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NATIONAL ADVISOEY CON_41T&_E FOR AERaNAUTICS

TEC_TICAL NOTE NO. 1364

STRFA_GTH ANALYSIS OF STIFFENED BEAM WEBS

By Paul Ku_m and James P. Peterson

SUMMARY

A previously published method for strength smalysis of stiffenedshear webs has been revised and extended. A set of formulas and

graphs which cover all aspects of strength analysis is given,

experimental data are presented, and the acctu_acy of the formulas

as Judged by comparison w!th these data Is discussed. Revisions

of some formulas have resulted in improved agreement with experi-

mental stresses and _zlth more rigorous theory, particularly for

low ratios of applied shear to buckling shear. The scope of the

experimental evidence has been _reatly increased compared with the

previous paper by incorporating the results of several investigations

undertaken since then.

INTRODUCTION

Many of the _shear webs used in aircraft strud_t_es are so thin

that they buckle at a fraction of the ultimate load. A purely

mathematical theory of basically simple form has been developed

for the limiting case of _ebs so thin that their resistance to

buckling is entirely negligible (reference 1). This theory ofII II

pure diagonal tension is too conservative for practical usebecause the resistance to buckling of practical webs - "incomplete-

diagonal-tension webs" - is far from being negligible. A mathe-

matical theory of incomplete diagonal tension has been developed

(reference 2), but it requires such extensive calculations that

its adaptability to stress analysis is questlonable) moreover, no

adequate check of its accuracy by comparing it with test results

over a wide range has been published 3 and It is not sufficiently

complete to explain upright failures, probably the most important

item in the design of web systems.

In the face of such difficulties, practical stress-analysts

have often resorted to entirely empirical formulas. There are

two objections to such a procedure: Without the benefit of some

2 NACA%_TNo. 1364

caiding theor_j, a very large number of tests is required to inside

the reliability of a given formula, and a formula established for

the strength of ons part of a beam is usually of little, if any,

help in establishing formulas for other items.

The method of analysis given herein constitutes an attempt

to avoid insofar as possible the objections to purely theoretical

or purely empirical methods. _le basis of the method is a semi-

empirical engineering theory of incomplete diagonal t_nslon, n_ade

as s_nple as possible by confining attention to over-_tll or average

effects. The theory is formulated in such _ way that the limiting

conditions of fully developed diagonal tension and of zero diagonaltension (so-called shear-resist_mt web") are _ncluded; it c_n

therefore be regarded as an aid for interpolating between these

limiting conditions.

_le analysis is divided into two parts. The presentation of

the theory and of the design fo_mulas is given in p_rt I and is

kept very brief in order to approach as closely as possible the

final form that it would take in a stress manual. Part II is

devoted to a discussion snd experimental verification of the

formulas; it incorpora+_s the results from a ntnuber of previous

_nvestigations. In order to keep the length of this part also

to a mln _imum, the discussion has been confJa_ed to items of decided

practical interest. Reading of part II is not necessary if

interest is confined to routine application of the design formulas

but is indispensable for anybody who wishes to interpret test

results or to extend or modify the fo_uulas in sny respect.

_le theory is basically the same as that previously published

in references 3 and 4, but it has been modified in some respects

o_nd therefore supersedes the material given in these references.

E

G

H

SYMBOLS

cross-sectlona! area, square inches

Yotmg's modulus, ksi

shear modulus, ksl

force in beam flange due to horizontal component of

disgonal tension, kips

moment of inertia, Inches 4

i_ACA _ No, 1364 3

L

q

R

S

d

e

]i

k

t

(Z

6

O

T

length of be_, inches

force_ kips

static moment about neutral axis of parts of cross

section as specified by subscript, inches 3

coefficient of ed_._erestraint (see fom_lla (7))

transverse shear force, kips

spacing of uprights, inches

dist,.nee from median plane of web to centroid of

(single) _q_rig_ht, inches

depth of beam, inches (see ST.)ecial Combinations)

diagonal- tension :[actot

thickness, inches (u_ed without subscript signifies

_hicLn_.ess of web)

angle between neutral axis of besms and direction of

diagonal tension, de(_rees

deflection of be_n, inches

normal strain

centroidal rsdius of gyration of cross s_ction of upright

about a-cis parallel to web, inches (no sheeb should

be include&)

normal stress, ksi

shear stress, ksi

DT

F

S

diagonal tension

flange

shear

Subscripts

4 _%CA TN No. ].364

U

w

cr

_lt

e

upright

web

critical

_Itimate

effective

PU

R 11

Rtot

d c

hc

he

kss

_o

cx!

Special Combinations

internal force in upright, kips

shear force on rivets per inch run, kips per inch

total shear strength (in single shear) of all rivets in

one upright, kip_

upright spacing measure_ as shown in figure 5(a)

_epth of _eb measured as shown in figure 5(a)

depth of be_u meas_tred between centroids of flanges, inches

depth of be_. measured between centroids of web-to-flange

rivet patterns, inches

length of upright measured between centroids of upright-

to-flan_e rivet patterns, inches

theoretical buckling coefficient for plates with simply

supported edges

"basic" allowable stress for forced crippling of uprights

(valid for stresses below proportional limit in

compression of upright material), ksi

tflange flexibility factor .Td _IC + iT)h e ,

where IC and IT are moments of inertia of compression

)flange end tension flange, respectively

NAC',A _ No. 1364 5

THEORY AND FORMULAS

_NG]_I9_R__.G _0RY OF INCOMPLETE DI&GGNAL THNSI@T

In a plate gird or subjected to a shear load less than the

buckling load, the web plate is .in a state of pure shear along the

neutral axis as indicated by the inset diagram in figure l(a).

Above e_nd below the neutral axis, normal stresses ezrlst in a

horizontal direction, but in the investigation of the web for the

present p_pose these stresses may be disregarded, and the stress

diagram may be assumed valid over the entire depth of the web.

The web stiff ethers carry no stress.

If the web is thin, it, will buckle at a certain critical

shear load. If the load is increased beyond the critical _alue,

the buckle pattern will _-adually approach a configuration con-

sisting of parol].el folds (fig. l(b)). In the theoretical limiting

case of an infinitely thin sheet 3 the web carries pure tensile

stresses In the direction of the folds as indicated by the inset

dla_r_,n in fidure l(b). The an_le _ which these folds include

with the horizontal axls of the beam is usually somewhat less

than 45 °. Simple statical considerations show that each upright

carries a load

PU = Ttd tan _ (i)

as reaction to the vertical component of the _:eb tension, and each

flang_e carries a compressive force

SH = cot _ (2)

as reaction to the horizontal component of the web tension.

Formulas (i) and (2) can be eveluated once the angle _ is Imuown.

The. theory of pure diagonal tension (reference l) shows that this

angle is given by the form_la

where _ is the strain In the web, _x is the strain in the

flanges due to the force H, and _ is the strain in the upright.

Elongation is considered as positive st_in.

6 I_ACA _ No. 1364

In a practical thin-web beam, the state of stress in the web

is intermediate between pure shear and pure diagonal tension. An

engineering theory of this intermediate state of incomplete diagonal

tension may be based on the assumption that the total shear force S

_n the web can be divided into two parts, a part SS carried by

pure shear and a part SDT carried by pure diagonal tension_ thus

S = S S + Szr 2

rJ2_Isexpression may be written in the form

(4)

where k is the "dla_onal-tension factor" which expresses the

degree to which the diagonal tension is developed at a given load.

},'ith this f_ctor, the state of pttre shear is characterized by k = O,

and the state of pure diagonal tension, by k = 1. 'lhe stress

condition of a web element is sho_n in figure 2 for the two

limiting cases k = 0 and k = 1 and for an intermediate case.

The factor k has been established empirically by evaluating

strain measurements on uprights because the stresses in the uprights

constitute the most sensitive criterion for the degree to which the

diagonal tension is developed. For loads less than five times the

buckling loads, for which the accuracy of the experimental results

is of ten poor, use _s made of the calculations made by means of

Levy Is large-deflection theory of plates (references _,_and 6).

