STRESSED-SKIN PANEL DEFLECTIONS AND STRESSES USDA FOREST SERVICE RESEARCH PAPER FPL 251 1975 U. S. DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY MADISON, WISCONSIN
ABSTRACT This paper presents a mathematical analysis based on "shear flow" and "shear lag" theories to determine deflections and stresses for stressed-skin panels wherein skins are rigidly bonded to stringers. Experimental examination of several panel constructions showed that the analysis can provide a rational basis for the design of stressed-skin panels. Designs based on the analysis will result in more efficient utilization of materials. CONTENTS Page Introduction............. 1 Theoretical Analysis......... 1 Experimental Examination........ 19 Test Results............. 26 Conclusions.............. 40 Suggested Design Procedure...... 41
STRESSED-SKIN PANEL DEFLECTIONS AND STRESSES By ED WARD W. KUENZI, Engineer and JOHN J. ZAHN, Engineer Forest Products Laboratory, 1 U. S. Department of Agriculture INTRODUCTION Forest Service Design procedures for stressed- skin panels (wherein thin facings are rigidly bonded to stringers thus producing lightweight, stiff panels) have been clouded by " rule- of- thumb"procedures regarding shear stresses in stringers. Also, the possible effects of " shear lag" which may cause the skin normal stresses to diminish with increasing distance from the stringer needed clarification and evaluation. This research study sought efficient utilization of materials by developing more rational design procedures for stressed- skin panels especially with regard to " shear flow" and " shear lag" effects. Included also in analytical work are complete derivations for panel deflections, effects of shearing deformations, and panel stresses. Experimental research was carried out to substantiate the analytical results and underlying assumptions. THEORETICAL ANALYSIS This analysis follows Reissner 2 except that the three components (top skin, bottom skin, and stringers) are made dissimilar and unequal. In addition, the shear deformation of the stringers is included since it can be a significant part of the total deflection for wood stressed- skin panels. Figure 1A shows the cross section of some stressed- skin panels. Figure 1B shows the smallest repeating unit of which general stressed- skin panel cross sections are composed. This repeating unit is shown crosshatched on the cross sections shown in figure 1A. 1 Maintained at Madison, Wis., in cooperation with the University of Wisconsin. 2 Reissner, E. Analysis of Shear Lag in Box Beams by the Principle of Minimum Potential Energy. Applied Math. Quarterly 4(3) : 268-278. Oct. 1946.
Figure 1. --Stressed- skin panel cross sections A and notation, B. M 142 555 Figure 2. --Illustration of how plane transverse cross section rotates and warps during bending in the presence of shear. M 142 563-2-
Coordinate axes x, y, and z form a right-hand set with the x axis along the span and the z axis vertical. The corresponding displacements are u, v, and w. The method of analysis is to assume a reasonable form for the displacements u and v and derive an equation for the displacement w from the principle of minimum potential energy. Assumed Displacements Previous analysis 3 has established that the displacement v can be ignored with little effect on the results for vertical deflection. For simplicity, then, it is assumed that and (1) (2) (3) (4) where z 1 equals c 1' z 2 equals -c 2, c 1 and c 2 are distances from neutral axis to centroids of skins; U 1, U 2, and U 3 are functions of x; and the α i are constants. The z coordinate is measured from the neutral axis. The location of the neutral axis is assumed to be unaffected by cross- section warping. 3 Reissner, E. Least Work Solutions of Shear Lag Problems. J. Aero. Sci. 8: 284-291. 1941. In other words, this is a beam analysis with special attention to shear deformations. In elementary bending theory the cross sections are assumed to remain plane (u = 0). The term " shear lag" implies a warping of the cross sections associated with the existence of shear strains in the skins. The result is a reduction in normal stress on the skin cross section from that predicted by elementary theory and an increase in vertical deflection. In this analysis the warping of the skin cross sections is assumed to be a parabola in the y direction (fig. 2). Let subscript i denote skin 1 (top), skin 2 (bottom), or stringer 3. Assume that the u displacement has the form -3-
Strains The motivation for the assumed displacements is more apparent when one examines the shear strain distribution. The shear strain in the stringers is (5) Note this is independent of z. The shear strain in the skins is (6) Note this is linear in y. This strain distribution is strongly suggested by elementary shear flow analysis in which the shear varies only very slightly in the stringers and varies linearly throughout the width of the skins. The extensional strains at the midplane of the skins are Continuity at Gluelines At the junction of stringer and skin we require continuity of displacement and of shear flow. For convenience, the junction will be taken to be at z = z i (centroid of skin) and the comer will be idealized to a point with no cross-sectional area. Then at y = b, z = c 1 (8) -4-
