4 AN ANALYTICAL AND EXPEREMENTAL ENVESTEGA’HON OF STRESS DESIRIBUTION EN THE PUNCHEQ METAL PLATE OF A 'E’EMBER iQQE‘iT Thai: for “to Degree of pk. D. MICHIGAN STATE UNIVERSITY Ram Daur Misra 1964 «meals ea, This is to certify that the thesis entitled An Analytical and Experimental Investigation of Stress Distribution in the Punched Metal Plate of a Timber Joint. presented by Ram D. Misra has been accepted towards fulfillment of the requirements for Ph D. degree mAgricultural Engineering m Major professor Date-MéIL/fgf ABST RACT AN ANALYTICAL AND EXPERIMENTAL INVESTIGATION OF STRESS DISTRIBUTION IN THE PUNCHED METAL PLATE OF A TIMBER JOINT by Ram Daur Misra The main objective of this study was to conduct a theoretical and experimental investigation of stress distribution in a punched metal plate of a timber joint. The theoretical investigation consisted of two different methods. One utilized the discrete approach of a difference equation while the other used the continuous approach of the principle of minimum com- plementary energy. A second order difference equation was derived and solved for a general case. The results for the particular case of the metal plate connector were calculated and plotted together with the experimental results for comparison. The principle of minimum complementary energy was used to derive a second order ordinary linear differential equation for an idealized case. The metal plate connector was treated as if glued to the surface of the wood by a fictitious adhesive of negligible thickness. The differential equation thus obtained was solved with appropriate boundary conditions. The results for the particular case of metal plate connector were plotted with the experimental results for compari- son. Two different methods of experimental stress analysis were used; namely, PhotoStress analysis and a strain gage technique. 1 Ram Daur Mis ra The PhotoStress analysis provided, qualitatively, the overall pattern of stress distribution in the entire plate. A set of bar graphs for the principal stress difference ( 61 - 62) along the various rows of the teeth in the metal plate connector were plotted. A symmetrical stress distribution was obtained. The isoclinic pattern for a portion of the plate was thoroughly examined. From this isoclinic pattern stress trajectories were drawn. The strain gage technique was used to obtain accurate and reliable values of strain in the metal plate to verify the theoretical results. From the results of this investigation the following conclusions and observations were made. 1. The results of the difference equation solution as well as that of the principle of minimum complementary energy predict reasonable agreement with the experimental results. Either method can be used to calculate stresses in the metal plate connector. 2. The difference equation solution can also be used with at least equal accuracy for riveted and bolted joints. Similarly the results of the principle of minimum complementary energy are equally applicable for adhesive as well as welded joints. 3. The stresses in the metal plate are not uniform as assumed in the normal design practices. The maximum calculated stress in the connector was 2.4 times the average value. 4. A set of distruction tests made in tension resulted in tearing failure of the plate in the center of the joint. If the middle part of the plate were not punched, the strength of the joint should be greater and also a more uniform stress distribution should result. Ram Daur Mis ra . The PhotoStress analysis provided an overall pattern of stress distribution in the entire plate. The variation between measured and calculated principal stress difference ranged from 9. 6% to 42.2%. The results of this analysis were, however, incomplete as the shape of the punched plate was too complicated for analytical separation of the principal stresses. The equipment for experimental separation (oblique incidence meter) was not available. Approved Major Professor AN ANALYTICAL AND EXPERIMENTAL INVESTIGATION OF STRESS DISTRIBUTION IN THE PUNCHED METAL PLATE OF A TIMBER JOINT BY Ram Daur Mi 5 ra A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCT OR OF PHILOSOPHY Department of Agricultural Engineering 1964 ACKNOWLEDGMENTS I wish to express my sincere thanks to Dr. F. H. Buelow, chairman of the guidance committee, for his pleasant counsel and many helpful suggestions. Sincere thanks are also expressed to Drs. J. S. Boyd and M. L. Esmay, ex-chairmen, for their inspiring guidance and constant supervision of this project. I am greatly indebted to Dr. A; W. Farrall, chairman, Department of Agricultural Engineering for making available the research assistantship which enabled me to undertake this study. The financial support of Troy Steel Corporation for this project is gratefully acknowledged. Grateful appreciation is expressed to the other members of the guidance committee; Dr. L. E. Malvern (Metallurgy, Mechanics and Material Science), Dr. R. K. Wen (Civil Engineering) and Dr. B. F. Cargill (Agricultural Engineering) for their guidance, suggestions and interest in this project. Special thanks are due to Dr. C. A. Tatro, formerly of the Department of Metallurgy, Mechanics and Material Science for his continual help in the experimental work. The assistance of Mr. J. B. Cawood and his associates of the Research Laboratory in mechanical construction and in supplying the necessary tools and equipment is duly recognized. >1: >‘,< >:< >;< >:< 3:: >,‘< >1: 2:: >:< 3:: fl: >‘,< * >:< >;< >:< >:< >:< ii TABLE OF CONTENTS Page INTRODUCTION........... ...... 1 Objective......................... 2 REVIEW OF LITERATURE . . . . . . . . . . . ...... . . 3 Timber Joints Connected by Metal Plates . . . . . . . . 3 AnalysisofJoints..................... 4 PhotoStress Analysis ....... . . . . . . . . . . . 6 THEORETICAL INVESTIGATION . . . . . . . . . . ..... . 8 Analysis by Means of a Difference Equation ..... . . 8 Solution of the Difference Equation . . . . . . . . . 12 Analysis by the Principle of Minimum Complementary Energy ..... O I O O O O O O O O C O C O O O O O 15 Derivation of Differential Equation. . . . . . . . . 18 Solution of the Differential Equation ........ 21 EXPERIMENTAL INVESTIGATION . . . . . . . . . . . . . . . 23 PhotoStress Analysis .......... . . ....... 23 TestSpecimen ....... 24 Testing Procedure . . . . . . . . . . . . . . . . . 27 Results and Discussion of PhotoStress Analysis . . 27 StressMagnitudes .. . 27 Stress Directions ........... . . . . 34 Isoclinics ..... . . . . . . . . . . 34 Stress Trajectories (Isostatics) . . . . 36 Strain Gage Analysis ......... . . . . . . . . . . 38 Test Specimen ....... . . . . . . ...... 38 Testing Procedure . . . . . . . . . . . . . . . . . 39 Results and Discussion of Strain Gage Analysis . . 39 RESULTS AND DISCUSSION. . . . . . . . . . . ...... . . 