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Evo Design - structural engineering
Calculation No.
001-RC Column
CALCULATION SHEET
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SAMPLE CALCULATION
Project Title:
Reinforced Concrete Column - interactive design spreadsheet
Calc. By
Date
Rev.
MN
16.04.2014
0
Subject
RC Column - M-N interaction diagram (EC2)
Checked By
Date
CN
16.04.2014
Input
Â
Output
Column dimensions
Moment capacity
Reinforcement
Materials (steel, concrete, bolts)
RC Column - Axial Force - Bending Moment Interaction
per EN 1992-1-1:2004*
Axial force - bending moment interaction -
Â
ultimate limit state
Section 3
Column dimensions
b
x
=
300
mm
(parameters that can not be
b
z
=
300
mm
modified in the demo version)
A
p
=
mm
2
(Element area = b
x
* b
y
)
Reinforcement
c =
mm
cover
d =
mm
(b
x
- c)
Tension side reinforcement
f
=
Â
mm
bars diameter
n =
no of bars
A
s.a
=
mm
2
area of tension reinforcement
p
reinf.a
=
%
percentage of tension reinforcement
Compression side reinforcement
f
=
Â
mm
bars diameter
n =
no of bars
A
s.b
=
mm
2
area of compression reinforcement
p
reinf.b
=
%
percentage of compression reinforcement
p
reinf.a+b
=
%
element percentage of reinforcement
Materials
Concrete class
C12/15
C16/20
C20/25
C25/30
C30/37
C35/45
C40/50
C45/55
C50/60
C55/67
C60/75
C70/85
C80/95
C90/105
per EN 1992-1-1:2004
f
ck
=
MPa
concrete characteristic
Â
Section 3 Table 3.1
cylinder strength
Reinforcement type
see reinforcement types here
f
yk
=
MPa
reinforcement yield strength
Partial factors for materials for ultimate limit states
per EN 1992-1-1:2004
Section 2 Table 2.1N
g
c
=
values for Persistent & Transient design situations
g
s
=
recommended by the Eurocode; values to be used
may be found in the Eurocode National Annexes
Design compressive concrete strength
per EN 1992-1-1:2004
Section 3.1.6 & Formula 3.15
a
cc
=
Coefficient taking account of long term effects
f
cd
=
a
cc
* f
ck
/
g
c
=
MPa
on the compressive strength and of unfavourable
effectsresulting from the way the load is applied
value may be found in the EC National Annex
References:
EN 1993-1-1:2004 - Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings
Calculation No.
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Ultimate concrete compressive shortening strain
e
cu3
=
per EN 1992-1-1:2004
1000
Section 3.1.3 Table 3.1
per EN 1992-1-1:2004
Section 3.1.7 (2) Figure 3.4
Bi-linear stress-strain relation
Design reinforcement strength
f
yd
=
f
yk
/
g
s
=
MPa
per EN 1992-1-1:2004
Section 3.1.7 Figure 3.8
Reinforcement ductility class
A
B
C
ductility class A, B or C
defining reinf. strain at maximum force
Characteristic reinf. strain at maximum force
e
uk
=
per EN 1992-1-1:2004
100
Annex C - Table C1
function of reinf. ductility class
Design reinf. strain at maximum force
e
ud
=
*
e
uk
=
%
per EN 1992-1-1:2004
Section 3.2.7 Note 1
The value of
e
ud
for use in a Country may be
found in its National Annex.
Â
The recommended value is 0.9*
e
uk
Reinforcement modulus of elasticity
per EN 1992-1-1:2004
E
s
=
200
GPa
Section 3.2.7 (4)
The design value of the modulus of elasticity
E
s
may be assumed to be 200 GPa
Stress strain relations for the design of cross-section
per EN 1992-1-1:2004
Section 3.1.7 (3)
A rectangular stress distribution is assumed
l
, defining the effective height of the compression
zone and
h
, defining the effective strength are
derived from formulas 3.19, 3.20, 3.21 and 3.22
l
=
h
=
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Points defining the axial force - bending moment interaction diagram
Case 0
The entire section is in tension
F
s.a
=
- A
s.a
* f
yd
=
kN
Reinforcement has yielded.
F
s.b
=
- A
s.b
* f
yd
=
kN
Concrete tensile capacity is ignored.
