Low-Relaxation Strand-Practical Applications in

Precast Prestressed Concrete

Low-Relaxation Strand-Practical Applications in

Precast Prestressed Concrete

Leslie D. MartinConsulting EngineerThe Consulting Engineers Group, Inc.Glenview, Illinois

Donald L. PellowSupervisor

Technical ServicesArm0 Inc.. Union Wire Rope

Kansas City, Missouri

The development of se.ven-wire. ..‘.stre?F reheved eand was one of themost important factors in the growth ofprestressed concrete as a standard ma-terial of construction. Just as the pre-cast, prestressed concrete industry hasmatured into using wider, deeper,heavier and more econotiical sections,so the seven-wire strand has alsochanged to meet this demand.

From a basic standard of 250 ksi(1720 MPa) in %, 7118, and ‘% in. (10, 11,and 13 mm) diameter 25 years ago,strand has developed to the pointwhere 0.5 and 0.600 in. (13 and 15 mm),270 ksi (1860 MPa) is the standard usedtoday. In addition, low-relaxation strandis now making its presence felt over theonce commonly used stress-relievedmaterial because it has significantlyless loss of initial tension. This can re-

sult in improved and more predictableservice performance, and in many caseswill allow a higher load carrying capa-bility.

In 1957, the American Society forTesting and Materials issued the first“Standard Specifications for UncoatedSeven-Wire Stress-Relieved Strand forPrestressed Concrete” (rlSTX1 A416).Today, the 1980 ASTM A416 standard’includes low-relaxation strand. Thisspecification requires that the low-re-laxation strand differ from ordinarystress-relieved strand in only two re-spects: first, it must meet certain re-laxation loss requirements, as measuredby ASTM E32Sz (note that ordinarystress-relieved strand has no such re-quirement); and second, the minimumyield strength, as measured by the 1percent extension under load method,

must be not less than 90 percent of thespecified minimum breaking strength,as opposed to 85 percent for normalstress-relieved strand. All other re-quirements are the same.

This paper presents the results of ananalytical study of the use of low-re-lax&ion strand in the most commonlyused precast prestressed products. Mostof the data were obtained from thecomputer program LODTAB, whichwas the program used to generate thedesig” load tables in the PC1 DesignHandbook3 and those used by severalprecast concrete producers in their owncatalogs. This program permits pre-stress losses to be input as a fixed per-centage of initial tension, “I calculatedby the method recommended by i?,ia etal.” In addition to comparing load capa-bility and service performance, somec”ncems expressed by potential usersand specifiers will be addressed.

It should be noted that this studyonly compares the maximum capabilityof the members, based “n the abovemethod “f loss calculation. The strandsavings shown are for those conditions.Overall strand savings to the precasterOI “n a project will be a function of theproduct mix and the number of mem-bers designed to approach maximumcapacity.

Loss of PrestressUntil the mid-1960’s, the most c”m-

man design practice was to assume alump sum valne of 35,000 psi (241 MPa)for prestress loss for pretensionedmembers. This value is based on the1958 report of ACI-ASCE Committee323,5 which served as the basis f”r de-sign for prestressed c”ncrete until thefirst AC1 Code provisions in 1963. Thisvalue is still mentioned in the .~CICode Commentary as giving “satis&tory results for many applications.”However, the performance of sum”long-span, heavily prestressed mem-bers seemed t” indicate that the lump

sum value underestimated the total10%.

In 1971, the PC1 Design HandbookCommittee selected a uniform value of22 percent of initial (iacking) tensionfor use in the load tables in the firstedition of the Handbook, which wasalso retained in the second edition. For270.ksi (1860 MPa) strand stressed to 70percent of ultimate, that value is 41,580psi (287 MPa).

‘llie AC1 Buildlug Code’ specifiesthe factors which contribute to prestressloss for pretensioned members. Theyinclude the long-term effects of creep,shrinkage and tendon relaxation, andthe immediate (upon release) effect ofelastic shortening. Considerable re-search has been done “n the sltbject,resulting in a variety of design rccom-mend&ions. Reference 4 includes a

bibliography of these research recom-mendations.

In 1975, the PC1 Committee on Pre-stress Losses presented a report whichincluded a general and a simplifiedmethod for computing losses.’ As withmany of the other methods mentionedabove for many members, this methodseemed to predict values that werehigher than experience could justify.