51 iii TABLE OF CONTENTS - Continued Comparison of Theoretical and Experimental Results. Evaluation of PhotoStress Analysis . SUMMARY 0 O O O I O O O O O O O O O O 0 CONCLUSIONS AND OBSERVATIONS . . SUGGESTIONS FOR FURTHER STUDY. . REFERENCES 0 O O O O O O O I O O O O 0 iv Page 51 55 58 60 60a 61 LIST OF TA BLES TABLE 1. Principal Stress Differences at a Network of Points in the Metal Plate Connector, Loaded in Tension. (System of Point Designation is Shown in Figure 5). . . 11. Comparison of Measured and Calculated Stresses on Strips Along Row 7 at a Tensile Load of 2500 lbs Per Plate 0 O O O O O O O O O O O O O O O O O O O O O O O O O Page 29 34 LIST OF FIGURES FIGURE Page 1. Schematic drawing of the metal plate connected timber jOi-nt. O O O O O O O O O O O O O C O O O O O O O O O 9 2. Deformed metal plate connector and the variation of force F as it increases from O at x = O to P at x = np. . 11 3. Actual and idealized joints used in the analysis . . . . . l7 4. Equilibrium of small elements across the idealized jOint O O O O O O O O O O O O O O O O O O O O O O O O O O O O 17 5. Shape of the punched teeth in the metal plate connector and location of strain gages . . . . . . . . . . . . . . . 25 10. 11. 12. 13. . Test specimen and principal stress difference variation alongcolumnSZandS.... ..... Test apparatus for PhotoStress study . . . . . . . . . . Test apparatus for strain gage investigation . . . . . . Histogram representation of principal stress differences alongcolumnslandZ. .. . . . . . . .. . . . . Histogram representation of principal stress differences alongcolumns3and4............. Histogram representation of principal stress differences alongcolumnsSandb.................. Isoclinics in the metal plate connector, loaded in ten- sion (The location of this portion on the plate is shown in Figure 5) O I O O O O O O O O O O O O O O O O O O O O 0 Stress trajectories in the metal plate connector, loaded in tenSion. O O O O O O O O O O O O O O O O O O O O O O O 0 vi 26 28 28 31 32 33 35 37 LIST OF FIGURES - Continued FIGURE 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. Load stress curves for points (1,1), l, 3), (1, 5) and (1,7) 0 e e e o e o o o o o e o e o e e o o oooooo e 0 Load stress curves for points (2,1), (2, 3), (2, 5) and (2,7). Load stress curves for points (3,1), (3, 3), (3, 5) and (3,7) Load stress curves for points (4,1), (4, 3), (4, 5) and (4,?) Load stress curves for points (5,1), (5, 3), (5, 5) and (5,7) Load stress curves for points (6,1), (6,3), (6,5) . . . Stress distribution in the metal plate connector at 5,0001bstension.................... Stress distribution in the metal plate connector at 5, 000 lbs tenSion O 0 O I O O O O O O O O O O O O O O O O Stresses at the bases of cantilever teeth in the metal plate connector at 5,000 lbs tension . . . . . . . . . . Stresses at the bases of cantilever teeth in the metal plate connector at 5,.000..1bs.tension. . . . . . . . . . . Comparison of experimental and theoretical results as predicted by the difference equation solution . . . . . . Comparison of experimental and theoretical results as predicted by the principle of minimum complementary energy 0 O O O O O C O O O O O O O O O O C O O O O O O O 0 Stress distribution in the metal plate connector as ob- tained from the analyses of the minimum complemen- tary energy and the difference equation . . . . . . . . 40 41 42 43 44 45 47 48 49 50 52 54 56 INTRODUCTION Wood trusses with spans of 20 to 40 feet have been used in the past 5 to 10 years for the construction of farm buildings, light storage structures and homes. A satisfactory and economical fastener to join the truss members continues to be the main problem in the design and development of light wood trusses. Considerable research has been done on ringc-bolt and glue-nail fasteners in the past few years. Both have limitations and disadvantages. Glue-nail fasteners, for example, cannot be used when the temperature is below freezing unless heated space is provided. In recent years various types of metal plate connectors have been developed for wood trusses. These connectors are made from com- mercial quality galvanized steel sheets ranging in thickness from 14 to 20 gauge. The method of fastening varies widely from manufacturer to manufacturer. It may range from rectangular teeth punched and bent 900 to the face of the plate to simply holes drilled in the plate in which nails are later inserted. The length of these punched teeth may vary from 1/4 to 3/4 inch. The general acceptance and preliminary per- formance of these connectors indicate a potential for farm use as well as wider application in house construction. There is a lack of basic information on the load transmitting characteristics of the metal plate connectors. A tangible and accurate method of analyzing and designing these connectors has not yet been developed. The stress distribution in a metal plate under field con- ditions or even idealized conditions of loading is not well-known. The design of the metal plate connectors is based on the assumption that there is a uniformly distributed load and that each tooth acts as a miniature cantilever beam. Also, it is assumed that the load is constant over the entire length and the width of the plate. These assumptions seem to be rather general and inaccurate. A few preliminary tests made before the beginning of this project indicated a sharp stress gradient in the metal plate in the direction of loading. This project was, therefore, undertaken to investigate accurately the stress distribution in a metal plate connector. Objective The objective of this investigation was to make a theoretical and experimental stress analysis of a metal-plate-connected timber joint in uniaxial tension. REVIEW OF LITERATURE Timber Joints Connected by Metal Plates Felton (1963) conducted an extensive study to determine the design loading values for various types of standard metal truss plates. He attempted to establish design loading values as follows: 1. Load per tooth or nail. 2. Load per square inch of plate area. 3. Load per ounce of metal plate and its fastenings. A total of eleven different types of plates were used for evaluating the performance of truss plate joints in tension and shear. Five repli- cations of each test were made. ‘ The test data were evaluated statistically by'an analysis of variance. It was established that the shear joints gave higher values of ultimate loads than the tension joints. The following conclusions may be drawn from his study: 1. The shear tests as used gave unrealistically high values for load carrying capacity of truss plates. 2. The tension test appears to give a realistic value for the load carrying capacity of truss plates. 3. The metal truss plates can be used to make satisfactory joints for trussed rafter construction. 4. The metal truss plates make relatively rigid joints. 5. Joints with a low net cross-sectional plate area failed in the plate rather than in the wood. 6. The 3/4" long rectangular teeth did not have a tendency to bend, but rather crushed the wood fibres on the sides toward the load application. 