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ENG5312 Mechanics of Solids II

Discontinuity Functions

Discontinuity functions are used to develop a single epression for the

elastic curve of a !ultiply loaded "ea! or shaft#

$se of a single epression %ill per!it evaluation of integration constants

fro! "oundary conditions only& as continuity conditions %ill "e satisfiedauto!atically#

Define the Macaulay function:

x a n

= 0

(x a)n,x < a

,x a

%here n 0

'nd

x a n

dx = x a

n+1

n +1+ c

(his function can "e used to represent a unifor!ly distri"uted load& wo &

starting at x = a )

w(x) = wo

x a 0

i#e# w(x)= 0 for x

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ENG5312 Mechanics of Solids II

V(x) = wo x a =

0

wo(x a)

,x

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ENG5312 Mechanics of Solids II

So shear force and "ending !o!ent are given "y)

V(x) = w(x)dx= P x a 0

M(x)= V(x)dx= P x a 1

' !o!ent& M o& positive counter/cloc%ise& is a li!it as 0of theapplication of t%o distri"uted loads#

w(x) = Mo x a

2=

0

Mo

,x a

,x =a

.here the /2 eponent is used to guarantee the units of w(x) areforce0length#

(he shear force and "ending !o!ent that result fro! this epression are)

V(x) = w(x)dx = Mo x a

1

M(x) = V(x)dx = Mo x a

0

.here the integrals are perfor!ed using operational calculus#

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ENG5312 Mechanics of Solids II

(he functions used for the different loadings are su!!aried in (a"le 12/2

of 4i""eler#

(he discontinuity functions provide a !eans of directly specifying the

"ending !o!ent as a function for a "ea!# (his function can then "eintegrated to deter!ine the deflection& and only t%o "oundary conditions%ill "e reuired to evaluate the constants of integration#

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