(9) and at y = b, z = -c 2 (10) (11) where b is half of stringer spacing, t i are thicknesses (fig. la), the G i are shear moduli of elasticity in the x-y plane of the skins or x-z plane of the stringer, and γ denotes shear strain. From equations (8) through (11), using equations (3) through (6), one easily obtains the following relations: (12) (13) (14) Thus we can drop the subscript on U 1 hereafter and seek two unknown functions of x, namely w and U. Potential Energy The potential energy of the load is (16) where M is the bending moment on the cross section as shown in figure 3 and L is the total length. Figure 3 shows the sign convention for M and V. - 5-
Figure 3.- - Sign convention for internal stress resultants on cross section. M 142 566 The strain energy of the stringer is (17) where E 3 equals modulus of elasticity in x- direction of stringer and (18) (19) The strain energy of the skins is (20) where E i equals modulus of elasticity in x- direction of skin i. Substituting equations (6) and (7) and integrating over y, equation (20) becomes (21) - 6-
where primes denote differentiation with respect to x and (22) The total potential energy is (23) Let (24) and use equations (14) and (15). written Then the total potential energy can be (25) where (26) (27) (28) (29) (30) -7-
Equilibrum Condition According to the minimum potential energy theorem, the condition of equilibrium requires that the first variation of the total potential energy vanish: (31) Using equation (25) and integrating by parts, equation (31) can be written (32) from which one obtains two differential equations (33) (34) and the boundary conditions (35) and the continuity conditions (36) at a concentrated load. (36) implies Since the curvature Q is continuous there, equation (37) at a concentrated load. These results reduce to agreement with Reissner 2 when the skins are similar and isotropic and G 3 is infinite. -8-
Differential Equation for U Assume A is greater than B 2. Then eliminating Q between equations (33) and (34) yields (38) where (39) (40) (41) Once U has been found, equation (33) gives Q, that is w", from which w can be obtained by double integration. Thus, writing B in terms of n and ρ from equation (40), equation (33) becomes (42) from which the deflection w can be obtained in the form of the elementary bending solution plus a correction for shear lag. The strains in the skins can be obtained from U and w as (43) (44) which are also in the form of the elementary solution plus a correction for shear lag. These strains are at the midplane of the skins. More accurate expressions for maximum strains can be obtained by replacing c i in the ele- mentary part of the solutions with the distance to the outer surface. Then equation (43) becomes - 9-
(45) using equation (42). Similarly equation (44) becomes (46) The shear strains can be obtained from U as (47) (48) (49) Application of this theory is next illustrated by two examples. Case 1. Uniformly Distributed Load and Simple Support Figure 4A shows the notation, Let the total load be W and the total span be 2a. The distributed load is (50) The bending moment is (51) Therefore equation (38) becomes (52) - 10-
Figure 4.--Notation for panel loadings, A, case 1, uniformly distributed load. B, case 2, loads at quarter- span points. M 142 560-11-
By symmetry (53) and since are zero at the supports, equations (45) and (51) imply (54) The solution of equation (52) subject to equations (53) and (54) is (55) Stringer and skin strains may now be computed by substituting (56) and equation (51) into equations (45) through (49). To obtain the center deflection, substitute equations (51) and (55) into equation (42) and get (57) The solution of equation (57) subject to the conditions (58) and (59) -12-
is (60) The center deflection is (61) Case 2. Quarter-Point Load and Simple Support Figure 4B shows the notation. is 4a, The bending moment is Here the total load is 2P and the total length (62) hence equation (38) becomes (63) whose solution, subject to the conditions (64) (65) (66) -13-
is (67) Stringer and skin strains may now be computed by substituting (68) and equation (62) into equations (45) through (49). Deflections can be obtained by solving equation (42) subject to the con d i t i on s (69) (70) (71) The resulting center deflection is (72) Remark : Since equation (38) possesses a nonzero homogeneous solution, U does not necessarily vanish when M' does. This means that cross- sectional warping, which is necessitated by vertical shear wherever M' is not zero, can " spill over" into regions where the bending moment M is constant and there - 14-