From these experimental and theoretical data, it ,_as found that k

can be given by the expression

(o ) (5)k = tar_q .5 IOgl0

As long as the web is resisting some compressive stress in a

diagonal direction: it ccn also resist some compressive stress in

the vertical direction and thus assist the uprights. If the

distribution of these vertical compressivestresses is assumed to

be sinusoi_al immediately after buckling as indicated in figure 3,

NACA_T I{o. 1364 7

the total effective _-Id.th of sheet cooperating with the upright

is 0.5d. _i_qeeffective width will decrease as the diagonal tension

develops and will become zero for fully developed diagonal tenn,.on

(k = i). Tf the effect i_-e _:idt.h is ass,_ued to decrease linearly

_,,_Ith k, the eCfective area contributed by the web to the upright

is

A_,_e = 0.5dt(l k) (6)

_ corresponding asstnuption could be made for the contribution of

the web to the flange area as f_,r as resistance to the force K

given by fo_uula (2) is concerned. Tbis refinement, howe_er, is

probably urunecossary in _du_-_, analysis of' besm webs.

Fo_uulas (4) to (6) are the fundamental foruulas which

generslize th_ theory of pt_e diagonal tension to cover the full

range of incomplete diagonal tension from the l_._[[ti_ case of pltre

shear to the !imitin,7_'case of fully developed diagonal tension.

qhey enable the stress _nualyst to make a reasonably acctlrate

estimate of the stresses in the uprights_ the necessity of estl-

mating thes_ stresses _rith a much better accttr_cy than that afforded

by the theory of p_e diagonal tension has been the predominant

reason for developing a theory of incomplete diagonal tension.

The theory expressed by formulas (4) to (6) defines only the

"over-all" state of stress in the median plane of the web. It

does not attempt to give an account of the detail distribution of

these stresses_ nor does it give any account of the bending stresses

in the web sheet induced by the shear buckles. Consequently, all

problem s that involve the details of the web action require,

additional ass_nptions or empirical data for their solution. For a

number of items (for instance, forces on the web attachuen,_ rivets),

the magnitude is ]_qown for the limiting cases of pure shear and

pure disgonal tens.ion; for any intermediate case of incomplete

diagonal tension, the magDitude can ?.hen be estimated by inter-

polating be!-_,_centhe l_niting cases with the fac 5or k as argument.

Straight-llne int_erpolation is used unless empirical data ortheoretical considerations indicate a different law of variation.

For some quantitles, straig_t-line interpolation is used for

simplicity and conservativeness although the theory indicates a

more complicated law.

A minor item from the theory of p1_re diagonal tension should

be mentioned here. The vertical component of tile diagonal tension

in the web will bend the beam flanges as indicated in figure 4.

8 NACA_L_NNo. 1364

As a result, the tension will be relieved in someparts of the weband correspondingly increased in other parts of the web as indicatedby the spacing of _2nediagonals in the finite. _le redistributionof web tension, in turn, will decrease the secondary bendingmomentsin the flanges. The theory of these effects is discussedin reference 1.

FOI_rfAS FOR_I_o_oo ANALYSIS

Limitations of Formulas

The fornmlas given herein are believed to give reasonableassurance of conservative strength predictions provided thatnormal desi_onpractices and proportions are used. The mostimportant poSzts under this provision are that the uprights should

(s_y _>_: 0.6) and that the upright spacing shouldnot be too thin

not be too much outside the range 0.2 < _< !.0.

webs _h <200)have not been explored adequately. For vebthick

systems in these ranges, some possibility of unconservative pre-

dictions may exist.

The accuracy that may be expected of the strength predictions

is discussed in part II, which presents the experimental evidence.

_e origin of the formulas and the references ar_ also given in

part II in order to keep part I free from details not necessary to

_:he routine _pplicetlon of the formulas.

Critical Shear Stress

In the elastic range, the critical shear stress of the sheet

between two uprights is calculated by the formula

Tcr kssE/_.)2 ,'-l._h+ 21(Rd Rh ) \hc' ]

t=

L-

(7)

NAC,\ TN No. 1_%64 9

_ ore

ks s theoretical buckling coefficient (6_von i_

fig. 5(a)) for panel of length h c and width

with simply supported edges

_C

_'C

h C

Rh

_rldth of _heet betwecm uprights measured as shown in

figtn_e 5(a), inches

depth of web measured as sho_m in figure 5(a), inches

restraint coefficient for edges of sheet along upright

(i rom .,,. ,. 5(b))

_d restraint coefficient for edges of sheet along flanges

(from fig. 5(b))

(If dc > he, substitute h c for dc_ dc for he, I]d for

a_'d P'h for )_d"

Ctu_ves of the critical she_r stresses for plates of 24S-T

almninum alloy w'l_h simply supported edges are Given in figure 6.

To the right of the dashed line, those ctu_ves are plots of the

_heore tical eguation

cr k_s E "t--_2T = (%]

Rh,

and may be used for most aluminum alloys. To hhe left of the

dashed line, the curves represent straight-line tangents to the

theoretical curves in a nonlogarithmic plot and are valid only

oI,c,,_,alloy.for r.,+,.- _

When the uprights are very thin, the value of Tcr obtained

by f3rmnla (7) may be less than that obtained by neglecting the

presence of the uprlghts. Web systems of such abnormal proportionsshould not be designed by the formulas of the present paper.

Loading I{atic

The loading ratio is the ratio T/Tcr where T is the

depth-wise av(___a{_eof nominal web shear stress, that is, of the

i0 _CA TN No. 1364

shear stress that would exist in the web if buckling wereartificially prevented (by external restraints acting _ the web).

Whenthe depth of the flanges is small comparedwith the depthof the beam, and the flanges are angle sections, the stress T maybe computedby the foln_ala

T _ (8)her

In beams with other cross sections, the average nominal shear

stress T should be com_uted by t_e formula

---To ?f@-/ (9)

where QF is the static moment ab_at the neutral axis of the flange

_mterial em__ %r is the static moment about the neutral axis of the

effective w_b _zteria] above the neutrel axis. For the computations

of i and Q, the oiToctivcness of the web must be estimated in

first approximation. As socon_ and final approximation, the

efi'ectivcncss ,_f tb_ web may bo _keu as equal to (i - k), where k

is the diagora!-tension factoi" dete_'_ned in the next step; in other

words, when I and Q are being computed, the effective thicknessof the web is taken as (i - k)t.

Diagonal-Tenslon Factor k

After the loading rstlo T/Tcr has been computed, the

di_gcnal-tension factor k can be computed by formula (5)or

read from figure 7.

Average Stress in Upright

The average stress aU in a double uprlg_t (average over

hhe length of the upright) is given by the formula

kT tan

_U =A U

d-Y+ o.5(i- k)

NACA T_T No. 1364 ii

which follows from formulas (1), (4.), and (6). It can be evaluated

with the help of figure 8 or figure 9 which give the ratio _U_

as a function of the ratio Au/dt and the loading ratio T/Tcr.

The stress _U is uniformly distributed over the cross section of

the upright tmtll buckling of the upright begins.

The stress qU for a single upright is obtained in the s_:me

manner, except that the ratio Au/at is replaced by AUe/dt where

AU

A% - ,e<2 (lO)1 + !$)

For the single upright, eU is still an average over the length

of the upright, but it applies only to the median plane of the web

along the line of rivets connecting4 the upright to the web. In any

given cross section of the uprig_t, the com!_rosslve stress decreases

with increasing dist_uce from the _zeb, because the upright is a

column loaded eccentrically by the web tcnslon. (For this reason,

formulas for local crlppl_ng based on the asst_nption of a uniform

distribution of stress over the cross section do not apply.)

Maximum Stress in Upright

The stress qU in an upright varies from a maximum at (ornear) the neutral axis of the beam to a _minim_nn at the ends of

the uprig_t ("gusset effect"). The ratio of the Nmximmn stress

to the average stress decreases as the upright spacing or the

loading ratio increases. The empirical formula for the ratio is

_Umax-l+ (n)

where \---_-U]°- is the value of the ratio when the web has just

buckled. The ratio is given graphically in figmre i0.

