Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

9.7Tension Field Beams 309

45 ◦,usuallyoftheorderof40◦and,inthetypeofbeamcommontoaircraftstructures,rarelybelow
38 ◦. For beams having all components made of the same material, the condition of minimum strain
energyleadstovariousequivalentexpressionsforα,oneofwhichis

tan^2 α=

σt+σF

σt+σS

(9.26)

in whichσFandσSare the uniform directcompressivestresses induced by the diagonal tension in
the flanges and stiffeners, respectively. Thus, from the second term on the right-hand side of either
Eq.(9.19)or Eq.(9.20),

σF=

W

2 AFtanα

(9.27)

inwhichAFisthecross-sectionalareaofeachflange.Also,fromEq.(9.23),

σS=

Wb

ASd

tanα (9.28)

whereASisthecross-sectionalareaofastiffener.SubstitutionofσtfromEq.(9.16)andσFandσSfrom
Eqs.(9.27)and(9.28)intoEq.(9.26)producesanequationwhichmaybesolvedforα.Analternative
expressionforα,againderivedfromaconsiderationofthetotalstrainenergyofthebeam,is

tan^4 α=

1 +td/ 2 AF

1 +tb/AS

(9.29)

Example 9.1
ThebeamshowninFig.9.12isassumedtohaveacompletetensionfieldweb.Ifthecross-sectionalareas
oftheflangesandstiffenersare,respectively,350mm^2 and300mm^2 andtheelasticsectionmodulusof
eachflangeis750mm^3 ,determinethemaximumstressinaflangeandalsowhetherornotthestiffeners
willbuckle.Thethicknessofthewebis2mm,andthesecondmomentofareaofastiffeneraboutan
axisintheplaneofthewebis2000mm^4 ;E=70000N/mm^2.

FromEq.(9.29),

tan^4 α=

1 + 2 × 400 /( 2 × 350 )

1 + 2 × 300 / 300

=0.7143

sothat

α=42.6◦

Themaximumflangestressoccursinthetopflangeatthebuilt-inendwherethebendingmoment
onthebeamisgreatestandthestressesduetobendinganddiagonaltensionareadditive.Therefore,
fromEq.(9.19),

FT=

5 × 1200

400

+

5

2tan42.6◦