A Study of Contact Stresses in Pin Loaded Orthotropic Plates

8/13/2019 A Study of Contact Stresses in Pin Loaded Orthotropic Plates

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Compurers .Sfru


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1068 E. K. Y~GE~WARENnd J. N. REDDYlimiting frictional resistance that can be sustained bya stationary node K is given by

Xi 1

component . (1)at node K

However, if there is already slip occurring in the lastload increment, the frictional capacity is given by

dynamic

i )(frictional = dynamic coefficientcapacity K of friction pd >

normal forceX

i 1component . (2)at node K

Although the above concept was discussed in [l], itwas left to the current study to implement this intothe program. Thus as a first step, the direct pathdiscussed in this section was chosen to check theusefulness of this notion.

Sticking contact occurs if the frictional capacity asdetermined by eqn (1) or (2) exceeds the tangentialforce at node K. he conditions of contact from thelast load increment determines whether eqn (1) or eqn(2) is used. Any free target segment that comes incontact will have a sticking contact since the tan-gential force is zero along the free surface. This is alsotrue when a segment re-establishes contact after aseparation. However, when the friction coefficient isvery low the frictional capacity may be so low thatsliding may occur at initial contact itself.

A sliding condition is brought about when thenodal tangential force exceeds the frictional capacityof the segment given by the appropriate eqn (1) or (2).The node is constrained to move only in the tan-gential direction and the frictional resistance opposesthe relative motion of the bodies. The dynamicalfrictional resistance opposing the motion changescontinuously as a function of the relative magnitudeof the tangential and normal forces acting at a givennode. As a first-order approximation, the value of theresistance at the beginning of the load increment isassumed to be opposite to the direction of motion. Itis to be noted that the global nodal forces are theexternal forces acting on every element to balance theforces due to the stresses. Frictional forces are exter-nally applied at the contactor nodes in the directionof tangential force and are given by

(external frictional forces),

if and only if

The frictional force is determined at all equilibriumconfigurations and applied along the target surface.When the coefficient of friction is very small thefrictional capacity of all the segments under contactis identically zero at all times. The contactor nodesfollow the fixed target surface and since the targetsurface need not be parallel to the global axes, a localcoordinate system is defined with the constraint ofmovement along one axis.

Separation occurs when the reactive contact forcesact in the negative direction of the unit outwardnormal to the contact surface. However, if the forcesas evaluated at the end of an equilibriumconfiguration become positive, then there is no con-tact force between the contacting bodies, and thesegment containing the node has separated. Thisnode is considered to be free and once again is apotential contactor node and checked for contactoverlap in subsequent load increments.

The finite element equations resulting from themixed formulation can be written as (see [l])

where

K =I

+ r_% $ xdy. (6)

It can be seen from eqn (6), for the first iteration ofthe first load increment, that [K ] 0 becauserXI= r,,. = 7Y v 0,hus resulting in an indefinite sys-tem of equations. Although eqn (5) can be solved forthis condition, by first solving for stress and then fordisplacements, a more direct way has been given byMirza [3]. He suggested that by premultiplying bothsides of the equation by the transpose of the globalstiffness matrix, one can obtain a positive-definitesystem of equations. For example, if eqn (5) can bewritten as

whereWI VI = PI (7)


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Contact stresses in pin-loaded orthotropic plates 1069then a positive definite system is given by

W[~l vt = WI {PI* (8)and the solution of eqn (8) gives the results of eqn (7).

This technique has been extended in the presentstudy by reforming eqn (5) into the type given by eqn(6) whenever leading diagonal terms are small andthen carrying out the operations described above.

3 FAILURE ANALYSIS OF MECHANICALJOINTS IN COMPOSITES

Calculation of the strength and failure mode of acomposite laminate containing a pin-loaded holerequires the knowledge of the load distribution insidethe surface of the hole. Frequently, a cosine loaddistribution is assumed since such a load distributiongreatly simplifies the calculations. The stresses insidethe laminate calculated by a cosine load distributionmay differ significantly from those which actuallyarise in the structure. As a result, those failure criteriawhich require an accurate knowledge of the stressdistributions near the surface of the hole, will predictfailure inaccurately when used in conjunction withcosine load distribution. Work on composite boltedjoints has been extensive with early studies concen-trating on empirical design and gradually progressingto analytical methods for stress analysis [4-l l] andthe search for appropriate failure criteria [12-171.Empirical design methods proved too costly and timeconsuming since the large number of variables injoint design require huge data bases for each materialand class of laminates. Thus analytical techniqueshave been attractive, although this requires vigilanceon the part of the designer towards the pitfallsnormally encountered.

