AN INVESflGATIC‘N OF THE MECHANICAL PRQPEM‘ES QF NA‘iL—GLUED WOOQJ’L‘I’WOQD TRU$SED GiRDERS ‘f‘hnsis fa: Hm Dogma of M‘ S. MICHSGAN STATE UNIVERSITY Richard M. Vodka @961 )5. E815; ' .051» ""¥“l. “' 1&5}! fr'» “l. J- L --_.__;| AN II‘NESIEATION OF TIE IJECFANICAL PROPERI‘ES OF NAILPGLUED HOOD-PLYHOOD TRUSSED GIRDERS by Richard M. Voelker AN ABSTRACT Submitted to the College of Agriculture Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements fer the degree of LuerR OF SCIENCE Department of Forest Products 1961 ABSTRACT The purpose of this study was to determine the mechanical proper- ties of nail-glued, wood-plywood, trussed girders, and to compare these with conventional nail-glued, wood-plywood girders. IA number of model beams of each type were constructed using four section depths. The deflection of these beams was measured under static bending loads and the empirical data gathered was utilized in conventional engineering equations to calculate the strength and stiffness properties of the beams. Similar tests were conducted for both types using fUll- scale teams of the same section depth and span. Stiffness factors ( a function of the modulus of elasticity and section properties) were calculated for the model beams and it was f ound that the Type-A beams (with diagonal stiffeners) were as much as 40 per cent stiffer than the Type-B beams (with vertical stiffeners only). This superior stiffness was also reflected in the full-scale testing where the Type—A beams were found to be 16 per cent stiffer than the Type-B beams. All the model beams failed due to horizontal shearing stresses in the plywood web. This failure occurred at a load far in excess of that producing the allowable design deflection at mid-span (generally accepted as being l/360th of the span). The horizontal shearing stresses in the plywood web, calculated at failure, were many times greater than the allowable design horizontal shearing stresses for plywood as given by the Forest Products Laboratory and the Douglas Fir Plywood Association. Because of this, it was concluded that the allowable design deflec- tion ofl/BéOth of the span should be used as the governing design factor, and not the allowable design horizontal shear stresses recom- mended for the plywood web. Buckling of the plywood web became more critical as the model beam depth increased. It was concluded that the test apparatus must be modified to prevent lateral buckling in future tests, in order to obtain valid test results. It is recommended that strain gauges be utilized in future testing of trussed girders to determine the amount of truss action within the diagonals. AN INVESTIGATION OF THE hECHANICAL PROPERTIES OF NAIL-GLUED WOOD-PLEFOOD TRUSEIED ORDERS by RIcmmD m. VOELKER A THESIS Submitted to the College of Agriculture Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Forest Products 1961 ACKNOWLEDGMENTS The author wishes to express his sincere thanks to Professor Byron Radcliffe, under whose inspiration, supervision and unfail- ing interest this study was undertaken. The author is also greatly indebted to Dr. Alan Sliker for his guidance and assistance in editing the manuscript. Sincere thanks is also due Mr. Harry Johnston for help in fabricating and to Mr. H. F. Law for his willing assistance in fabricating and testing. TABLE OF CO NTENTS AC KNOYI‘ILEDGEI‘JE IE S o o o o o o o 0 LIST OF TABLES . . . . . . . . . LIST OF FIGURES . . . . . . . . INTRODUCTION . . . . . . . . . . History of "Built-Up" Purpose of the Study . . FABRICATION . . . . . . . . . . Model Beams . . . . . . . . Full-Scale Beams . . . . . TEST IIJG O O O O O O 0 O O O O 0 Model Beams . . . . . . . . FUIl-Scale Beams. . . . . . AblALYS IS OF DATA 0 O O O O O O 0 Model Beams . . . . . . . . Beam Test Performance Shear Deflection . . . Modulus of Elasticity Moment of Inertia . . Stiffness Factor . Extreme Fiber Stress in the Flanges Horizontal Shear Stresses Full-Scale Beams . . . . . Beam Test Performance Shear Deflection . . . Modulus of Elasticity Mbment of Inertia . . Stiffness Factor . . . Extreme Fiber Stress in the F1 nges Horizontal Shear Stresses DISCUSSION OF RESULTS. . . . . . Model Beams 0 o o o o Full—Scale Beams CONC LUS IONS AND REC OLME NDAT IO NS BIBLIOGRAPHY . . . APPENDIX . . . . . . . . . . . O Plywood I-Beams O O O o o 0 iii Page ii iv O‘NN N] 13 13 18 22 22 LIST OF TABLES Table No. Page I l’IOdel Beam Performance 0 o o o o o o o o o o o o o O 24 II Shear Deflection Component of Total Measured Deflection (Model Beans) . . . . . . . . . . . . . . 26 III Mbduli of Elasticity of Flange Compression salnples (I‘VIOdel BeaInS) O O O O O O O O O O 0 O 0 O O O 61 IV Theoretical Moments of Inertia (Model Beams) . . . . 28 V Comparison of Stiffness Factors (Mbdel Beams) . . . 35 VI Extreme Fiber Stress in Flanges at the Allowable Deflection (1/360th of the Span) (Model Beams) . . . 