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AN INVESTIGATION INTO THE TREATMENT OF
AMPLIFICATIONS OF MOMENTS IN TIMBER BEAM
COLUMNS
Harrison Wallen
1st
May 2015
Dr Mi...
Harrison Wallen
2
Declaration
I, Harrison Wallen declare that this thesis and the work presented in it are my own and
has ...
Harrison Wallen
3
Acknowledgments
I would like to thank Dr Michael Byfield for his support and guidance throughout the dur...
Harrison Wallen
4
CONTENTS
CONTENTS..........................................................................................
Harrison Wallen
5
C3 – Effect of Increasing the Elastic Modulus..............................................................
Harrison Wallen
6
Mc Moment resistance
Mel Elastic Moment resistance
N British Standards reduction factor for the Perry Ro...
Harrison Wallen
7
ABSTRACT
The presence of second order effects induced in beam-columns through load eccentricity
intrinsi...
Harrison Wallen
8
1 INTRODUCTION
When analysing beam-columns it is important to include a factor, 𝛼 in order to account fo...
Harrison Wallen
9
deflections. For members that are compressed centrically as well as having point loads or end
moments th...
Harrison Wallen
10
2 ELASTIC SECOND ORDER ANLYSIS OF BEAM-COLUMNS
2.1 The Gordon Rankine Method
The Gordon Rankine method ...
Harrison Wallen
11
Figure 2 - Deflection and interaction diagrams for an eccentrically loaded column.
By considering momen...
Harrison Wallen
12
As this equation is a function of a sec curve, asymptotes at π/2, π, 3/2π etc. bound the solution
below...
Harrison Wallen
13
Eq.4 has two parts; the first order analysis for a pin-ended beam with two equal end moments,
and a fun...
Harrison Wallen
14
using the amplification of moment’s factor. The data points were calculated using a value of
elastic mo...
Harrison Wallen
15
Nσ
σy
=
1
2
+
(η+1)σe
2σy
-√[(
1
2
+
(η+1)σe
2σy
)
2
-
σe
σy
]
which can be written as,
𝑁𝜎
𝜎 𝑦
=
1
2
+
...
Harrison Wallen
16
𝑀|| = 𝑀| 𝜇
where || and | denote first order and second order moment analyses respectively and μ is the...
Harrison Wallen
17
means the equations only remain safe if the column is restrained against lateral torsional buckling
(Po...
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18
These investigations were based on stochastic and mechanical models in order to determine the
ultimate ...
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19
stability of the member. It is apparent that reports on a statistical analysis for numerous timbers
rea...
Harrison Wallen
20
3 DESIGN EQUATIONS COMPARISON – A PREREQUESIT TO THE
EXPERIMENTS
Using calculations detailed in Appendi...
Harrison Wallen
21
It can also be seen that, as expected, increasing the eccentricity of axial load and the
slenderness ra...
Harrison Wallen
22
3.2 Conclusion
The Eurocode method appears to be the most efficient way of designing timber columns for...
Harrison Wallen
23
4 EXPERIMENT METHODOLOGY
This section describes the experimentation testing carried out to assess the a...
Harrison Wallen
24
Table 3 - Testing matrix describing the number of tests carried out for each sample configuration.
Ecce...
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25
The column was held in place with a minimal amount of pre-load and the dial gauge was
placed in positio...
Harrison Wallen
26
All but one sample were measured to have moisture content below 20% and by averaging
sample C and D a v...
Harrison Wallen
27
variable under displacement control and the load was recorded until failure. Figure 11 describes
the te...
Harrison Wallen
28
the member being tested that was not identified or the fact that the sample was considerably
shorter th...
Harrison Wallen
29
5 COMPARISON WITH EXPERIMENTAL RESULTS
5.1 Material Properties
The material properties obtained from th...
Harrison Wallen
30
5.2 Code Comparisons with Results
5.2.1 British Standards Comparison
The factors of safety achieved by ...
Harrison Wallen
31
5.2.2 Eurocode Comparison
A similar approach was taken using the Eurocode guidance with tabulated data ...
Harrison Wallen
32
material properties investigated by Blaβ (1987) and TRADA (1991). Slender columns behave in a
different...
Harrison Wallen
33
Both the Gordon Rankine method and the British Standard use this approximation, confirming
that they ar...
Harrison Wallen
34
the column would theoretically not be limited by its value of Pcr in the approximation formula
used by ...
Harrison Wallen
35
5.5 General Method Comparisons Using Design Equations and Standard Material
Properties
By plotting the ...
Harrison Wallen
36
slenderness value of 40. The Eurocode remains consistently more efficient than the other two
methods ho...