is no net vertical shear force. This effect was experimentally verified in this study as can be seen by examining the case of quarter - point loading in tables 1 through 4. Note that the skin strains at x = 12 inches (where bending moment is constant) are smaller between stringers than they are at stringers. However this theory contains one incongruous result. By equation (49) the stringer shear strain is nonzero wherever the cross-sectional warping amplitude U is nonzero. Yet surely the stringer strain must vanish wherever there- is no net vertical shear force. The above- mentioned spill- over thus puts stringer shear strain where there should not be any. This apparent contradiction is the result of only approximating equilibrium by minimizing the potential energy rather than imposing exact equilibrium equations. To do a better job it would be necessary to introduce more degrees of freedom in the assumed displacements than was done here. However, this theory does an excellent job of modeling the maximum values of strains and deflections in stressed- skin panels and is entirely adequate for design purposes. Limiting Cases--Reduction to Elementary Theory For simplicity, only the case where all E values are equal is considered here. There is an elementary theory of shear deflections of beams which uses Castigliano s theorem. The shear stress energy is obtained by integrating the elementary shear stress distribution given by (73) where τ equals shear stress, V equals total shear force on cross section, and A'z equals first moment about neutral axis of area bounded by free surfaces and longitudinal cutting plane. In the skins (74) so that (75) -15-
and in the stringer A'y is nearly constant so that, approximately, (76) The shear stress energy is therefore (77) where (78) and by Castigliano's theorem the shear deflection is (79) where Q is a dummy force at the point where deflection is desired, and δ s is the shear deflection. Applying this theory to cases 1 and 2 above yields (80) where δ b equals elementary bending deflection at center and (81) - 16-
It will now be shown that the Reissner- type shear lag analysis of this report reduces to the elementary result above under appropriate limiting assumptions. In cases 1 and 2 the correction for shear lag has the form (82) where (83) and (84) Unless the span a is extremely short and G is extremely low, the quantity Ψ(ka) will be legs than 0.05 and can be neglected with error less than 5 -percent. Using the definitions of k and n (eqs. (40) and (41)) and neglecting Ψ(ka) equation (82) becomes (85) where B and C are given by equations (28) and (29). Assuming that (86) - 17-
it follows that β = 1 and B and C become (87) (88) where (89) The contribution of the stringer to shear deflection can be obtained by letting G approach infinity. Then 1 (90) (91) and (92) which essentially agrees with the first term of equation (80) except for a factor which is within the approximation error of these approximate theories. That they are different is not surprising, for the theories differ in small respects. For example, one matches shear flow at the junction of stringer and skin and the other does not. To obtain the contribution of the skins to the shear deflection, let G 3 approach infinity. Then from equations (87) and (88) -18-
(94) and (95) which agrees exactly with the second term of equation (83). Thus the theory presented here reduces to elementary theory under appropriate limiting assumptions. It should be noted that without these limiting assumptions the theory is considerably more accurate than elementary theory. The only reason for making these reductions is to check the form of the theory and to increase confidence in it. EXPERIMENTAL EXAMINATION An experimental examination of stressed- skin panels was conducted to determine whether the theoretical analysis could be utilized to predict panel deflections and stresses or strains. Panel Materials and Fabrication Six panels were fabricated with plywood skins and plywood stringers. Details of construction are shown in the sketches of figure 5. Panels were 96 inches in length, thus avoiding splicing of thin, tension skins for longer panels. The panel depth, however, was determined by bending deflection criteria for panels 14h inches long; namely that a panel should deflect no more than l/360 of the span under a uniformly distributed load of 40 pounds per square foot. The effects of shearing deformations and shearing stress were thus somewhat accentuated by testing panels with rather thin stringers on a short 93- inch span. All plywood used to construct the panels were grademarked as sanded, A- C, exterior, Group 1 of U.S. Product Standard PS 1-66. The thin 1/4- inch skins had three veneers of Douglas- fir 4 of equal thickness. The 1/2-inch plywood stringers were five- ply with face veneers about 0.08 inch thick and crossband and core veneers each about 0.11 inch thick. Douglas- fir veneers were used for panels 1, 2, 3, and 4. Stringers for panels 5 and 6 had one face ply and the core ply of Douglas- fir; the other face ply was of western hemlock; and the crossbands were of ponderosa pine. 4 Veneer species were identified by Dr. B. Francis Kukachka of the Forest Products Laboratory. -19-
