to their construction and have a polaris-ing axis. They orientate the light since they only permit those field lines to pass through whose vectors vibrate parallel to the polarising axis. For linearly polarised light, two sucessively arranged polarisers are employed whose polarisation axes are rotated at 90° to each other (cf. Fig. 1.7). By means of this, an orientation exists for the first polariser perpendicular to the second polariser's (the analyser) polarisation axis for all the electric field lines of the electromagnetic vibrations. The analyser then inhibits the transmission of light. Besides this, the photoelastic equipment (cf. Fig. 1.6) can be operated with cir-cular polarised light.

Figure 1.7: Generation of linearly polarised light

1.3.2 Fundamentals for analysing stresses on plane models (PPE)

Photoelastically analysing a planar model, which is arranged between the two polarisers (cf. Figs. 1.6 and 1.8), is most easily performed using linearly polar-ised light. In its stress-free state, the model behaves optically isotropic and in its stressed state optically anisotropic. The device for loading the model is not de-picted. The loaded, optically anisotropic model resolves the light waves, which are ori-entated according to the polariser, into components so that their electric fields are aligned parallel to the principle stresses s1 and s2 (cf. Fig. 1.8). Since planar models are involved, for which the thickness b is smaller than the model's sizes

l

E1

E2

E3

Light

Polariser

Analyser

0°

90° polarised waves vibrate in only one plane.


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in the other two dimensions, one can, as a first approximation, neglect the third principle stress . (1.6) According to the magnitudes of the principle stresses s1 and s2, one component of the light wave is faster than the other so that a path difference (1.7) occurs beyond the model. Where, v1 and v2 are the light speeds of the two light waves in the model. The time for the light to pass through the model of thick-ness b can be calculated with sufficient accuracy using the following equation:

, (1.8)

where v0 is the speed of light in the stress-free model.

Using the Maxwell-Wertheim equations (1.9) (1.10) and equation (1.8) inserted into Eq. (1.7), one obtains

(1.11)

3 0s »

1 2 bg (v v )t= -

b0

btv

»

1 0 1 1 2 2v v k k= + s - s

2 0 1 2 2 1v v k k= + s - s

1 2 1 20

bg (k k )( )v

= + s - s

Light

Polariser

b

s1 s2 g

1 2

2 1

2 1 g

Analyser

Model

Figure 1.8: Planar photoelasticity (PPE) in a model using linearly polarised light

90°

0°


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By combining the material constants k1, k2 and v0 as

(1.12)

the light wave's path difference is then given by (1.13) On propagating further, the light waves (1 and 2) which are specifically influ-enced by the model's stress state, reach the analyser. The analyser's polarising axis is rotated through 90° to the polariser. From the two light wave's compo-nents (1 & 2), in each case, only those portions which are parallel to analyser's polarisation axis pass through the analyser. Here, the path difference g is pre-served. The observer, who is located beyond the analyser, perceives that mono-chromatic light is emitted from the light source and, according to the model's stress state, dark lines occur with alternating bright regions. Fig. 1.9 shows a PPE image of meshing teeth for a gearwheel. The dark lines are the so-called isochromatics; they each represent a specific mechanical stress state. In the tooth's contact region, one discerns a particularly high line density which indi-cates here a strongly inhomogeneous stress state with a stress peak.

The light wave's interference (1 & 2, cf. Fig. 1.8) leads to the light being extin-guished by the analyser. That is the dark lines, the isochromatics, emerge when the path difference between the two waves is a multiple integer (n) of the wave lengths l (monochromatic light, cf. 1.8) (1.14) For the isochromatics of the n th order, one obtains the following relationship by equating Eq. (1.13) and (1.14) (1.15) and, after transposing the relation,

1 2

0

k kKv+

=

1 2g K( )b= s - s

g n= l

1 2K( )b ns - s = l

Figure 1.9: Illustration of the stress state by means of PPE during tooth meshing


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. (1.16)

By introducing the photoelastic constant

, (1.17)

one finally obtains the general expression for transmitted light photoelasticity:

(1.18)