In 1979, a working group of ACI-ASCE Committee 423, PrestressedConcrete, developed a calculationmethod which was largely based onearlier work, tempered by the experi-ence of members of the gro~p.~ Thiscalculation method is the one used inthis study. One of the primary differ-ences between this method and othersis that upper limits are included. How-ever, no lower limits are specified, andsubsequent use of these equationssometimes yields values which are sus-picioosly low. Therefore, the programLODTAB also places B lower limit of35,000 psi (241 MPa) for stress-relievedstrand, and 30,000 psi (207 MPa) forlow-relaxation strand.

It should be noted that prestress losshas virtually no effect on the ultimatestrength of the member, at the level ofprestress normally used in preten-sioned products.

Initial (Jacking) Tendonstress

It has been the practice in the pre-cast, prestressed concrete industry toapply an initial tension of 70 percent ofthe nominal strength of the strand whenusing ordinary stress-relieved strand.Stress relaxation losses in ordinarystrand have been shown to be propor-tionally higher when an initial stresshigher than 70 percent is used, to thepoint that the gain in actual prestress-ingofthe concrete is minimal.

With low-relaxation strand, the relax-ation loss remains more or less propor-tional up to an initial stress of 7.5 per-

cent of ultimate, so there is often adefmite advantage of higher tensioningforces. Furthermore, all manufacturersoflow-relaxation strand approve the useof the higher stressing force at 75 per-cent ofnominal strength.

Comparison ofLoad-Carrying Capability

The flexural capacity of prestressedconcrete members is limited by twocriteria: (1) Stresses at service load and(2) Ultimate design strength. In addi-tion, the AC1 Building Code and theAASHTO specifications limit the con-crete stress at the time of transfer ofprestress.

When the ultimate design strength ofmembers is calculated using compat-ibility of strains, there is some indica-tion that low-relaxation strand may pro-vide somewhat greater capacity thanordinary stress-relieved strand becausethe minimum yield strength is higher.However, typical stress-strain curvesshown in the PC1 Design Handbookmake no distinction between the twomaterials. The carves in the Handbookwere checked against about 25 actualcurves of both types from differentmanufacturers, and little difference wasfound. The computer program LOD-TAB is based on these typical stress-strain cwves, so no difference in ulti-mate design strength will he indicated.

The actual area of both stress-re-lieved and low-relaxation strand may hesomewhat different from the areashown in ASTM A416. However, thestandard specifically exempts cross-sectional area from any tolerance limi-tation. The minimum breaking strengthis specified as a specific force, which isthe product of the strand grade 1250 or270 ksi (1720 or 1860 MPa)J times thenominal or specified strand area.

For example, while the actual cross-sectional area of Vz in. (13 mm) 270 ksi(1860 MPa) stxmd is, say, 0.158 sq in.(102 mmz), the minimum breaking

strength is 41.3 kips (270 x 0.153) (285MPa). Therefore, the nominal area ofthe strand [e.g., 0.153 sq in. (99 mm2)for v2 in. (13 mm) diameter] should al-ways be used for design and for theinitial force to be applied to the strand.

The AC1 Building Code permits amaximum tensile stress under serviceload of 124x (1.0 ,tE) provided certaindeflection criteria are met. It is com-mon practice in the prestressed con-crete industry to design to this maxi-mum in stemmed deck members suchas double tees and single tees. For flatdeck members, such as hollow-coreslabs, and for beams, the most comrrmnpractice is to limit the tensile stress to 6vE (0.5 ,E). This is the procedure fol-lowed in the PC1 Design Handbookand in the analyses that follow.

In addition to flexural criteria, it iscommon practice to also limit loadingin hollow-core and solid-slab deckmembers to the shear strength of theconcrete, since it is very difficult toreinforce for shear. Since the amount ofprestress in a slab influences the shearstrength, slabs which use low-relaxationstrand may have a slightly higher ca-pacity at heavily loaded short spans.This effect is minor, howwer.

A variety of precast, prestressed con-crete sect ions commonly used inbuilding and bridge construction wereinvestigated in this study, as listedbelow. The building sections andstrand patterns were taken from thesecond edition of the PC1 DesignHandbook.‘The number in parenthesesrefers to the page from that publicationon which the section appears. (Note: forthe hollow-core section investigated,actual strand numbers and sizes wereused rather than the “strand designa-tion code” used in the Handbook.)

