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Prediction of Compressive Strength of Columns with Stub
Column Properties
by
Bharath Surendra
Guided by
Dr. Yuner Huang
in the
Institute for Infrastructure and Environment
School of Engineering
UNIVERSITY OF EDINBURGH
June - August 2016

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CONTENTS
ABSTRACT ………………………………………………………………………. 3
1. INTRODUCTION ………………………………………………………….4-6
1.1 Background
1.2 Current Design Model
1.3 Stub Column Tests
2. EXPERIMENTAL DATA ……………………………………….....................7
3. DATA ANALYSIS ……………….……………………………………....8-10
4. RESULTS ...……………………………………………………………11-12
5. CONCLUSION ……………………………………………………………….13
REFERENCE …………………………………………………………………..14

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Abstract
A column is a structural element that transmits, through compression, the weight of the structure
above to other structural elements below. In other words, a column is a compression member. In
the design of a column the most important information needed is the maximum compressive load
it can bear before failing due to buckling. This report presents an innovative way of predicting this
compressive capacity of a column of any length using its stub-column properties. A set of 36
specimens were used as data and a new design model has been proposed.

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Chapter 1
Introduction
1.1 Background
Columns made of varied materials are used extensively all around the world for structural purposes
in the construction, automobile and aerospace industries among many others. Every column has a
compressive capacity, loading beyond which will result in failure due to buckling. In order to
determine this compressive capacity experiments are carried out which are very expensive and
hence numerical methods are a rather practical alternative which yields results with an acceptable
accuracy. The compressive capacity of a column is a function of its material and cross-sectional
properties having accounted for the residual stresses and imperfections present.
Some of material properties typically used are:
 Young’s Modulus (E)
 Yield Strength (Fy or σ0.2)
 Ramberg Osgood Parameter (n)
Some of cross-sectional properties typically used are:
 Radius of Gyration (r)
 Slenderness (λ)
Several tests are present to find out these properties but the most popular is the Tensile Coupon
Test and Stub Column Test.
1.2 Current Design Model
Current formula for predicting the maximum compressive load by Eurocode3 is as follows
N = χ A Fy (for class 1, 2 & 3) (1.1)
N = χ Aeff Fy (for class 4) (1.2)

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𝜆 =
𝐿
𝜋∗𝑟
√
𝐹𝑦
𝐸
(1.3)
𝜙 = 0.5(1 + ŋ + 𝜆2
) (1.4)
χ =
1
𝜙+√(𝜙2−𝜆2)
(1.5)
Where,
N – Compressive capacity in kN
λ – Non-dimensional Slenderness (χ vs λ plot is called a column curve shown in fig1)
χ – Non-dimensional Strength or Reduction Factor
Fy – Yield Strength in MPa
E – Young’s Modulus in GPa
ŋ – Imperfection Parameter
r – Radius of gyration of the cross section in mm
L – Effective Length
A – Area of Cross section
The Fy and E is the yield strength from tensile coupon test and parameters like ϕ & ŋ are used to
account for residual stresses and imperfections since the results from coupon test don’t reflect the
afore mentioned losses.
Fig1: A typical column curve proposed in this method

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1.3 Stub Column Tests
A stub column is a column whose length is sufficiently small to pre- vent failure as a column, but
long enough to contain the same residual stress pattern that exists in the column itself. Column
capacity may be expressed as a function of the tangent modulus and yield strength determined from
the stress-strain relationship of the stub column test.
The difference between the Young's modulus and the tangent modulus at any load level, determined
from a compression test on the complete cross section, essentially reflects the effect of residual stresses.
The presence of residual stresses in the cross section implies that some fibers are in a state of residual
tension while others are in a state of residual compression. The fibers in a state of residual compression
are the first to reach the yield point under load.
Stub Column tests also account for imperfections and change in strength due to corners present in the
cross section.
In this report an effort is made in the direction to employ these stub column test results to find the
compressive capacity of long columns by introducing a new reduction factor and column curve.

