Evaluation of Concentrated Oblique Load at the Apex of a Wedge by the Method of Caustics

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1 90 The Open Mechanical Engineeing Jounal, 0, 6, Open Access Evaluation of Concentated Oblique Load at the Apex of a Wedge by the Method of Caustics G.A. Papadopoulos * National Technical Univesity of Athens, Depatment of Mechanics,5, Heoes of Polytechnion Av., GR-57 73, Zogafou, Athens, Geece Abstact-The expeimental method of caustics was used fo the evaluation of the stess-singulaities ceated by concentated oblique load applied at the apex of a wedge o on the hoizontal staight bounday of a half-plane. The wedge was consideed to be elastic, isotopic and unde genealized plane stess conditions. Accoding to the method of caustics, the light ays impinging nomally at the thin plate ae patly eflected fom eithe the font o the ea faces of the plate. The eflected ays ae deviated because of the impotant constaint of the plate at the vicinity of the applied concentated load and the significant vaiation of the efactive index thee. The deviated light ays, when pojected on a efeence sceen, ae concentated along a singula cuve which is stongly illuminated and foms a caustic. It was poved that the shape and size of the caustic depend on the stess singulaity at the point of application of the load. The paametic equations of caustics ceated by such a singulaity wee studied in elation with the loading mode of the wedge. Thus, by measuing the dimensions of the caustic, one can evaluate the state of stess at the singulaity. The expeimental evidence, based on the above method on Plexiglas (PMMA) wedges, cooboated existing elastic theoies concening the ode of singulaities fo concentated loads. The elations fo the calculation of the load, the components of stess and the concentation facto based on the geometical chaacteistic of caustics, which wee eflected fom the font face of the wedge, wee given. Keywods: Caustics, wedges, concentated load, concentation factos, contact poblem. I. INTRODUCTION The optical method of caustics is suitable fo the expeimental study of singulaities in stess fields ceated eithe by geometic discontinuities o by loading. Fo many cack poblems, the stess intensity factos at the cack-tip wee calculated by caustics [-3]. Contact and wedge poblems have many pactical and impotant applications [4]. The stess distibution poblem in a semi infinite plate unde genealized plane stess conditions, which is loaded by a concentated load applying at a point of the staight bounday was solved by Flamant [5], while Boussinesq [6] solved the same poblem fo the case of any inclined load. Fo the solution of such poblems, besides mechanical analysis, the expeimental method of caustics can also be applied. Caustics has been applied to the study of the singula stess fields developed nea concentated o unifomly distibuting loads which ae applied along staight boundaies [7,8], the load shaing in olle-beaings [9, 0], in multiple gea tooth contact [,] and in study of vetical concentated load in wedges [3]. Also, thee-dimensional wedge poblems, bimateial wedges in antiplane shea defomation and ubbe wedges in lage stain analysis wee studied [4-6]. The optical method of caustics has been applied to study of the singula stess fields nea holes [7, 8] and in the contact poblems of anisotopic mateialas [9]. *Addess coespondence to these authos at the National Technical Univesity of Athens, Depatment of Mechanics,5, Heoes of Polytechnion Av., GR-57 73, Zogafou, Athens, Geece; Tel: ; Fax: ; gpad@cental.ntua.g The aim of the pesent wok is to give the paametic equations of the caustics in wedges made of isotopic mateials fo the case of a concentated oblique load at the apex of the wedge. Also, the elations of calculation of the concentated oblique load at the apex of wedge, the components of stess at any point and the concentation factos of the stesses based on the diamete and the opening displacement of the caustic wee given. II. CONCENTRATED OBLIQUE LOAD AT THE APEX OF THE WEDGE Concentated oblique load P pe unit thickness of the wedge (compessive load) was applied on the diection which is limited by the angle with the axis of the wedge (Fig. ). The thickness of the wedge was taken as unity. Resolving the oblique foce P into two components, P cos vetically and P sin hoizontally (Fig. ), the conditions along the faces, =±, of the wedge ae satisfied by the stess components which ae deived fom the stess function [4]: = P Asin P B( +)sin( +) () with: cos sin A = +, B = sin () sin whee is the angle of wedge and, ae the pola coodinates X / 0 Bentham Open