Angle of Diagonal T'ens{on

The angle a between the direction of the diagonal tension

and the axis of the beam can be found with the aid of figure ii

after k and aU/V have been determined.

Allowable Stresses in Uprights

The following four types of failure of uprights areconceivable :

(i) Column failure

(2) Forced crippling failure

(3) Natural crippling faih_e

(4) General elastic instability failure Of web and stiffeners

Column failure.- Col_mm failures in the usu_l meaning of theword (failt_re due {o instability, without previous bowing) are

possible only in double uprights. When column bowing begins, theuprights will force the web out of its original plane. _he web

tensile forces will then develop components normal to the plane

of the web which tend to force the uprights back. This bracingaction is ta!_n into account by using a reduced "effective" column

length of the upright Le, _ich is given by the empirical formula

hULe -- (12)

\l+k -2

The stress aU at _ich column failure takes place can then be

found by entering a stDndard coltmm curve for the upright _ththe slenderness ratio Le/_ as argument.

The problem of "column" failures in single uprights has not

been investigated to any extent, and test results are greatly at

variance with theoretical results. The following two criterionsare suggested for strength design:

NACA T_No. 13o4

(a) The stress _U should be no greater than the cohmm

yield stress for the upright material.

(b) _ne stress at the centroid of t_e upright (which is the

average stress over the cross section) should be no greater than

the allowable col_m stress for the slenderness ratio hu/20.

The first criterion accounts for the upright acting as an

eccentrically loaded compression member_ the second one is an

attempt to take into account a two-wave type of buckling failure

that has been observed in w_ry slender uprights.

Forced crlpplin6 failtu'e.- __e shear buckles in the web will

force buckling of the upright in the leg attached to the web,

particularly if the uprig_t is thinner than the web. These buckles

give a lever arm to the compressiv e force acting in the leg and

thereby produce a severe stress condition. _u_ buckles in the

attached leg will in turn induce buckling of the outstanding legs.

in single uprights the outstsa_ding legs are relieved to a con-

siderable extent by virtue of the fact that the compressive stress

decreases with distance from the web; the allo_ble stresses for

sing_le upri_jhts are therefore somewhat higher thea_ those for double

uprights. Because the forced crippling is of a loc,_l natur(_, it is

assumed to d,_pend on the i_eak value CUmax of the upright stress

rather than on the average value.

The upright stress at which final collapse occurs is obtained

by the following empirical method:

(!) Compute the allo_mble value of o_Jmmx for a perfectly

elastic upright material by the formula

(for single uprights) (!3a)

25k ,,.itu/t (for double uprights) (13b)

(2) If qo exceeds the proportional limit for the upright

material, use as allowable value the stress corresponding to the

compressive strain qo_"

(3) If k < 0.5_ use an effective value in formula (13a)

or (13b) given by the e_ression

ko = 0.15 + 0.7k (13o)

14 IK_.CATN No. 1364

Natural cri_olin_ fail_Lre.- The term "naturalcrippling failure"

is used herein to denote a crippling failure resulting from a

compressive stress uniformly distributed over the cross section of

the upright. By this definition, it can occur only in double

uprights. To avoid natural crippling failure, the peak stress CUmax

in thc_ upright should be less than the crippling stress of the

section for L _ O. Nat_a! crippling failure does not appear to0

be a controlling factor in actual designs.

General elastic instability of web and stiffeners.- Test

experience so far has not indicated that general elastic instability

need be considered in strer_th d_sign. Apparently, the web systemis safe against general _lastic instability if the uprights are

designed to fail by col_2_n action or by forced crippling at a shear

load not much lower than the shear strengt2_ of the web. It should

be borne in mind, however, that the test experience available at

present is not adequate [!'orvery_ _thin_and,,for very.,thick webs..")(See section of present paper _nti,_le_ Limi_a hions of Formulas

Web Design

For desi_ pu__poses, the p_ek value of the nominal ,,_b shearstress within a bay is t:_Icenas

(liO

where CI and C2 are the factors giw_n in figt_ros 12 and 13,

respectively. The factor C1 constlhutes a correction factor to

allow for the angle _ of the diagonal tension differing from 45° .

The factor C2 makes allowance for _le stress concentration in

the web brought about by flexibili1_ of the flanges as discussed

in connection with figure 4..

T_le allowable value of Tma x is determined by tests and

depends on the wlue of the diagonal-tension factor k as well

as on the details of the web-to-flange and web-to-upright fastenings.

Figure 14 gives empirical curves for two aluminum alloys. It shouldbe note_, that these curw_s contain an allowance fox' the rivet

factor_ inclusion of this factor in these curves is possiblebecause tests have sho_m that the ultimate shear stress based on

I_ACA_T No. 1364 15

the gross section (that is, without reduction for rivet holes) isalmost constant within the normal range of rivet factor (CI_ >0.6).A separate check must,be made, of course, to insur(,, that theallo_._ble bearing stresses between rivets and sheet are notexceeded.

Rivet Design

The load per inch rtn%acting on the web-to-flange rivets istaken as

R" S!'T(l+ o. 14k)

With double upriff_hts_ the web.,to-upri_ht rivets must provide

sufficient lon_]itudinal shear strength to make the two uprights

act as an integral _mit until column failure occurs. The total

rivet shear strength (single shear strength of all rivets) required

for an upright is

2(_coQ hU

RSot - b Le (16)

where

_CO coltmm yield strength of upright material (if _co is

exp_ressed in ksi, Rto t will be in kips)

static moment of cross section of one upright about an

axis in the medis_n plane of the web, inches3

width of outstqnding leg of upri_it, inches

ratio obtainable from formvZLa (12)

The rivets must also have mtfficient tensile strengd_ to

prevent the buckled sheet from lifting off the stiffener. Tqle

necessary strength is given by the tentative criterion

Tensile streng_ (per inch) of rivets > O.15tdul t (17)

where qult is the tensile strength of the web.

16 NACATN No. 1364

For web-to-upright rivets on single uprights, the required

tensile strength is given by the tentative criterion

Tensile strength (per inch) of rivets > 0.22t_ul t (17)

(The tensilestrength of a rivet is defined as the tensile lo_d

that ceuses _[failure} if the sheet is thin, failure will consist

in the pulling of the rivet through the sheet.)

No criterion for shear strength of the rivets on single

uprights has been established; the crit_rion for tensile strength

is probably adegt_te to instLre a satisfactory design.

The pitch of the rivets on single uprights should be small

enough to prevent inter-rivet buckling of the wsb (oi"the upright,

if thinner than the web) at a compressive stress equal to _U_a x.

The pitch should also bs less thon d/4 in order to Justify the

asstLmption on edge support used in the determination of V cr. The

two criterions for pitch are probably al_ys fulfilled if the

strength criterions are fulfilled and normal riveting practicesare used.

The uRright-to-flange rivets must carry the load existing inthe upri_it into the flange.

PU = _uAu (for double uprights) (19)

PU-- qUAUe (for single upri@_ts)(2o)

These formulas neglect the _isset effect (decrease of oU towards

the ends of the upright) in order to be conservative.

Secondary Bending Moments in Flanges

The secondary bending moment in a flange, caused by the verticalcomponent of the diagon21 tension (fig. 4), may be taken as

1 kvt&2C3 (21)

where C3 is a factor given in fi_nxre 13. The moment given byfo_nula (21) is the maximum moment in the bay and exists at the

NACA TNNo. 1364 17

ends of the bay over the uprights. If C 3 and k are near unity,

the moment in the middle of the bay is half as large as that given

by for_rala (21) and of opposite si@u. (See fig. 4.)

Shear Stiffness of _Jeb

The theoretical effective shear modulus of a web C e in partial

diagonal tension is given by figure 15. This modulus is valid only

in the elastic range.

II - DISCUSSION OF FORMULAS AND

EXPERIMENTAL EVID_ZN CE

In the fo!lo_ingpart of the present paper, the formulas and

the e_erimental evidence are discussed in the sequence in which

the formulas appear in part I. The experimental evidence presentedis based on results from manufacturers' tests and tests made in the

Langley Struc_n_es Research Division of the NA_. All test

evaluations are based an actual material properties insofar as

possfble.