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Chapter 2
Experimental Data
A set of 5 stub columns made of duplex steel of different cross-sectional and material properties
and each stub column having set of 5 columns of different length (200mm, 550mm, 900mm,
1200mm, 1550mm) was collected from [1] and is tabulated as shown in table1.
Column
No.
Fy (MPa) E (GPa) σy (MPa) n √(E/σy) Dimensions Radius of
Gyration Ry (mm)
C2 610 194 515 4 19.4087 50×50×1.5 19.8095
C3 635 202 690 3 17.1100 50×50×2.5 19.4200
C4 613 204 621 3 18.1246 70×50×2.5 20.2409
C5 625 207 545 4 19.4889 100×50×2.5 21.0176
C6 664 202 430 8 21.6741 150×50×2.5 21.7540
Table1 : Column sample details of material and sectional properties
Further 4 more stub columns were collected from [2] and are as shown in Table2. The SHS stub
columns have set of 5 columns of lengths 650mm, 1000mm, 1500mm, 2000mm, 2500mm each
and the RHS stub columns of lengths 1400mm, 2200mm, 3000mm. Effective length being half of
the nominal length.
Column
Name
Fy (MPa) E (GPa) σy (MPa) n √(E/σy) Dimensions Radius of Gyration
Ry (mm)
SHS1 707 226 757 3 17.2785 40×40×2.0 15.5350
SHS2 622 200 608 4 18.1369 50×50×1.5 19.8095
RHS1 486 214 441 6 22.0286 140×80×3.0 33.5855
RHS2 536 213 390 9 23.3699 160×80×3.0 34.0342
Table2: Column sample details of material and sectional properties from [2]

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Chapter 3
Data Analysis
Data of 25 samples as mentioned in the table1 were used to calculate non-dimensional strength
(RF) using (1.3) and non-dimensional slenderness (λ) from (1.1) and RF and λ were plotted against
each other for the 5 individual columns as shown in fig2.
Fig2: Individual Column Curves from Table1
It was observed during the course of the study that the individual column curves are varying with
respect to 2 parameters
1. Ramberg Osgood Parameter (n) – Greater the value of n, lower is the RF for a given λ.
2. √(E/Fy) – Greater the value of this, lower is the RF for a given λ.
In order for the individual solutions to converge we have to propose a new Reduction Factor as
given below

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χ’ = [χ (
√ 𝐸
√Fy
) ]
𝑛
𝑛+1 (3.1)
Where E and Fy are and n are material properties from stub column test. Using the above
mentioned relation for RF we see the individual curves converging as shown in the Fig3 (Right).
Fig3: Individual best fit line plots (left) and the converged solution with the proposed RF(right)
However after employing the same for the data in Table2 the results did not exactly converge as
expected amd is show in Fig4 below.
Fig 4: All the 9 samples are plotted with new RF (χ’) vs Slenderness (λ)

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With Fig 4 in hand we observe that curves do not exactly converge and reason for this would be
the radius of gyration (Ry) of the cross section of the column and the Reduction Factor χ’ as
proposed in (3.1) (also in the plot above) can be made to converge using another Reduction Factor
χ” as given by the equation below:
χ’’ = χ′
/(
𝑅𝑦
20
)
𝑛−2
𝑛 (3.2)
The χ” (RF”) vs λ plot was generated with a 3rd
Order Polyfit line using Matlab® and is as
shown in fig-5 below.
Fig 5: χ” vs λ with Best Fit Line

