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3.1. A 16 × 20 in. column is made of the same concrete and reinforced with the same six No.
9 (No. 29) bars as the column in Examples 3.1 and 3.2, except t hat a steel with yield strength
f y = 40 ksi is used. The stress-strain diagram of this reinforcing steel is shown in Fig. 2.15 for
Ec
Part a The solution is identical for grade 40 and grade 60 reinforcement
Part b
fy
fy1
Problem 3.1
Part c
Pu
Comments
1. There is no difference at fc = 1200 psi and elastic assumptions are used
3.2 The area of steel, expressed as a percentage of gross concrete area, for the column
of Problem 3.1 is lower than would often be used in practice. Recalculate
the comparisons of Problem 3.1, using f y of 40 ksi and 60 ksi as before, but
for a 16 × 20 in. column reinforced with eight No. 11 (No. 36) bars. Compare
your results with those of Problem 3.1.
16"
Ec
Part a The solution is identical for grade 40 and grade 60 reinforcement
P0.246The steel carries 25 percent of the load
Part b
fy
fy1
2/2
Problem 3.2
Part c
Pu
Comments
3.3.Asquareconcretecolumnwithdimensions22×22in.isreinforcedwith
atotalofeightNo.10(No.32)barsarrangeduniformlyaroundthecolumn
perimeter.Materialstrengthsarefy=60ksiandfc =4000psi,withstressstrain
curvesasgivenbycurvesaandcofFig.3.3.Calculatethepercentages
oftotalloadcarriedbytheconcreteandbythesteelasloadisgradually
increasedfrom0tofailure,whichisassumedtooccurwhentheconcretestrain
reachesalimitvalueof0.0030.Determinetheloadsatstrainincrementso
f
0.0005uptothefailurestrain,andgraphyourresults,plottingloadpercentages
vs.strain.Themodularratiomaybeassumedatn=8forthesematerials.
UsingConcretedatafromFigure3.3
0.00%
10.00%
20.00%
30.00%
40.00%
50.00%
60.00%
0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035
Pc/Ptotal
Ps/Ptotal
3.4. A 20 × 24 in. column is made of the same concrete as used in Examples 3.1
and 3.2. It is reinforced with six No. 11 (No. 36) bars with f y = 60 ksi. For
this column section, determine ( a ) the axial load that the section will carry at
a concrete stress of 1400 psi; ( b ) the load on the section when the steel begins
to yield; ( c ) the maximum load if the section is loaded slowly; and ( d ) the
maximum load if the section is loaded rapidly. The area of one No. 11 (No. 36)
bar is 1.56 in 2 . Determine the percent of the load carried by the steel and the
concrete for each combination.
Reinforcement Areas
Given Properties
Part (a) Compute the axial capacity of the section loaded below the elastic limit.
Solution: The axial capacity is based on the gross area of the column plus the effective area of
the steel. Since we count the holes where the steel is removed, the additional effective area of
the steel is (n-1)Ast.
PsfcnAst
Ps105 kip 100
P13.7
Part (b): Compute the capacity of the column when the steel begins to yield εy
fy
Es
εy0.00207or 2/10 of one percent
2/2
Part (c): Compute the maximum load capacity of the section if loaded slowly
Part (d): If we reexamine the problem with a fast loading, then the concrete stress would be
3.5 A 24 in. diameter column is made of the same concrete as used in Examples
3.1 and 3.2. The area of reinforcement equals 2.1 percent of the gross cross
section (that is, A s = 0.021 A g ) and f y = 60 ksi. For this column section, determine
( a ) the axial load the section will carry at a concrete stress of 1200 psi;
( b ) the load on the section when the steel begins to yield; ( c ) the maximum
load if the section is loaded slowly; ( d ) the maximum load if the section is
loaded rapidly; and ( e ) the maximum load if the reinforcement in the column is
raised to 6.5 percent of the gross cross section and the column is loaded slowly.
Comment on your answer, especially the percent of the load carried by the
steel and the concrete for each combination.
Reinforcement Properties
Given Properties
Part (a) Compute the axial capacity of the section loaded below the elastic limit.
Solution: The axial capacity is based on the gross area of the column plus the effective area of
PsfcnAst
Ps91 kip 100
P14.6
Part (b): Compute the capacity of the column when the steel begins to yield εy
fy
Es
Part (c): Compute the maximum load capacity of the section if loaded slowly
Examining Figure 1.16, we are beyond the elastic portion of the concrete stress strain curve
PsfsAst
Ps570 kip 100
P27.5
Part (d): If we reexamine the problem with a fast loading as would occur in a building, then the
concrete stress would be
Note: the total increase is in the concrte contribution. 100
P24.3
Part (e): Determine the capacity for a slow loaded column with the steel changed to 6.5%
Ast 0.065 Ag
Ast 29.4 in2
3/3
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