Figure 5.--Construction details of stressed- skin panels tested. M 142 562
Note that a double stringer of two nail- glued 1/2- inch plywood strips was used in panels 5 and 6. Panel 6 was cut from panel 5 after panel 5 had been tested to determine stiffness and strains without loading to failure. The thick, 5/8-inch, five- ply skin of panel 1 had 0.08-inch face plies and a 0.16- inch core ply of Douglas- fir, one crossband of 0.15- inch white fir, and the other crossband of 0.15- inch ponderosa pine. The thick skin of panel 2 was seven-ply, 5/8- inch, with face plies of 0.06-inch Douglas-fir, outer crossbands and core of 0.10-inch white fir, and inner plies of 0.10- inch Douglas- fir. The thick, 3/4- inch skin of panels 3, 4, 5, and 6 had face plies about 0.08 inch thick and core and crossbands about 0.19 inch thick. All veneers were of Douglas-fir in the thick skins of panels 3, 5, and 6. Panel 4 had face and core veneers of Douglas- fir and crossbands of ponderosa pine in the thick skin. The thick skin of panels 1, 2, and 3 were spliced at midlength with inner splice plates of plywood the same thickness as the thick skin. The splice plates were 16 inches long in the dimension parallel to the panel length. These nail- glued splices were made prior to assembling the stressed- skin panels. Blocking of 3/4- inch plywood was placed between stringers at midspan and each end of the stressed- skin panels. The stressed-skin panels were glued with a urea- formaldehyde 5 glue. assembled panel was allowed to cure overnight in a cold press. Each Thickness and width dimensions of the panels were measured and these data are included in tables 1 through 6. Panel Testing Panels were supported on a span of 90 inches and loaded under three types of load --uniformly distributed load, line loads applied at outer quarterspan points, and concentrated point load applied at stringers. Panels under line loads at quarter-span points were first tested inverted (thin skin in compression) and then tested right side up to failure. (Panel 5 was not loaded to failure.) Reaction points for the panels were of 3- inch steel pipes supported on a heavy timber framework in a large-platen testing machine. The pipes and portions of the framework can be seen in figures 6, 7, and 8. Uniformly distributed loads were applied to the panel through air pressure in a large plastic bag placed between the panel and a sturdy " strongback" fastened to the movable head of a testing machine. Figure 6 shows one end 5 This glue was chosen as a convenient one to produce a strong rigid bond in the test panels. A more durable glue must be used in actual panel construction. - 21-
Figure 6.--Stressed- skin panel under uniformly distributed load applied through an inflated plastic bag. The bag is placed between the panel and an upper "strongback" attached to movable head of a testing machine. (M 141 738-6) Figure 7. --Concentrated load applied through a 1- inch diameter bar under a load cell placed at a panel stringer and 3 inches from reaction. (M 141 563-10) - 22-
of the panel with the inflated plastic bag on the panel. Pressure in the bag was measured with an electronic pressure transducer having a pressure calibration of 150.31 microvolts per inch of water. Concentrated point loads were applied with a 1- inch-diametersteel bar loaded by an electronic load cell through a steel ball. The load cell was fastened to the movable head of the testing machine. Figure 7 shows apparatus used to apply the point load. Line loads at quarter-span points were applied to the panels through 3- inch pipe pieces extending across the width of the panel. The pipes were loaded at two points through a framework which was loaded at its center with a load cell and ball fastened to the upper, movable head of the testing machine, The apparatus is shown in figure 8. Panel midspan deflections were measured with a dial gage micrometer fastened to the stem of a long- top T- bar supported at its ends by nails at the panel reactions. The movable stem of the dial gage micrometer was attached to a nail driven into the panel edge. Deflections were measured at both panel edges and the readings were averaged. Deflections of one side of the panel were within a few thousandths of an inch of deflection of the other side. Strains at various points in the panel were measured with 1- inch electricalresistance strain gages. Skin strains in panels 1, 2, 3, and 4 were measured 12 inches from midspan. Single gages for these panels were placed midway