TEST SPECI_!S

The analysis covers about 90 beams tested by fot_ manl_acturers

and 32 beams tested by the NACA. Some of the manufacturers' tests

could not be fully analyzed because the data were incomplete. The

range of the tests can be defined as follows:

Ratios

hltdlhAu/dttu/t

Range

300 to 2500

0.18 to 0.91

0.039 to 1.2

0.42 to 8.4

The NACA tests are discussed in greater detail than the

manufacturers' tests becsuss the strain measurements taken in

these tests served as the main basis for establishing the

diagonal-tension factor k. The essential data on all beamstestedby .the NACA are givGn herein in condensed, form in order to obviate

the necessity of referring to references 4, 7, and. 8.

Each beam of the NACA series is given a code designation such

as I-2}-4D, with the following meaning:

I designates the test series of reference 4 (Series II ix

from reference 7, serles III from reference 8, series IV

from recent tests not previously published.)

25 is the approximate depth of the beam !n inches

4 is the nun_er of the beam within the series

D strands for double uprights (S for single uprights)

The basic data on the bo_ms are given by fi_?_e 16 and table i.

_e main results of the strength tests and of the c_lctulations

ar_ given in table 2.

CR!TICkL SB]!'AR $_.{ESSES

Fos.nnulas for the critical shear stress of a flat plate _rith

simply supported edlses ma,7 be found in r,3ference 9. Fol_m_.la (7)

for plates In which the edge conditions on one :eeir of ed[_es

differ from those on the other palr of e_.g%es was obtained by

fitting an empirical curve to theoretical results for plates with

one pair of edges simply supported (E = l) and on_: pair of er,]i,es

clamped (R = 1.62). The theoretical results were taken from

references lO to 13. Some of the results given in reference ll

were shown to be in error by Moheit, whose rest,]_ts are given in

reference 13, but correctc_d values were not given for all cases

that may be in error. A d_finitc statement on the accuracy of

formula (7) for the case of two edges clemloed an@. two edges simply

supported can therefore not be made, but it is b_lieved that +he

foi._nula has a maximum error of about 4 percen",:..

The res_..rain+. coefficients R 61ven by f!_ure 9(b) are based

on incidental detc_rminations of b1_ckllnc stresses made in some

beam tests.. It shoul_ be real.lzed_ first of _;ll, that representing R

as a function of only tu/t constitutes a rather extreme simplifi-

cation of a we__V complex problna and, f_,trthermore, the experimental

de1_crmination of the cr!t!cal stress in a beam web is a dif'ficult

}L_.CA_,[ No. 1364 19

problem. :_e curves given should, therefore, not be interpreted

as me_s for a very exact determination of the critical stress,

but as means for obtaininc 8n approxim_te value of the critical

stress adequate :for the pttrpose of obtaining the diagonal-tension

factor k. The upper c1_rvo of _igure 5(b) is believed to be

reasonably reliable because existing test data a_oree fairly well

with it. Considerable doubt exists about the lo_rer curve,

particulorly for 4.-_-< 1.2, because the test data are not only

very meager but also difficult to interpret.

__e part of the curve near the origin is sho_ as a dashed

llne to indicate two facts. One is that no expGrimental evidence

was available for this region. _he other one is that the appli-

cation of fon_mXa (7) in this rezion m_y give buckling stresses

lower than those that would be obtained if the presence of the

upri_<hts were disrcL, arded entirely and the web werc_ considered

as a plate framed by the beam fl_nges ar.d the tip and root uprights.

This obviously erroneous res_rlt ms caused by the simplifying

ass__.npt_ons implied by forn_11a (7); forttunately, it appears very

_uprobab!e that the region in question will be epproached in any

actual web system desi_ed to develop a strength somewhere near

the shear strength of the web.

D]_G<_AL-_VA_;SION FACTOR k

Formula (5) for the factor k was obtained by fitting an

empirical curve to values of k calculated from the strain

_ueast_rements on the uprights of the N_CA test beams. A direct

comparison of th_ calculated values of k _nd the e_irical

e:qoression is not given because it is of much less interest than

the comparison of The experimental upright stresses with those

predicted with the aid of the empirical k-ctu_ve.

ANALYSIS CK.%RT?o

_ne analysis charts (flgs. 8 and 9) were calculated from the

formula for _U given in part I under the heading "Average Stress

in Upright." .This for_ula must be evaluated by successive approxi-

mations because tea_ m is a function of aU according to formula (3).

The fl_n_ge area was ass_uned to be so large that _x could be

neglected in formula (3).

20 _%CATN No. 1364

__neuse of the ana].ys_,s._harts to determine the stresses insingle uprights by moansof the _ff3ctive cross-sectlon area AUe(fo_.qula (lO)) implies sew_ral simplifying assumptions. Formula (lO)is obtained from the f_¢i].iar formula for eccentrically loadedcompression members

_}+ ee_AcI

by setting c = e and I = _2A. r±_neimplied assumptions are:

(a) The eccentricity e of the load is constant.

(b) The ratio e/P is not ch_,tn@ed appreciably if the

contribution of the _,_ebto the effective cross section of t/_e

upri_it is neglected.

Assumption (a) is plausible if the upr!gj_ts are very closely

spaced, becs_use hhe web then moves with the uprights (reference l).

In general, however, both asmunptions can be Justified only by the

fact that "t_.eyhave yielded good ag_eement with test results.

Thick webs are 1._kely to require more refined assumptions. If

eit/_er ass_nn!,_tion (a) or (b) is dropped, the analysis charts carmot

be used for webs with single uprig_its.

'_IFICATIC_ OF S%_IESS FO_\_LAS

In this sec't_on, results obtained by means of the engineering

t]leory of incomplete dJa_onal tension will be comp_:,red with

(a) Exper_uuentel stresses deduced from NACA strain meast,rements

on upr i_h t.s

(b) Upright forces calculated by Levy's theory (references 5 and 6)

(c) Diagonal-tension factors k deduced from the tests of Lahde

and Wagner (reference 14)

Outline of _roced_tre in NACA tests.- Ln the NACA tests, the

strains in the upriehts were measua_ed with electrical gages. In

order to obtain a reasonably represent_.tive average, a fairly large

number of (_age stations _as used on each upright (9 gage stations

on the 2D-inch be._ns of series II and III) and measurements were

NACA TN No. 1364 21

taken on two or three uprights depending on the total number of

uprights in the beam. After the strains h:_,dbeen converted to

stresses with the help of stress-strain curves obtained from coupon

tests, all the stresses for one beam were averaged to obtain the

final value of cU shown .in the plots.

Comparison with ex_erlmental stresses i_ doubl_ u_rig_.- On

double uprights, a pair of gages was used at each strain stationin order to average out bonding stresses. The efficacy of this

device depends on the def_-ee to which the two stiffeners comprising

the uprights act as an integral_it. A high degree of integral

action was probably achieved in all the beams discussed her_in,

and will probably be achieved in any beam in which the upri_hts

are not thinner than the _b, provided that the rivet connection

is adequate to prevent inter-rivet buckling.

Inspection of figure 17 sho_ that _le calculated val_s of _U

for double uprights are general_, conservative or in close agreement

with the experimental stresses; the only case of a decidedly uncon-

servative prediction is be_n IV-72-3D at high loads, K somewhat

curious phenomenon is sho_nby beam IV-72-1D, for which aU begins

to decrease with increasing load at a lomd well below the 1_tlmate.

The phenomenon eppears to be linked to somc extent with the effectsof local buckling or forced crippling. _e be_ in guestion had

next to the lowest ratio of tu/t of all the dm_le-upright beams

tested, and the same phenomenon vas e_?ibited to a much greater

degree by the single-upright beam IV-72-4S, which had an even lower

rstio of tu/t. Some other double-upright beams with a somewhat

higher ratio of tu/t (beams 111-25-4D _nd II!-25-7D) appear toindicate a slight tendency toward the same phenomenon. It is,

therefore, debatable whether _U really begins to decrease or

whether the strain readings are falsified by local buckling stresses.

