Chapter 9 Stability of column ??????

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? 9 ? ??????

??????
? ?????? ? ????????? ? ???????????
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Chapter 9 Stability of column??????

• 1 Introduction ?? 2 Critical loads of
  slender column with hinged ends ?????????????3
  Critical loads for slender columns with various
  support ???????????4 Critical stresses for
  columns with small and inter-mediate slenderness
  ratio ??????????5 Stability analysis and
  design of columns???????????

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1 Introduction ? ?

• ? Three types of balance
• ???????
• ? Concept of stability of column ??????
• ? Other forms of stability problems ?????????

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? Three types of balance?????????
?????????
Stable ????
Unstable?????
???? (?????)
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Stability of Rigid bar-Spring Structure ??-???????
??
????
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?????
19???,?????
19??????,??????
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?????????,?????????????
?????????????,?????????????
?????????????????
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????
Rigid bar-Spring Structure??-????
1.???
The system is slightly disturbed??????, ???????

• ? ???????????
• Fd- ??????
• kdl- ???????????

• ? ????
• ? Fd lt kdl ? F lt kl
• ??????????
• ? Fdkdl ? F kl
• ??????????????
• ? Fd gt kdl ? F gt kl
• ????????????

Fd-????deviation moment

• kdl-????recovery moment

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Three types of balance???????
???? ???? F lt kl ???? F kl
???? F gt kl ?????
F???,????????
Fcr kl - Critical load ????
????????????,????????,? FFcr ?,????????
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2.??? ????,?????,???????
??
????
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? Concept of stability of column??????
Stability of Column?????
An ideal slender column which is perfectly
straight and compressed by a centrally applied
load (??????????)
????
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Concept of stability of column???????
F lt Fcr The equilibrium is stable ????(The
deflection will disappear, and the bar will
return to its straight form??????) F gt Fcr The
equilibrium is unstable?????(The deflection will
increase continuously, and the column will
collapse??????,????) F Fcr A condition of
neutral equilibrium????(the deflection will be
maintained in a slight bent form???????????????)
Critical load????- axial force which is
sufficient to maintain the column in a slightly
bent form????????????????????(??????????,?????????
????????)
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Other forms of stability problems?????????
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? ????,AB????,BC????,???????????P??,????????.?BC?
???EI???.
?? ??????????????
????????????
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????
????
Mr MB
???????????
???????????,??????
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? Examples ??
Rigid rod AD is attached to springs (K 4.5 kN/m)
at B and C and subjected to equal and opposite
loads at each end. The rod is in its equilibrium
position as shown. Determine the critical load at
which the rod will no longer be horizontal.
Rod AD will rotate through an angle ??.
Pcr 4.41 kN
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2 Critical loads of slender column with hinged
ends ?????????????

• ? Euler's Formula ????
• ? Discussion about Concept of Stability???????????
  ?
• ? Examples ??

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? Euler's Formula ????
????
Fcr-????????????????????
???????,?????????????
????
??M(x) , w - ???
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??????
?n1
Euler's Formula????
????????Euler?16????
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??
-??????
? ????????????-????
????????????????????????-???????, ???????
???????? (1)The deflections are assumed
small??? (2) Linear elastic????
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Discussion about Concept of Stability????????????
? ??????Large Deflections of Columns ,
?????????,?????,??????,???????????????????????????
????, ??????????????????????
?????
? ????? Fcr ???, ????????????????????? ??,
???, ??????
?????
? ?? AB ?A ??????, ?F ??? Fcr ?, ????????,
????????
? ? F?Fcr , ????? wmax ????, ??Fcr????????,
?????????????????Fcr , ???????
Load-deflection diagram
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? ??????,l 0.8 m, d 20 mm, E 200 GPa, ss
235 MPa,?Fcr ?
?
????????,??????????
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9-22 ????,???C????F100kN??????????,??????Q275,??
???s180Mpa,?????????
? ???C???
AC????,
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BC????,
???????,
?
??
?
??AC?????
BC?????
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? Examples ??
????????????????Determine the maximum allowable
force P that can be applied to the frame shown
without causing either pinned-pinned member to
buckle.
Members AB and BC both have a rectangular cross
section with dimensions as shown, and
E 70 109 N/m2
Solution
Free-body diagram of joint B
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Member BC is in tension and does not buckle
therefore, we only consider the compressive
member AB.
IAB (0.400)(0.200)3/12 266.67 10 6 m4
1.75P
P 537.12 kN
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3 Critical loads for slender columns with
various support ??????????????