2 Evaluation of Concentated Oblique Load at the Apex of a Wedge The Open Mechanical Engineeing Jounal, 0, Volume 6 9 and fo =/ (hoizontal foce at O, Fig. ) the stess function and the adial stess ae: = P B ( +)sin( +) (8) sin = k with: B = and k = PB = sin P sin (9) Fig. (). Geomety of a symmetical wedge of angle. The stess components ae deived fom equation () as: = + cos sin = k + k = The constants: = PA cos + PB sin = 0, = = 0 P cos k = PA = + (4) sin k = PB P sin (5) sin ae the stess concentation factos. By taking = / and =0 the solution fo a semiinfinite plate was given which has been studied by the method of caustics and photoelasticity method [9, 0]. Fo =0 (vetical foce at O, Fig. ) the stess function and the adial stess ae [3]: = P Asin (6) cos = k with: A = + and k = PA = sin P + sin (3) (7) III. EXPERIMENTAL METHOD OF CAUSTICS The light ays eflected fom the ea o font face of the specimen o tansmitted though the specimen fom wave fonts [-3]: S(x, y, z) = const. (0) If s(x,y) expesses the optical path of the light ay between two planes paallel to the middle plane of the loaded plate and lying at the faces of the plate, then the elation: S(x, y, z) = z s(x, y) = const. () is valid and: gad S(x, y, z) = k s s i x y j () whee i, j, k ae the unit vectos of the (Ox,y,z) Catesian coodinates system. Fo optically isotopic mateials and fo the fist light ay eflected fom the ea face () o fom the font face (f) of the specimen the deviation of the light ays at a distance z 0 fom the middle plane of the specimen, is expessed by the vecto w,f on the plane z=z 0 and accoding to the theoy of Econal [0], is given as: w, f = z 0 dc, f m gad( + ) (3) whee d is the thickness of the specimen, c,f ae the stessoptical constants, = fo eflected light ays fom the font (f) face and = fo eflected light ays fom the ea () face of the specimen and. m. is the magnification atio, which fo divegent light beam is given by the elation: z0 + zi m = (4) z i whee z 0 is the distance between efeence-plane and specimen and z i is the distance between specimen and the light beam focus. The stess-optical constant c f is c f =/, whee is the Poisson s atio and E is the modulus of elasticity of the specimen mateial. The sum of the stesses + is given by: + = + = PA cos + PB sin cos sin = k + k (5)

3 9 The Open Mechanical Engineeing Jounal, 0, Volume 6 G.A. Papadopoulos Then, the Rel.(3) becomes: w, f = C, f gad (x,y) [A cos + B cos( +)] = C, f A cos i + sin j + B (6) [sin i cosj] whee: C, f = z 0 dc, f P m, c f = E The paametic equations of the caustic ae: W, f = + w, f = cos + i sin + w, f o: (7) X, f = cos ± C, f [ A cos Bsin ] (8) Y, f = sin ± C, f [ Asin + B cos ] (9) whee is the adius of the initial cuve of the caustics and takes values in the egion +. Fo divegent light beam, the sign (-) coesponds to the caustics () (caustics which ae fomed by the eflected light ays fom the ea face () of the specimen) and the sign (+) coesponds to the caustic (f) ( caustics which ae fomed by the eflected light ays fom the font face (f) of the specimen). The initial cuve of the caustic is detemined by setting the functional deteminant Jacobian J equal to zeo: J = (X Y ), f, f = 0 (0) (,) Then, the initial cuve is: 0 = = C, f A + B () By substituting equation () into equations (8), (9) the paametic equations of the caustics becomes: a) caustics (f): X f = m C f A + B Y f = m C f A + B b) caustics (): X = m C A + B Y = m C A + B cos + ( A cos Bsin ) A + B sin + ( Asin + B cos ) () A + B cos ( A cos Bsin ) A + B sin ( Asin + B cos ) (3) A + B Consideing the elations (4) and (5), the equations () and (3) can be witten, espectively: X f = m C f + Y f = m C f + and: X = m C + Y = m C + whee: cos + ( cos sin ) + sin + ( sin + cos ) (4) + cos ( cos sin ) + sin ( sin + cos ) (5) + = k = B + sin k A = tan (6) sin and: C, f = z dc k 0, f (7) m Fig. () illustates the atio = k / k of the stess concentation factos vesus the wedge angles fo angles of loading =,0,45,60,80. Fo =0 (B=0), concentated vetical load at O [3], the initial cuve of the caustics and the espective paametic equations of the caustics, ae: 0 = = ( C, f A) /3 (8) X f = m C f A Y f = m C f A cos + cos sin + sin X = m ( C A) /3 cos cos Y = m ( C A) /3 sin sin fo + (9) (30) Fo =/ (A=0), concentated hoizontal load at O the initial cuve of the caustics and the espective paametic equations of the caustics, ae: 0 = = ( C, f B) /3 (3) X f = m C f B cos sin