The three basic failure modes associated withbolted joints in composites are the bearing failure, theshearout failure and the net tensile failure (12-141and [15-201. It has been found that the shearout andthe net tensile failure can be adequately modelled by

Number of nodes = 286Number of el ements = 236E, = 10. 400 ksiE2 = 10, 400 ksiG12 = 3,910 ksi12 = 0. 33

(l i near)

a plane elastic stress analysis, with a point stressfailure hypothesis and a macroscopic failure modelsuch as the maximum stress or the maximum straincriterion. Recently, there has been a trend towardsstudies incorporating ply-by-ply failure analysis ofbolted joints, in order to assess the damage toindividual plies (see e.g. Reddy and Pandey [21]). Thisrequires some form of macroscopic criterion such asthe Tsai-Hill or Hoffmans criteria to predict failure.

An altogether different approach has been adoptedtowards failure by Hyer et al. [4,5], who have usedthe maximum radial and circumferential stresses asindicators of the capacity of a bolted joint in orderto study the effects of pin elasticity, pinfit-interference and friction on the capacity of a joint.They concluded that all three factors are detrimentalto the capacity of the bolted joint. The presentcomputational algorithm is used to study the pin-loaded plate problem considered in [4, 51.

4. APPLICATIONS4.1. A pin-l oaded alumi num plat e

Three different meshes were used to model thealuminum plate. All meshes took advantage of thesymmetry of the problem and modelled only half theplate. Further simplification was carried out on MeshA (Fig. l), where only a quarter of the pin wasmodelled as was discussed in the earlier study [I].However, both Mesh B and Mesh C did not adoptthis simplification and they modelled half the pin asshown. The contact nodal locations of Mesh A arereiterated here for completeness. These locations arethe following (degrees): 0.0, 1.0, 2.0, 4.0, 6.0, 8.0,10.0, 12.5, 15.0,20.0,25.0, 30.0, 35.0,40.0,45.0, 54.0,63.0,72.0,90.0,99.0, 108.0, 117.0, 126.0, 135.0, 144.0,153.0, 162.9, 171.0 and 180.0. In Mesh B theselocations were spaced at 9 intervals and in Mesh Cthese were at 2.5 intervals for the first 90 and at 9intervals for the second 90 as shown in Figs 2 and3 respectively. The number of elements and the nodes

Radi us of t he hol e = 0.375 i n.Radi us of the pi n = 0. 3745 i n.a = 1.495 i n. , b = 7.005 i n. ,c = 1. 5 i n. , pl ate thi ckness = O. i )6 n.

I b1Fig. I. Mesh A for the pin-loaded aluminum plate (pin is also made of aluminum).


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E. K. YCJ GESW RENnd J. N. REDDY

Number of nodes = 391Number of el ements = 332 ( l i near)

Fig. 2. Mesh B for the pin-loaded aluminum plate.

Mesh A: 236 elements, 286 nodesMesh B: 332 elements, 391 nodesMesh C: 452 elements, 540 nodes.

in the meshes are given as follows: was constrained in both directions. The load appliedwas distributed along the shorter edge away from thepin, in the lengthwise direction. A dynamic coefficientof friction pLd 0.25 and a static coefficient of frictionp, = 0.35 were used in the analysis, these numericalAll nodes along the line of symmetry were con- values having been estimated from the hybrid

strained to move only in the lengthwise direction and technique.the center of the pin (numbered 1) in all three meshes The load was applied in 14 steps closely following

Number of el ements = 452Number of nodes = 540

I _- Pi n -2(Fig. 3. Mesh C for the pin-loaded aluminum plate.


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Contact stresses in pin-loaded orthotropic plates

2 -10..CPDz2 -20m

z- Experi mental [ 2]

5 **. FEM (Mesh C)- 40 I 5. I I, 8. 1, I. I.

0 10 20 30 40 60 60 70 80 so

Radi al Angl e ( i n degrees)Fig. 4. Comparison of the experimental and finite element results (the FEM scheme is the modified version)for the load decreasing phase at load level 1210 b.