Harrison Wallen
37
By analysing the method design curves on Figure 17 and the distance the experimental plots lie
above th...
Harrison Wallen
38
6 CONCLUSIONS AND RECOMMENDATIONS
6.1 Conclusions
The amplification of moments factor is required in so...
Harrison Wallen
39
cases the material properties can be increased for that particular grade of timber. It is hard to tell
...
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40
APPENDIX A
A1: Derivation of the Moment Amplification Factor (Adapted from, Timoshenko and
Gere, 2009: ...
Harrison Wallen
41
y = yI + yII
and M = -P(yI + yII)
following a similar process as before yields,
EI
𝛿2
𝑦
𝛿𝑥2
= −𝑃(𝑦𝐼 + 𝑦...
Harrison Wallen
42
−𝑀
𝐸𝐼
=
𝛿2
𝑦
𝛿𝑥2
𝐸𝐼 ∫ 𝛿2
𝑦𝛿𝑥 = ∫ −𝑀𝛿𝑥
𝐸𝐼𝛿𝑦 = −𝑀𝑥 + 𝑐1
where, 𝛿y = 0 at x = L/2 ,
0 = −
𝑀𝐿
2
+ 𝑐1
giving...
Harrison Wallen
43
APPENDIX B
B1 - Derivation of N for the Gordon Rankine Method
𝑁
𝑁𝑏
+
𝛼𝑀
𝑀𝑒𝑙
= 1
Where Mel = Zσm,adm,||
...
Harrison Wallen
44
𝑁2
𝑒
NZσm,adm,|| −
1.5𝑁2Zσm,adm,||
Aσe
×K12
+
𝑁
Aσc,adm,||
=1
𝑁𝑒
a − bN
+
𝑁
𝑐
= 1
which has the solutio...
Harrison Wallen
45
B4 – Eccentricity Effects on Different Lengths
Figure B.1 – Effect of eccentricity on the critical load...
Harrison Wallen
46
APPENDIX C
C1 - British Standard Results Comparison
Table C.1 – Material properties and factors used fo...
Harrison Wallen
47
Section size
(mm)
Grade
Length
(mm)
Eccentricity
f(b)
Nexp
(KN)
NBS 5268,m
without
factors, with
measur...
Harrison Wallen
48
C2 - Eurocode Results Comparison
Table C.4 – Material properties and factors used for NEU 1995,s: with ...
Harrison Wallen
49
Section size
(mm)
Grade
Length
(mm)
Eccentricity
f(b)
Nexp
(KN)
NEU 1995,m
without
factors, with
measur...
Harrison Wallen
50
C3 – Effect of Increasing the Elastic Modulus
Figure C.3 - Chart showing the approximate values of defl...
Harrison Wallen
51
C5 – Comparison of Results for Different Section Sizes and Eccentricity
Figure C.5 - Comparison between...
Harrison Wallen
52
Figure C.7 - Comparison between the effects of slenderness on the design load for different
calculation...
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53
Figure C.9 - Comparison between the effects of slenderness on the design load for different
calculation...
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54
Figure C.11 - Comparison between the effects of slenderness on the design load for different
calculatio...
Harrison Wallen
55
(
𝑀𝑒𝑙
𝑒
1
𝑁𝑐𝑟
) + (
𝑀𝑒𝑙
𝑒
1
𝑁𝑝𝑙
) +
1
1 −
𝑁
𝑁𝑐𝑟
=
𝑀𝑒𝑙
𝑁𝑒
𝑀𝑒𝑙
𝑒
𝑁𝑐𝑟
+
𝑁𝑐𝑟
𝑁𝑐𝑟 − 𝑁
=
𝑀𝑒𝑙
𝑁𝑒
−
𝑀𝑒𝑙
𝑁𝑝𝑙 𝑒
w...
Harrison Wallen
56
APPENDIX D
D1 – Risk Assessment
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57
D2 – Method Statement
Method Statement:
Crushing of timber and blockwork columns to failure in the heav...
Harrison Wallen
58
10/11/2014
The timber props and dry laid blockwork columns must be tested in accordance with the
follow...
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59
REFERENCES
Arya, C., 2009. Design of Structural Elements: Concrete, Steelwork, Masonry and Timber
Desig...
Harrison Wallen
60
Ross, C., T., F., Case, J., Chilver, A., 1999. Strength of Materials and Structures, [e-book].
Availabl...
