Unformatted text preview: wice using rules for the disconRnuity funcRons to get M. No constants of integraRon needed. • Integrate M twice to ﬁnd dυ/dx and υ. Two constants of integraRon will be needed • Apply boundary condiRons to determine the constants of integraRon. Be very careful to evaluate the disconRnuity funcRons properly. < xa >n = 0 for x < a! • Write a ﬁnal expression for υ in terms of the disconRnuity funcRons including the values of the constants. Macaulay FuncRons • Macaulay FuncRon DeﬁniRon: where n ≥ 0 • Its integral: • Examples: a
x w0 • C = 0 in both integraRons here. m a
ax
x Macaulay FuncRons a
x • Examples: 1
1 m
m
a
ax
x w0
a
L x • Construct equivalent loads that can be represented by Macaulay FuncRons! • How? w0 w0 w0 a
a
x
x L
L a
x w0
w0
P w0 w0 Singularity FuncRons w a
x 0 • For handling point forces/couples we use singularity func(ons and P M
a
x a
x
M0
a
x • Its integral: where n = 1
or n = 2 Boundary CondiRons Support Type: Pin or roller at end of beam Fixed end Boundary CondiRon: Displacement: and Moment: Displacement: and Slope: Only condiRons on displacement or slope are needed to ﬁnd constants of integraRon. 16 28) Hibbeler 4th ed The beam is subjected to the load shown. Determine the equaRon of the elasRc curve. EI is constant. 3 kip/ ft
5 kip ft 5 kip ft A B x
4 ft 8 ft 4 ft Given: applied loads, EI Find: υ(x) Plan: Use disconRnuity funcRons to ﬁnd M(x) then integrate to ﬁnd υ(x). 24 kip
5 kip·ft 5 kip·ft
3 kip/ ft 16 28) Hibbeler 4th ed EnRre FBD 5 kip ft 4 ft
Ay 24 kip
5 kip·ft By 5 kip·ft
4 ft
Ay Load: 4 ft
Ay 3 kip/ft
8 ft 5 kip·ft B y 4 ft 3 kip/ft B 4 ft 8 ft 4 ft By symmetry 5 kip·ft By x
4 ft 8 ft 5 kip ft 4 ft A 5 kip·ft
4 ft
Ay 8 ft 3 kip/ft
8 ft 5 kip·ft B y 4 ft 3 kip/ft 3 kip/ ft 16 28) Slope and DeﬂecRon Hibbeler 4th ed 5 kip ft 5 kip ft A B x
4 ft Boundary CondiRons: 8 ft 4 ft 3 kip/ ft 16 28) Slope and DeﬂecRon Final Result Hibbeler 4th ed 5 kip ft 5 kip ft A B x
4 ft 8 ft 4 ft General State of Plane Stress • The element below shows a general state of Plane Stress: • Two normal stresses, sx and sy, and one shear stress txy. • The stress components are shown in their posiRve senses. • The components sx, sy and txy deﬁne the stress state. • Their values are dependent on the orientaRon of the element. Rotated State of Plane Stre...
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 Winter '08
 OSAMA
 Force, Stress, Trigraph, Ed 00

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