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the experimental loading values. In the load in-creasing phase the values of load were 20, 23, 520,1240, 1460, 1670 and 19801b respectively with thedecreasing phase values being 1800, 1600, 12 10, 1070,535,210 and 40 respectively, giving 14 load steps. Thenumber of iterations required for each load incrementwere 8, 8, 3, 3, 3, 1, 2, 2, 2, 3, 2, 3, 3 and 3 for thecase illustrated in Fig. 4, where Mesh C was used, andresults are plotted for load step 4, where the totalload is at a value of 12101b.The case shown in Fig. 5 was analyzed with themodified computational procedure discussed in Sec.2. The results show a closer agreement with theexperimental results of Joh [2], than the results shownin Fig. 6, which were obtained using the originalprocedure of[l]. It is interesting to note that aconstant value for the friction coefficient of 0.15 givesa closer agreement in Fig. 6 than the constant frictioncoefficient of 0.30, these values being chosen arbi-trarily. It is possible here to be misled easily to

conclude that a better choice of value of frictioncoefficient might be 0.15, until the full picture isrevealed by Fig. 5, which indicates that betterresults are given by a dynamic/static friction model.Indeed it is possible to conclude here that yet bettermodelling can be achieved by a continuous frictioncoefficient variation such as that given by a power lawalthough the implementation of this may be morecumbersome. It can also be seen that the angle ofseparation is more realistic in Fig. 5 than in Fig. 6.The load decreasing phase results shown in Fig. 4compare favorably with the experimental results,despite some 66% difference in the values of trOfrom40 to 75. This can be considered fair compared tothe results presented in [I].4.2. A pin-loaded orthotropic plat e

An analytical solution to the problem of pin-loaded composite laminate has been obtained byHyer et al . [4, 51 based on a complex Fourier series

- - - FEM (Mesh 8)

-I0 10 20 30 40 10 60 70 80 90Radi al Angl e ( i n degrees)

Fig. 5. Comparison of the experimental and finite element results (the FEM scheme used is the modifiedversion of the scheme developed in [I]; load increasing phase at load level 1240 lb).


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1072 E. K. YOGESWAREN and J. N. REDDY

- Experi mental [Z]

Radi al Angl e ( i n Degrees)Fig. 6. Comparison of the experimental and finite element results for load-increasing phase at load level1240lb. (the FEM scheme used is that originally developed in [I]).

method and a collocation technique which enforcedboundary conditions at discrete locations around thehole boundary. Results were obtained by this ana-lytical method for infinite orthotropic plates loadedby pins. These results were chosen to be comparedwith the finite element results since both methodscapture the idealized conditions of the model to thesame degree.

The mesh shown in Fig. 7 was selected to idealizethe infinite plate and the pin, with 426 elements and509 nodes. The nodes along the line of symmetry wereconstrained to move only in the lengthwise direction

in the same way as the nodes along the longerboundary whereas the nodes along both shorterboundaries were constrained to move along the y-direction. Normalized circumferential, radial andshear stresses along the hole boundary were plottedagainst the radial angle and found to compare wellwith the analytical results (Figs 8 and 9).4.3. Appl icati on of t he hybri d t echnique to estimat estati c, and dynami c coefi cients of ri ction

This technique basically consists of applying theloading, prescribing the displacements from Moire

E, = 12, 400 ksi Radi us of pl ate hol e = 1 i n.Pl ate Radi us of pi n = 1 i n.properti es: E2 = 3, 730 ksi a = 4 i n. , b = 2 i n. , thi ckness of pl ate = 1 i n.

52 = 3.210 ksi Pin i s made of steel

12 = 0. 66 Number of nodes = 509Number of el ements = 426

Fig. 7. Mesh D for the pin-loaded orthotropic plate.


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Contact stresses in pin-loaded orthotropic plates 1073- Analytical lYer, et al. [4,51)___ FEM (normalized circumferential stress). . . FEM (normalized radial stress)Flesh D i s used.

40 10 20 30 40 60 20 70 20 20 100Radi al Angl e ( i n degrees)

Fig. 8. Comparison of the analytical and finite element results for the pin-loaded orthotropic plate. Thestress is normalized with respect to the average bearing stress (81.9 ksi ).0.4

0.3

0.2

0.1

0.0

-0.1

-0.2

-0.3

-0.4

. :.,~...