N
cap
=
F
s.a
+ F
s.b
=
kN
M
cap
=
F
s.b
* (b
x
/2 - c) - F
s.a
* (b
x
/2 - c)
M
cap
=
kN*m
Case 1
Concrete strain:
e
cu3
Concrete has reached ultimate concrete design
Tension reinforcement strain:
e
s.a
=
e
ud
compressive shortening strain and
Â
reinforcement
x
1
=
Â
e
cu3
* d / (-
e
s.a
+
e
cu3
)
=
mm
has reached design reinforcement strain
a
1
=
Â
l
* x
1
=
mm
F
b
=
h
* a
1
* b
z
* f
cd
=
kN
F
s.a
=
-A
s.a
* f
yd
=
kN
e
s.b
=
e
cu3
* (x
1
- c) / x
1
=
/1000
Compression side reinforcement strain
f
yd
/E
s
=
/1000
Reinforcement yield strain
F
s.b
=
Â
- f
yd
* A
s.b
compression side reinforcement is in tension and it has yielded
F
s.b
=
Â
- E
s
*
e
s.b
* A
s.b
compression side reinforcement is in tension and has not reached yield point
F
s.b
=
Â
+ f
yd
* A
s.b
compression side reinforcement is in compression and it has yielded
F
s.b
=
Â
+ E
s
*
e
s.b
* A
s.b
compression side reinforcement is in compression and has not reached yield point
F
s.b
=
kN
N
cap
=
F
b
+ F
s.a
+ F
s.b
=
kN
M
cap
=
F
b
* (b
x
- a
1
)/2 + F
s.b
* (b
x
/2 - c) - F
s.a
* (b
x
/2 - c)
M
cap
=
kN*m
Case 2
Concrete has reached ultimate concrete design
Concrete strain:
e
cu3
Â
compressive shortening strain and
Â
reinforcement
Tension reinforcement strain:
e
s.a
= -f
yd
/ E
s
has reached yield reinforcement strain
f
yd
/E
s
=
/1000
x
2
=
Â
e
cu3
* d / (-
e
s.a
+
e
cu3
)
=
mm
a
2
=
Â
l
* x
2
=
mm
F
b
=
h
* a
2
* b
z
* f
cd
=
kN
F
s.a
=
-A
s.a
* f
yd
=
kN
e
s.b
=
e
cu3
* (x
2
- c) / x
2
=
/1000
Compression reinforcement strain
f
yd
/E
s
=
/1000
Reinforcement yield strain
F
s.b
=
Â
- f
yd
* A
s.b
compression side reinforcement is in tension and it has yielded
Â
- E
s
*
e
s.b
* A
s.b
compression side reinforcement is in tension and has not reached yield point
Â
+ f
yd
* A
s.b
compression side reinforcement is in compression and it has yielded
Â
+ E
s
*
e
s.b
* A
s.b
compression side reinforcement is in compression and has not reached yield point
F
s.b
=
kN
N
cap
=
F
b
+ F
s.a
+ F
s.b
=
kN
M
cap
=
F
b
* (b
x
- a
2
)/2 + F
s.b
* (b
x
/2 - c) - F
s.a
* (b
x
/2 - c)
M
cap
=
kN*m
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Case 3
Concrete has reached ultimate concrete design
Concrete strain:
e
cu3
Â
compressive shortening strain and
Â
the height
The entire section is in compression
x
3
=
Â
b
x
=
mm
of the compressed zone is equal with the section height
a
3
=
Â
l
* x
3
=
mm
F
b
=
h
* a
3
* b
z
* f
cd
=
kN
e
s.b
=
e
cu3
* (x
3
- c) / x
3
=
/1000
Compression side reinforcement strain
e
s.a
=
e
cu3
* (x
3
- d) / x
3
=
/1000
Tension side reinforcement strain
f
yd
/E
s
=
/1000
Reinforcement yield strain
Compression side reinf. is in compression
F
s.b
=
Â
+ f
yd
* A
s.b
compression side reinforcement is in compression and it has yielded
F
s.b
=
Â
+ E
s
*
e
s.b
* A
s.b
compression side reinforcement is in compression and has not reached yield point
F
s.b
=
kN
Tension side reinf. is in compression
F
s.a
=
Â
+ f
yd
* A
s.a
compression side reinforcement is in compression and it has yielded
F
s.a
=
Â
+ E
s
*
e
s.a
* A
s.a
compression side reinforcement is in compression and has not reached yield point
F
s.a
=
kN
N
cap
=
F
b
+ F
s.a
+ F
s.b
=
kN
M
cap
=
F
b
* (b
x
- a
3
)/2 + F
s.b
* (b
x
/2 - c) - F
s.a
* (b
x
/2 - c)
M
cap
=
kN*m
Case 4
The entire section is in compression,
Concrete strain:
e
cu3
concrete has reached ultimate concrete design
The entire section is in compression
e
s.b
=
e
s.a
=
e
cu3
compressive shortening strain and
Â
reinforcement
is in compresiion
f
yd
/E
s
=
/1000
Reinforcement yield strain
F
b
=
h
* b
x
* b
z
* f
cd
=
kN
Compression side reinf. is in compression
F
s.b
=
Â
+ f
yd
* A
s.b
compression side reinforcement is in compression and it has yielded
F
s.b
=
Â
+ E
s
*
e
s.b
* A
s.b
compression side reinforcement is in compression and has not reached yield point
F
s.b
=
kN
Tension side reinf. is in compression
F
s.a
=
Â
+ f
yd
* A
s.a
compression side reinforcement is in compression and it has yielded
F
s.a
=
Â
+ E
s
*
e
s.a
* A
s.a
compression side reinforcement is in compression and has not reached yield point
F
s.a
=
kN
N
cap
=
F
b
+ F
s.a
+ F
s.b
=
kN
M
cap
=
F
s.b
* (b
x
/2 - c) - F
s.a
* (b
x
/2 - c)
M
cap
=
kN*m
Data for the M-N interaction graph:
N
cap
M
cap
kN
kN*m
Case 0
Case 1
Case 2
Case 3
Case 4
N
eff
M
eff
kN
kN*m
CO1
CO2
CO3
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