Comparison with experimental stresses in single u_)rif:hts.-The

predicted stresses _j for single uprights are valid only for the

median plane of the web at the upright in guestion, whereas the

stra_ measurements were t_ken on the exposed face of the attached

leg of the upright. In order to permit a direct c_nparison, the

predicted, stresses were corrected to the plane of measurement,

assumin_ linear stress variation in the uprights.

In single uprishts it is not possible to average out bending

stresses by using pairs of gages. Theoretically, they could be

almost averaged out if gages were located on each crest and in each

trough of the buckles, but the buckle patterns cannot be predicted

22 N_&CAkgl No. 1364

with sufficient accuracy to achieve such a £istribution of gages.

In order to give somf_ idea of the matuuitudo of the bending stresses,

figure 17 shows for all bea_ with single upriF_ts not only the

average stress _U but also the lowest an_ t_hehighest in&ividusl

stress measured at any one _go in any of the uprights of the beam

at each load. Inspection of the figt_re shows that the range from

the lowest to the highest stress is quite large. In spite of this

fact, however, the experimental average stress I_U agrees quite

well with the predicted stress in most cases, much closer in fact

than for doubl_ uprights conslder_ng all tests, except that on the

sinE<le-u__right beams the st:'esscs GU at the highest loads show a

tendency to exceed the calculate& stresses in a number of cases.

The phenomenon of a reversal in the curve of _U against load,

noted for beam IV-72-1D, is exhibite_ very markedly by beam IV-72-4S.

This beam had a ratio of tu/t of uni+_', the lowest of all single-

upright beams tested. The rsn_e of ._ndividual stresses is extremely

large. Tensile stresses of large magnitude appear, _hereas in all

other beams the st:tess remained compressiw_; that is, t_he tensilestress due to local buckling vas al_ays less than the column

compressive stress in the _q0right (with minor exceptions for

beam IV-T2-2s).

comparison with Levy's theory.- On most of the beams testedthe critical shear st'_'_ss_s too low to permlt comparisons st low

loading ratios V_c r. The only exceptions were the beams of

series IV, and, as noted, the rem<Lts on two of these were presumably

partly invalidated by local bending effects. Fortunately the regi_

of low T/Tcr is reasonably amennble to theol_etical calculations.

Levy has developed a suitable theory of plates with large deflections

end has carried through calculations for several specific cases

(referenc(;s 5 and 6). C_mp:Lriso.nsbetween the results obtalncd by

the present semlempirical method and Lewj's theoretical results are

sho_n in figure 18.

The upright load PUmax rather than the upri_j:t stress is

sho_n in figure 18 in order that the result for the theoretical

limiting case of infinite upric_:t area mlL_ht be included. Thed AU

agreement for _ = 0.4 and _ = 0.25 is very close. For the two

dcases with _ = 1.O, the agreement is not so close. For the finite-

f Au 9)size upright <_ = 0.2 , the upright loa_ predicted by the present

NACA TN No. 1364 23

theory is less than t2_at pr_,di_ted by Levy's theory 3 with a maximum

dovistion of about 30 percent e t _ 1.o_ for the infinitely largeTcr

upright, the deviation is of the same sign _m_[ somewhat larger.d

Although the agreement for K = 1.O is not so good as might be

ddesired, the agreement for this case as well as for _ = 0.40 is

very much better than that sho_m in references 5 and 6, which were

based on lho expression for k given in reference 3.

Kmo_Tledge of lhe st_rength of be:_ns designed to fail (by

simul+aneous foih_e of the uprights and the ",_b) at low ratios

of v_c r is very incomplete at present. There are indications,

hoover, that for efficient deoii_as the spacing ratio d/h will

probably be about I/2. T._e se_'_.Iompirical method of predicting _U

appears to hold promise, therefor_, of g_ving reasonable accuracy

the r_nz_ most Lmpor_a_ _ for de_ign even for webs that arecommonly called "shear-resistant" webs rather th_ "incomplete-

diagon_, l- ten sion" _eb s.

Comparison wii_1 tests of !_ah_e and _>Taf_er.- The comparisons with

experiments presented so fnr have be_n nmde for the stresses oU

rather then for the dia6onsl-tension factor k because the stress

is the directly meamu'_d quantity and is of _-_e_ter direct interest.

For the tests reporte_ in r_fer_nce 14, it is more convenient to

compare _lues of the diagonsl-te_sion factor k. %_qese tests were

not made on actual b_ams, but on plates _n a special test Jig. __?.e

test conditions wer_ somewhat artificial and tony not represent the

conditions existing in a beam w_ry well. There _'e also doubts

concer_ing the evaluation end interpretation of the test results.

As a matter of some interest, hog,ever, fig_re 19 shows the gG.aph

for k obtained by a comparison between formula (5) and e_:srimental

values of k deduced from these test recruits. In view of the

factors of doubt mentioned, the agA'eement is perhaps as good as

can be e.xp_.ected. The fact that th(_ expe_mental points lie con-

sistently above the curve _5y be explained in p_rt by the fact that

the critical siamese has been taken at the theoretical wlue for

clamped edges; it is well kno_ that the, fully clamped condition

can be realized only imperfectly in tests, and lack of initialflatness would cause a further reduction of the critical stress.

Max_uum stresses gUnmx in uprights.- Formula (!i) for the

ratio _Umaxi_U is based on Levy's theoretical results (references 5

,! dand 6) for _ = 1.O and _ = 0.40 and on the assumption of linear

24 NACA TI_ No. 1364

variation with d/h. Linear decrease of the ratio with k toward

unity at k = I was also assumed.

On _ingle uprights, experimental values of _Umax_U cannot

be established with satisfactory aocuracy because the local bending

stresses cannot be eliminated from the measured total stresses.

Figure 17 shows, therefore, calculated and experimental values of

_Umax only for double uprights. For a number of beams the cal-

culated curves of _Umax differ but little from the curves of eU;

for these beams the experimental values of C_Jmax were so close to

the experimental values of _I that it was not feasible to show them

on the plots. For the other beams the e_)erimental ratios _UmaxlqU

m_gn!tude the calculatedare _enerally of about th_ s_me order of _ as

values. A close comparison is hardly warrented in viev of the experi-

mental scatter.

Distribution curves for _U are given in figure 20. The test

points represent the average of corresponding stations on three

uprights as well as the average of corresponding stations to eitherside of the neutral axis of the beau; they are, therefore, shown

only in the lower half of the bc_-am. The crave in the lower half isfaired throug_ the test points; the curve in the upper half is simply

the mirror image of the faired c1_ve in the lo_rer half. The curves

for the series IV besms show a very pronounced influence of the web

splice plate along the neutral axis of the beams. The splice plate

tends to act like a beam flange in producing gusset effect_ in

addition, the splice plate adds directly to the cross-sectional areaof the upright. All these effects were neglected in the analysis,as was the increase in critical shear stress caused by the splice

plate. For purposes of.comparison, however, a second analysis wasmade for six beams with upright failures, based on the assumption

that the web plate was simp]y supported along the splice line. This

assumption is obviously optimistic and yielded predicted failing

loads 7 percent g_eater (average for all six beams) than the analysis

naglecting the presence of the splice plate. Of the six beams, three

were from NACA series IV, and three were from mam_facturers' testson similar beams.

ANGLE OF DIAGONAL TENSION

In a fully developed diagonal'tension field, the direction of

the diagonal tension coincides with the direction of the folds in

the _eet (reference 1). IWnen the diagonal-tension field is

NACA TN No. 1364 25

incompletely developed, as it is in any actual beam, the diagonal

tension constitutes a component of stress which is separated out

of the total stress purely mathumatically, not physically. Moreover,

the diagonal tension thus separated from the total actual stress is

only an average for the ._ntlre bay and does not describe the details

of the stress variation _thin the bay. The angle of diagonal

tension consequently bears only a very loose r_lation to the physical

folds in the sheet. A study of the contour mao of a sheet Just after

buckling (reference 9) shows that _e "folds" _re c_rved to sucll an

extent thet an average directlGn cannot be defined to a_v degree of

accuracy. No attempt has been made, therefore, to compare computed

sa_gles of diagonal tension with observed anglc;s of folds.