4 Evaluation of Concentated Oblique Load at the Apex of a Wedge The Open Mechanical Engineeing Jounal, 0, Volume 6 93 Fig. (). Vaiation of the stess concentation factos atio k / k vs the wedge angles fo angles of loading =,0,45,60,80. Y f = m C f B sin + cos X = m ( C B) /3 cos + sin Y = m ( C B) /3 sin cos fo + (3) (33) Fo values of between and +, the caustic (), fom the ea face of the plate and the caustic (f), fom the font face of the plate, ae obtained aound the point of the concentated load P. Fig. (3) illustates the shapes of the espective caustics fo angle loading =45 o and fo (a)wedge of =45 o, (b)wedge of =90 o, (c) wedge of =0 o and (d) wedge of =35 o. The espective positions and the magnitude of the two caustics ae depended on the values of the espective adii of the initial cuves and the stess optical constants c, c f. Also, Fig. (4) illustates the shapes of the espective caustics aound the point of the concentated load fo wedge of =90 o (semi-infinite plate) and fo angles of loading (a) =0 (vetical load [3]), (b) =60 o, (c) =80 o and (d) =90 o (hoizontal load). IV. EXPERIMENTAL SET-UP In ode to veify the theoetical esults of the concentated load and the concentation factos evaluation at the apex of the wedges by the method of caustics, specimens made of Plexiglas (PMMA) wee used. The thickness of the specimens was d = 3x0 3 m. The expeimental set-up was simple, Fig. (5) [-3]. A coheent divegent light beam fom a He-Ne gas lase impinged on the face of the specimen close to apex of the wedge. Sceen in font the specimen was placed at distance z 0 and paallel to specimen mid-plane. On this sceen the caustics fom eflected ays wee fomed and ecoded by a camea. The optical set-up had z 0 =.79m, z i = 0.30m and the magnification atio m of the optical system was m = 6.97 (Rel. (4)). Fo a positive value of z i (the focus lying in font of the specimen) a double caustic is fomed on the sceen due to eflections fom the font and the ea face of the specimen (caustics of Figs. 3, 5). The mechanical popeties of the specimen mateial (Plexiglas) wee: Poisson s atio = 0.38 and elastic modulus E =.MPa The stess-optical constant of the specimens mateial was c f = E =.7x00 m / N. The specimens wee loaded by a unifom concentated load at the apex of the wedges. Fig. (6) illustates the expeimental eflected caustics (f) and () which wee caied out by divegent light beam, fo a Plexiglas specimens of thickness d = m. The loading angle was =0 o and the wedge angles wee (a) =45 o, (b) =90 o and (c) =0 o. Also, Fig. (7) illustates the theoetical (a) and the expeimental (b) eflected caustics (f) and () fo a Plexiglas specimens of thickness d = 30 3 m. The loading angle was =5 o and the wedge angles was (a) =35 o, We can see good quality ageement between theoetical and expeimental caustics. The elative position of the two caustics, () and (f), is depended on the values of the stess optical constants c and c f. Fom Figs. (3, 4) it is appeaed that fo oblique loading the diametes of the caustic can not be calculated. In these cases the load P and the concentation factos can be calculated fom the opening displacement L of the caustic (Fig. 3b). The hoizontal and the vetical components of the opening displacement L ae given:

5 94 The Open Mechanical Engineeing Jounal, 0, Volume 6 G.A. Papadopoulos (a) (a) (b) (b) (c) (c) (d) Fig. (3). Caustics () and (f) at the apex of wedge fo angle of loading =45 o and fo wedges of (a) =45 o, (b) =90 o, (c) =0 o and (d) =35 o. (d) Fig. (4). Caustics () and (f) at the apex of wedge of angle =90 o (semi-infinite plate) and fo loading angles of (a) =0 o (about vetical foce), (b) =60 o, (c) =80 o and (d) =90 o (hoizontal foce).