between stringers and directly above or below inner stringers. At outer stringers the strain gages were placed 1 inch from the panel edge. Skin strains in panels 5 and 6 were measured with 1- inch strain gage rosettes placed at 27-1/2 inches from midspan. Widthwise, these rosette gages were placed midway between stringers or 6-1/2 inches either side of the midpoint between stringers. Strain gage rosettes were also placed on stringers near one reaction. The gages were centered at stringer midheight and in panels 1, 2, 3, and 4 rosettes were placed 35-1/2 inches and 41-1/2 inches from midspan. On panels 5 and 6 rosettes were placed 35-1/2 inches from midspan. Uniformly distributed loads and loads at quarter- span points were applied to the panels in 200- pound load increments to about 1,400 pounds for panels 32 inches wide and 2,400 pounds for panels 48 inches wide. At each load increment, load cell and strain gage readings were recorded with a multichannel digital data acquisition system having a scanning rate of about 4 gages per second. A few readings of all the data were also taken during removal of load. Concentrated point loads were applied at 100-pound increments to 500 pounds and strain gage readings on the stringers were recorded with the data acquisition system. - 23-
Figure 8.--Stressed- skin panel under load applied at quarter- span points. the left foreground are strain gage reading and recording equipment. In (M 141 738-8) Figure 9.--Type of shear test to determine shear stiffness of the stringers. (M 125 798) - 24-
Panels and coupons from them were tested within the laboratory during the winter months. Conditions recorded during the testing period showed a relative humidity of 35 ± 10 percent and a temperature of 72 ± 6 F, to result in a wood moisture content of 7 ± 1-1/2 percent. 6 Graphs were constructed of load-deflection and load-strain data. Except for the final tests to failure these graphs showed a linear relationship between load and deflection and load and strain for nearly all data. A few nonlinear data graphs were able to be represented by a straight line fitting the general slope of the data. The slopes of the lines on the graphs were used to compute the data given in tables 1 to 6. Shear strains were computed from the strain gage rosette data. Coupon Testing The elastic properties of the skins and stringers of the panels were determined by testing small coupons cut from the panels after they had been failed. Two compression skin coupons 1 inch wide and 4 inches long and two tension skin coupons 1 inch wide and 10 inches long were cut from the skins in each space between the stringers of the stressed-skin panels. The long dimension was parallel to the length of the stressed- skin panel. Compression coupons were loaded through a spherical-seated loading head in the movable head of the testing machine. Tension coupons were gripped in self- alined wedge grips in a testing machine. Deformations of compression and tension coupons were measured by a mechanical gage that caused motion of the core in a differential transformer. The signal from the transformer was combined with the testing machine load transducer signal to produce loaddeformation graphs on an x- y recorder. The deformation gage had knife edges spaced 2 inches apart and was fastened to the edges of the specimen. One stringer shear coupon 4 inches wide and 10 inches long was cut from each stringer. The specimen was cut so that the strain gage rosettes were included in its length. The type of shear test conducted is shown in figure 9. Average values of the elastic properties are given in the lower portions of tables 1 through 6. These properties were combined with the theoretical analysis to compute the stiffnesses of the repeating element (fig. 1) of each stressed-skin panel. Several panels did not have a true repeating element. For these panels an average stringer thickness was used to compute bending stiffnesses and deflections. Actual stringer thickness was used to compute local strains. Stiffness values are also given in the lower portion of tables 1 through 6. 6 Determined from data in table 38 of Wood Handbook, Agr. Handb. No. 72. 1955. - 25-