(1) 8 in. Hollow-core, normalweight4HC8 (p. 2-27)Strand patterns: 7%. 4%, S/z,6X, 7% in.Straight strands at 1% in. frombottom

(2) 12-in. Hollow-core, normalweight4HCl2 (p. 2-31)Strand patterns: 6%, 7%, 8% in.Straight strands at 1% in. frombottom

(3) 12-i”. Hollow-core, normalweight , with 2 in . normalweight composite topping-4HC12 + 2 (p. 231)Strand patterns: 6%, Wz, 8% in.Straight strands at 1% in. frombottom

(4) 24- i” . Double tee , normalweightGDT24 (p. 2-16)Strand patterns: 68-S, 108-Dl,128-Dl, 148-Dl

(5) 24.in. Double tee , normalweight , with 2 in . normalweight composite topping-8DT24 + 2 (p. 2-16)Strand patterns: 68S-108-Dl,128.Dl

(6) 24-i”. Double tee, lightweightconcrete--8LDT24 (p. 2-17)Strand patterns: 68-S, 108-Dl,128.Dl, 148-Dl

(7) 32- i” . Double tee , normalweight-8DT32 (p. 2-18)S t r a n d p a t t e r n s : 168-Dl,188-Dl, 208-Dl, 228.Dl

(8 ) 36 - i” . S ingle tee , normalweight--BST36 (p. 2-22)S t r a n d p a t t e r n s : 168.Dl,188-DA, 208.Dl, 228-Dl

( 9 ) 3 6 - i ” . S i n g l e t e e , light-weightdLST36 (p. 2-23)S t r a n d p a t t e r n s : 168.Dl,188-Dl, 208-Dl, 228.Dl

(10) Inverted tee beam 24IT36 (p.262)Strand patterns: 12, 14, 16, 18,and 20 straight strands

(11) Type IV AASHTO girders with24, 30, 33, 36, and 42 strands.

For each of the sections and strandpatterns, three separate load tableswere generated: one with ordinary 270ksi (1860 MPa) stress-relieved strand,

Note: 1 in. = 25.4 mm.

initially tensioned to 0.70 of the guar.anteed ultimate strand strengt, v;,):one with 270 ksi (1860 Mfa! low-re-laxation strand initially tensioned to0.70/,.; and one with 270 ksi (1860MPa) low-relaxation strand initiallytensioned to 0.75 J,..

This variation enabled not only acomparison OS the load carrying capr-ity ofmembers made with each type ofstrand, but also how mucl~, olthe diner-ence in capability is attrilrutable~to therelaxation characteristics, and howmuch is caused by the difterence ininitial tension.

1. Hollow-core slabs-The llse oflow-relaxation strand slwws an increasein load capacity of generidly less than10 txrcent, all OS whicl, is due to lessstrand relaxation. Increasing initial ten-sion does not increase capacity. normany-probably most--applications,the maximum span is limited by deadload drflrction (loss rllcmnher~.

2. 24.in. Double tees-This is themost commonly used stemmed section,and significant strand savings can berealized in many applications by usinglow-relaxation strand. For example, at aspan OS70 St (21.3 m), with a superim-posed load of 40 psi’ (1.9 kl’a) [lo psi(0.5 kl’a) dead load, 30 txf (1.4 kPa) liveload], 14 ordinary stress-relieved strandwould be required, wtlereas 12 low-relaxation strands would be adequate,even witho~~t increasing the initial ten-

sion. Also with the Sewer strand, alower release strength is required [lessthan 3500 psi (24 MPa! vs. 4100 txi(28.3 MPa )I. Similar sa\ ings are shownin dollhle tees with topping or thick-ened flanges in the span ranges mostcommonly used in parking stnwtnres.

3. Long span roof members-Forlong span members typified by the 32in. (813 mm! double tee and 36 in. (914mm) single tees investigated, the in-creased capacity becomes even moresignificant, as more prestress is re-quired.

4. Beams-The investigation of the36 in. (914 mm) deep inverted tee beamindicated typical strand savings of’16 to25 percent for these members with theuse of low-reln.xation strands.

5. AASHTO girders-The AASHTOgirder sections were inx;estigated as-suming a spacing OS 6 St 6 in. (2.0 mj,with a 6% in. (165 mm! thick compositeslab. A Sew otller girder spacings werealso checked. In order to make com-parisons, the moment requirements forStandard HS20-44 loading Sor the spansshown, including impact and load dis-tribution, were converted to an equiv-alent reqllired load per foot.

A summary OS the comp,&er stltdy isshmvn in TaLle 5 and Fig. 5. The re-snlts indicate that strand savings ofabout 20 percent wonId he typical inthe span ranger indicated. Based onlosses calculated by KeSe,-ace 4, whencaladated by the method given in tlieAASHTO Specifications,* savings aresomewhat less. .4n even greater overallsavings is possible on some multi-lanebridges by increasing the spacingenough to reduce the total nnmber OSgirders required.