L, L lenses of focus-distances f, f, espectively. (a) (a) (b) (b) Fig. (7).

6 Evaluation of Concentated Oblique Load at the Apex of a Wedge The Open Mechanical Engineeing Jounal, 0, Volume 6 95 Fig. (5). The optical set-up fo eflected caustics. L, L lenses of focus-distances f, f, espectively. (a) (a) (b) (b) Fig. (7). Theoetical (a) and expeimental (b) caustics () and (f) at the apex of wedge fo angle of loading =5 o and fo wedge of =35 o. L H = Y f = Y f = = m C f A + B A sin + sin A + B (34) The vetical component of the opening displacement L is given: (c) Fig. (6). Expeimental caustics () and (f) at the apex of wedge fo angle of loading =0 o and fo wedges of (a) =45 o, (b) =90 o and (c) =0 o. L V = X f = X f = B = m C f A + B sin A + B and the opening displacement L of the caustic is: L = L H + L V = m C f A + B whee: (35) R (36)

7 96 The Open Mechanical Engineeing Jounal, 0, Volume 6 G.A. Papadopoulos A R = sin + sin A + B B + sin A + B / (37) By compaing equations (7) and (36) the oblique concentated load P is calculated by the elation: P = L 3 6z 0 d m c f R 3 A + B (38) Because of the P is the load pe unit thickness, the total load is: Q = Pd = L 3 6z 0 m c f R 3 A + B (39) The expeimental stess concentation factos k ex, k ex, accoding to elations (4), (5) and (39), can be calculated: L 3 cos k ex = 6z 0 m c f + (40) sin R 3 A + B L 3 sin k ex = 6z 0 m c f (4) sin R 3 A + B Consideing the elations (4), (5) and (40), (4), a coection factos f, can be calculated []: f, = k ex, (4) k, V. ESTIMATION OF THE STRESSES IJ On the hoizontal plane nn at a distance fom the apex of the wedge, the Catesian stess components at any point M (Fig. ) ae calculated fom the pola stess component : xx = cos = PA cos3 + PB sin cos cos 4 sin cos 3 = k + k yy = sin = PA cos sin + PB sin3 cos sin sin 3 cos = k + k xy = sin cos = PA sin cos + PB sin cos sin cos 3 sin cos = k + k (43) (44) (45) with: = /cos = 0 /cos (46) Fo a hoizontal plane nn close to the apex of the wedge at a distance equals to the adius of the initial cuve of the caustic, = 0, and by substituting elations (), (36) and (38) into equations (43)-(45), the components of stess ae obtained fom the opening displacement of the caustic: L ( ) (47) xx = A cos 4 Bsin cos 3 8z 0 d m c f R A + B L ( ) (48) yy = A cos sin Bsin 3 cos 8z 0 d m c f R A + B L ( ) (49) xy = Asin cos 3 Bsin cos 8z 0 d m c f R A + B fo values of between and +. Fig. (8) illustates the distibution of the non-dimensional components of stess, (Eqs (47)-(49)): ij M L = f () with M = 8z 0 d m c f R A + B (50) on the hoizontal plane at distance 0 (singula egion of concentated load) fom the wedge apex, (Fig. ), fo wedges of angle up to = 3 /4 and of angle of loading a) = 0 o, b) = 45 o and c) = 90 o. VI. EXPERIMENTAL RESULTS AND DISCUSSION In ode to veify the theoetical esults of the concentated load and the concentation factos evaluation at the apex of the wedges by the expeimental method of caustics, specimens made of Plexiglas (PMMA) and the expeimental set-up of Fig. (5) wee used. Figs. (6, 7) illustate the eflected caustics (f) and () which wee fomed at the apex of the wedges. Fom the opening displacement (L) of the caustics (f) (Fig. 3b) the concented load P (o ex ex Q=Pd) and the concentations factos k and k wee calculated by the Eqs (38)-(4). Fig. (9) illustates the expeimentally calculated load Q (by Eq. (39)) fom the opening displacements L of the caustics (f), (Fig. 6), fo wedges angles =45, 90 and 0 and loading angle =0 (vetical concentated load). As it can be obseved that the calculated load Q depends on the opening displacement (L) of the caustic. The opening displacement (L) of the caustic depends on the wedge angles and the size of the initial cuve ( 0 ). The size of the initial cuve is smalle than the theoetical one because of the bounday defomation of the specimen. The vaiation of the load with the theoetical opening displacement is paabolic