TEST RESULTS Detailed data from tests of the stressed- skin panels under concentrated point load applied 3 inches from a reaction at one stringer are not included. The load- strain data were linear up to 500 pounds. The shear strains at stringers adjacent to the loaded stringer were about 10 percent of strains measured on the loaded stringer. Strains in stringers two or three spaces from the loaded stringer were only about 3 percent of strains measured on the loaded stringer. Thus when concentrated point loads occur near a reaction very little of this load is transferred to other stringers. Experimental values given in tables 1 through 6 show that data for panels inverted (thin skin in compression) are in general agreement with those or panels tested right side up. Deflections were within a couple of thousandths of an inch of each other. Strains were not in as close agreement but the general pattern was similar whether the panel was inverted or not. The skin normal strains, ε and ε were at most under 400 microinches per inch, thus 1 2' representing stresses less than about 700 pounds per square inch. The shear strains, γ, were at most under 1,600 microinches per inch, thus representing stresses less than about 160 pounds per square inch. Thus the elastic behavior of the panels was not greatly dependent upon their orientation (inverted or not) even though the 1/4- inch tension skin was noticeably buckled inward even without load on the panel. Experimental stringer shear strain data given in tables 1 through 6 show that inner stringers are strained more than outer stringers of the same size, the difference being as much as 3 to 1 in a few cases. Panels with inner stringers double the thickness of outer stringers had stringer shear strains more nearly equal. Five of the stressed- skin panels were loaded to failure under load applied at quarter- span points. Failures occurred suddenly in shear at the stringerskin bonds. Details of the type of failure are given in table 7. For panels 1, 2, 3, and 6 the load- deflection curve was linear to about 95 percent of the maximum load. For panel 4 the load- deflection curve was linear to about 60 percent of maximum load. Maximum shear stresses at skinstringer bonds ranged from 300 to 779 pounds per square inch as computed by elementary shear- flow theory and 230 to 680 pounds per square inch as computed by shear- lag theory. The elementary theory was about 20 percent conservative for the panels tested. - 26-
Table 1.--Stressed-skin panel No. 1-27-
Table 2.--Stressed-skin panel No. 2-28-
Table 3.--Stressed-skin panel No. 3-29-
Table tressed-skin No. 4-30-
Table 5.--Stressed-skin panel No. 5-31-
Table 6.--Stressed- skin panel No. 6-32-
Table 7.--Failure of stressed- skin panels - 33-
Comparis on o f The ore ti cal an d Experimental Results Data given in tables 1 through 6 include theoretical values of deflections and strains determined by substituting dimensions and elastic property values in the pertinent formulas given in the Theoretical Analysis. The deflections and strains were computed for the total panel load indicated in the crossheadings. Experimental values are given for comparison, A comparison of theoretical and experimental results can be had by constructing graphs showing theoretical values as ordinates and experimental values on the abscissa. Perfect agreement between the theoretical and experimental values would result in all points lying on a 45º line from the origin. Graphs of figures 10 to 16 were placed in the order of importance of panel characteristics to adequate structural design. Figure 10 shows that theoretical and experimental values of midspan deflection agreed well; hence the theoretical analysis can be used to closely predict the prime design characteristic of panel deflection. The computation of deflection by the theoretical analysis involving " shear lag" includes effects of shearing deformations in the stringers and skins to produce shear deflection of the panels. The amount of midspan shear deflection of panels under uniformly distributed load as a percentage of total deflection of the panels tested is given as: Panel (Pet of Shear deflection panel midspan deflection) The shear deflection was determined by subtracting the computed bending deflection from the theoretical deflections given in tables 1 through 6. The shear deflection appears relatively large but this is because effects of shear were emphasized in these panels by choosing short spans and thin stringers. An important difference between these results and current practice can be seen in the values of stringer shear strains. It is current practice to reduce the allowable design rolling- shear stress for plywood skins at the glueline with the outer stringer, on the assumption that stress concentrations are produced by a panel edge- effect. However, if all stringers are the same size, elementary shear flow theory shows that the inner stringers carry twice as much shear stress as the outer stringers. The data for panels 1 through 4 confirm that inner stringers are more highly stressed, though not quite by a factor of 2. All of these panels failed at an inner stringer glue bond. A more rational panel design would be to have inner stringers double the thickness of the outer stringers. Then all stringers would be stressed nearly alike. This is confirmed by the data for panel 6 in table 6. - 34-28 28 49 25 34 32
Figure 10.--Comparison of theoretical and experimental midspan deflection. δ. (M 142 565) Figure 11.--Comparison of theoretical and experimental shear strains, γ 3, of inner stringers, (M 142 564) - 35-