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An Investigation into the Treatment of Amplifications of Moments in Timber Beam Columns

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An Investigation into the Treatment of Amplifications of Moments in Timber Beam Columns

  1. 1. AN INVESTIGATION INTO THE TREATMENT OF AMPLIFICATIONS OF MOMENTS IN TIMBER BEAM COLUMNS Harrison Wallen 1st May 2015 Dr Michael Byfield Word Count: 9819 This report is submitted in partial fulfillment of the requirements for the MEng Civil Engineering and Architecture, Faculty of Engineering and the Environment, University of Southampton
  2. 2. Harrison Wallen 2 Declaration I, Harrison Wallen declare that this thesis and the work presented in it are my own and has been generated by me as the result of my own original research. I confirm that: 1. This work was done wholly or mainly while in candidature for a degree at this University; 2. Where any part of this thesis has previously been submitted for any other qualification at this University or any other institution, this has been clearly stated; 3. Where I have consulted the published work of others, this is always clearly attributed; 4. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work; 5. I have acknowledged all main sources of help; 6. Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself; 7. None of this work has been published before submission.
  3. 3. Harrison Wallen 3 Acknowledgments I would like to thank Dr Michael Byfield for his support and guidance throughout the duration of this work.
  4. 4. Harrison Wallen 4 CONTENTS CONTENTS.........................................................................................................................4 SYMBOLS...........................................................................................................................5 ABSTRACT.........................................................................................................................7 1 INTRODUCTION ............................................................................................................8 1.1 The Amplification of Moments Factor...................................................................................8 2 ELASTIC SECOND ORDER ANLYSIS OF BEAM-COLUMNS ...............................10 2.1 The Gordon Rankine Method...............................................................................................10 2.2 The Secant Formula .............................................................................................................10 2.2.1 Comparison of exact and approximate formulae for deflections..................................13 2.3 British Standard Design Guidance.......................................................................................14 2.4 Eurocode Design Guidance..................................................................................................15 2.4.1 Method 1.......................................................................................................................15 2.4.2 Method 2.......................................................................................................................16 2.5 Conclusion............................................................................................................................18 3 DESIGN EQUATIONS COMPARISON – A PREREQUESIT TO THE EXPERIMENTS................................................................................................................20 3.1 Effect of Eccentricity on the Maximum Design Load .........................................................21 3.2 Conclusion............................................................................................................................22 4 EXPERIMENT METHODOLOGY ...............................................................................23 4.1 Measurement of Material Properties....................................................................................25 4.1.2 Densities of Samples and their Moisture Content.........................................................25 4.2 Measured Values of Bending Stress, Compressive Stress and Modulus of Elasticity.....26 5 COMPARISON WITH EXPERIMENTAL RESULTS.................................................29 5.1 Material Properties...............................................................................................................29 5.2 Code Comparisons with Results ..........................................................................................30 5.2.1 British Standards Comparison ......................................................................................30 5.2.2 Eurocode Comparison...................................................................................................31 5.2.3 Conclusion ....................................................................................................................31 5.3 Comparison of the Approximation Model and Experimental Data......................................32 5.4 Reasons for the Extra Strength Observed ............................................................................33 5.5 General Method Comparisons Using Design Equations and Standard Material Properties 35 6 CONCLUSIONS AND RECOMMENDATIONS .........................................................38 6.1 Conclusions..........................................................................................................................38 6.2 Recommendations for Future Work.....................................................................................39 APPENDIX A....................................................................................................................40 A1: Derivation of the Moment Amplification Factor (Adapted from, Timoshenko and Gere, 2009: 31-32),..............................................................................................................................40 A2: Derivation of the first order maximum deflection for a beam with two equal end moments ....................................................................................................................................................41 APPENDIX B ....................................................................................................................43 B1 - Derivation of N for the Gordon Rankine Method..............................................................43 B2 - Derivation of N for the British Standards Method.............................................................43 B3 - Derivation of N for the Eurocode Method .........................................................................44 B4 – Eccentricity Effects on Different Lengths.........................................................................45 APPENDIX C ....................................................................................................................46 C1 - British Standard Results Comparison ................................................................................46 C2 - Eurocode Results Comparison ...........................................................................................48