AI_OWABLE ST_]ES_S FOP UPPIGHTS

Col_mm failt_e .- Out of a total of 19 bean,s with double-uprlght

failures analyzed, nine were predicted to fail by coltmm failtu-e.

In fi6utre 21, th(_ values of qU calculated for these becms at their

respective faillnc loads are plotted against t/_o effective slender-

ness ratio. _ slenderness ratios were sufficiently high to make

the Euler c_u'w' applicable in all cases_ it was, therefore, possible

to include upri_hts of 24S-T alloy as well as of 758-T alloy.

The plot .indicates that formula (12) for estimating the effective

slenderness ratio is som<what conservative considering all tests, but

a sufficient number of points lie so close to the ctu-ve that a less

conservative formula does not appear edvisable.

Forced criopli_ng failure.- Test observation has shown that the

shear buckles in the web will force bucklln6 or "crippling" of the

upright in the leg attached to the web. Th_ amotmt of the forced

cripplin_ will obviously depend primarily on the relative sturdiness

of the upr!_t and the web. _le s_mplea't parachuter expressing

relative sturdiness is the ratio tu/t , and in reference 4 empirical

formulas for allowable stress in the upright wer_ given based on this

parameter. Strength predictions based on these formulas, however,

showed a rather large scatter, which indicated that additional

parameters were necessary to define the failing stress more accurately.

A considerable reduction in the scatter was effected by using the

parameter k _tu/t instead of tu/t , and at the sarae time using the

maxlmttm instead of the a-_erage stress in the uprights. (See

formulas(13a)and (13b).)

26 _"_CA TN _o. 1364

Figt_e 22 is a plot of values of _Umsx computed from the

failin8 loads with the aid of the analysis chart, for all beams

of 24S-T alloy presumed to have failed by forced crippling. Omitted

from the plot are three points for a series of three 10-inch beamswhich fell from I00 to 150 percent above the average curve. This

discrepancyis so large tl_at it Justifies doubts as to the accuracyof the test data. It will be noted that all the points are fairly

uniforr_dy distributed about the average curve

The curve recomnended for design (form,±!a 13a), which is the lower

edge of the scatter band, lies 20 percent below this curve. Only a

single point lies appreciably below l_ledesign ctlrve, _md only fivepoints lie distinctly above the upper e_ge of the band, which

is 20 percent above the avera{_e curve. Two of these points are for

beams having a value of k < 0.5; this r_nge will be discussed

presently.

In accordance with formula (13c), an effective value of k was

used _hen the actual w lue was below 0.5. It _!l be noted in

figurs 2° that the point for one of "&he three 80-inch beams which

fell in this r_mge lies considerably above the upper edce of the

scatter bend. For _he beam represented by thls hlgh point, the

ratio of actual to predicted failing load is above 1.4l} _n exact

va].ue cannot be given because the actual failt_re was web failure,

•ailur_ (It may b<_noted that the ratio 1.41 of actualnot, upright _ _.

to predicted failing load is appreciably less l:han the ratio 1.57

obtained by comparing th_ plotted point for _Umax _-ith the average

curve for _GUmax , because the relation bet_'een _Umax and P is

not linear.) Analysis of Incomplet_ t_st data on a few 12-inch beans

_Ith V_c r from 1.5 to 2.5 (at failure) indicated very close agree-

ment in some ca_es and near]j 50-percent conservativeness in othor

cases. _ese values indicate that strength predictions for boams

designed to fail at T-L-< 4 may be considerably more conservativeTcr

than indicated by the upper ed_e of the scatter band in figure 22.

Figure 23 shows that the available tost data ca beams of

75S-T alloy with sing_le uprights agree with form%tla (13d) t_ithin the

same scatter limits as those fo_' 2_-T alloy. Fi_trc 24 shows that

NACATNNo. 1364

t

27

none of the points for double-uprig_ht beams fall below the recommendedaesign c_trve for such beams (formula 13b); the average curve is givenby

= 3ok (1Be)

_ich is about 14 percent lower th,_u t_e corresponging value forsingle uprights given by formula (13d).

General elastic instability of web end sbiffeners.- Experience

_rith simple elem_nts such as plstes _ud stiffeners tested ss

Indivianal coltunns has shown tJhat elastic instability beginning at

low stresses is not immediately followed by ultimate failure; theultimate load z_,ybe several times the critical load. Only _hen

elastic _stability occurs at a fairly high stress does the ultimatefailure follow soon after.

A similar condition appears to exist r_garding _he general

elastic inst_bility of a web _rlth double stiffeners. Analysis of

test data by means of existin8 theories of buckling of orthotroplcplates has shown in a ntu;_berof cases that the ultimate load vas

several times the calculated critical load. Because of experimental

difficttlties, very little effort has been nmde to determine experi-

mentally the load at which instability begins. It should also be

noted that existing theories hsve not dealt adequately _rltlithe

problem of general elastic inst_billty in web systems where the webhas buckled betveen the stiffeners.

Analysis of prel_iinary dat_ on thlck-web beams (h/t about i00)has indicated [hat perhaps some correlation between ultimate load

and th_,oretical critical load may be established if the critical

stress is definitely above the proportional limib of the material.

Because w_b systems with these proportions h_ve been studied very

little, it is not possible to state at present whether analysis for

column failure end forced crippling failure automatically coversth_ possibility of general inst_Jbility failure over the entiredesi_ range.

DESIC_

Formula (14) for the peak web stress Tma x is simply a formula

for linear interpolation between the limiting cases of ptLre shear

(k = O) and pure diagonal tension (k - 1). The factor C1 isdefined by

1Cl - sin 2m i (22)

28 _CA 'I_,TNo. 1364

According to the theory of pure @ziagonal tension (reference I)

the diagonal tension stress is

(23)= sin 2c_

According to form_la (3), _ = 4 _° if the flanges and the uprights

are infinitel_r !aa_C,_ (_'x = Ey = 0); in this case, sin 2_ = 1.

'_le factor CI expresses, therefore, the exccss stress, caused.r, ._ _ ., LeOby _ dif_zlng from ,,._ , as it. will in any actt_! structur_ with

members of finite size. The factor C2 is sJz_ilarly equivalent

to the theoretical factor C2 given in reference 1.

The allow_ble values for T max given in flg_re 14 were based

chiefly on tests of long w_bs subJec_:ed to loads approximating p1_e

• ,_-tsheer (reference 15) q_ese tests showed _ha, the ultimate value

of Tma x was _ndependeni, of the rivet factor in the practical

range (CR > 0.6) as lon¢_ as the bearing stresses did nol exceed

the allowable values. These t_:Jts yielded values of allowable

stress at k = 0 end k = 0.3 as sheba, in fi,_ure 25. (The test

points shown represent +he avera{_r_ of all ',:,est;sover the range of

rivet factor covered.) The al].o_.nble stresses at k = l.O were

estim_:ted as follows: For fully developed diagonal tension, the

tension stress in the _eb _s given by for e_ula (23); the ttltlr4at,e

nominal shear stress is i]:e._'efore

T = _- sin 2¢_

or s_mewhat less than _/2. This stress must be reduced to take

into acco_ni_ the rc.duction of section caused by the presence of

the rivet holes and the stress-concentration effects caused by

these holes. The combined effect of ib,_se two factors _s

'" c, theestimated as 0.75 for 24S-T alloy sn_ as 0._i for 75,_-T alloy,

latter having a smaller factor of stress concentration.

Figure 25 sho_._salso points obtained from tests on a square

picture-frame Jig (reference 16). Th,_;higShc_r stresses developed

in this Jig may have been higher bec:-_us,_.friction between the sheet

and the frame angles relieved hb.e rlveted Jo_t. _e effect of

friction can be quite high: but it shov&d probably not be relied

on to operate under service condition_:.