Camber ComparisonsCamher in prestressed concrete

members is calxd by the prestressingforce being applied eccentrically withrespect to the center of g*a\ity of’ themember. This is o&t by the dead load

Table 1. Uniform load capacity (psf) for hollow-core slabs

I

30

31

32

33

34

35

36

36

38-41

41

43

44

45

46

47

48

50

37

38

39

40

41

42

43

43

45-

SP/\N, FT. Sl%IN, FT.

Fig. I. Load capacity comparison (hollow-core slabs).

Table 2. Uniform load caoacitv [DSf) for 24-in. 610 mm) deeD double tees

10

12

14

-8

55-74

76

76

-43

44

44

57

66

66

-82

86

86

-

-60-56

57-57-67

73

73

-

-

-

-

-

-36-44

46

49-57

63-63-68-68-

-.

-

-

-

-<Pa 1 ft = 0.305 m; 1 i”. = 25.4 nun

span, fi

i&&i

60 65 70 75

SPAN, FT.

LOW-RELAX*TION STRAND

STRESS-RELIEVED STRAND

60 65SPAN, FT.

Fig. 2. Load Capacity comparison [24-h. (610 mm) double tees].

Table 3. Uniform load cawacitv fwf) for low swan members- -

18

20

22

18

20

22

18

22

84

86

89

86

88

92

88

90

j

94

93

96

100

96

99

103

99

102

1034106

1104106

1094113

1094112

n

bm

Table 4. Uniform load capacity (plf) for inverted tee beam.

14

16

18

c

LR 70 9103 7623,6503 55YY 4860 4247 3733 329x 2927

75 965X 8109 6922 5964 5180 4530 39X6 3525 3132

SR 70 7130 65Y4 5677 492i 4306 3785 3344 2966

Lri 7” 6241 7036 6W4 5269 4610 4057 3590 319"

75 X777 7498 6466 5622 4922 4336 3639 3416

SH I 70 I I 8466 7217 6210 5385 4702 4129 3649I I I I I I I3244

SR 70

LR 70

I751 I I 8891 7668 6666 5837 5141 4553I I 1~ I I I4051

Fig. 4. Capacity of Type IV AASHTO girdercompared with highway load requirement

Fig. 5. Capacity of Type IV AASHTO girder compared withhighway load requirement.

deflection of the member. Creep of theconcrete canses each of these compo-nents to increase with age, but becauseof losses, the prestress force is beingconstantly reduced, so the downward(deflection) component increases fasterthan the upward (camber) component.With low-relaxation strands, there isless difference in the rate of increase.

With proper plant quality controlover such items as concrete releasestrength, strand placement (especiallywhen strands are depressed at midspan)and initial tension, the camber at thetime of release can be predicted withreasonable accuracy. There are manymore and less easily controlled factorswhich influence long-time camber,however, and predictions of what thecamber will be at critical times, such asat the time of erection, are at best ap-proximations.

The computer program LODTABuses the equations suggested by Mar-tin9 for predicting the change in camber

over time. His paper is also the basis forcamber predictions used in the PC1Design Handbook.

In general, most precast, prestresseddeck members used at their optimnmspan and prestress level will show Bcamber increase for the first fewmonths, and then a gradual decrrase forthe remainder of their life. This de-crease levels off after a year or so andthen there is usually no discernablechange. For roof members on “flat”roofs it is, of course, desirable to havesome upward camber remaining in themember so ttrat pending is prevented.

Thus, it is common and desirablepractice to limit the span to lengths thatindicate no worse than level at the“final” condition. When low-relaxationstrand is used, this will permit some-what longer spans for deck membersbefore this limiting criterion is reached.Increase of initial tension further in-creases the span range. Tables 1through 5 show these suggested limits,

Table 5. Uniform load capacity (plf) for AASHTO girders 6 ft 6 in.(2.0 m) spacing-6% in. (165 mm) thick composite deck.

based on the assumptions described For a given number of strands andearlier. prestress level, it is apparent that the