Figure 12 shows shear strain data for the outer stringers. The comparison between theory and experiment is not good here. The experimental values average about 30 percent higher than theoretical as indicated by the dashed line on figure 12. It is not known why there is disagreement between theory and experiment but this is not a critical matter since the design must be based on possible greater stresses in inner stringers. A comparison of thin skin normal strains at the stringers is shown in figure 13. Although there is data scatter the general agreement between theoretical and experimental strains is good. The skin stress level represented by the largest strains in the graph was less than 800 pounds per square inch. The thick skin normal strains at the stringers are compared in figure 14. More scatter of the data points is exhibited than for the thin skins but general agreement is shown between theoretical and experimental values. Thick skin stresses at the strains shown on the figure were less than 700 pounds per square inch. Figure 15 shows a comparison of theoretical with experimental skin shear strains at stringers. In spite of large scatter there is some agreement between theoretical and experimental data, The stresses corresponding to these strains were less than 90 pounds per square inch. Skin shear strains midway between stringers theoretically should be zero. The measured strains, as given in tables 1 through 6, were small but not equal to zero. Since they are small strains they are of no real consequence in design. The smaller strains between stringers than at stringers show that " shear lag" exists in both thick and thin skins. Figure 16 shows values of skin normal strains between stringers. Again, there is a great deal of scatter in the data points but theoretical and experimental values show some agreement. Stresses represented by these strains would be under 600 pounds per square inch. Figure 11 compares theoretical with experimental values of stringer shear strains in the inner stringers. Although there is considerable scatter in the data points, the trend shows good agreement between theory and experiment. The data scatter could be attributed to local variation of elastic properties of the face veneers and also to inaccuracies in measuring maximum strains. The strains shown on the graph represent shear stresses less than 200 pounds per square inch. -36-
Figure 12. --Comparison of theoretical and experimental shear strains, γ, of outer stringers. 3 (M 142 561) Figure 13.--Comparison of theoretical and experimental thin skin normal strains, ε 2, at stringers. (M 142 559) - 37-
Figure 14.--Comparison of theoretical and experimental thick skin normal strains, ε 1, at stringers l (M 142 558) - 38-
Figure 15.--Comparison of theoretical and experimental skin shear strains, γ, at stringers. (M 142 557) Figure 16.-- Comparison of theoretical and experimental skin normal strains, ε, between stringers. (M 142 556) - 39-
CONCLUSIONS 1. The theoretical analysis presented can be utilized as a rational design procedure for stressed- skin panels. 2. Elementary shear- flow theory can be used to predict stringer shear stresses. From this theory it is seen that inner stringers of stressed- skin panels should be twice as thick as outer stringers to equalize the shear stresses in the stringers. Experimental results confirmed this quite closely. 3. The theory presented here is the simplest possible theory which still incorporates the essential features of shear lag and is still capable of agreement with experimental data. U S GOVERNMENT PRINTING OFFICE 1975-650-253-25 - 40-
The usual stressed- skin panel involves skins with different stiffnesses and also stringers different from the skins. This generality results in too many parameters for simple graphs. Thus it precludes solving the design problem, namely working the analysis in reverse to determine dimensions of components for given performance criteria. The design problem can be solved by an iteration procedure that can be easily adapted to a programable calculator, The following steps are suggested: a. Choose skin thicknesses and elastic properties to comply with deflection performance under concentrated load between stringers or other criteria such as customary usage, etc. (This step is beyond the scope of this paper.) b. Determine required panel bending stiffness (EI) from design loads and deflection limitations, including an estimated 10 percent shear deflection. Utilize elementary mechanics of materials to determine a stringer depth to satisfy panel bending stiffness (EI) required. c. Determine stringer widths based on shear stress calculations by elementary mechanics. d. Analyze theoretical panel performance--deflections and stresses--by theoretical analysis given in this paper. e. Compare theoretical panel performance with design deflections and stresses. f. Iterate, if necessary, after choosing a new set of component dimensions. - 41-