Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

15.3 Deflections due to Bending 449

Fig.15.20

Macauley’s method for the deflection of a simply supported beam.

deflectionproblemsbyMacauley,in1919andhencethemethodisfrequentlyknownasMacauley’s
method.
Wenowintroduceaquantity[z−a]anddefineittobezeroif(z−a)<0;thatis,z<a,andtobe
simply (z−a)ifz>a. The quantity [z−a] is known as a singularity or half-range function and is
definedtohaveavalueonlywhentheargumentispositive,inwhichcasethesquarebracketsbehave
inanidenticalmannertoordinaryparentheses.

Example 15.9
Determinethepositionandmagnitudeofthemaximumupwardanddownwarddeflectionsofthebeam
showninFig.15.20.

Aconsiderationoftheoverallequilibriumofthebeamgivesthesupportreactions;thus,

RA=

3

4

W(upward) RF=

3

4

W(downward)

Using the method of singularity functions and taking the origin of axes at the left-hand support, we
writedownanexpressionforthebendingmoment,M,atanysectionZbetweenDandF,theregionof
thebeamfurthestfromtheorigin.Thus,

M=−RAz+W[z−a]+W[z− 2 a]− 2 W[z− 3 a](i)

SubstitutingforMinthesecondofEq.(15.32),wehave

EIv′′=

3

4

Wz−W[z−a]−W[z− 2 a]+ 2 W[z− 3 a](ii)

IntegratingEq.(ii)andretainingthesquarebrackets,weobtain

EIv′=

3

8

Wz^2 −

W

2

[z−a]^2 −

W

2

[z− 2 a]^2 +W[z− 3 a]^2 +C 1 (iii)