8 Evaluation of Concentated Oblique Load at the Apex of a Wedge The Open Mechanical Engineeing Jounal, 0, Volume 6 97 (a) (b) (c) Fig. (8). The stess components vaiation vesus fo wedge angles up to = 3 /4 and fo loading angle a) = 0 o, b) = 45 o and c) = 90 o. (Rel. (39)). We can see a good ageement between the theoetical vaiation and expeimentally calculated values based on the caustics (Fig. 9). Fig. (0) illustates the vaiation of the expeimental concentation facto k ex which was calculated fom the

9 98 The Open Mechanical Engineeing Jounal, 0, Volume 6 G.A. Papadopoulos expeimental caustics of Fig. (6). As it can be obseved that the concentation facto inceases as the wedge angles decease. The vaiation of the theoetical stess concentation k (Rel. (4)) is linea with the load. We can see a good ageement between the theoetical vaiation and expeimentally calculated values based on the caustics (Fig. 0). poblems. Using this expeimental analysis of the caustics, stess infomation may be obtained in singula points of the contact poblems. It can be concluded that: The concentated load, the concentation factos and the stess vaiation at the singula apex of the wedges may be expeimentally calculated by the expeimental method of the caustics. The expeimental method of caustics which is a poweful tool fo discussing fields aound a cack, is also a poweful method fo the contact poblems and wedges with concentation loads. Using this expeimental method the existing wedge theoy may be eviewed with measuements and taking into consideation othe influential paametes such as the wedge angles and the loading angles. Fig. (9). Vaiation of the expeimentally calculated load Q vs the caustics opening displacement L fo wedge angles =45, 90 and 0 and loading angle =0. Fig. (). Vaiation of concentation factos k ex ex, k and the atio = k ex ex / k vs the load Q fo wedge angle =35 and loading angle =5. ACKNOWLEDGEMENT Declaed none. CONFLICT OF INTEREST The authos confim that this aticle content has no conflicts of inteest. Fig. (0). Vaiation of the concentation facto k ex vs the load Q fo wedge angles =45, 90 and 0 and loading angle =0. Fig. () illustates the vaiation of the expeimental concentation factos k ex ex, k and the atio of concentation factos = k ex ex / k which wee calculated fom the expeimental caustics of Fig. (7b) (wedge angle =35 and loading angle =5 ). As it can be obseved that the concentation factos incease as the load inceases, while the concentation factos atio is emain constant. The concentation factos k ex, k ex ae depend on the load P (Rels (4) and (5)), while the atio = k ex ex / k is independent on the load (Rel. (6)). We can see good ageement between the theoetical vaiation and the expeimental values based on the caustics (Fig. ). VII. CONCLUSIONS An expeimental method of caustics has been poposed to calculate expeimentally the load and the concentation facto at the apex of the wedges. A detail analysis of the eflected caustics is poved fo the study the contact REFERENCES [] P. S. Theocais, Local yielding aound a cack-tip in Plexiglas. Jounal of Applied Mechanics, Tans. ASME, vol. 37, pp , 970. [] J. F. Kalthoff, Shadow optical method of caustics, in Handbook on Expeimental Mechanics, A. S. Kobayashi, Ed. USA: Pentice- Hall, 987, chapt. 9, pp [3] G. A. Papadopoulos, Factue Mechanics, The Expeimental Method of Caustics and the Det.-Citeion of Factue, G.A. Papadopoulos Ed., London: Spinge-Velag 993. [4] S. P. Timoshenko and J. N. Goodie, Theoy of Elasticity, New Yok: McGaw Hill, 970. [5] A. Flamant, De l influence su la flexion des poutes de la position supeficielle de la chage Annales des Ponts et Chaussees, vol. VI, pp. 8-4, 893. [6] J. Boussinesq, Petubations locales que poduit audessous d ell une fote chage epaie unifomement l ong d une doit nomale aux debods a la suface supeieue d une poute ectangulaie et de longueu indefinie posee de champ soit su un sol hoizontal, soit su deux appuis tavesaux equidistants de la chage, Comptes Rendus de l'académie des Sciences, vol. 4, pp , 89. [7] P. S. Theocais, Stess-singulaities due to unifomly distibution loads along staight boundaies, Intenational Jounal of Solids and Stuctues, vol. 9, pp , 973. [8] P. S. Theocais, Stess-singulaities at concentated loads, Expeimental Mechanics, vol. 3, pp. 5-58, 973.