NACA _i_No. 1364 29

In six NACA beam tests _ which web failure _as predicted and

observed, the ratio of actual to predicted failing load _mngedfrom 0.95 to 1.06, with an average of 1.01. In six manufacturers'

tests, the ratio was 1.14 ± 0.06. The allowable stresses obtained

from figure 14 are therefore somewhat conservative on the average.

All these comparisons are based on actual material properties. The

t_o premature web failures sho_ in table 2 _ere due to damagecaused by shop accidents _nd should be disregarded.

RI_T DESIGN

TJeb-to-fl_nge rivets.- Failures of web-to-flange rivets were

observed in flve manufacturers' tests. _u_n actual rivet strengths

as determined by special tests were used as allowable values, the

strengths developed in th_ beam tests rT_n_;edfrom 3 percent lowerto 16 percent higher than predicted, with an average of 7 percent

higher. _hen nominal rivet strengths _Jere usod as allo_._ble

strengths, the actual values ranged from 37 percent to 60 percenthigher than pr_Bdicted. These values reflect the _Tell-Scaownfact

that nominal rivet strcngths are usually quite conservative.

Formula (15) for the load (per inch run) on the flange rivets

is a formula for strai_ht-line int,srpolation between the limitingcases k = 0 and k = 1.0. If the engineering theory of incomplete

diagonal tension were interpreto__ literally, separate rivet loads

• would be computed for the shear component and _he diagonal-tensioncomponent of the shear load, and these leeds _ould be added

vectorially. _E_e resulting formula would be from 7 percent to

9 percent !_ss conservative than formula (15) in the range of the

five tests under discussion. Consequently, if actual rivet strengthshad been ussd as allowable strengths, the strength predictions

based on this more rational formula would have been about 2 percent

unconservative on the average and up to 12 percent unconservativein the extreme case'. The _more rational foz_nula is therefore

unconservative by a sufficient mar_in to give preference to

fornn_la (15). It should also be r_alized that the greater ration-

ality is largely spurious; the engingering theory of incomplete

diagonal tension claims only to represent the average stresscondition in a bay, and these average conditions do not exist

along the edges of the bay where the rivets are located.

Web-to-u_riE_t rivets.- Formula (16) for the rivet shear

strength required in double uprights is a semiempirical formulaand was taken from reference 17. _e tests described in the

reference showed that the column strength developed does not depend

30 _&CA T_i _o. 1364

very critically on the rivet strength; fig_re 4 of the reference

sheens, for instance, that a reduction of the rive_ strength

to 50 percent of the l'equirod _lue reduces the column strength

on the average to 92 percent of the v_lue obta_ublo with adequate

r_vetlng. The average curve in this flg1_e _as used for evaluating

the early NACA be&m tests in which the rivet strengths were

generally less than required by form_la (16).

Formulas (17) and (18) for the required tensile strengths of

the rivets represent an attempt to provide a criterion for safe-

guarding against a type of failt_re sometimes observed in tests.

It is especislly importan t for single uprights_ because no

criterion pr_wlously existed to &el:,ermine the required rivet

strengOl. Because no t_._stshave b,zen made to check specifically

on this item, th_ available evidenc_ is raLher frag_mentary end

largely negative; _h_t is, in most tests no failures _erc observed

(or at least none were recorded). An additional difficulty isthat rivet failures are often found after the failure of the beom_

and it is then i_zpossible to state whether the rivet failure _as a

primary one responsible for hhe beam failume el" a secondary one

that took place _ile the beam _._asfailing for other reasons. In

view of all these unc_rta_nties, the coefficients {%iven in

formv&as (17) and (l_) should be cons!de red c_!y as tentative values.

For single upri_9__ts, the analysis _as bssed on a total of 21 tests.__ree failures were observe& _ith the coeff'_ciont in formu!e (!8)

ranging from O.lO to 0.13. No failures were observed in the

remaining 18 bo_ns, for which the coefficient ranged from 0.18

to 0.31. Ther_ were only two tests in the range from 0.18 to 0.22,

however, the remaining 16 becms having coefficients above 0.22.The coefficient for the design fo_naula _ras therefore taken _,s 0.22

in order to be conservative. IS" more Cests had been available in

the range from 0.13 to 0.18, a lo_,,er coefficient in the design

formula might have been Justified. Althou@_h the coefficient 0.22

may appear to be more conservative than nec,Dssary if the tests are

taken at face value, it is not considered to be tuuduly severe. Allthe manufactt_er's beams analyz_o ful_illed this criterion_ and

presumably they represented acceptable riveting prscticos. _ _e

lower rivet str(_ng_hs incorporated in _ number of the NACA b_ams

arose from the fact that these beams w_re intended pri_marily for

strain-gage tests to determine the dia_onal-tonsion factor k. In

order to accomo_te the strain _ages, I.ho rivet pitch _._s increased

in some cases; in other cases cot_ntersunk rivets were used which

have a relatively lo__ tensile strength.

On beams _i_h double uprig_hts, t_o failures were observed with

coefficients of 0.09 to 0.13. No failures were observed on four

beams _dth coefficients above 0.12 and on two beams with coefficients

m\CAI_[ No. 1364 31

of 0.07 and 0.08. The su_,gesteddesign coefficient of 0.15 istherefore probably conservative. Failttres wore observed on severalbeamswith coefficients ranging from 0.07 to 0.27# but the shear

strengths of the rivets on these beams were from 25 to 75 percent

below the strengths regulred by formula (16); _lese fall1_res were

therefore attributed to shoal" rather than to tensile loads.

[_i_ht-to-fl_m_o rivets.- Al+/qough tests on a rather large

ntmtber of beams _Tero available for analysis_ there were almost no

records of failln-e in upright-to-flan6_e rivets. This lack of

failures c_in probably be att_,'ibu+ed to the use of very conservative

design formulas based on ptu_e-diogonal-tonsion theory or slight

modifications of this theory. In the NACA bean,s of series II

_md III, the excess strength arose from the fact that the beam

flan6es were used for a n_nb_r of-te_ts, and bolts instead of

rivets _ere employed for all flangr_ cora%ections in order t;o

facilitate _isassemb3.y after a test.

i .... uprightsFor beams , _ double one rivet failure _,s recorded.

The exJ.stJ.n_ nominal flyer stren6th _ras only about 5 percent bo!ow

th_ required strc,nglh. The existing act'_l l'ivet strength -_s

t}_ereFore probably well above the required strength, but the

analysis _,_s very uncertain because of a peculiar desi_ feature

(reinforc,_d u-,,?riiJlt). In -three beams, no failures were recorded,

although the existin_ nominal rivet, strengtAs ranged from 0.47to 0.70 of the required stren{Lth; these values are so low that the

exis1_in_ actual rivet, streng/:hs were probably below the required

strengths in at least two cas_s. The aw_llable evidence appears,

therefore, to Justify the conclusion that fo!_nula (19) for the

rivet strength required on the ends of double u-pri{_ts would

gen._rally be safe even if actual rivet strengths were used as

allowable values.

For beaus _._d_ sincle _._-_.>_ _, there _;ere two racer{is of failure

a!thou_9<h the rivet strengths were apprcciablF cl'eater than required.

_e analyses were extremely uncertain_ however_ because several

_]nortant i.imensions of the begums were not _iven and had to be

est±meted or i_'erred.. Against these two records ,)f failture there

are 13 records of suceessftul Joints in _hich the rat_o of existing

nominal rivet strength to required Stren_;tl] was less than v_/lity,

the two lo_..,estratios being 0.68 and 0.53. It appears therefore

reasonably s_fe to draw the same conclusion as for double uprig:hts,

that !s, that formula (20) would probably be safe, in @eneral, even

if actual rivet stren_%ths %,e,re used as allowab]._ value::_. An

appr_ociable mar_oin of safety should exist in-practice because the

32 NACATN No. 1364

allo_ble strength values for rivets are likely to be well belowthe ac _u_l strengths. It mig_t be pointed out also that formulas (19)and (20) are inherently conser%_tive because they neglect t/_e _asseteffect; however, this eff_.ct is small in manybeams-_udmaybeovershadowedby unpredic tabl_ irre6_llaritles.

