FOR AERONAUTICS

The enalogy between the distribution of stresses in flat stiffened panels snd the distribution of electric current in a ladder-type resistance network is used as the theoretical basis of an electrical computer for the rapid solution of shear-lag problems. The computer, c..rmistingof variable resistors and multiple-current sources. is described; and typtcal examples are given-of its use. The joint problems, and an e&l.ogy is extended b-include bolted-exsmple is given also.


INTRODUCTION
When a load is applied along the flange at tue edge of a flat stiffened panel (fig.1), some of the load is transmitted to the various stiffeners through the mediumof the sheet, which is placed in shear by the action.Similarly, when two plates are connected by a bolted strap (fig.2), the load is transmitted from the plates to the strap through the medium of the bolts.As shown in reference 1 end in appendix A, an analog can be drawn between the distribution of forces in the cases previously described and the distribution of electric current in a ladder network.This analogy is very useful in obtaining a numerical solution, particularly in complicated cases which ~not readily amena%le to mathematical soluticn.The time saved in such cases is of appreciable importance.
Zn reference 1 the solution of a shesr-lag problem was obtained by mesns of an enalogous network consisting of properly chosen fixed resistors.The power supply consisted of a single battery; the proper input currents were obtained from a voltage divider by trial and error.
The present paper describes a pflot model of a more elaborate setup.
The model uses variable resistors and several souroes of N3CA m Not ,.l$?m current, both of which can be introduced independently into the netwrlc.Relng readily adaptable -to the eolutlon of 'more than one problem, thi SIconstruction aonverts the apparatus into a ccmputer.The front panel.of the computer supparts the variable resistors constituting the network, togethe~with the resistor ' control knobs and dials, as shmm in Figui'e3.In order to set any element to a given resist~mce, the dial is turned to the proper reading as determined by the calibration constant for that element since the dials are g~'aduatedin 100 erbitrq divisions rather than in ohms.The dial constants were .indivi.dually determined; consequently, $t.was possible to get the resistance elements to ~1.percent or ~2 ohms, whichever unit wem the gz'eater.
The resistor elements are connected to form a ladder-type network as shown in the wiring diagram in figure 4. Each element has a maximum and minimum resistance of appi"oximately1000 ohms NACA TN NO. 12?il and 10 ohms, respectively.Turning the dial to the extreme low end short-circtits the mi.nimvmresistance cf the element; whereas.tvrn!ng the dial to the hi@ end open-circuits the element after the 1000-ohm limit is ~eached.
Current may be independently introduced into the network at eight points along the two sides opposite the ground connections.The current is suppliedby electronic regulating circuits (fig.5), which are designed to @ve negligible reaction upon each other and to be Independent of the reststence setting of the network.The effect of line-voltage variation is also negl.igi%le.The output of the regulating circuits cem he varied from about 1 to 15 milliamperes hy means of control dials on the front of the penel, Figure 6 is a rear view of the computer and shows the arrangement of the resistors and the current supplies.
The current in any element, ae well as the output of the electronic supply un~.ts,is determined M plugging in a meter to a current-measuring jack associated with the desired element.A high-grade milliammeter is used and allows the current to %e measured with an accuracy of about l/~4percent of full scale.
The chief sou~ces of error in the apparatus itself arise from inaccuracy in resistor values and in the current measurement.In the present model, en attempt was made to malcea reasonable compromise between absolute minimization of all sources of error on one hand and cost and convenience on the other hsnd.For instance, the variable reststors are of comparativ&ly high resistance; thus, the errors due to variable contact resistance and meter-insertion loss azzereduced."If the apparatus is allowed to warm up for about 15 minutes, errors due to further heating are negligible.
The errom Inherent in the method, when used for shear-lag problems, are due to the division of'the structure fnto a finite number of lays (corresponding to an electrical transmission line with lumped constants} and due to the inclusion of the sheet area with the stringer ana to account for the axial stress in the sheet.These errors have been discussed and evaluated in reference 1.

ANALOGY FOR SEEAR-LAG PROBLEMS
As shown in reforenc~1 the correspondence between the mechanical struoture and the electrical network ts given in the notation of the present paper by the following table: R="L %A." (2)

V=aflu (5)
The boundary conditions arenot included in these equations end, therefore, have to be determined from the individual problem, in which current corresponds to foroe and potential.differencescorrespond to displacements.At a fixed boundary no displacement occurs; hence, there must be no potential difference song that part of the" network representing the bounda~, The electrical resfst~ce corresponding to a fixed edge is therefore zero, which means that any amount)of current may flow without producing a potential difference.
It is interesting to note that at a loaded edge or point the force against displacement characteristics of the applied load may very from a dead weight, where the force M Independent of my motion of the end of the specimen, to a screw-t~e Ioading machine, where the applied force varies greatly with slight motions of the specimen."Between these two extremes would be a structure which is loaded by another, the stiffness of which is of a corresponding order of magnitude.The axial force in the flange gradually "leaks off" as shear force into the sheet; the.,shearforce h turn is transmitted.as Wisl force into the stringer.
The amount of force leaking off will be determined by the shear stiffness of the sheet and the axial stiffneeses of the flenge and'stringers.
The smooth curve shmm in figure '? thus represents tie stress in the flange.
The chrve is high at the loaded end and decreases smoothly as the distance from the end increases." ,, In order to set up the equivalent electrical networ~the structure must be broken into a finite number of sections so that & qectlon of resistor network may be ad~usted to.correspond to each eect$on of the s,trtiture.; As.the,dt@mce increases ~ifor?iLY .@m.@',the @qctr$c~ne$y~rk, "t.he, cmmt in the.resisters representing the flenge,,r~ns q.onsttit in any one resistor &d then abrupt~de,breaaes.to.,the,he~+ower vaue as a $mction potnt Is r6ached.This c,cndititi is graphioaly tllustratedby the series'of, steps, iti fim 7P .At the points where.thgstep function croaees the emootkcurve there 1s, .of,course,ne error due to the finite Steps.
At "al other pdnts a relatlve e,rror will exist between the two curves~this error may generally be red~ed ti.magalt@e by ad~usting the steps ,toI$e half above and halt belo~the amoth curve. . . ...' ":n ok'er ~o approach this "conditlcm, the stiffness of each section" is ,considere~to lie at its center, and.the sections are shifted so tha$,,their cent'erk come over'the points .LO, Ll, and .