7. The smooth shanked nail had a tendency to bend and withdraw and to crush the wood fibres on the sides toward the load application. It has been well-established that the timber joints connected with metal plates and nails slip with respect to wood when loaded. This is also an indication of the rigidity of these joints. The amount of slip increases with the magnitude of load as well as with accelerated aging. Joy (1960) conducted a comprehensive study of metal plate con- nectors for wood trusses. He particularly investigated the slip character- istics of these connectors at different loads and a series of humidity conditions. A photographic record of load slip data was made. From this permanent record the average slip was plotted against load. The study indicated that the slip of plates, loaded while aging,was very marked. At about half ”Plastic Flow Start“ load the slip of joints being aged was 2 to 5 times the slip of unaged units. The author accounts . a part of this increase to humidity effect while the rest due to time alone or creep. Analysis of Joints It appears that there has been no attempt to analyze the metal- plate-connected timber joints analytically. The literature surveyed here deals primarily with the analysis of forces in riveted, bolted and welded connections. Certain similarities of these joints to metal plate connectors make it relevant to review these investigations. Hrennikoff (1932) used the work of rivets in riveted joints to determine the distribution of load among various rivets. He divided his study into three parts. Part I consisted of a qualitative study with some observations on conventional design methods. Part II included the derivation of formulas in a few simple types of joints based on the assump- tion that a rivet develops a force proportional to deformation. In Part III the formulas of Part II were applied to the determination of the numerical values of the coefficients found analytically. The author drew the following conclusions on. the basis of his analysis. 1. The standard practice of dividing the force in proportion to the shearing areas of the rivets and disregarding the deformation of the plates leads to results that are very inaccurate. 2. Actually, the total force acting on a riveted joint is not distributed equally among the rivets; a larger portion of the total work is done by the outer rivets. 3. The proportion of the total work in the outer rivets increases as the pitch and diameter of the rivets increase and the cross- section of the plates decreases. 4. If there are many rivets in a longitudinal row, the inner rivets are inefficient, and an increase in the number does not improve the value of the joint appreciably. Muckle (1949) used the principle of minimum strain energy to determine the distribution of load in riveted joints. He made the follow- ing assumptions for simplicity of his analysis. 1. The plate between any two rows of rivets is in a state of uniform stress; and 2. A portion or the whole of the shank of the rivet is in a state of uniform shear stress. The investigation covered treble, quadruple and quintuple-riveted lapped joints, and treble-riveted double cover butt joints. The author concluded that (except in double riveted joints) the load is not uniformly distributed over the various rows of rivets; the outer rows take more and the inner rows less than the average load. He further pointed out that because of the nature of assumptions made in developing the theory the results should be regarded as being qualitative only. The friction between plates, for instance, had been entirely ignored. The author also made an interesting observation: that except for treble- riveted lapped joints, the modulus of elasticity of the joints examined exceeded that of solid plate, and it is possible that, in practice, the modulus of elasticity would be greater in all cases. The reduced value of modulus of elasticity in large riveted structures should not therefore be employed for calculations. In the analysis of Hrennikoff and Muckle reviewed above, very laborious calculation is involved, and whenever the number of con- nectors is to be changed a new problem must be solved. To overcome this difficulty Harris (1962) used the difference equation approach for analyzing parallel-type structural connections. Harris developed a second order difference equation relating the increment of force transmitted by connectors and the stiffness factors of members and connectors. The equation was solved by a trial and error process with appropriate boundary conditions. The results were plotted for various ratios of stiffness factors. PhotoStres 3 Analysis PhotoStress is a trade name for the birefringent coating technique of experimental stress analysis. It is essentially a photoelastic tech- nique except that no model analysis is required. The actual specimen to be stress analyzed is coated with a special transparent plastic that exhibits temporary birefringence (double refraction) when strained. This birefringence is directly proportional to the intensity of strain. When a polarized light is passed through the strained plastic, black and colored fringe patterns corresponding to the direction and intensity of principal strains can be observed and measured by a reflection polari- scope. The black lines called isoclinics, connect points of the same principal stress direction on the specimen. The colored fringe patterns called isochromatics, connect points on the stressed specimen that have same magnitude of principal stress difference. The effects of strain gradient and the curvature of the surface under load on the photoelastic pattern were studied by Duffy (1961). He used a Fourier Series solution for the displacement to represent surface strain gradients for a one-dimensional problem. From his series solution he showed that the fringe order is directly proportional to surface strain only if strain is uniform or varies linearly with dis- tance. In all other cases the results will be erroneous. Even for thin coating thicknesses errors may be as high as 35%. He c oncluded that Experimental results are influenced considerably by two factors not previously investigated; namely, a gradient in the strain at the metal surface, and the curvature of this surface under load. Neglecting either of these factors may lead to large errors and, under certain circumstances, it is possible for one or the other to produce a greater birefringence than does the surface strain. Post and Zandman (1961) made an extensive study of effects of Poisson's ratio and coating thickness on the accuracy of PhotoStress technique. They made the following conclusions on the basis of their expe rim ental inve stigation. 