37 VII Horizontal Shear Stresses in the Plywood Web (I‘ibdel BeamS) O O O O O O O 0 O O O I O O O O O C O 39 VIII Comparison of Stiffness Factors (Full-Scale Beams) . 45 IX Extreme Fiber Stresses in the Flanges at the Allowable Deflection (l/360th of the Span) (Ml-scale Bealns o o o o o o o o o o o o o o o o o o 45 X Horizontal Shear Stresses in the Plywood Web at the Allowable Deflection (l/BoOth of the Span) (Full-Scale Beams) . . . . . . . . . . . . . . . . . 46 Figure No. l. 13. 15. 16. 17. 18. 19. Model Beam Full-Scale Full-Scale Full-Scale Model Beam Mbdel Beam Model Beam LIST OF FIGURES Details . . . . . . Beam Details. . . . Type-A Beam . . . . Type—B Beam . . . . Test and Apparatus Testing . . . . . . Buckling Under Load 8-inch Flange Sample Compression Hydraulic Testing Floor Full-Scale Beam Test and Apparatus . . . . . . . Test thal Rods and Blocking Restraining Buckling . . Load-Deflection Curves for Model Beams . . . . . Shear and Bending Moment Diagrams for l/L point Loading Of LIOdel Beams 0 o o o o o o o o o 0 Theoretical Equivalent X-Section . . . . . . . . . . . . Model Beam Stiffness Factors . . . . . . . . . . . . . . Load Deflection Curves for Full-Scale Beams Shear and Bending Moment Diagrams for Full-Scale Beams . Shear Failure of Model Beam. . . NIOdel Beam Failllre o o o o o o o Page 10 11 ll 14 15 16 17 19 2O 21 23 30 32 36 40 42 51 52 AN INVESTIGATION OF TEE MECHANICAL PROPERTIES OF NAIL-GLUED NOOD-PLYNOOD TRUSSED GIRDERS by Richard M. Voelker INTRODUCTION History of "Built-up" Plywood I-beams The use of the "built-up" or laminated structural wood members was first conceived in the early 1900's in Europe with Otto Hetzer of Weimar, Germany being credited as the originator.ll* Although Hetzer construction, as it was known then, dealt mostly with laminated arches, the advantages this method offered were soon realized and adapted to use in laminated beams with rectangular, I and double-I cross-sections. It was soon discovered by Stoy, Egner and Erdmann that the use of the I-beam section resulted in a savings of 35 per cent or more in construction materials when compared to conventional rectangular sec- tions. In 1937 Wills published design information which enabled engin- eers and builders to determine the required dimensions of wooden I-beams for various support and load conditions. In most of these earlier in- stances the adhesive used to laminate these beams was casein glue.ll There was little interest in this country in built-up beams until shortly after the First World War. ‘In 1919 the National Advisory Committee for Aeronautics sponsored a number of investigations on the use of wood. One of these projects dealt with the use of laminated beams in airplane construction. These beams were constructed by gluing pieces of wood together and then machining the assembled beams to the 11 desired I cross-sectional shape. The emphasis in this application *Literature cited in Bibliography was to provide adequate strength where needed and to reduce the amount of material and weight. The earliest structural use in a building reported in the United States for plywood I-beams was in 1942. The RCA Manufacturing Company of Camden, New Jersey had a 125,000 sq. ft. warehouse constructed which utilized 198 plywood girders 36 feet long. These girders were fabricated on the building site. Cement-coated nails were used to attach the plywood webs to the lumber flanges. When the government occupied the warehouse in 1952, after ten years of service, the beams were found to be in excellent condition, had not sagged and had not required any maintenance.9 With the advent of the Second World War, and the resulting shortage of solid structural materials, interest in laminated wood products was further stimulated. The Forest Products Laboratory of the United States Department of Agriculture, under the supervision of the Aero- nautical Board, was charged with the responsibility of formulating de- sign equations, substantiated by test data, that would facilitate the use of plywood and built-up sections in structural members. In 1943- 1944.the Forest Products Laboratory conducted extensive studies on 6 7 - 5’ ’ Based on these studies recommen- plywood box-beams and I-beams. dations were made concerning face-ply grain orientation, stiffener spacing, buckling and cross-sectional design. It was found that significant increases in web shear resistance were obtained by re- ducing the spacing of the stiffeners; and it was recommended that the minimum stiffener spacing, compatible with economy, he used.5 The conclusion was made that I-beams generally use plywood web material more efficiently than box-beams. Test results indicated that for equal panel sizes and section properties, an I-beam with a single web was usually significantly stronger