• ? Fixed-Pinned Slender Column
• ????????????
• ? Analogy method to determine the critical loads
  ?????????
• ? Effective length ????
• ? Examples ??

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Fixed-Pinned Slender Column?????????????
?? A,B,FR ????????
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??????
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Analogy method to determine the critical
loads?????????
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? Effective length ????
(?????????)
ml -????-??????????????
m -????-??????????????
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? Axis of Buckle ??????
???Pcr,min?????

• ??

(1)??????Imin?????
(2)??????(I/µ2)min????
y-z?????, x-z???????, µ?0.5?1???
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? Examples ??
?3-1 ??????,????? Fcr?
?1. ??????
????????????????
2. ??????
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? 3-2 ??????,? Fcr ?
?
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??(a),(b)???(1),(2) ,?
??(a),(b)???(3),(4),(5),?
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A1, A2 ? d ??????
??????
???
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(?????????)
?????,?????????????????????????????
??????????b,??hgtb???,???????????I?
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4 Critical stresses for columns with small and
inter-mediate slenderness ratio ??????????

• ? Critical stresses and slenderness ratio???????
• ? The conditions of Euler's formula to be valid
  ????????
• ? Empirical formula of critical stresses????????
• ? Examples ??

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? Critical stresses and slenderness ratio ???????
- Critical stresses????
??????????????????
-radius of gyration??????
- slenderness ratio ??????
????,??????( l )?????(µ)???????( i )????????
- Euler's Formula ????
?????????, ?????????, ????, ??????
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The conditions of Euler's formula to be
valid?????????
?????????
???-????????
??,Q235?,E200 GPa, sp196 MPa
?s gtss ?,??????????
?s??sp?ss???,?????????????
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? Empirical formula of critical stresses????????
l0 ltlltlp ???-????,?????????
1. ???????
? a, b ??????
? ?????????????????
?
????
????
????
? ??????
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2. ????????
? a1, b1??????
? ??????????????
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? Examples ??
? 4-1 ???????, d 40mm, E 210GPa, lp 100,
?Fcr?
?
????
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? 4-2 ????????Fcr, A 70 mm2, Iz 6.5?104
mm4, Iy 3.8?104 mm4, ???????????
?
?x-y????
?x-z????
?????, ??x-y????
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5 Stability analysis and design of
columns???????????

• ? Stability analysis of columns??????
• ? Method of reduction coefficient?????
• ? Design of columns ??????
• ? Examples ??

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? Stability analysis of columns ??????
? ??????????
nst-??????
gt??????n
Fst-??????
? ??????????
sst-??????
? ??Fcr?scr?, ???????????
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? Method of reduction coefficient ?????
? ??????
s-?????
f-?????????
? ??????
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? Design of columns ??????
? ??????
??????? E ?????, scr ??
??????(??????)? E ????
??????? ?? ?????, scr ??
??????? ??? ??????
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? ??????
??????????, l??, scr ??
? ????????????
? ????????
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? ???????????
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? Examples ??
? 5-1 ??????????F 12 kN, ???D 45 mm, ???d
36 mm, nst 2.5, ???Q235?? scr235
MPa-(0.00669 MPa) l2 lp100
?
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(No Transcript)
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??E210GPa, sp200MPa, ????a12.510-6(oC)-1
,D10cm, d8cm,?7m,??????????
?
? (1)???????
????
???????
????,?Euler??
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(2) ???????
??????T,
????
?


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??s160MPa,?l2m,?????,??b150mm,?AB???????d36
mm,?l0.8m,?p99.3,?057,
????scr304-1.12?,nst3,????,?????F?
???????
?(1) ?AB
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(2) ?AD
????
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? ?????????
I ???? ?
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? ???????????
???? ???? ????
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? ?????????????
????? ??????? ??????? ??
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? ???????????
??????? ???
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Thanks ? ?