1. For the case of plane stress problems and equal Poisson's ratio of structure and coating, the influence of coating thick- ness on birefringence developed along free boundaries is almost identically zero. For unequal Poisson's ratio and simply connected structures in plane stress, birefringence developed along free boundaries is almost exactly independent of coating thickness. For simply connected structures in plane stress, very thick coatings behave essentially as independent bodies subjected to prescribed end displacements, i. e. , as photoelastic models. In this case birefrigence is independent of Poisson's ratio. In order to minimize effects of dissimilar Poisson's ratio and local reinforcement, thin coatings are preferable. For most engineering problems 1/8 in. coating should be adequately thin. THEORETICAL INVESTIGATION The theoretical analysis of the metal plate connector was made by two different methods. One used the discrete approach of a difference equation while the other utilized the continuous approach of the principle of minimum complementary energy. The validity of these analyses, of course, depends upon the soundness of the assumptions underlying them and the mathematical limitations, as a theory describing a physical phenomenon can be no better than the assumptions on which it is based. Neither of these analyses can, therefore, be claimed to describe the exact physical behavior of the joint. They are only approximations to the actual behavior. Analysis by Means of a Difference Equation By definition, a difference equation relates the values of a function y and one or more of its differences Ay, Azy, . . . for each x-value of some set of numbers S (for which each of these functions is defined). Or, in brief, a difference equation is a relation involving differences. The analysis made here by means of a difference equation is similar to one used by Harris (1962) for parallel-type structural con- nections. The metal plate connected timber joint is shown schematically in Figure 1a. It consists of two punched metal plates driven into the wood, one on each face of the joint. Because of symmetry, it is sufficient to consider only a quarter portion of the joint for analysis. This part is shown in Figure 1b. The load from member A, the wood part, is transmitted to member B, the metal plate, by connectors c. The force at any point 8 $50.6. Menage con—02800 mama H.308 623 we wag/map oflpmsofiom .H .mfm A3 3 we maux mwux mmux mmux aux oux h4+hl lib .m L n _ mi ; u Am<+h7ml It 1] _ a. “To __ r C I E 1.. __ _ _ __ .. __ a _ _" Z in _ u m_ : _ : __ _ __ m r I r I C l u I z 1 c I z I .s m II.il..Hl.|J||.IH. . -l . i. .1 .7. il...JI|-..JI-.|.l.| Hill.|l1.|.-l| I _: __ _ __ __ __ : _ __ _ __ _ __ _ __ __ __ __ : _ : __ : Z __ Z _ __ 10 in member B is F and corresponding force in member A is P-»F. Figure 2b shows the variation of force F in member B as it increases from 0 at x = 0 to P at x = np. The increase in F from Fx—p at x = x-p to Fx at x = x is the first difference of F, and is represented by (AF)X. Thus, (AF)X=F -F (1.1) X x—p The second difference of F is (AZF)X = (AF)x+p - (AF)X (1.2) = (Fx+p - FX) - (Fx - Fx-p) = (Fx+p - 2FX + qup) or (AF)X+p = (AZF)X + (AF)x The smaller portion of the connector is shown in Figure 1d. Here, the force in the left-hand portion of member B is F and the force on the right-hand is F+AF; the force transmitted by the connector is AF, which is the first difference of the force in member B. Figure 2a, shows that the force in the left-hand connector c; is AF and the force in the right—hand connector c; is AF + AZF; where AZF is the second difference between forces on adjacent connectors. Supposing that k1 and k; are the stiffness factors for connectors c1 and c; respectively, the deformation of these two connectors can be expres sad as follows, . Force in CI AF Deformation of connector c1 = , = — stiffness of Cl k1 Force in ca _ AF + AZF Deformation of connector c . - Z stiffness of ca k2 .mnnxummouo nxumo Eon“ mommouocfi fi mm .m mono“ mo comedies, 05 new nouooccoo mama H.305 poEHOHoQ .N .mfim 3. ll ark uh mux r A —fir NM Quhxnmm m . , at" m at. .— 3 m A S xifl a e m - 1.4% if. bdrm Allfi _ llwlltudé , a _ / X . . x I L ._ 4r x g p / / , 4 / , ,. , , no , i , , E<+Ed , e i, Bind L a _ _ L L. x 1, $4.33 12 Again from Figure 2a it is seen that member A stretches by (P-F)/kA and member B stretches by F/kB; where kA and kB are the stiffness factors of members A and B respectively. In addition to stretching of members A and Band deformation of connectors c1 and c2, there is a relative slip between the wood and the metal plate. Assuming that this slip is negligible which implies that the joint is rigid; it must be then, that the sum of dimensions in upper part of the joint shown in Figure 2a less the sum of dimensions in the lower part must vanish, or mathematically AF F P-F AF + AZF (— + —)—(———-+., ) =0 (1.3) k1 kB kA .. k3 Assuming that the connectors have the same stiffness factor (which they do in this case) k1 = k2 = k, Eq. (1.3) becomes i _ P-F AZF _ 0 kB kA k or AZF - (1./1.3)}? + (k/kA) (12.1.?) = o (1.4) or AZF - (k/kA + k/kB)F = - (k/kA) P or AZF - (92]? = flail; (1.5) where (k/kA + k/kB) = .02 and (k/kA) = (of Equation (1. 5) above is a difference equation relating the force in member B to the total force P and the various stiffness factors. Solution of the Difference Equation The solution of the difference equation (1. 5) may be broken down into two parts, analogous to ordinary differential equation theory; namely, the complementary solution and the particular solution. 13 The complementary solution of the difference equation (1. 5) is the solution of the homogeneous difference equation AZF-wzF=0 (1.6) To obtain the complementary solution one may use the method of trial. Keeping in mind the similarity of the difference equation (1.6) with certain ordinary linear differential equations, one may guess that the solution would be a combination of hyperbolic cosine and hyperbolic sine with two constants, as the difference equation is of second order. Supposing that F = A Cosh mx, where m = an unknown constant then (AZF)x Fx+p - ZFX + Fx_p = A Cosh m(x+p) - 2A Cosh mx + A Cosh m(x-p). Substituting AZF and F into Eq. (1.6) gives A Cosh m(x+p) - 2A Cosh mx + A Cosh m(xop) - wZA Cosh mx = 0 (1.7) U sing the identity Cosh(ai b) = Cosh a Cosh biSinh a Sinh b in Eq. (1.7) reduces it to: 2 Cosh mx Cosh mp - 2 Cosh mx - wzCosh mx = 0 or Cosh mx (2 Cosh mp - 2 - (oz) 8 0 Since Cosh mxyf 0, one obtains 2Coshmp-2-w2=0 z or mp = Cosh"l 10—5—2- 2 or m = (l/p) Cosh'l M (1.8) 2 14 Therefore, F = A Cosh mx, where m is given by Eq. (1.8). is a solution of the difference Eq. (1. 6). In a similar manner one can show that F = B Sinh mx, is also a solution of Eq. (1.6). Thus, the _ general solution of Eq. (1.6) is a linear combination of above two solutions, i. e. , FC=ACoshmx+BSinhmx (1.9) To obtain the particular solution of Eq. (1. 5) an analogy with differential equations is again made, in which case, one proceeds as follows: Let F = a, a constant Thus AZFP = 0 Substituting this value of AZF in Eq. (1. 5) yields or F =(..{-/..Z)P - (1.10) Therefore, the final solution of the difference equation (1. 5) becomes F=FC p A Cosh mx + B Sinh mx + (oi/mp (1.11) Where A and B are to be obtained from the boundary conditions. 