in shear than a box-beam with two webs, each half as thick as the single web of the I-beam.6 It was fur- ther demonstrated that box-beams or I-beams having the face grain of the webs at an angle of 45 degrees with the axis of the beam were more effi- cient than those with either 0 or 90 degree grain orientation. It made little difference whether the grain orientation was vertical or hori- zontal as the ultimate shear stresses were nearly equal.7 Buckling of the plywood web proved to be a critical design factor.7 A portion of the research done for the National Aeronautical Board concerned shear deflection. It was fbund that shear deflection was an important consi- deration in designing plywood I-beams and box-beams. The magnitude of the deflection attributable to shear was found to be inversely prepor- tional to the unsupported length of span. Formulae were derived by which shear deflection could be calculated.12 After the War was concluded, further research by the U. S. Forest Products Laboratory was conducted to obtain data on the effect of var- iations in lamination thicknesses, joints within the laminations and the location of various size and types of defects on the strength and failure characteristics of plywood beams.ll There was also published at this time, methods for calculating the strength and stiffness of plywood and suggested working stresses fer plywood design.lo Based on the pioneering efforts made by the Forest Products Laboratory, other publications soon appeared from various sources. Because of the vital interest the Douglas Fir Plywood Association has in built-up construction, it soon published a design handbook which pre- sented to the engineer and architect useable formulae and design cri- teria.2 The DFPA also published a set of design specifications embody- ing the latest design procedures and methods for plywood I-beams and box-beams.3 With the proven use and general acceptance of plywood I-beams as structural members, new methods were developed to facilitate fabrica- tion. Nailing proved a practical means to secure adequate glue line pressure while fabricating.14’l7 With the advent of "nail-gluing", the plywood I-beam became practical enough to be used in residential con- struction. While a significant amount of work has been done on built-up ply- wood beams, the incorporation of diagonals to produce truss action is an idea which merits further investigation. The alteration of the basic plywood I-beam design in such a manner, indeed opens a relatively new area for plywood beam research. Purpose of the Study The purpose of this study was to evaluate the strength and stiffness properties of plywood I-beams with diagonal stiffeners as compared to conventional plywood I-beams with vertical stiffeners. Nbdel beams were constructed and tested in an attempt to deter- mine (1) if there was any increase in stiffness resulting from the use of diagonals; (2) what, if any, relationship existed between the beam depth and the strength and stiffness properties of the beams; and (3) to gather empirical information on the type of beam failures encountered, the loads at failure and the behavior of the two types of beams when loaded in excess of the proportional limit. Full-scale beams were constructed and tested in an attempt to verify the stiffness properties observed in the model beams. Because of limitations of the testing apparatus used, no information could be gathered on the proportional limits of the full-scale beams or the type of failure. FABR ICAT IO N Model Beams Sixteen, eight fect span model beams were constructed. Of these, eight were Type-A and eight were Type-B. (Fig. 1). Within each group of eight, two beams were constructed with a nominal sec- tion depth of eight inches, two with a nominal depth of nine inches, two with a nominal depth of ten inches and the remaining two were a nominal eleven inches deep. The flange, diagonal and stiffener members were constructed of No. 1 grade Douglas-fir and western hemlock. Grade A-A exterior type, l/L inch, sanded Douglas-fir plywood was used for the webs. The nomi- nal 1 x 2 inch members were cut from 2 x 6 inch stock and surfaced. All the lumber used contained the typical defects found in construc- tion grade lumber. However, those members containing a great number of defects were restricted to use as vertical stiffeners or diagonals. The per cent moisture content of the l x 2 inch members was determined at the time of fabrication and was found to vary from 6% to 9%. After the components were cut to size, they were ready for assem- blance. The assembly process was essentially the same for both types of model beams. The l x 2 inch members were first laid-out and tacked together. Next, casein glue, meeting U. S. Specification iflM-A-125 and mixed in accordance with the manufacturer's specifications was brushed on the l x 2 inch members. The plywood web was then lightly tacked to this framework. The l x 2 inch members of the opposite side of the beam were then spread with glue and set in place. F oasw E m I WQKN Q3 6.9.93de J I? L L 5-1% 4.! a \\ t. y. .t .r til.