10 Evaluation of Concentated Oblique Load at the Apex of a Wedge The Open Mechanical Engineeing Jounal, 0, Volume 6 99 [9] G. A. Papadopoulos, Expeimental estimation of the load distibution in beaings by the method of caustics, Expeimental Mechanics, vol. 44, pp , 004. [0] G. A. Papadopoulos, Expeimental study of the load distibution in beaing by the method of caustics and the photoelasticity method, The Jounal of Stain Analysis fo Engineeing Design, vol. 40, pp , 005. [] V. A. Spitas, G. K. Sfantos, Th. Costopoulos, C. A. Spitas, and G. A. Papadopoulos, Numeical and expeimental Analysis of load shaing in multiple gea tooth contact. 7 th National Congess on Mechanics, June 4-6, 004, Chania, Geece, vol. I, pp [] V. Spitas, G. A. Papadopoulos, C. Spitas and T. Costopoulos, Expeimental investigation of load shaing in multiple gea tooth contact using the stess-optical method of caustics. Stain, vol. 47, pp. e7-e33, 009. [3] G. A. Papadopoulos, Study of concentated vetical load at the apex of a wedge by the expeimental method of caustics and photoelasticity, Advances and Applications in Mechanical Engineeing and Technology (AAMET), vol. 3, pp. 59-8, 0. [4] V. M. Aleksadov and D. A. Pozhaskii, Poblems of a cut in a thee-dimensional elastic wedge, Jounal of Applied Mathematics and Mechanics, vol. 70, pp , 006. [5] A. R. Shahani, Some poblems in the antiplane shea defomation of bimateial wedges, Intenational Jounal of Solids and Stuctues, vol. 4, pp , 005. [6] Y. C. Gao, Lage stain analysis of a ubbe wedge compessed by a line load at its tip, Intenational Jounal of Engineeing Science, vol. 36, pp , 998. [7] B. G. Badalouka and G. A. Papadopoulos, Range of applicability of the method of caustics aound holes. Stain, vol. 4, pp , 006. [8] D. N. Pazis, Z. Agioutantis and S. K. Koukoulis, S. K., 00, The optical method of eflected caustics applied fo a plate with a cental hole: Citical points and limitations. Stain, vol 47, pp , 00. [9] A. Bacic, D. Semenski and S. Jecic, The caustics method in the contact poblems of anisotopic mateialas, EPJ Web of Confeence, vol. 6, pp.-8, 00. [0] M. Bon and E. Wolf, Pinciples of Optics, London: Pegamon, 970. [] P. S. Theocais and G. A. Papadopoulos, The influence of geomety of edge-cacked plates on K I and K II components of the stess intensity facto, Jounal of Physics D: Applied Physics, vol. 7, pp , 984. Received: June 6, 0 Revised: July 6, 0 Accepted: July 7, 0 G.A. Papadopoulos; Licensee Bentham Open. This is an open access aticle licensed unde the tems of the Ceative Commons Attibution Non-Commecial License ( which pemits unesticted, non-commecial use, distibution and epoduction in any medium, povided the wok is popely cited.

Title. Author(s)NOMURA, K.; MOROOKA, S. Issue Date Doc URL. Type. Note. File Information

Title. Author(s)NOMURA, K.; MOROOKA, S. Issue Date Doc URL. Type. Note. File Information Title CALCULATION FORMULA FOR A MAXIMUM BENDING MOMENT AND THE TRIANGULAR SLAB WITH CONSIDERING EFFECT OF SUPPO UNIFORM LOAD Autho(s)NOMURA, K.; MOROOKA, S. Issue Date 2013-09-11 Doc URL http://hdl.handle.net/2115/54220