15 The boundary conditions are 1. F 0atx=0 2. F P at x = np, where n = total number of connectors. Using the first boundary condition one obtains 0 = A +(wiz/wz)P or A =(-wf/wz)p ‘ (1.12) and the second boundary condition gives P: A Cosh m up + B Sinh m np +(w‘13/wz)P or B = [P - A Cosh m np - (of/w?” P]/[Sinh m np ] (1.13) Analysis by the Principle of Minimum Complementary Energy The analysis presented here utilizes the principle of minimum complementary energy. This principle can be stated as follows: when true state of stress is varied by an infinitesimal amount in a system in such a manner that the new state again constitutes equilibrium with the given set of external loads, the first order change in the complementary energy less the work done by the increments of the reactions while traveling through the actual displacements of the supports is equal to zero. Or, mathematically 6U' - Zrnéano where 6U' the first variation in the complementary energy r n displacement of the nth reaction 6 Rn: the first variation in the nth reaction. 16 In applying this principle to the metal plate connected timber joint, the following assumptions were made: 1. The metal plate connectors instead of being driven into the wood are glued to the wood surface by a fictitious adhesive of negligible thickness, thus, forming a continuous contact between the metal plate and the wood. 2. The wood consists of two parts; one concentrated mass that takes all the normal stresses while the second thin part takes the shearing stress alone. 3. The joint behaves as a linearly elastic material. Since the joint is Symmetrical about the middle, only one—half portion is necessary for analysis. This part of the joint is shown in Figure 3. Considering equilibrium of an element dx across the joint, one obtains the state of stress as shown in Figures 4a, b and c. From Figure 4b, considering the equilibrium of forces in the x-direction one obtains, (T-l- '13:); Ady)tdx - Ttdx = O (2.1) or 33: = 0 which implies that T is independent of y. BY From Figure 4a, considering the equilibrium of forces in the x-direction one obtains dUC dx AC+ Ttdx =0 dx d U C t C X t X or dc . = - K'- T dx C 0 ACTUAL JOINT IDEALIZED JOINT plate connector Fictitious adhesive of negligible ood thickness T I I I | l | I I l d7 § ‘ L 3; h [u ) l x i l .1). .1 / i WLij>-\_ - —-9- dc) (TX (2.61)) EL 0 ZAL AL and 2 U - 2[ dc d ] dc .- v 2EC v _ _A_. L 2. _ EC 0 Ole dx (2.6c) A change 6 UT in the total strain energy corresponding to a change in the stress do by 6 do can be expressed as = + . 6UT 6UT +6U0’L 6U0’C (2 7) The individual variations can be evaluated as follows, if terms involving squares and products of the infinitesimal variations and of their derivatives are neglected. bt .53). [L [ ,Cdd dx+50’:)2_(51219;).J dx =% (Ate) L 14([j_gg/)2H7f%:2di®d50’ _(;C)Z)]d L bt A d d6 =.<—:—>Z f 2 .SEC .36 w 20 In a similar manner AL [L AC P 6UUL=2T 0 (AL UC'TXL) 60} dx (2.7b) L and A [L 6UdC = 2 fi— 0 6C 60; (IX (2.7C) . . . dédc . . Now in order to eliminate dx from Eq. (2.7a), the use of integration by parts is made, i. e. , L L L d O’C— déO; dx = uv / - j v du dx dx o o o d d whereu=—gc-, dv= —6—Q’-9-e dx dx L L -_- Lg}; 5 0’ / ‘ f __O_’£‘... 5 O’c dx dx C ,o 0 dx3 but 6 do = 0 at x = 0 and x = L because both the actual and the varied stress states must be in equilibrium with the applied loads there. Therefore, L j d dc C160": dx O/L-d—TC— 66 dx 0 dx dx Thus L z 6UT:%21)G—chc%zqc 5024.. L +2 ——f(—— -—)60;dx O L E] dcédcdx 0 21 L bA2 c12 0’ 2AC A 0' P = I -2 --—C- ——E* + ( C C - ) Ac + 2 0’.) 5 chdx (2.8) EC Now making use of the principle of minimum complementary energy which state s that 6 (U' + V') = 0 for equilibrium, and noting that 6V' = 0, as the reactions are unyielding, and U' = U, as the materials are assumed to be lineary elastic, one obtains 6U=0 Therefore, for any arbitrary variation 6 0; it must be that bAé dz O’C A P Ac - __c____ _ __ = 2 Gt dx. + 2 ELAL (Acdc 2 )+ ZEC dc 0 dz Uh 3 _ Gt 1 1 Gt where k2 = ( + ) ELAL ECAC b Solution of Differential Equation (2. 9) The differential equation (2. 9) can be solved by well-known methods of ordinary linear differential equation theory. The solution can be written in two parts: (TC/complementary and (ye/particular. Then do = Oé/complementary + Og/particular dC/complementary = A cosh kx + B sinh kx 0’ E. C 2(ALEL 'l' ACEC) /particular = + 22 Therefore, Ec 2(ALEL = A + ' + 0:: cosh kx B Sinh kx + AcEc) where A and B are constants of integration and are determined from the following two boundary conditions 1.0’ C O atx=0 P 2. O’C-Z—A'; atx-L From the first boundary condition: Ec P 2(ELAL + EcAc) o=A+ Therefore Ec 2(ELAL + ECAC) A = p (2.10) From the second boundary condition 1 [ _§>_ + Er Sinh k L 2AC 2(ELAL + ECAC) (Cosh kL - 1)] Finally ' P EC Sinh kx 1 EC = _[ -————— Coshkx+ . —+ (Cosh kL-l) 0; 2 ETAT Sinh kL AC ETAT l + E A ] T T where (ALEL + ACEC) = A E EXPERIMENTAL INVESTIGATION Two techniques of experimental stress analysis; namely, PhotoStres: anda strain gage technique were used to determine the stress distribution in the metal plate. The PhotoStress analysis was used to obtain an overall picture of stress distribution in the entire plate while strain gages were used to give accurate and reliable strain measurements at a network of points. PhotoStres 5 Analysis The PhotoStress technique of stress analysis has been extensively used in the past few years to determine the surface strains on metals as well as non-metallic surfaces. It has also been demonstrated by Agostino e_:_t ail. (1955) that this technique yields a good measure of interface strains for both elastic and plastic deformation of the metal part. The PhotoStress technique employs the birefringent properties exhibited by certain materials when strained. This birefringence is directly proportional to the intensity of strain. The surface to be stress analyzed is coated with a special transparent plastic. The bi— refringence is observed and measured with polarized light in a specially designed instrument called a reflection polariscope. An important advantage of this technique over other means of strain measurement is that the strain is obtained over the entire coated area simultaneously. * Trade name for birefringent coating. 23 24 Test Specimen The test specimen consisted of two pieces of nominal 2" x 4" structural grade Douglas fir, butted together, and fastened by two l6—gauge, 2%" x 7%" metal plate connectors, one on each face of the joint. The shape of the teeth punched in the plate is shown in Figure 5. The overall length of the test specimen was 26". The details of specimen are shown in Figure 6. The joint was made in the laboratory by pressing the teeth in with a hydraulic press. Two PhotoStress sheet plastics (sheet type S, K factor 0.083 and thickness .121 i . 