‘ 4 [-1 Pl .1 .ll‘, .3 ‘ N\ - I-.. :.o|ll\\ W\ '0!!! y 4 . -il'nullb‘ll‘l'l III III .5 Illlllutfolrl.|l|l‘nlj.|' :I‘ulllllou'.llllil .11 Il.‘ II.:|}I||£|‘lllllij 23¢qu News? , \SRMRJ . A. . a o n _ .s a _ _ L-,-l.----.rl\ wee seesaw at m. _ e / l o J m _ \\ all ONCSKVKKBQ) N... \ \d W e .l . _ W. o o 4!; ~|\ \NQ\§oor\ MERPN L46 \Cmq f V m .3.er Eek $4»ka l. kogxmmn SST W. m .\ “p.39 Nam? \\ \3 cl : P A r - Rum? 7 amt 3e. dfikawxx A m\\\) _ 7\\\\_\ Z4 Z 5x CEQQDNQ r ,, we - wm % , we. _ w 3. - -- %- :t w, -1 \_ i- n \R.NQ ”VAQRVQQV NVSX‘QP‘Q‘ \ \yllM‘Il \‘Whfi \‘Rv W\\k NR“. d . -¢»—-—- 3 Figure 6 Model Beam Testing 15 -._._.—. —_— — _--_- w-._~___._. H—__ _ _ _ _ -_l-. Figure 7 Mbdel Beam Buckling Under Load 1" *""-o 16 17 i .- r-c-/”~-—e~;’/4;—~/'-—~ ‘ , . (swans may #40 I y 3 2 -——~-»- 1114/l/1"/Llf//1/ ijj/fz/ 4WPI/4p1/an0Z/w . ' 3 3 3 l. . ‘ '7‘ l . ., A. ,, 3' —\ r“ i __- _ flu—-.-—.- v——_—a——._———c.-—~-_—-—-u——-_*e v ~'-h-— ’ l_ ~ . n v yvfirfrrirfr f—fT Alanv [Em/v7 V/[u 5/05 ”EU Figure 8 45’ ” flan/6! waz [ _CO/%0A7[IJ/O/\/ 7257 18 Full-Scale Beams The ten full-scale 16 foot beams were tested using a hydraulic testing floor. This is shown in Figure 9 and is graphically illus- trated in Figure 10. A hydraulic cylinder was located every two feet along the length of the beams with the load being applied directly over the stiffeners. There was no load applied over the two end supports. The cylinders were connected in series to assure uniformity of pres- sure. The load was applied in increments of 25 p.s.i. up to 400 p.s.i. of cylinder ram, the maximum output of the testing machine. The piston head area was 2.94 in.2, thus, each p.s.i. represented 2.94 lbs. per cylinder. Deflection readings at mid-span were taken for each increase in load of 25 p.s.i. An Ames dial deflection gauge was placed against the lower flange and the readings taken to the nearest .001 inch directly as the beam deflected under load. The beams were restrained from lateral buckling by means of metal straps secured to the test floor. Steel rollers and blocking were used to assure freedom of movement of the beams as they deflected. (Fig.11) Because the capacity of the hydraulic system was limited to 400 p.s.i. of ram per cylinder, it was impossible to gain information or data as to the proportional limit of the beams or the type of failure that would have occurred. However, the allowable deflection (1/360th of the beam span) of .533 inches was exceeded in each instance and the data obtained gave a good relationship between load applied and deflection measured. This was sufficient to make a stiffness comparison between the two types of beams. Figure 9 Hydraulic Testing Floor 19 2O SENSE a NW ESQ ddw3. ask OF omsmam 1...... . «a.-. _ . ._.. .... .7 .1 53\\ — L, ..\\ : :/I3\ I \ .\\ _M/ll\. 8/ Q33 vb R SENS. 3.36%. gxkxukhwq Rd. N - t _werQF?QJ¥s§§§&§\Qw56NV A; V w. .7 I § 1 ///. fieumcfitpu l\\\ Ill... fwsamvuw§uwkfl Nrrdomfi§uu§\\r Figure 11 Metal Rods and Blocking Restraining Buckling 21 22 ANALYSIS OF DATA Model Beams Beam Test Performance The data obtained from the static bending tests was used to plot a load vs. mid-span deflection curve for each of the 16 model beams tested. A composite plot is shown in Figure 12 with the average curve for each beam type superimposed. Loads producing the allowable deflection (l/BoOth of the span), the proportional limit and failure in each beam are recorded in Table I. The type of failure occurring for each beam was also noted. Shear Deflection Because a gross deflection was measured at the time of testing, it was necessary to compute the amount of deflection attributable to shear in order that the amount due to bending alone could be found. The formula used to compute shear deflection is that advocated by the Douglas Fir Plywood Association3 and verified experimentally by the U. S. Forest Products Laboratory and is expressed as: 2 = mm c (1) GI ds where: D; II shear deflection, in. II total load on beam, lbs. span length, in. a factor determined by the beam cross-section 9 D‘NHV u depth of beam, in. . .l . \1. x- .. n: . x. a. .. .2 V .. .. . \ 4 . .I . . . .1 ..\l . . ... \. \ . N x. l. . . . .. . e a r L .\ .. 2.. i . l . .Mx. .Iu. . .e .L z . f‘ 1 r..- I‘. x .. r _ 37R. .2. SR new NQQ r~ \ 'J —-O '3 O '3 3 . . . , 2 \ {at . . I! x x J J \Jeriixxax \. .l\ .\\ rustlx Cx \(A‘wap\ul.n ll ‘1'. \\ \\ \‘ . ., I i w, w . w -08, udXNWMN%.LNN+ - QCQx _ 09? 