EBT 252-Lecture 5
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Transcript of EBT 252-Lecture 5
EBT 252/4: STRENGTH OF MATERIALS - J. B. JOHNSON FORMULA - AISC COLUMN FORMULAS
By
DR. SRI RAJ RAJESWARI MUNUSAMY
PPK BAHAN, UNIMAP
E-mail: [email protected]
2/12/2013 1
J.B.JOHNSON FORMULA
The Euler formula does not apply for the
intermediate columns.
Hence, many semi-empirical formulas
have been developed.
J.B.Johnson formula is used extensively in
steel structure design and machine
design.
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- J. B. Johnson formula is the equation of parabola with its vertex at the
point on the vertical axis with ordinate equal to y.
-The parabola is tangent to the Euler curve at the transition slenderness
ratio kL/r = Cc, which equals to ยฝ of the yield stress, y of the steel.
kL/r
The value of transition slenderness ratio, Cc can be
determined as follows :-
Thus,
The J. B. Johnson formula is:
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๐๐ =๐๐๐๐
๐๐ฒ ๐๐ช. ๐
๐๐๐ซ =๐๐๐ซ
๐= ๐ โ
๐ค๐๐ซ
๐
๐๐๐๐ ๐๐ฒ ๐๐ช. ๐๐
The Euler formula applies when kL/r is
greater than Cc and the J.B.Johnson formula
applies when kL/r is less than Cc.
For kL/r = Cc, both formulas give the same
result.
The Euler formula applies to all materials,
whereas the J.B.Johnson formula applies
mainly to ductile steel.
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Example 1
Determine the allowable compressive
load of a 4-in., standard weight steel
pipe that is 25ft long. The column is
made of A36 steel with y = 36 ksi and
is welded to fixed supports at both
ends. Use F.S. = 2 and E=29 x 103 ksi.
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Solution:
From Table A-5(a) in Appendix 1, for a 4-in., standard
weight steel pipe,
A = 3.17 in.2
r = 1.51 in
The slenderness ratio is:
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From Eq.9, the value of the transition slenderness
ratio Cc is :
Since kL/r < Cc, the J.B.Johnson formula applies.
From Eq.10, we find,
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Thus,
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THE AISC COLUMN FORMULAS
The American Institute of Steel
Construction (AISC) manual gives formulas
for calculating the allowable compressive
stresses to be used in steel column design.
The AISC column formulas are essentially
the critical buckling stresses from the Euler
and J.B.Johnson formulas divided by the
factor of safety.
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The AISC formulas are :-
1. For long columns :
2. For intermediate and short columns :
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๐๐๐ฅ๐ฅ๐จ๐ฐ =๐๐๐/(๐ค๐/๐ซ)๐
๐ . ๐=
๐๐๐/(๐ค๐/๐ซ)๐
๐. ๐๐ ๐๐ช. ๐๐
๐๐๐ฅ๐ฅ๐จ๐ฐ =
๐ โ๐ค๐/๐ซ ๐
๐๐๐๐ ๐๐ฒ
๐ . ๐ ๐๐ช. ๐๐
Where the factor of safety, F.S. is computed from:
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* F.S varies from 5/3 (or 1.67) when kL/r = 0 to 23/12(or 1.92)
when kL/r = Cc
๐ . ๐ =๐
๐+
๐๐ค๐๐ซ
๐๐๐โ
๐ค๐๐ซ
๐
๐๐๐๐ ๐๐ช. ๐๐
Table 1 shows the value of the AISC recommended
effective length factor k for steel column design
when the end-supporting conditions are
approximated.
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End Conditions Pinned
Ends
Fixed Ends Fixed,
Pinned Ends
Fixed, Free
Ends
Theoretical k
value
1.0 0.5 0.7 2.0
AISC
recommended
k value
1.0 0.65 0.8 2.10
Table 1: AISC Recommended k Values
Values of the allowable compressive
stress computed from the AISC
formulas corresponding to y = 36 ksi
and y= 50 ksi are tabulated for kL/r
values from 1 to 200 in Tables 2 and 3.
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Table 2: AISC Allowable Compressive Stress for Steel Columns for
y = 36 ksi (250 MPa)
Table 3: AISC Allowable Compressive Stress for Steel Columns for
y = 50 ksi (345 MPa)
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Example 1 Determine the allowable axial compressive load
for a 10-ft long standard L6 X 4 X ยฝ steel angle of
A36 steel if the supporting conditions are (a)
pinned at both ends or (b) fixed at both ends.
Use the AISC formulas and the recommended k
values.
Solution:
From the Appendix 2, Table A-4(a), for an L6 X 4 X
ยฝ steel angle, A= 4.75in2 and the least radius of
gyration is rz=0.870in.
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Since kL/r >Cc, Eq.11 applies. Thus,
Or from Table 19-2, for y=36ksi and kL/r =138(rounded to the
nearest whole number for use in the table, interpolation is not necessary), the allowable compressive stress is allow =
7.84ksi, the same as calculated above. Thus,
๐๐๐ฅ๐ฅ๐จ๐ฐ = ๐๐๐ฅ๐ฅ๐จ๐ฐ๐ = (๐. ๐๐๐ค๐ข๐ฉ๐ฌ/๐ข๐ง๐)(๐. ๐๐๐ข๐ง๐)= ๐๐. ๐๐ค๐ข๐ฉ๐ฌ
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Table 2
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Table 2,
Example 2 A 3-m column having an L127 X 127 X12.7
angle section (Refer to Appendix 3) is
made of A242 steel with E=200GPa and
y=345MPa. The column is fixed at both
ends. Calculate the allowable axial
compressive load using the AISC formulas
and the recommended k values. Use the
allowable stress listed in Table 19-3 to verify
the computations.
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๐ = ๐. ๐๐ ร ๐๐โ๐๐ฆ๐; ๐ซ๐ฆ๐ข๐ง = ๐ซ๐ณ = ๐. ๐๐๐๐ฆ
๐ ๐จ๐ซ ๐๐๐๐ ร ๐๐๐ ร ๐๐. ๐.
Table 3,
Solution :
๐ ๐จ๐ซ ๐๐ข๐ฑ๐๐ ๐๐ง๐๐ฌ, ๐ค = ๐. ๐๐
๐ค๐
๐ซ=
(๐. ๐๐)(๐๐ฆ)
๐. ๐๐๐๐ฆ= ๐๐
๐๐ =๐๐๐๐
๐๐ฒ=
๐๐๐(๐๐๐ ร ๐๐๐)
๐๐๐ ร ๐๐๐= ๐๐๐. ๐ = ๐๐๐
๐ค๐
๐ซ< ๐๐ ; ๐๐ก๐๐ซ๐๐๐จ๐ซ๐ ๐. ๐. ๐๐จ๐ก๐ง๐ฌ๐จ๐ง ๐๐จ๐ซ๐ฆ๐ฎ๐ฅ๐ ๐๐ฉ๐ฉ๐ฅ๐ข๐๐ฌ
๐ . ๐ =๐
๐+
๐๐ค๐๐ซ
๐๐๐โ
๐ค๐๐ซ
๐
๐๐๐๐ =
๐
๐+
๐ ๐๐
๐ ๐๐๐โ
๐๐ ๐
๐ ๐๐๐ ๐
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Table 3,
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Appendix 1 2/12/2013 26
Appendix 2 2/12/2013 27
Appendix 3 2/12/2013 28
Appendix 4 2/12/2013 29