002 in.) of approximately the same size as the metal plate were bonded with reflective cement onto each of the plates. The outer surfaces of the metal plates had been sandblasted before they were attached to the wood. Sandblasting was found necessary to allow the cement to transmit the surface strains from the plate to the plastic. This fact was revealed by a test made on a specimen bonded with a clear cement on an aluminum painted surface. Reflective cement was used to give a better reflecting surface and hence a clear fringe pattern. The effect of sand blasted surface with reflective cement was remarkably good in revealing the necessary details of isochromatics and isoclinics. The PhotoStress plastic was made in approximately the same shape as the plate with punched holes similarly located. Considerable difficulty was encountered in making the intricate holes precisely. It was found in preliminary trials that the plastics must be made to con- form to the shape of the specimen precisely for accurate measurements. 25 II 21 Cozlumn3N bers5 1 Milt “iii \\~Wire-gages ' te ate 2 Zirfwlrliilc lifi slbclliiiic s \l \7 2:25:12.” . MM] 1 1 n S l \bFoil-gages 4 . fluguguguflngufl 6 i7 fluUnflu nfluUnfl 7.7;" E Z (3) 8 m 9 l U l U H U l 10 u M 1311 U 1 12 13 MM ll U fl Fig. 5. Shape of the punched teeth in the metal plate connector and location of strain gages. 26 5,000 LBS 2 7/d'X1 7/d‘ METAL- PLATE counscron mm 21 EFFECTIVE TEETH -3 (O;- QIXIOJSL -3 (0;— Oémo, 23.1 so 40 so 20 0 " :TRucTuIm. GRADE oooeus rm 5, 00 LBS FIG. 6 TEST SPECIMEN AND PRINCIPAL STRESS DIFFERENCE VARIATION ALONG COLUMNS 2 AND 5 27 Te sting Pr oc edure The specimen was mounted in a Baldwin testing machine and loaded in uniaxial tension. Figure 7 shows the equipment used for testing. The load was applied in increments of 500 lbs. The iso- chromatic (locus of points with constant principal stress difference) patterns were observed through a large field PhotoStress meter as loading proceeded. The loading was continued until good isochromatic patterns were observed, which was around 5000 lbs. (approximately 5/8 of the ultimate load). The data were taken photographically. Six isochromatic pictures, with o. (angular parameter of analyzer) ranging from 0 to 750 were taken at intervals of 150. To obtain a good isoclinic (locus of points with same principal stress directions) pattern, nine isoclinic pictures with parameters ranging from 0 to 800 were taken at intervals of 100. The PhotoStress meter was placed at a distance of 5 feet from the specimen. Ektachrome, type-B, high-speed film was used in a 35 mm camera. The shutter was opened for 11 seconds at an f 22 lens opening. A 135 mm telephoto lens was used. The photographs thus obtained gave the necessary detail. A point by point measurement of isochromatic fringe order was also made at a network of 78 points on the plate. The data from the photographs were transferred onto sheets of paper by tracing fringe as well as isoclinic patterns. Results and Discussion of PhotoStress Analysis Stres s Magnitudes The results of stress distribution at a finite network of points as determined by the PhotoStress Analysis are given in Table I. The location and numbering system of the points is shown in Figure 5. 28 Fig. 7. Test apparatus for PhotoStress study. Fig. 8. Test apparatus for strain gage investigation. 29 Table 1. Principal Stress Differences at a Network of Points in the Metal Plate Connector, Loaded in Tension. (System of Point Designation is Shown in Figure 5) Columns Rows 1 2 3 4 5 6 1 16,900 14,810 14,810 14,810 23,202 16,900 2 16,900 16,900 -6,350 -6,350 16,900 16,900 3 14,810 14,810 14,810 14,810 14,810 14,810 4 -10,590 -10,590 -10,590 -10,590 —10,590 -10,590 5 40,250 40,250 40,250 40,250 40,250 40,250 6 -2,118 -2,118 -2,118 -2,118 -2,118 —2,118 7 46,500 46,500 46,500 46,500 46,500 46,500 8 -2,118 -2,118 -2,118 —2,118 -2,118 -2,118 9 40,250 40,250 40,250 40,250 40,250 40,250 10 16,900 -10,590 -10,590 -10,590 -10,590 16,900 11 16,325 19,000 19,000 19,000 16,650 14,620 12 16,900 -8,490 -8,490 -8,490 -8,490 16,900 13 14,810 14,810 14,810 14,810 14,810 14,810 This table was prepared using photographic as well as point by point measurement data. The following equation as given by Zandman (1959) in his booklet Photostress--Principles and Applications, was used to calculate the principal stress differences. (6' dz): 5—_L l + p. 0.). . . where 6 = n). :l: T8_0- = relative retardation n = the fringe order 0. = angular parameter of analyzer 30 A = wave length of the monochromatic light = 2. 27 x 10-5111. E = Modulus of elasticity of steel = 30 x 106 psi. u = Poisson's ratio for steel = 0.25 t 2 thickness of plastic = 0.1195 in. k = strain optical coefficient of the plastic = 0. 09. Thus substituting in the known values, (61 - Ugl=1119.9x106 6 psi. The principal stress differences are plotted in Figures 9, 10 and 11 at the network of selected points. For direct comparison, ( 61 - 0;) are also plotted in Figure 6 along the plate length. From Table I and Figures 9, 10 and 11, it is clear that principal stress differences are nearly symmetrical about the axis of loading and normal to it. The variation of ( 61 - 62) in the transverse direction is negligible. This indicates that there was no appreciable eccentricity in loading. The principal stress differences at the central strips are maximums and decrease rather rapidly towards each end along the'axis of loading. The maximum stress difference was 46, 500 psi at the central strips while minimum was 14, 810 psi closer to the ends. The stresses at the central strips along row 7 were calculated by dividing the total axial load on one plate by the effective cross- sectional area of the plate. In Table 11 these stresses are compared with the measured stresses. It is clear from this table that measured stresses are influenced by slight variation in the E and u combination. The percent difference between measured and calculated values ranged from 9. 61 to 42. 2% assuming that the measured values were correct. N 024 _ wZZDJOU 02.0.3. wmozmmwmua mmmmhm .._m30 mmwmkm 954 m. .07.. b.x.55 was... .8 .3: 8%.? m. n. m. m m n :W _.._«_E.T ,. 66 8.9 :6. 0| X (SQI)OVO'1 46 are discussed here, as very close agreement in stress distribution existed between the two tests. Figures 2.0 and 21 present the same data in two different ways as obtained from the wire-type gages. Figure 20 shows the stress variation along four different rows as indicated. Each row, normal to the direction of loading, consisted of a set of six gages. The first set at the end of the connector indi- cated very small strains which were in tension for smaller loads and became compressive for higher loads. This was attributed to the warping tendency of the metal plate as the load was increased. Figure 21 shows the same data as presented in Figure 20 except that the stress plot is made at six different colmnns along the direction of loading. This plot indicates the existence of eccentric loading in testing which in most tension tests is unavoidable. The stresses varied from maximums along column one to minimums along column six. The stresses obtained from the strain measurements by foil- type gages mounted at the bases of the miniature cantilever teeth are presented in Figures 22 and 23. Here also the stresses increase gradually from minimum near the ends to maximum along the third row. It is also apparent from these curves that teeth along the sides of the joint take comparatively less stress. A special note should be made here of different scales used in representing the stresses as obtained from wire-type and foil-type gages. Generally, the stresses at the bases of cantilever teeth were higher than those at the corresponding strips. 