Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

440 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams

SinceMx=1500NmandMy=0,wehave,fromEq.(15.19),

σz=1.5y−0.39x (i)

inwhichtheunitsareNandmm.
ByinspectionofEq.(i),weseethatσxwillbeamaximumatFwherex=−8mm,y=−66.4mm.
Thus,

σz,max=−96N/mm^2 (compressive)

In some cases, the maximum value cannot be obtained by inspection so that values ofσzat several
pointsmustbecalculated.

15.2.5 Load Intensity, Shear Force, and Bending Moment
Relationships, General Case

Consideranelementoflengthδzofabeamofunsymmetricalcrosssectionsubjectedtoshearforces,
bendingmoments,andadistributedloadofvaryingintensity,allintheyzplaneasshowninFig.15.14.
Theforcesandmomentsarepositiveinaccordancewiththesignconventionpreviouslyadopted.Over
thelengthoftheelementwemayassumethattheintensityofthedistributedloadisconstant.Therefore,
forequilibriumoftheelementintheydirection
(
Sy+

∂Sy

∂z

δz

)

+wyδz−Sy= 0

fromwhich

wy=−

∂Sy

∂z

TakingmomentsaboutA,wehave
(
Mx+

∂Mx

∂z

δz

)

−

(

Sy+

∂Sy

∂z

δz

)

δz−wy

(δz)^2

2

−Mx= 0

Fig.15.14

Equilibrium of beam element supporting a general force system in theyzplane.