Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

16.2 Shear of Open Section Beams 485

Fig.16.6

Shear loaded Z-section of Example 16.1.

ThesecondmomentsofareaofthesectionhavepreviouslybeendeterminedinExample15.14andare

Ixx=

h^3 t

3

, Iyy=

h^3 t

12

, Ixy=

h^3 t

8

SubstitutingthesevaluesinEq.(i),weobtain

qs=

Sy

h^3

∫s

0

(10.32x−6.84y)ds (ii)

Onthebottomflange12,y=−h/2andx=−h/ 2 +s 1 ,where0≤s 1 ≤h/2.Therefore,

q 12 =

Sy

h^3

∫s^1

0

(10.32s 1 −1.74h)ds 1

giving

q 12 =

Sy

h^3

(

5.16s^21 −1.74hs 1

)

(iii)

Hence at 1 (s 1 =0),q 1 =0, and at 2 (s 1 =h/2),q 2 =0.42Sy/h. Further examination of Eq. (iii) shows
thattheshearflowdistributiononthebottomflangeisparabolicwithachangeofsign(i.e.,direction)
ats 1 =0.336h.Forvaluesofs 1 <0.336h,q 12 isnegativeandthereforeintheoppositedirectiontos 1.
Intheweb23,y=−h/ 2 +s 2 ,where0≤s 2 ≤handx=0.Then

q 23 =

Sy

h^3

∫s^2

0

(3.42h−6.84s 2 )ds 2 +q 2 (iv)