- Anal yti cal sol uti on [4,5]. FEM (Mesh D)

0 10 20 30 40 20 20 70 20 90Radi al Angl e ( i n degrees)

Fig. 9. Comparison of the analytical and finite element solutions for the pin-loaded orthotropic plate(stress is normalized with respect to the average bearing stress, 81.9ksi).

analysis to the hole boundary and prescribing otherboundary conditions as before. Thus only the plate isdiscretized for this analysis, without the pin, and themesh used is shown in Fig. 10. The stresses along theboundary are obtained from the analysis and areshown in Figs 11 and 12 plotted against the radialangle. Thus the shear stress r, and the radial stresscd and the radial stress 7, , show remarkable resem-

blance to the stresses given by Joh [2] at a load levelof 1840 lb.In order to assess the friction coefficient values, tobe used for the regular finite element analysis, theT&,, ratio was plotted against the radial angle forload increasing and decreasing phases and the resultis given in Fig. 13. In the load increasing phase nearlyall contact is associated with slip and the maximum

Number of nodes = 307Number of el ements = 254

Fig. 10. Mesh E used for hybrid (experimental/numerical) study of the pin-loaded plate problem.


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1074 E. K. YCXESWAREN and J. N. REDDY

Experi mental [ 2]- - - - Hybri d (Fl eshE)

-I0 16 30 42 60 72 a0Radial h[lle (in degrees)

Fig. 1 . Comparison of the shear stress distributions obtained in the experiment and hybrid analysis ofthe pin-loaded aluminum plate (load increasing phase at load level 1840 lb).

ratio of ~, /r,, cannot exceed the dynamic frictioncoefficient and thus pLd 0.25 is a rational choice. Thenegative ratio is maximum between 55 and 60, wherethe transition from slip to stick occurs and thusp8 = 0.35 is chosen to be a static friction coefficient.4.4. Anal ysis of ai lur e i n mechanical joi nts

A composite laminate (V/+45/90), with laminaeof the following properties has been used for thesestudies:

E, = 19.1 x lo3 ksiE2 = 2.0 x lo- ksi

G,, = 0.9 x lo3 ksi

V,J= 0.3X, = 229.4 ksiY,= 10.1 ksiX, = 252.1 ksiY, = 32.0 ksiT = 17.3 ksi.

It has been established that certain configurationsfavour certain modes of failure [15]. This fact hasbeen used in determining the plate configuration forstudying bearing failure (see Mesh F in Fig. 14),

- Experi mental [2]- - - - Hybri d (Mesh E)

12 30 42 20 72 20Radi al Angl e ( i n degrees)

Fig. 12. Comparison of the radial stress distribution obtained in the experiment and hybrid analysis ofthe pin-loaded aluminum plate (load = 1840 lb).


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Contact stresses in pin-loaded orthotropic plates0.41

.EuI 0.3.z2 0.2.Yul

2 0.1..Cc2 0.0.

i 1i -0.1., ' -0.2.

Load i ncreasi ng phase (1040 l bs)

Load decreasi ng phase (1070 l bs)

0 10 20 30 40 60 60 70 60 90Radi al Angl e ( i n degrees)

Fig. 13. Variation of the ratio of shear stress-to-radial stress for load increasing and load decreasing phasesof the pin-loaded aluminum plate (results obtained using the hybrid technique).

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shearout failure (see Mesh G in Fig. 15) and tensilefailure (see Mesh H in Fig. 16).

The analysis has been carried out and normalizedradial and circumferential strain curves have beenproduced for each failure mode. Failure is indicatedby the increased strains given by a nonlinear modelincorporating Tsai-Hill criterion as compared to alinear elastic model. Results are shown in Fig. 17 forbearing failure, Fig. 18 for shearout failure and Fig.19 for tensile failure. The Tsai-Hill criterion used inthis study is

where X, Y are either compressive (XC, Y,) or tensile(Xr, Yr) strengths and T is the shear strength in thexy-plane.

5. SUMMARY AND CONCLUSIONSThe mixed finite element model developed in [I]has been modified by incorporating a realistic inter-

Number of nodes = 723; sl umber of el ements = 625Fig. 14. Mesh F used for the bearing failure in a mechanical joint.

Number of nodes = 619; Number of el ements = 529Fi g. 15. Finite element mesh G used for shearout and bearing failure in composite joints.

I I I I INumber of nodes = 420; INumber of el ements = 376Fig. 16. Finite element mesh H used in tensile failure of composite joints.


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1076 E. K. YOGESWAREN and J. N. REDDY20

19r -

Nonl i near model*** Li near model10 *

z Ci rcumw enti al. r24-lYI 0.F. ,?