It shmld be noted that it can be very else of low-relaxation strand will resultdangerous to depend on camber for roof in more camber. Howwer, the previousdrainage. A positive slope of at least section showed that fir the same loadI:100 (and preferably more) should be capacity, in many cases fewer strandsprovided in designing any roof system. are required, even without an increase

in initial tension. Doing this will result are negligible. The member used in thein less camber of the members, and the example represents the most extremepredictions are likely to be more accu- case that could be found among stan-rate. dard products. In fact, the effects shown

are probably even more severe than

Release StrengthStrand tensioned to 75 percent of nl-

timate will obviously require the con-crete at time of transfer to be of higherstrength than the same number ofstrand tensioned to 70 percent of ulti-mate, under the criteria imposed bycodes and specifications. However, aswith cambers, equivalent load capacityrequires fewer strands, often withoutrequiring increased initial tension, sorelease strength requirements are less,resulting in further savings. In beamswith straight strands, it is commonpractice to reinforce for end tension at,release. With fewer strands for equiva-lent capacity, less reinforcement is re-quired.

would actually occur, since the differ-ence in prestressing force occurs overtime and creep would tend to neutral-ize the differences.

Optimizing Strand Usage

relaxation strand, and if the plant ismanufacturing products designed for

Effect of Mixing

low-relaxation steel, it is highly rec-ommended that the prestressing plant

Low-Relaxation Strand With

convert to it for all products. This

Ordinary Strand

would avoid the possibility of inadver-tent use ofordinary strand in a memberdesigned for low-relaxation strand.However, when using more than one

If the decision is made to use low-

strand supplier, occasions may arisewhen the two types of strands would bemixed in a member.

maximum force difference, P, would be

In double tees, additional strandsavings could be realized by “un-balancing” the strand patterns. If only

at the end at release. If a ‘/z in. diame-

even numbers of strands are used,

ter, 270 ksi (13 mm, 1860 MPa) strand

statistically 50 percent of the memberswould have one more strand than re-

were tensioned to 75 percent of ulti-

quired. Producers have been reluctant

mate, this difference (neglecting losses

to use odd numbers of strands, such assix in one stem and seven in the other,

which occur before release) would be:

because of the same potential effectsillustrated in the calculations of theprevious section.

In the case of unbalanced strand, the

Some specifiers have expressed aconcern that the different propertieswould have some detrimental effectswith regard to possible lateral deilec-tion, cambers or torsion even if theywere designed for ordinary stress-

P = co.1531 (270) (0.75)= 31 kips (138 kN)

using a modulus of elasCcity at releaseof 3300 ksi (2.3 x lOa MPa), corre-sponding to a release strength of ap-proximately 3000 psi (21 MPa), themaximum lateral sweep (LS) would be(see Appendix):

LS = 0.050 (31.0116.8) (215013300)= 0.060 in. (1.5 mm)

The torsional stress (TS) would be:

TS = 5.6 (31.0/16.8)= 10.3 psi (71 kPa)

and the flange tension (FT):

relieved strand. The calculations in the FT = 92 (31.0116.8)Appendix show that any such effects = 170 psi (II70 kPa)

This is less than the cracking stress,even when the compression caused bygravity loads is not included.

CONCLUSIONS

The analyses made in this paper in-dicate that the we of low-relaxationstrand can result in significant strandsavings for virtually all types Of precast,prestressed standard products for de-signs approaching maximum capability.This is especially true in longer,heavier structural members. For exam-ple, the savings will be less with hol-low-core slabs than with bridge girders.Low-relaxation strand can also provideimprovements in deflection and crack-ing control.

It was also shown that low-relaxationstrand can be mired with stress-re-lieved strand in a design based onstress-relieved strand properties with-out harmfill eilects. With the much im-proved properties of low-relaxationstrand over stress-relieved strand, andwith the price of it being competitivewith stress-relieved, low-relaxationstrand will undoubtedly become thestandard of the industry.

ACKNOWLEDGMENT

This paper is based upon B studyconducted hy The Consulting Engi-neers Group of Glenview, Illinois,under contract with Armco, Union WireRope, Kansas City, Missouri.

REFERENCES

1. ASTM Standard A41580 “Specificationfor Uncoated Seven-Wire Stress-RelievedStrand far Prestressed Concrete,” Ameri-can Society for Testing and Materials,Philadelphia, Pennsylvania, 1980.

2. ASTM Standard E328-78, “Recom-mended Practice for Stress-RelaxationTests for Materials and Structures,”American Society for Testing and Materi-als, Philadelphia, Pennsylvania, 1978.

3. PCI Design Handbook -Precast and Pm-stressed Concrete, Second Edition, 1978,Pmstressed Concrete Institute, Chicago.

4. zia, Carl, Preston, H. K., Scott, N. L., andWorkman, E. B., “Estimating PrestressLosses,” Concrete International, v. 1,No. 8, June 1979, pp. 32-38.

