Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

9.7Tension Field Beams 317

Fig.9.17

Determination of plastic hinge position.

Then,

Sxccφ= 2 Mpc′φ+σtctwsin^2 θ

c^2 c

2

φ

TheminimumvalueofSxisobtainedbydifferentiatingwithrespecttocc,thatis,

dSx

dcc

=− 2

Mpc′

c^2 c

+σtctw

sin^2 θ

2

= 0

whichgives

c^2 c=

4 Mpc′

σtctwsin^2 θ

(9.56)

Similarly,inthetensionflange,

c^2 t=

4 Mpt′

σtttwsin^2 θ

(9.57)

Clearly,fortheplastichingestooccurwithinaflange,bothccandctmustbelessthanb.Therefore,
fromEq.(9.56),

Mpc′ <

twb^2 sin^2 θ

4

σtc (9.58)

whereσtcisfoundfromEqs.(9.52)and(9.53)atthemidpointofWX.
TheaverageaxialstressinthecompressionflangebetweenWandXisobtainedbyconsideringthe
equilibriumofhalfofthelengthofWX(Fig.9.18).
Then,

Fc=σcfAcf+σtctw

cc

2

sinθcosθ+τmtw

cc

2