47 29sz mmaoood E mobmzzoo Eqdbflwz mi 2. 20.59.55 mmmEm ON .0... N n n _ J— ma gnaw! monbs Jed son) seams 48 zo.mzw._.. mm..000.m ._.< mOHowzzoo 9215.44th m1... 2. zo.._.Dm_m._.m_o wmwmhm _N .OI m 0 ¢ g<_wt(u;:u!810mm 19d SQI) $33313 49 205sz 8.. 80.... E mobmzzoo mzduflmz m1» 2. EB» 65:23 no mwmqm m5 2 mmmmmEm NN .9“. 7 J _ _ 4.]: .3 2 313::E a E: c c C . O m SQI) 88381.3 0 (D ‘N ooowlfl — WNW Tooow o m omqaug 310an ad 50 29sz mmjooow E mobmzzoo 33345: ME 2. EH: $53.55 no 82m NE 2 3.0.8um mm .0... 4_ Z 4 m _ O to 8 gum! ambs Jed SQI) ssaais RESULTS AND DISCUSSION Comparison of Theoretical and Experimental Results The results of the Difference Equation solution for the no-slip case are presented in Figure 24. A set of average experimental points as obtained by strain gage measurements at four different locations along the connector are also shown. Theoretical curve for the stress in the connector increases from zero at x = 0 to a maximum at x = L. The experimental points are widely scattered on both sides of the theoretical curve. This probably can be expected as the analysis is only an approximation to the actual behavior of the joint. A part of this discrepancy might be attributed to the facts that the joint is not rigid as has been assumed but semi-rigid and. that the friction between connectors and wood, grain orientations of wood, and moisture content of wood have not been considered. It must also be mentioned here, however, that a difference equation assuming equal slip of all the teeth in the connector was derived and solved in a similar way. The slip function was obtained by fitting a least squares polynomial to a set of experimental points obtained from load-gross slip tests. The result of this consideration, however, did not improve the analysis, but on the contrary predicted rather absurd results. This probably was due to the erroneous assumption that each connector slips the same amount. Actually the connectors do not exert the same force on the wood and hence the slip is not the same for each connector. 51 .co330m ”83.350 028.833 9.3 .3 wouofluoum mm 33mm.” Hmofiouooau paw Hmucoefiuomxo mo GOmEmmEoU .wm .mfim mogog 5 ugon 05 mcofim x oocmumflfl a: no mm ma» mm mm Q 9 .fi. \ N .O :35 £\o "Q d: .. . . .Qm .QN d "X pom If x ”Egon .35 um oonoh n .m J‘l 3.310 + x mm; .35 33.9. x am; :80 33.0. n a? z a s m o .o e \ Imr Imwl ml 53 The theoretical curve obtained from the analysis of the principle of minimum complementary energy is shown in Figure 25. The equation obtained by this analysis for the normal stress 0:, at any point in the connector wa s 0’. P : -1.875 Cosh 1.155 x+1.94 Sinh1.155 x + 1.875 Four sets of readings as obtained by strain gage measurements are also shown. These points are plotted from the strain gage data taken randomly at different loads and averaged over each row. Here. again the experimental points are widely scattered on both sides of the curve. The interesting point to note, however, is that even with the idealized assumptions that had been made in carrying out this analysis, it gives results that are not too much different from those obtained by strain gage measurements. It is, of course, clear from Figure 25 that the stresses in the metal plate are not uniform as assumed in normal design practices. The maximum calculated stress was 2.4 times the average stress. The theoretical curves in both cases predicted the maximum stress in the middle of the joint, however, the experimental readings gave the maximum at the second gage point from the middle (row five). This was true only up to a certain load. As the load was increased up to the ultimate, it was observed that the maximum stress did occur in the middle. This could have been due to the relaxation in the load trans- ferring characteristics of such connectors. Since the fourth row of the connectors are not as properly imbedded into the wood (because of their proximity to the end of the wood pieces) as are the rest of the connectors, this row would be more susceptible to slip than the rest. Consequently, it would not be able to take its full share of the load. But as the load is increased to a certain value near the ultimate, the slip progresses from the middle towards the end. Thus the stress in. middle row eventually becomes more than the one next to it. '54 .>m.~o:o >nmuco83m§oo SEEMS mo oawocwum 23 >3 pmuoficmum mm. gamma Hmofiouooafi pad 33083098 «0 GOmCmmEoU .mN .mfm maxed“ CH “Eon 05. wcoam on 005339 03v m.m o.m m.N o.N m4 OJ m.o \ H 2...; J +£24.26 8.. +5.42; :80 m3..- uniob Room; .N u mmmnv>< d/°p Ill'Allllll]|l||l[l| |' Hun O "N 1 W W\/\ 55 The results of both of the theoretical analyses are plotted in Figure 26. It is interesting to note that both theoretical curves are rather close and predict a similar stress distribution. The equation of the result predicted by difference equation solution was 02: P = - 0.04526 Cosh 1.59 x + 0.04551 Sinh 1.59 x + .04526 while that predicted by the principle of minimum complementary energy was -Q—I:-C- = -1.875 Cosh 1.155 x +1.94 Sinh 1.155 x + 1.875 It must be emphasized here that these equations are applicable only within the elastic range. Their use, therefore, must be limited to this range. The maximum calculated stress, which occurred in the middle was 2.4 times the average stress. A more uniform stress distribution might be obtained by not punching the middle row and reducing the length of the connectors with a proportionate increase in their number. Evaluation of PhotoStress Analysis The results of the PhotoStress analysis as presented in Table I and Figures 9, 10 and 11 for a uniaxial load of 5, 000 lbs indicated that the stress in the metal plate has a rather sharp stress gradient. The princi- pal stress difference varied from a maximum of 46, 000 psi at the center row to a minimum of 14, 810 psi closer to the ends. An approximate separation of the principal stresses when compared with the calculated stresses at the middle row indicated a wide variation (9.61 to 42. 20%) between the measured and the calculated values. This variation was jointly accounted for by stress gradient effect, stress concentration effect, experimental error and possible variation in the E and P values of the plate. 56 dogma—Um 023.33% 0:» can >muoco >nmucogo~mnfioo SEEMS 05. Ho $53.95 05 go: begun—no mm nouoonaoo 3.3m #305 on» a. mofiafifihmg mach—m .om .mwh meson“ Etufion 05 mcoam x condumwfl 04V m.m o.m m.~ o..~ m." cg m.o Gofimavm 0 one numflnv {illilllll 1\«\\ l ll.lll!