9,7/ 0.7/0 7 7549/ I 7" ‘-. ../ 4009V 699w MODEL BEAM PERFORMANCE Load at Allow- Load at Propor— Load at able Deflection tional Limit Failure Beam No. (Lbs.) (Lbs,) (Lbs.) Type of Failure l-A-8 2100 2400 3300 Horizontal Shear 2-A-8 2700 2900 3700 " " l-B-8 1970 2400 3700 " " 2-B-8 2190 2700 4100 " " l-A-9 4060 4300 5200 Horiz.Shear & Buckling 2-A-9 3020 3900 —--- Buckled* l-B-9 2590 3400 4500 Horizontal Shear 2-B-9 2400 3600 4900 " " 1-A-10 3740 3500 4300 Horiz.Shear &.Buckling 2-A-lO 3680 3000 4200 " " " 1-B-10 3840 ---- ---- Buckled* 2-B-10 2820 3000 5300 Horiz.Shear & Buckling l-A-ll 4010 2800 ---- Buckled* 2-A-ll 4810 3400 6000 Horizontal Shear l-B-ll 4250 3800 4900 Horiz.Shear & Buckling 2-B-ll 3300 3800 4500 Buckled *Extreme buckling caused beam to "spring out" of testing machine Table I 25 a coefficient depending on the manner of loading 0 n' C) II the shearing modulus of the Douglas-fir plywood webs, determined empirically to be 117,000 p.s.i. if the face grain of the plywood is either parallel or perpendicular to the span at 15% moisture content,3 p.s.i. I = gross moment of inertia of the section, in.4. The calculated amounts of deflection attributed to shear correspon- ding to a given load for the various section depths are recorded in Table II. These shear deflection values were subtracted from the gross observed deflection corresponding to the total load on the beam (P) in order to arrive at the deflection due to bending alone, for that load (P). These corrected bending deflection values were used to calculate the stiffness factors (BI) for the beams. Modulus of Elasticity Using the data obtained from the compression tests of the flange sections, as previously described, the modulus of elasticity was computed using the equation: E = £23, (2) Derived as follows: Modulus of elasticity (E) = sires: E2? S ran. and: stress (CV) = :22: i Mbdulus of elasticity (E) = deflzction QQQ SHEAR DEFIECTION COI‘IIPOIDJNI‘ OF TOTAL MEASURED DEFLECTION (MODEL BEAMS) Nominal Ac tual Depth Depth Gross* Shear Deflection (h) (h) K I (d5) in. in. in.4 in. 8 7.88 1.36 77.73 .089 9 8.88 1.37 105.86 .084 10 9.88 1.37 138.78 .079 11 10.88 1.34 176.61 .074 *Includes total thickness of plywood web Table II 27 therefore: where: E = modulus of elasticity, lbs. per in.2 P = total load, lbs. A = cross-sectional area, in.2 4C>= deflection, in. The empirical E-values calculated by using the above formula are reported in Table III (Appendix). These values are later used to calculate the stiffness factors (EI) of the beams. Moment of Inertia A theoretical moment of inertia was then calculated for the four beam sections using the formula: 3 3 I = 31.9.1. - 132‘12 (3) 12 12 , where: I = moment of inertia, in.4 t1: total flange thickness including two plys of the 1/4 inch plywood web. in. d1= total depth of the beam section, in. t2= total flange thickness, not including the plywood web, in. d2= total depth of the beam section, minus twice the depth of the flanges, in. These calculated I-values are recorded in Table IV and are used in the calculation of the theoretical stiffness factors. THEORETICAL Ex-TOTIENTS OF ITERPIA (Model Beams) Neminal Actual Section Section I Depth Depth Values* (Inc) (In') (In-4) 8 7.88 74.31 9 8.88 100.96 10 9.88 132.04 11 10.88 167.61 * Includes only two plys of the plywood web Table IV 29 Stiffness Factor Three comparative stiffness factors were next computed for each beam using the previously calculated data. The first of these is termed the theoretical stiffness factor (EI) and was calculated by using the theoretical moment of inertia values recorded in Table IV. These I-values were multiplied by 1.95 x 106, the average modulus of elasticity for coastal type Douglas-fir at a moisture content of 12 per cent.4 These theoretical EI-values are recorded in Table V. The second group of stiffness factors or EI—values might best be termed the actual EIdvalues. They were calculated using the actual de- flection recorded at the time of testing and they reflect the true or actual performance of the beams as loaded. These values were computed from the equation: EI=£A§c A—A S (4) where: E = modulus of elasticity, p.s.i. I = moment of inertia, in.4 2A3: the sum of 1/2 the individual areas located within the bending moment diagram shown in Figure 13, multiplied by the distances (i) of the centroids of these areas from the left edge of the diagram, in.3 (Sometimes referred to as the second area moment theorem) .4 = total deflection measured at mid-span, in. .AS= calculated shear deflection, in. 3O T T .4 ..... 