S_CO_!Y_%_YB_D]]_C_MO_gKNT$]I_ I,_Lr_C_,

E_mperimentalevidence on secondary bending stresces in theflanges is confined to a few measurementsgiven in reference 3.Mo_t of these moas_rementswere made.on beaus with very flexibleflanL_es b_yond the ran{Leof ,:_ffici-_nt design an_ showed the. pro-dictions to be w_,'ryconservativo. In the range of normal flangeflexibilities, the predictions yore somewhatconservative.Formula (21) is probably alw_:ys conserva_ivc b_causc ,_t neglects

the fact that a web in _z:complete dingona! tcn'._ion contributes to

the section modulus of the bc_m fh_ges.

The def!ec+,ion of a centi!evcr bee_u is calculated by adding

the so-cal]e,_ bcnd.in_ -_" _t"_ . , ......._I_,_ _._o.__and the shear (_ef!ection accordingto the formula

_L J i_L

N.'II' netG e

where Q e is the effective shear _uodu!uc given by figure !5. This

cffect_ve shear m.odulus _ac calculated as follows: According to

for_mla (4), the shear load .S c'_n be divided into a shear

component end a diagonal-ten_ion component. Th_ total shear

deflection is the s'_ of the deflections causc,d by thc_o two

componenbs; the cffective shear mod.ulus for .incomy_le_e dia,gonal

tension is therefore defined by th_,:,_relstion

_ -O e C GDT

_ere GDT is the ef'fec,n..e modulus of a web in pure

diagonal tension. 5Fne value of GDT can b-_ calculated, by the

NACA T_._I'[o.].3,C4 33

relations given Jn reference 1. A convenient formula _,,._ichcan bedcrivec! from thcs_ ro!_tions is

GDT = 2

on the simplifying assttmotlo.u t.hat th, b_am fl,'_nges are sufficiently

].argo to permi5 n_,{_.!._cti_gthe strain _x c,sus'_,_db7 hhe horizontal

component of the dia6_onal-t,_nsion :;force.

_.fnen the, web stresses o:.-c._,-_,d1:he proportlons! limit, a corrected

value c_e must be use,}.. A tentative curve for _,-,e/_e ,,_s _ in

reference 4. _is ctuwe was based on e sn_]..lntuabcr of unpublished

tests and is rather m_certain. Tinc,__.the, shape of the sJ_r,')ss-,straiu

curve is kuqo_,_,to bs qt',..itcvariable _:_th the e._:.istingtolerances

of composition of _.he, ri.Q. aria boa; brea',::._:ent_ it is not likely

that a I_i._1 accva-acy can be achleved in !2red.ic",:ing de:i:_owmc_tions at

stresses near or beyond the yio!,'). _;t.ress. l:'or;mnately_ there appears

±.o be no practical need for such. _-cct_racy.

Fi_a'e 26 sho_.m e_._.,erim_nta! sm.d cr!cu!ated 4_,Sq.ectJ.ons :for the

beams of series i. q_e agreement is very sat.infacl;ory.

Langley Memorial Aoronatrhical F_.bora i.,ory

Nationa]..^.dvise-_y Com_uil,t:,_ofor Ac_:onautics

[ang!uy Field, Va., _q:ri! 2, 1947

34 _T_;_CA'l'_ No. 1364

R E F E R EN C E S

i. Wagner, Herbert: Fiat Sheet Metal Girders with Very ThinMetal Web. Part I - General Theories and Assumptions.

NACA TM No. 604, 193].

Wagner, Herbert: Flat Cheat Metal Cirders w_.th Very _"_In

Metal Web. Part Ii - Sheet Metal Girders with Spars

Resistant to Bending_. Ob!igue Uprights - Stiffness.

I_ACA TM No. 605, 1931.

Wagner, Herbert: Flat Sheet Mete.l Girders with Very '!_ninMetal Web. Part II! - Sheet Metal Girders with Spars

Resis#_nt to Bending. The Stress in Uprights - Diagonal

Tension Fields. NACA qM No. 606, 1931.

2. Denke, l aul H.: Strain Er_erl_, Analysis of Incomplete Tension

Field Web-Stiffener Combinations. Jeur. Acre. Sci., vol. II,

no. l, Jan. 1944, pp. 25-40.

3. Ktuhn, Paul: Invest__g<ations on the Incompletely Developed

Plane Diagonal-Tension Field. NACA Rep. No. 697, 1940.

4. Kuhn, Paul, and Chiarito, Patrick T. : The Strength of l'!ane

14eb Systems in Incomplete Diagonal Tension. NACA ARE,

Aug. 1942.

5. Levy, Samuel, Fienup, Kenneth L., and Woollcy, Ruth M.: Analysis

of Square _!]hear Web above Buckling Load. NACA qN No. 969-,

1945.

6. Levy, Samuel, Woolley, Ruth M., and Corrick, Josephine N.:

Analysis of Deep Rectangle!at Shear _,;ebabove Buckling Load.

NACA _II{No. 1009, 1946.

7. Peterson, James F. : _Strain Me;_surements and (3trengt/_ Tests

of 25-Inch Diagonal-Tension Beams _nLth Sir cElloUpri_ts.

I_ACA A/_R No. LSJO2a, 1945.

8. Peterson, James P. : Strain MeastLrements and Strength Tests

of 25-Inch Diagonal-Tension Beams of 75S-T Aluminum Alloy.

NACA _IN No. 1058., 1946.

9. Timoshenko, S.: Theory of Elastic Stability. McGraw-Hill

Book Co., Inc., 1936.

NACA_ _To.1364 35

I0. Leggett, D. M. A.: The ]_uckllng of a Square Panel under Shear_:ThenOnePair of Opposite Edges is C!ampo¢_, s_ndthe OtherPair _s Simply Supported. __. & M. No. 1991, British A.R.C.,1941.

!i. I6_chi, S. : Die Ifniclm_ngdot rechtecklgen Platte dutch_c. !ng.-Archiv, Bd. IX, Ileft !, F_b. 17383 pp. 1-12.

12. Smith, ?,. C. T.: _'_e Buckling of' ?lywood i?l.ates in Shear.

Rop. SH. 51, Co_cil for Sci. and Ind. Res._ Div. Aoro.,

Com_nonwealth of Ausbrmlia, Aug. 1945.

13. l£romm, A. : S_;abilitat yon homocenen Plattsn _und Schalen J_

e].astischer_ Bereich. _in%buch der L_tfc_r_h. toc!mi];,

D'I. i!, 'r%. f_lO, M_," 191:0.

14. Iah_le, _., and Wa#ner, H.: Tests _"or th_ Determination ofthe Stress Condition in Tension I_ields. }TAC_ 'I_4No. 809,

1936.

15. Levin, L. i_oss, and Nelson; David H. : _f'iv_ot of P,ivet or Bolt Holeson the Ultimate J trength Developed by 2417_-T an& Alcl_d 75S-T

Shs_t in Inco_a]?lo_e D!a_onal To_sion. NACA _llq_]o. 1177,

1947.

16. l['0_hn,7'anl: Ultim_fio S tress_:_s Developed by 24S-T f_heot in

Inco_:_pleto Diagonal Tension. NACA _97 i\]o.833, !941.

17. Knkm, Paul, and Mo,s{_io, Ed_:in M. : .The Longitudinal Shear

Strength Ecguirod in Donblc-_.n_io Columns of 24S-T All nninum

Alloy. NACA RB No. 3E08, ].943.

NACA TN No. 1364 36

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o--= z'-" - - d: sin 2o,: cr= sinP_

k=O O<k< 1 k=!

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Figure 5.-Assumed distribution of (vertical}ncrmol stress in web _rr,medi-

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Fi{jure 4.- Diagonal-tension beamwith flexible flanges.

NATIONAL ADVISORY

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"_Ci_ COMMITTEE _ AERONAUTICS

Figure I!7. Diaqonol tension factors k obtained by presenl method and

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