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Chapter 4
Results
Using the Graph as shown in Fig 5 the compressive load N was found using the proposed design
rule and the experimental load was taken from [1] and [2] and is tabulated as shown below.
Column Χ’’ Χ N (kN) Nexp (kN) Nexp/N
C2 9.6032
9.0814
7.7090
6.2612
4.7286
0.8658
0.8074
0.6579
0.5073
0.3571
152.27
141.99
115.70
89.21
62.81
153.400
139.300
120.800
92.300
65.400
1.0075
0.9810
1.0441
1.0347
1.0413
C3 9.6014
8.7392
6.9507
5.3082
3.9969
1.1772
1.0384
0.7652
0.5341
0.3659
343.86
303.32
223.51
156.03
106.88
355.300
302.100
242.800
146.100
103.900
1.0333
0.9960
1.0863
0.9364
0.9721
C4 9.6054
8.9839
7.4822
5.9601
4.4549
1.1326
1.0359
0.8117
0.5994
0.4066
396.15
362.34
283.93
209.66
142.22
373.100
352.800
270.700
211.200
148.300
0.9418
0.9737
0.9534
1.0074
1.0428
C5 9.5967
9.1953
7.9902
6.6531
5.1285
0.8940
0.8475
0.7110
0.5655
0.4085
398.22
377.51
316.71
251.91
181.95
370.100
372.300
335.200
249.000
193.700
0.9294
0.9862
1.0584
0.9884
1.0646
C6 9.5710
9.3866
8.5130
7.4358
6.0515
0.6287
0.6151
0.5511
0.4733
0.3754
397.58
388.98
348.49
299.29
237.38
404100
353.200
333.500
284.500
230.000
1.0164
0.9080
0.9570
0.9506
0.9689
SHS1 9.5105
8.9045
7.4070
5.6818
4.3019
1.0423
0.9547
0.7469
0.5245
0.3619
211.50
193.72
151.55
106.42
73.44
222.800
197.800
136.000
106.300
71.300
1.0534
1.0211
0.8974
0.9989
0.9709

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SHS2 9.6041
9.3929
8.5945
7.4266
6.1285
0.9266
0.9012
0.8065
0.6719
0.5285
167.15
162.57
145.48
121.20
95.33
181.000
175.100
156.800
124.700
95.100
1.0829
1.0771
1.0778
1.0288
0.9976
RHS1 9.5997
9.3143
8.6813
0.9508
0.9179
0.8455
580.83
560.73
516.53
553.100
525.100
513.500
0.9523
0.9365
0.9941
RHS2 9.6053
9.3996
8.8745
0.8368
0.8051
0.7477
585.77
563.57
523.37
537.100
515.300
439.400
0.9169
0.9144
0.8396
Table3: Tabulated Results of 9 samples
The results are tabulated as shown above and it’s observed that almost all the samples have yielded
a result with less than 10% deviation from the experimentally calculated strength. Moreover,
the average Error Percentage was calculated to be ± 4.6375% and Co-efficient of Variance
(COV) of about 0.057 over the entire 41 samples employed to validate the proposed design
formula. Also the Mean Nexp/N over the entire data was found to be 0.99.
The proposed model yielded more accurate results than any of the existing models as seen from
[1] and [2]. Given below is a result table of the sample C4 from [1] and results can be compared
with the table presented above to see the difference in the accuracy.
Table 4: Comparison of test strengths with design strengths for series C4.

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Chapter 5
Conclusions
The idea of using Stub Column test data for predicting the compressive strength of different
columns was mathematically realised by proposing altered Reduction Factors such as χ’ in 3.1 and
χ” in 3.2 which was validated for the existing data and was found to be applicable to columns with
varied, cross-sectional and material properties and end-supports. Having said that, the proposed
design model should be applicable to all kinds of columns having a square or a rectangular cross-
section and under pure compression and its compressive strength can be found well within ± 5%
error from experimental values.
Employing the stub-column test data instead of the coupon test data has allowed us to get rid of
imperfection and residual stress parameters, yielding a more intuitive formula to calculate the
compressive strength of the given column. However this proposed formula can undergo further
mathematical alterations in-order for it to yield more accurate result and we believe it’s just the
beginning.

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References
[1] Y. Huang, B. Young, Tests of pin-ended cold-formed lean duplex stainless steel
columns 2013.
[2] L. Wing Man, Design of cold formed high strength stainless steel tubular columns
and beam-columns 2004.
[3] EC3. Design of steel structures — part 1.1: general rules and rules for buildings.
European Committee for Standardization, EN 1993-1-1. Brussels: CEN; 2005.
[4] Rasmussen K J R, Rondal J, Column curves for stainless steel alloys. Journal of
Construction Steel 2000; 54:89–107.
[5] Rondal J and R. Maquai, Stub-column Strength of Thin-walled Square and
Rectangular Hollow Sections 1985. Thin- Walled Structures 0263-8231/85/$03 .30.
[6] Proposal for Stub Column Test Procedure, Document No. X -282 – 61. New York
April 1961.