5. ACI-ASCE Committee 423 (formerly323), “Tentative Recommendations forPrestressed Concrete.” ACI ~Ioumnl, Pro-ceedings V. 54, No. i, Jan& 195b, PP.545678.

8. ACI Committee 318, “Building Code Re-qnirements for Reinforced Concrete (AC1318.77),” (including Commentary), Amer-ican Concrete Institute, Detroit, Michi-gan, 1977.

7. PCI committee on Prestress Losses,“Recommendations for Estimating Pre-stress Losses,” PC1 JOURNAL, V. 20,No. 4, July-August 1975, PP. 43-75.

8. Standard Specifications for HighwayBridges, Twelfth Editioq American As-sociation of State Highway and Trans-portation Otficials, Washington, D.C.,1977.

9. Martin, L. D., “A Rational Method forEstimating Camber and Deflection ofPrecast Prestxssed Concrete Members,”PCI JOURNAL, V. 22, No. 1, January-February 1977, pp. 100-108.

Note: A design example is provided in theAppendix on the following two pages.

APPENDIX - DESIGN EXAMPLEThis design example shows the effect were spaced the maximum lateral dis-

of mixing low-relaxation strand and tance apart. From the PC1 Designstress-relieved strand. The effect would Handbook, select a lODT32 sectionbe greatest in a double-tee member with 22 ‘/-in. (12.7 mm) diameter, 270where the maximum number of strands ksi (1860 MPa) strands.

IO’-O”(3.05m)

+

5’-o”11.5 m)

II STRESS-RELIEVED STRANDS

I I LOW RELAXATION STRANDS

The section properties and stranddetails are as follows:

I, = 626,791 in.4 (1.03 x 1O’O mm4)I, = 59,720 in.4 (9.79 x lo8 mn+)

Strand eccentricity:e, = 5.57 in. (141 mm)e, = 17.48 in. (444 mm)e = 11.91 in. (303 mm)

Stress to 70 percent ultimate:0.70 x270 = 189 ksi (1303 MPa)

Strand arei, per stem:11 x 0.153 = 1.683 iaz (109 mmz)Assume a maximum prestress loss

(according to the recommendations inReference 6).Final strand stress:

Low-relaxation strand = 189 40 = 149 ksiStress-relieved strand = 189 - 50 = 139 ksi

= 10 ksi (69 tiPa)

The difference in the final prestress 1. Potential lnterzal cl~fl~ctinnforce is: (sweeol:

P = 10 x 1.68 = 16.8 kips (74.7 kN)Assume that: &.?= 16.8 (30) (86 x 12)*

j’; = 5000 psi (34.5 MPa)SE1 8 (2150) (626,791)

E, = 4300 ksi (3.0 x IO’ MPa) = 0.05 in. (1.3 mm)

To acconnt for creep, “se the long- which is a negligible quantity.ten” value:

I$, = 2 = 2150 ksi (1.5 x lo4 MPa)

Span = 86 ft (26.2 m)

2. Potential torsional stresses:The theoretical difference in opward

camber of each stem is:

A Peel2 + _P&2SEZ 12EZ

in which

Therefore:

~ = 16.8 (86 x 12)*2,150 x 29,860 1

= 0.471 in. (12 mm)

The equivalent uniform load to causethe sane deflection is found from theequation:

384EZAw=51'

384 (2,150) (29,860) (0.471)-5 (86 x12)"

= 24.6 lb per ft (359 N/m)

u utR

3. Potential flange stressWith reference to the sketch:

!A4 = 24.6 (30) = 738 in:lb per ft

Section modulus of.tlange:

from which the weight

W = 24.6 x 86= 2,113 lb (9,345 N)

The torsional moment equals:

T = 2,113 x30= 63,390 in.-lb

The polar moment of inertia is foundfrom the sum:

J =I, +I,

= 626,791 + 59,720

= 686,511 im4

The distance from the neutral axis is:

c = b/Z = 60 in. (1524 mm)

The torsional stress is obtained from:

Tc _ 63,390 x 60

.I 686.511

= 5.6 psi (38.6 kPa)

which is obviously negligible.

Maximum tensile stress:

Ms = F = 92 psi (634 kPa)

The computed tensile stress is muchless than that which would cause

Ld2 12 (2)”- = ~ = Sin.3perft cracking even neglecting compre.ssion6 6 from gravity loads.

* * l

nknnc’o “MNv W I R E

ROPE