45" v... 24 - 96" Q‘ a p/a Ra: p/e + B? 6% _ + _ _ f/Aé'flfl a Q; . A} — .. P/d_ [£22 Figure 13 Jaw/7 # flaw/M6 /’/O/‘/[/\/f 0mm [0A7 ’/¢ flow/f [Mp/M 0/" Max flaw 31 These EI—values are recorded in Table V. The third and final group of stiffness factors calculated are based on the moduli of elasticity determined earlier from the eight inch compression samples taken from the beam flanges. Because of the variability of these E-values, adjusted moments of inertia (I-values) were next calculated for each beam. These ad- justed I-values would theoretically compensate for the variances en- countered within each beam between the modulus of elasticity of the compression flange and that of the tension flange. This is generally referred to as the method of equivalent sections. An equivalent thickness for the tension flange was computed based on the modulus of elasticity of the compression flange. This was ac- complished by using the fermula: (See Fig. 14) b2 = :33: b (5) where: b2 = equivalent thickness of the tension flange member, not including the plywood web, in. E0 = the modulus of elasticity of the compression flange, p.s.i. Et = the modulus of elasticity of the tension flange, p.s.i. b = the actual thickness of the compression flange member, not including the plywood web, in. Because of the change in section-geometry and the resulting shift of the neutral axis, (Fig. 14) a new theoretical location of the neutral axis had to be computed us ng the formula; if = g “I; 2:: (6) —4 p-a .4 n _ hr,“ Ii -1 _ L“ l 1 . 1 E 9‘ #5 C dl d‘ . (metr-ir—"{ L z _[ A " '18 i -‘Q g ‘4' a g v V g I 17 :7 -_ “pa/4:05PM gm 7 __.___..__ _ _ ___-.___._-______ -V _ I--. .3____.1--' Y "VJ—Kil- .MhflWML4WUJ , LC! at” N I i “1" . ; j ; ‘ A 2 ; d8 d2. h——~—-—- ’ - x l . ,1 x X i i I V ' " ‘b ’7 I” " L: ” ”7 Er flaw/’94 X— JErr/o/v 0; 77/[0057/(44 [ 4311/4/on /- [Err/M 0; Mona 8.54M Mona Him, In usmpflw 72.4: NON- 5VM£727lCflA 650”: my ”#0 72/5 055m rwa JH/rr 0; 77.4: Met/mm. ,0”: Figure 14 721409; W641 [Q 0/ M04 [/W' X— Jar/0N 33 where: y = distance to the neutral axis from the x-axis, in. fhy = the sum of the areas of the flanges multiplied by the distance of their respective centroids from the x-axis, in.3 iA.= the sum of the cross-sectional areas of the actual com- pression and theoretically equivalent tension flange members, in.2 Using the equivalent thickness calculated for the tension flange and the new location of the neutral axis, the adjusted equivalent I- values were next computed for each beam, ignoring the plywood web, using 'Wefmmfla: 3 I = 3:11) + A1(§1)2 + __b.2::2)3 +A2 (5’2)2 (,7) where: I = moment of inertia, in.4 bl = thickness of the compression flange, in. d1 = depth of the compression flange, in. A1 = x—sectional area of the compression flange, in.2 fl = distance to the neutral axis from the centroid of the compression flange, in. b2 = equivalent thickness of the tension flange, in. d2 = depth of the tension flange, in. A2 = x-sectional area of the tension flange, in.2 y2 = distance to the neutral axis from the centroid of the tension flange, in. 34 These calculated adjusted equivalent I-values were then mul- tiplied by the moduli of elasticity of the compression flange, as determined in the ASTM compression tests, to compute the adjusted equivalent EI-values of each beam. These are recorded in Table V. A graphical comparison is made between the theoretical stiffness fac- tors and the actual stiffness factors in Figure 15. Extreme Fiber Stress in the Flanges To determine the stresses developed in the extreme fibers of the flanges, when the beams were loaded to the allowable deflection at mid-span (1/360th of the span or .267 inches), the flexure formula was used. o’= 31-9 (8) where: CF" extreme fiber stress, p.s.i. M = bending moment at allowable deflection, in.-lbs. c = distance from the neutral axis to the extreme fiber, in. I = the adjusted equivalent moment of inertia, as previously computed, in.4 These computed fiber stress values are recorded in Table VI. COMPARISON OF STIFFNESS FACTORS (Model Beams) 35 EI Values (Lbs.-In.27x 106) Beam No. Theoretical : Actual : Adj. Equiv. l-A-8 144.90 264.94 142.33 2-A-8 l .90 2 2. 1 118.47 Average: ... 144.90 248.73 130.40 1-B-8 144.90 139.25 132.56 2-B-8 144.90 162.46 1 6.2 Average: .... 144.90 150.86 144.43 l-A-9 196.87 491.89 203.23 2-A-9 196.87 272.52 188,15 Average: .... 196.87 382.21 195.69 l-B-9 196.87 206.05 192.26 2-B-9 196.87 183.65 202.21 Average: .... 196.87 194.85 197.24 l-A‘lo 257048 389091 2440144 2-A-10 2 7. 8 389.91 2 7.16 Average: .. 257.48 389.91 250.65 l-B-lO 257.48 415.48 211.67 2-B-1O 227.48 2 0. 0 267.22 Average: .. . 257.48 322.94 239.60 l-A-ll 326.84 436.97 314.41 2-A-11 326.84 704.00 399.18 Average: .... 326.84 570.49 356.80 l-B-ll 326.84 496.94 348.65 2-B-ll 326.§4 288.00 263.82 Average: .... 326.84 392.47 306.24 Table V. (fix/[v 2x 89 ff/ [TA/[IJ flan/Rf ___.____ A——..————— ....