|l . Hummuocrb haamfi .N \ \ />muocm >hd§053m500 d/So The direction of the principal stresses at the central unpunched strips were in close agreement with the expectation. However, they were rather complicated,away from the center. A number of points in hydrostatic state were also observed. The magnitude of the stresses and their directions were found symmetrical about the axis of loading. For PhotoStress analysis to be really effective as a tool of stress analysis in a multiconnected body of intricate shapes, experimental separation of principal stresses seems to be highly desirable. For this purpose an oblique incidence meter is a necessity. SUMMARY The main objective of this study was to conduct a theoretical and experimental investigation of stress distribution in a metal-plate- connected timber joint. The theoretical investigation consisted of two different methods. One utilized the discrete approach of a Difference Equation while the other used the continuous approach of the principle of minimum comple- mentary energy. A second order difference equation was derived and solved for a general case. The results for a particular case of the metal plate connector were calculated and plotted together with experimental results for comparison. The principle of minimum complementary energy was used to derive a second order ordinary linear differential equation for an idealized case. The metal plate connector was treated as if glued to the surface of the wood by a fictitious adhesive of negligible thickness. The dif- ferential equation thus obtained was solved with appropriate boundary conditions. The results for a particular case of metal plate connector were plotted for comparison with the experimental results. Two different methods of experimental stress analysis were used; namely, PhotoStress analysis and a strain gage technique. The Photo- Stress analysis provided,qualitatively, the overall pattern of stress distribution in the entire plate. A set of bar graphs for the principal stress difference ( 61 - 0;) along the various rows of teeth in the metal plate connector were plotted. A symmetrical stress distribution was obtained. The isoclinic pattern for a portion of the plate was thoroughly examined. From this isoclinic pattern stress trajectories were drawn. 58 59' The strain gage technique was used to obtain accurate and reliable values of strain in the metal plate to verify the theoretical results. The particular results of both of the theoretical analyses predicted reasonable agreement with the experimental values, and similar patterns of stress distribution. The maximum stress as calculated by comple- mentary energy method was 2.4 times the average stress. The general results of the difference equation solution can be used to calculate stresses in metal plate connectors as well as riveted and bolted joints. Similarly, the results of complementary energy method can be used to calculate stresses in metal plate connector as well as in adhesive and welded joints. CONCLUSIONS AND OBSERVATIONS The following conclusions and observations are based on the results of this investigation. 1. The results of the difference equation solution as well as that of the principle of minimum complementary energy predict reasonable agreement with the experimental results. Either method can be used to calculate stresses in the metal plate connector. 2. The difference equation solution can also be used with at least equal accuracy for riveted and bolted joints. Similarly the results of the principle of minimum complementary energy are equally applicable to adhesive as well as welded joints. 3. The stresses in the metal plate are not uniform as assumed in the normal design practices. The maximum calculated stress in the connector was 2.4 times the average value. 4. A set of distruction tests made in tension resulted in tearing failure of the plate in the center of the joint. If the middle part of the plate were not punched, the strength of the joint should be greater and also a more uniform stress distribution should result. 5. The PhotoStress analysis provided an overall pattern of stress distribution in the entire plate. The variation between measured and calculated principal stress difference ranged from 9. 6% to 42. 2%. The results of this analysis were, however, incomplete as the shape of the punched plate was too complicated for analytical separation of the principal stresses. The equip- ment for experimental separation (oblique incidence meter) was not available. 60 SUGGESTIONS FOR FURTHER STUDY 1. Carry out theoretical analysis using Plastic Analysis by assuming a criterion of yielding and a mechanism of hinge formation. 2. Determine the dynamic behavior of the joint under different cycles of loading and aging. 3. Establish long term aging characteristics of such joints. 60a 10. 11. REFERENCES . Boyd, J. S. (1954). Secondary Stresses in Trusses with Rigid Joints, Special Application to Glued Wooden Trusses. Ph. D. thesis Iowa State Univ. (unpublished). Boyd, L. L. (1959). Design of Semi-Rigid Timber Joints. A. S. A. E. Paper No. 59-828, presented at 1959 Annual Meeting. Brown, H. P., A. J. Panshin and C. C. Forsaith (1952). Text Book of Wood Technology, Vol. II. McGraw-Hill Book Co. , Inc. , New York. . Duffy, J. (1961). Effects of the thickness of birefringent coatings, pp. 74-82. Experimental Mechanics, March 1961. . Durelli, A. J., E. A. Phillips and C. H. Tsao (1958). Introduction to the Theoretical and Experimental Analysis of Stress and Strain. McGraw-Hill Book Co. , New York. . Felton, K. E. and H. D. Bartlett (1963). Punched Metal Truss Plates Used in Timber Joints. A. S. A. E. Paper No. 63-431, presented at 1963 Annual Meeting. . Freudenthal, A. M. (1950). The Inelastic Behavior of Engineering Materials and Structures. John Wiley and Sons Inc. , New York. . Frocht, M. M. (1941). Photoelasticity,Vol. I. John Wiley and Sons Inc. , New York. . Goldberg, S. (1958). Introduction to Difference Efluations. John Wiley and Sons Inc. , New York. Granholm, H. (? ). A Summary of Report on Composite Beams and Columns with Particular Regard to Nailed Timber Structures. Harris, C. O. (1962). The Analysis of a Parallel-Type Structural Connection by Means of a Difference Equation. Unpublished manuscript. 61 12. 13. 14. 15. l6. 17. 18. 19. 62 Hetenyi, M. (1950). Handbook of Experimental. Stress Analysis. John Wiley and Sons Inc. , New York. Hoff, N. J. (1956). The Analysis of Structures. John Wiley and Sons, Inc., New York. Hrennikoff, A. (1932). Work of Rivets in Riveted Joints. Am. Soc. of Civil Engineers Proceedings Nov. 1932. Joy, F. A. (1961). Methods for Testing Metal Connector Plates Used in Trussed Rafter Construction. Engineering Experiment Dept. , Penn. State Univ. Kuhn, Paul (1956). Stresses in Aircraft and Shell Structures. McGraw-Hill Book Co. , Inc. , New York. Muckle, W. (1949). The distribution of Load in Riveted Joints. The Shipbuilder and Marine Engine-Builder, pp. 225-228, April 1949. Post, D. and F. Zandman (1961). Accuracy of Birefringent Coating Method for Coating Arbitrary Thicknesses, pp. 21-32. Experi- mental Mechanics. Jan. 1961. Zandman, F. (1959). PhotoStress Principles and Applications. Instruments Division, The Budd Company.