-8.8 - 10.

-2040 10 20 30 40 90 00 70 90 90

Radi al Angl e ( i n degrees)Fig. 17. Radial and circumferential strains in an orthotropic plate (the linear and nonlinear models showseparation indicating bearing failure from 0 = O-35).

20 - Nonl i near model19. . . . . Li near model10.

-207 I I I I , 1 10 10 20 30 40 90 90 70 90 99

Radi al Angl e ( i n degrees)Fig. 18. Radial and circumferential strains in an orthotropic plate (bearing failure: 0 = &35; shearout

failure: 0 = 6CrSOq.

Radi al Angl e ( i n degrees)Fig. 19. Radial and circumferential strains in an orthotropic plate (tensile failure: H = 75-90).


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Contact stresses in pin-loaded orthotropic plates 1077face friction condition and a solution procedure. Adynamic as well as a static friction coefficient are usedto analyze a pin-loaded plate problem for whichexperimental results are available. The new solution

*,algorithm not only provides flexibility in numberingthe nodes but also avoids the halt of computation dueto the appearance of small terms on the leadingdiagonal, during the analysis. An accurate contact

9,stress analysis is essential in some applications suchas the study of failure in bolted joints of laminatedcomposites and some example problems have been 10.studied in this area. 11.Acknowledgements-The research reported here is sup-ported by the Mechanics Division of the Officer of NavalResearch through Contract No. NOOO14-84-K-0552. The 12.encouragement and support of the research by Dr AlanKushner is gratefully acknowledged. Thanks are also dueto Mrs Vanessa McCoy for the skillful typing of this 13.manuscript.

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pin joints in composite plates. Aeronautics Researchand Development Board, Govt. of India, New Delhi,Rept. ARDB-STR-5014 (1980).D. W. Gplinger and K. R. Gandhi, Analytical studiesof structural performance in mechanically fastenedcomposite plates. Army Materials and Mechanics Re-search Center, Watertown, MA, Report M574-8, pp.221-240 (1974).D. W. Oplinger, On the structural behavior of mechan-ically fastened joints in composites materials. Pro-ceedings of th Conference of Fi ber Composites in S~ruc-rural Design, pp. 575-602. California (1978).S. G. Lekhnitskii. Anisotrooic Plates (Translated fromRussian). Gordon & Breach, New York (1968).N. I. Mushkhelishvili, Some Basic Problems of theMathematical Theory of Elasticity (Translated -fromRussian bv J. R. M. Radok). Noordhoff. Gronineen.Netherlands (1963). _F. K. Chang, R. A. Scott and G. S. Springer, Strengthof mechanically fastened composite joints. UniversityofMichiaan, AFWAL-TR-82-4095 (1982).T. A.-Callings, On the bearing strengths of CFRPlaminates. J. Camp. Muter. 13, 241-252 (1982).S. P. Garbo and M. Ognowski, Effect of variances andmanufacturing tolerances on the design strength and lifeof mechanically fastened composite joints. McDonnellDouglas, AFFDL-TR-81-3041, St Louis, MO (1981).J. L. York. D. W. Wilson and R. B. Pines. Analvsis ofthe net tension failure mode in compos;te bolted joints.J. Reinforced Plast. Camp. 1, (1982).J. M. Whitney and R. J. Nuismer, Stress fracturecriteria for laminated composites containing stress con-centrations. J. Camp. Mater. 8, 253-265 (1974).D. W. Wilson and R. B. Pipes, Analysis of the shearoutfailure mode in composite bolted joints. Proceedings ofthe I nr ernati onal Conference on Composite Structur es(Edited by I. H. Marshall). Applied Science Publishers,Barking, U.K. (1981).T. H. Tsiang, Damage development in fiber compositesdue to bearing. Sc.D. Thesis, Department of MaterialsScience and Engineering, MIT Cambridge, MA (1983).T. H. Tsiang and J. F. Mandell, Bearing/contact forantisotropic materials. AZAA Jn f 23, 1273-1277 1985).T. H. Tsiang and J. F. Mandell, Damage developmentin bolt bearing of composite laminate. AIAA Jnl 23,1570-1577 (1985).J. N. Reddy and A. K. Pandey, A first-ply failureanalysis of composite laminates. Compuf. Struct. 25,371-393 1987).

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A Study of Contact Stresses in Pin Loaded Orthotropic Plates