___ _._ 700 d a w 600 500 Il-BvH (EV-0-9 , /-;0‘ // [iv—640 0 400 Ox4(/..4,_ /0 (2—fi*49 300 , , , / E11‘i5‘5‘h’ GLI'Q’B 0‘8'9~9 / K / : / Qa-a-a /-5-9 ' / / / EBB—Eva 200» ”:E:6 _. . _. Z'Bva‘vm I / / / ' '8'9 — “ — “Tfié'OPtr/ca [I (any: I '8 ”a D frag-[3 554M |OO . r e , 8 9 to I I _J£C.sz/v_._ Dray/4M Figure 15 M0022 1354M J f/Ff/VA’JJ Maw? r EXTREME FIBER STRESS IN FLANGES AT THE ALLOWABLE DEFLECTION (l/360th) of the span) (Model Beams) Load Flexure Beam No. (Lbs.) (F.s.i.) 1-A-8 2100 2083 2-A-8 2700 1841 1-B-8 1970 1405 2-B-8 2190 1397 l-A-9 4060 2152 2-A-9 3020 1540 1-B-9 2590 1536 2-B-9 2400 1284 1-A-10 3740 2023 2-A-10 3680 1884 l-B-lO 3840 1613 2-B-1O 2820 1461 l-A-ll 4010 1375 2-A-ll 4810 2145 1-B-ll 4250 1568 2-B-11 3300 1640 Table VI 38 Horizontal Shear Stresses The norizontal shear stresses occurring in the plywood web when the beams were loaded to the allowable deflection and failure were next computed using the formula: ’7“ 3%- (9) where: vertical Shear, lbs. {I zjy'= horizontal shear stress, p.s.i. V Q the statical moment about the neutral axis, in3 I the adjusted equivalent moment of inertia, as used previously, in.4 b the plywood web thickness, in. A comparison is made of the calculated shear stress values of the plywood webs of each beam and is presented in Table VII. Full-Scale Beams Beam Test Performance A load vs. mid-span deflection curve was plotted for each of the ten 16-foot beams tested. A composite plot is shown in Figure 16 with _ the average curve for each beam type superimposed. Shear Deflection Because a gross deflection was observed at mid-span at the time of testing, equation (1) was again used to calculate that portion of the measured deflection attributable to shear. This value (ds) was found to be .149 inches for the 16 inch deep beams. 39 HORIZONTaL SHEAR STRESSES IN THE PLYNOOD HEB (model Beams) Shear Stress at Shear Stress : Allowable Deflection* : at Failure Beam No. (p.s.i.) : (p.s.i.) l-A-8 809 1271 2-A-8 954 1307 l-B-8 728 1367 2-B-8 724 1355 l-A-9 1164 1491 2-A-9 833 ----- * l-B-9 831 1444 2-B-9 695 1419 1-A-lO 1138 1308 2-A-10 1058 1208 1-B-10 906 ——--** 2-B-10 821 1543 l-A—ll 796 ----** 2-A-1l 1245 1553 1-B-1l 909 1048 2-B-11 951 1297 * 1/360th of the span ** Extreme buckling caused test beam to "spring out" of the testing machine. Table VII 40 8001- -_ fi -.. .-. , _ .. - ./ / , . /;/// / _ // j/ / 7001.-.-1.W _ _V . ‘ ,1 o - j" / / f / / i 5. / / / / / // / / eoo——sn n .4 : 1,. , . ,c A; , /,/ . 400111.-.. [0,40 flap 0/ 41/0217 -2/7.' 0 C @553 1‘9: 3001 ¢ , / , #4740: 3:4»! ”7 ,j/ ——-— - - —-— 10- 7'50: flame: ///’// - - - - - - ~B-7'm: BEAM 2001 /«/ - - B-fi’PE flue’pxi f // ‘ / i I I I \oo-m~ » ,1 i i . 1 . 1 4 1' 1 i I 'l .2 . 3 .4 5 .6 -T 6 .0272 a: new (1w) Figure 16 [0,40 D[/Z[(7/0/V (ax—mar 404 flu ~ 1041; [BM/w 41 Modulus of Elasticity A theoretical modulus of elasticity was next calculated. This calculation was made using formula (4) and using the bending moment diagram as given in Figure 17. The deflection due to bending was calculated by subtracting the above calculated shear deflection from the average deflection observed for the five B-Type beams at a load of 500 lbs. per two feet. The deflection due to bending was found to be .312 inches. The Type-A beams were not included in this calculation be- cause of the presence and uncertain influence of truss action within the diagonals. The theoretical modulus of elasticity calculated as described was found to be 1.15 x 106 lbs. per in.2 This compared favorably with the average E-value for ponderosa pine of 1.26 x 106 lbs. per in.2 (4) Moment of Inertia The theoretical moment of inertia was next computed using formula (3). As with the model beams, only the plys of the plywood web parallel to the span were included in this calculation. The theoretical I-value calculated was 1017 in.4 Stiffness Factor Two stiffness factors were calculated, the theoretical and the actual. The theoretical EI-value was found by first computing the theo- retical I-value using the formula: 3 3 b d _ b d I =- _1_...3.L__ __2___2.__ (10) 12 12 30 WI If“ U1 ‘0 h) Y1h-r—wo :k 1 -v-‘-—— ~—-—n & H—-— 1 F4— ,‘__.. H——U 35P— asp- ’_ L -_ , .1“ AIM/=9 2511—1 3.5m amp/Na MOME/W' _. , 7-133’ . - .. ..-..— p lap—1 , 4 / /. {\‘\ \ ”P"... admin"--. ..., \,\\ / . \ /,g \ l2P—4 , 3 \\ 4 - 72‘ 3.33 .— \ / ? '47 \ m, / -. \\ SJJ // ’42 #6 . \ 1 7p-—-4 _ ’ K ..x 133.. \ Q / \ / 225 \ E 2 , \ / i \ 7 1 / 2; \ / , . o— ' 1‘2 \ -—¢ Y :3 .- —- '— I H x7.- -, 7 Figure 17 SAM/P 2 23.