1.3.3 Determining the photoelastic constant The constant have to be determined prior to every photoelastic analysis. This is carried out using simple models for which the stresses can be calculated via giv-en equations. Using these known stress values and the isochromatic's n th orders measured on the model, the required constant can be established according to Eq. (1.18). Here, it is important that the model, used for determining the constant s, is taken from the same plate material as that used for all the other models which serve to investigate or, as the case may be, to optimise the component ge-ometries. In the laboratory test, a smooth, simple 4 point bending bar (4-PB-bar) is em-ployed to measure the photoelastic constant. As Fig. 1.10 shows, the bar is load-ed to a constant and specified bending moment (1.19) between the applied forces F. Using the section modulus

(1.20)

the bending stress s1 in the edge fibres can be calculated

(1.21)

1 2n

K bl

s - s =

sKl

=

1 2nsb

s - s =

bM Fa=

2K

bbdW6

=

b1 b 2

b K

M Fa 6W bd

s = s = =


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Figure 1.10: 4-point bend loading of a 4-PB-bar The second principle stress s2, which is oriented parallel to the direction of the force, drops out at the edge's free surface. (1.22) Inserting Eq. (1.21) into the general expression (1.18) of the PPE gives the fol-lowing relationship

, (1.23)

from which one obtains the photoelastic constant

. (1.24)

The parameters F, a, b, dK and the order n must be measured. 1.3.4 Determining the stress concentration factor In order to be able to determine the stress concentration factor

(1.2)

of a notch bar or component, planar photoelastic models, which possess the cor-responding geometry, are manufactured and investigated using the photoelastic equipment. On applying a tensile load using the force F, one calculates the nom-inal stress for the ligament's cross-section

(1.1)

By employing equation (1.18), the maximum stress in the notch root is given by using the isochromatic's order n

02 =s

1 b 2K

6Fa nsbbd

s = s = =

2K

6Fasnd

=

maxK

n

sa =

s

nK

FA

s =

a F

a F

b

dK


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. (1.25)

From the experimental results, and Eq. (1.1) and (1.25), one finally obtains in Eq. (1.2) the required stress concentration factor.

1.4 Assignment Firstly, the photoelastic equipment must be set-up by employing monochromatic light. Linearly and circular polarised light are to be set on the equipment and the observed differences are to be elaborated. One 4-PB-bar and several notched bars are cut from a polycarbonate plate. First-ly, the photoelastic constant is determined using the 4-PB-bar. By means of the notched bar model, the stress concentration factor aK is to be subsequently determined and the notch effect is to be discussed. The transmission of the force in all the models is observed and assessed.

1 2 1 maxn0 sb

s - s = s - = s =


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1.5 Results Table 1.1: Stress concentration factor aK in notched bars

Notched bar 1 Notched bar 2 Notched bar 3 Thickness b in mm Width D0 of the bar in mm

Notch separation dK in mm

40 40 40

Notch radius r in mm

12 8 4

Ligament cross-section AK in mm2

Notch flank angle w in °

- - -

Nominal stress sn in MPa

Maximum stress in MPa

stress concentration factor aK,PPE

stress concentration factor aK,approximation


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1.6 Protocol Group: Date: Recorded by: Testing machine:

1.6.1 Determining the photoelastic constant s n Scale units F in N Mb in Nmm Wb in mm3 in

MPa s in N/

(mm order) 1 2 3 4

bs

0 1 2 3 4 5 n

5

4

3

2

1


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1.6.2 Photoelastic investigation of notched bars

Notched bar 1: n Scale units F in N AK in

mm2 in MPa in

MPa

1 2 3 4

Photoelastically determined stress concentration factor

stress concentration factor


mm2 in

MPa in

MPa

1 2 3 4




mm2 in

MPa in

MPa

1 2 3 4



ns maxs Ka Ka

K,PPEa

K,approximationa

ns maxs Ka Ka

K,PPEa

K,approximationa

ns maxs Ka Ka

K,PPEa

K,approximationa


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1.6.3 Approximation of the stress concentration factor

(According to: W. Beitz und K-H. Küttner: Dubbel Taschenbuch für den Ma-schinenbau, 19. Auflage, Springer Verlag, Berlin, Heidelberg, New York, 1997)

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