L2 in flgum~.
The first sectfcn must therefore be made of half length with the sttffness concentrated at the losded end) the last section will also be of half length but with the stiffness concentrated at the point L3, or root.Since the end conditions fix the shear at the root as zero, this last half secticm does not ., the., qtiffnesses are ad@sted in thrpugh the midpoint of each s%ep will clcmeti approach the exatit" soltit ion...
The exact-method curve of longitudinal stress does not cross the last step exactly in the middle because of the rapidly changing slope of the curve in this region.The plot for shear stress by the ccmputer method is shown as a set of 'fstendpfpesn inasmuch as the transverse current correspondtig to the shear can be measured only at the tremsverse junction points, The mex$mumvalue of stress given by this method does not agree very well with the maximum value of the exact method.The results cen be improved by adjusting the length of the hays; that is, by using shorter lengths in the regions of steeper slope.In order to note the effect of this adjustment, the computer was set up with bay len@@ of 50, 20, 15, .10,and '5 inches; and a new analyfiis was ms.det agreercent with the axial.stresses and The results are shown in figure Il.The theoretical curve i.snow excellent, %oth for for shear stresses.

Analysis of aBox Beam
When the cover of a lox been is to W analyzed, the resistances are detemined and set up exactly as for the panel.The load, however, is now lntro&~ced into the flange 'byehesr forces in the web.When the beam is loadeiiat the tip, the shear is uniformly distrilm.rhed..In the network, however, it is, of oourse, necessary to introduce the cuyrent in discretg smounts at finite intervals.As for the previous case of the panel in which "the @hear resistances of the sheet were distributed,,the cmrrent introduced at.the free end of the box beam is made one-half of the amount introduced at succeeding points so as to obtain a series of steps which straddle the smooth curve.

As a numericel example the panel previously
considered is used as the cover of a box beam.(See fig.12.) Because the %eem is symmetrical about the center.>line,onlyone-half of the beem need be considered.The'depth h is,3 inches, and a load of 250 pounds is assumed on each side at the tip, Also, the %eam is divided into five equal-length bays of 20 Inches.The running shear per unit length of the flange i-s . .P 250 ' -= A-.

pounds per inch h 3
For a bay Iength of 20 inches the total shear force per bay is 83.3 x m = 1666 pounds.As before, a convenient vslue of 10 milliamperes is chosen for the current, and the constant then is

$.XL = oJ30t50 1666
The equivalent network Is shown in figure 13, and the corresponding stress-distribution plots are shown in figure 14.Since the slope of the exact-method curves does not change very NMA TN NO. 1281 rapidly, the u~e of equal-length bays is shown to give very good agreement between calculated and test results.1% should he noted that with the structure divided into equal bays with a half step at the loaded end, a half step is left,unaccounted for at the root end, This omi~slon may be disregarded without appreciable error.

ANALOGY FOR BOLTED-JOINT PROBLEMS
As shown in appendix A, the correspondence between the elements of a bolted joint (fig.15) end the computer network (fig.I-6) is given in the following table: Written in equation form with ez%itrary constants a end @.the following relations hold: ..
(lo) l?rom a comparison of figures 15 anfi 16, the resistor network will be noted to consist of a single row of longitudinal resistances R6 which simulate the two straps of the %olted joints A single set of transverse resistances rn which simulate the two shear ~eas of the bolt is also usedt These apparent discrepancies am accounted for by the appearance of the factor 2 in equation (8).

V=c$u (U)
\%en bolted-~oint pro%lems are to be solved, it should be noted that the load is transmitted through the lolts in discrete amounts acting at definite points, rather than through infinitesimal elements uniformly distributed along a sheet ae in shear-lag problems.
There is, therefore, no "finite-length increment" to cause errors when the network is used to simulate a bolted ~oint.

T
Vri%ten in equation .formwitharbitrary oonstants" B and cc. the following relations hold:.

015 .
for convenience at l? milliamperes; to the load of 1000 Tounds.Then by$ = *" 0.These values of R, r, and p were set up on the computer network and the corresponding currents were recorded as shown in figure9.Substitution of the current values in equations (1) and (3) then gave the desired stress values.

Figure 12
Figure IO.-Stresses in panel of equal bays.

Fig. 16 NACA
Fig. 16 voltagegen4rator, md a generator with an internal resistance ~ntermediate between infinity and zer%.The present computer is built with constant-~urrent generators ,wd, }herefore,obtains solutions whtch apply to the dead-weight type of lceding since this type of loading agrees more,cl.oselywith the loading of the M%WiL 'dnnwtllre.
,,, MKM 'm No. l.ml current f3uppm9a by a constant-current generator, a constantshok loaded.at one end of the flange.

Sufficient acc~acy Is obta~ed to wamant
the use of the computer in ordinary engineering stress analysls.The use of a rcoreextended electrical network capable of simulating more complicated structures would appreciably reduce the time necessary for their enalysis.