}; (2)/Na fl’o/Vwfl 04mm [0,42 / c J (422 2322726 43 where: I = the theoretical moment of inertia, in.4. bl= the total thickness of the flanges, including two plys of the 3/8 inch plywood web, in. d1: total depth of the beam, in. b2= total thickness of the flanges, excluding the plywood web, in. d2: total depth of’the beam minus twice the depth of a flange, in. However, because the flange members were of ponderosa pine and the plywood was Douglas-fir, an equivalent thickness of ponderosa pine was calculated to compensate for the difference in E-values of the two species. This was accomplished by dividing the E-value for Douglas-fir by the E- value of ponderosa pine (4) and multiplying this calculated equivalent section factor times the thickness of two plys of the Douglas-fir ply- wood web (.25 in.). The equivalent thickness of ponderosa pine calcu- lated was.388 inches. This value was then used in formula (10) to compute the theoretical I-value of the beam. The theoretical I-value calculated was 1047 in.4. The theoretical EI—value was found by multiplying the above theo- retical Idvalue times the average modulus of elasticity of ponderosa pine. (4) This stiffness factor or theoretical EI—value was 1318.89 x 106 lbs. per in.2. The second set of stiffness factors, or the actual EI—values, was calculated using formula (4). The bending moment diagram (Fig. 17) was used to compute 1A5? in this formula. The actual EI-values calcu- lated may be found in Table VIII, which also includes the average EI-values by beam types. Extreme Fiber Stresses in the Flanges The extreme fiber stresses in the flanges occurring at mid- span corresponding to the load producing the allowable deflection of 1/360th of the span or .533 inches, was calculated using the flexure formula (8). These values are recorded for each beam in Table IX. Horizontal Shear Stresses The horizontal shear stresses occurring in the plywood web when the beams were loaded to the allowable deflection were again computed using formula (9). The statical moment used in this formula was calculated using the equivalent thickness of ponderosa pine rather than the thickness of Douglas-fir plywood actually tested. This equi- valent thickness, as reported earlier, was .388 inches and the result- ing statical moment about the neutral axis was 85.67 inga. The horizontal shear stresses occurring in the plywood web are given in Table X. COMPARISON OF STIFFNESS FACTORS (Full-Scale Beams) 45 Theoretical . Actual E -Values EI-Valuegx 6 , Lbs.-In. x 100 Beam No. Lbs-‘In- 10 : Type - A : Type - B 1 1318.89 1571.94 1358.97 2 " 1665.14 1217.57 3 " 1726.56 1423.24 4 " 1548.82 1478.18 5 " 1607.94 1300.25 Average: 1624.08 1355.64 Table VIII EXTREHE FIBER STRESSES IN THE FLANGES AT THE ALLOHABLB DEFLECTION (1/360th of the span) (Full-Scale Beams) : Flexure in Flanges (p.s.i.) Beam No. : Type-A : Type-B l 936 851 2 973 789 3 998 878 4 929 902 5 924 828 Table IX HORIzomeL SHEAR STRESSES IN THE PLEIOOD "..EB AT TEE ALLOWABLE DEFLECT ION (l/360th of the span) (Full-Scale Beams) Horizontal Shear Stress (p.s.i.) Beam No. Type-A : Type B 1 487 443 2 506 411 3 519 457 4 484 470 5 497 431 Average: 499 442 Table X 47 DISCUSSION OF RESULTS fiodel Beams Figure 12 shows graphically that the Type-A beams (with diagonals) were stiffer than the Type-B beams (without diagonals). The load vs. deflection curves for each beam tested are plotted and the average curve for each beam type is superimposed. It can be noted that the stiffness of the beams increases with the section depth. Also, within each depth category, the average curve of the two Type-A beams is stiffer, in all instances, than the average curve of the two B-Type beams. By comparing the EI-values, or stiffness factors, (Fig. 15 and Table V)it can be readily observed that the Type-A model beams were significantly stiffer than the Type-B model beams. The Type-A beams having a section depth of eight inches are approximately 40 per cent stiffer than the Type-B beams of the same depth. This stiffness trend is also apparent in the nine, ten and eleven inch deep beams; however, as the section depth increases, the per cent increase in stiffness decreases. Because the Type-A beams were considerably stiffer in all instances than the expected or theoretical EI—values (Table V), it was assumed that the diagonal stiffeners impart truss action to the beam, thus re- sulting in a more rigid construction. The deviation of some of the points plotted in Figure 15 (BI-values vs. section depth) from the apparent average curves can be explained by considering the buckling and twisting observed in the beams having sec- tion depths greater than eight inches. Because of the relatively long laterally unsupported span, there was a decided tendency for the beams 48 to buckle under load. This buckling and twisting was noted at the time of testing and was reflected in the plotted EI—values of Figure 15. Erroneous deflection readings undoubtedly resulted as the beams twisted about the neutral axis and the deflection-recording nail at the neutral axis rotated in the direction of twist. Figure 7 is a photograph taken of a representative beam twisting during testing. It was found that the adjusted equivalent EI—values (Table V), computed using the eight inch compression sample data (Table III in Appendix), agreed favorably with the computed theoretical EI—values. The close agreement of the actual EI—values computed for the eight and nine inch deep B-Type beams to the theoretical EI-values for the same beam