  5. 5. Harrison Wallen 5 C3 – Effect of Increasing the Elastic Modulus...........................................................................50 C4 – Method Comparison using Standard Material Properties and Factors...............................50 C5 – Comparison of Results for Different Section Sizes and Eccentricity ................................51 C6 – Determination of E for the Gordon Rankine Method ........................................................54 APPENDIX D....................................................................................................................56 D1 – Risk Assessment ................................................................................................................56 D2 – Method Statement..............................................................................................................57 REFERENCES ..................................................................................................................59 SYMBOLS α Amplification of moments factor 𝜂 Brittle/ductile material constant 𝝀 Slenderness 𝜎 Applied stress 𝜎c,0,d Eurocode design applied compressive stress 𝜎c,a,|| British Standard applied compression stress 𝜎c,adm,|| British Standard permissible compression stress (including K12) 𝜎𝑐,𝑒𝑥𝑝 Experimental measured compressive stress 𝜎e Euler critical stress 𝜎m,a,|| British Standard applied bending stress 𝜎m,adm,|| British Standard permissible bending stress 𝜎m,exp Experimental measured bending stress 𝜎m,y,d Eurocode design applied bending stress in the y axis 𝜎m,z,d Eurocode design applied bending stress in the z axis 𝜎y Maximum compressive stress a First order central deflection A Cross-sectional area b Width D Depth e Eccentricity E Elastic Modulus fc,0,d Eurocode design permissible compressive stress fm,y,d Eurocode design bending stress in the y axis fm,z,d Eurocode design bending stress in the z axis F Axial load Fu Axial load resistance I Second moment of area kc,y Eurocode instability factor in the y axis kc,z Eurocode instability factor in the z axis km Eurocode adjustment factor kmod Eurocode load duration and moisture content factor K2 British Standard modification factor for moisture content K3 British Standard modification factor for load duration K5 British Standard modification factor for notching K7 British Standard modification factor for depth K8 British Standard modification factor for member interactions K12 British Standard modification factor for compression members L Length Lcr Effective Length M Applied moment
  6. 6. Harrison Wallen 6 Mc Moment resistance Mel Elastic Moment resistance N British Standards reduction factor for the Perry Robertson formula Nb Buckling capacity, Gordon Rankine NBS 5268,m British Standard design load using measured material properties and no modification factors NBS 5268,s British Standard design load using standard material properties and modification factors Ncr Euler critical buckling load Nexp Experimental critical design load NED Applied axial load NEU 1995,m Eurocode design load using measured material properties and no modification factors NEU 1995,s Eurocode design load using standard material properties and modification factors Npl Plastic load capacity P Axial load Pc Axial load resistance Pcr Euler critical buckling load PED Applied axial load yo Central deflection yx Deflection at any point x yI First order deflection at any point x yII Second order deflection at any point x Z Section modulus
  7. 7. Harrison Wallen 7 ABSTRACT The presence of second order effects induced in beam-columns through load eccentricity intrinsically causes greater deflections and therefore lower acceptable loads for the design of timber columns. The purpose of this project is to investigate how theoretical and semi-empirical methods treat the amplification of moments due to secondary effects on natural timber beam- columns subject to these eccentric axial loads. A literature review has been carried out to determine how numerous methods model and apply factors to design calculations in order to treat second order effects. Experiments were carried out to determine the ultimate axial load of graded timber columns using a number of different slenderness ratios and load eccentricities. Tests were also carried to measure the bending strength, crushing strength, density, and elastic modulus of the grades used. Results show that methods presented by the British Standard and Eurocode guidance produce overly conservative design loads for beam-columns with a high slenderness. The Eurocode equations provide the best fit with general experimental trends across slenderness ratios and any under-estimation in strength is likely to come from the use of standardised values of material properties as opposed to the underlying mathematical model being implemented.
  8. 8. Harrison Wallen 8 1 INTRODUCTION When analysing beam-columns it is important to include a factor, 𝛼 in order to account for the amplification of moments in members that experience an axial load combined with a bending moment. As the column deflects, the external axial force on the member creates an additional moment, and hence increased deflection, over the standard value calculated through a first order analysis. Therefore a second order analysis is required to reflect this phenomenon. Tests need to be carried out in order to observe the effects of combined axial and compression on timber columns and compared with the results obtained using common construction standards and previous research findings on the subject to analyse the second order effects. A literature review has been carried out with research relevant to the buckling of timber beam-columns to determine whether there is any current precedence on the use of an amplification of moments factor for timber. It will compare findings on how the factor can be obtained and how it is currently applied in timber design standards. 1.1 The Amplification of Moments Factor When the design of members subject to combined axial compression and bending moments are being investigated second order analyses suggest that the effect of secondary moments caused by eccentric loading during deformation should be considered. The amplification of moments factor takes the form, 𝛼 = 1 1 − 𝑃𝐸𝐷/𝑃𝑐𝑟 where, Eq. 1 𝑃𝐸𝐷 = the applied axial force 𝑃𝑐𝑟 = 𝜋2 𝐸𝐼 𝐿 𝑐𝑟 2 where, 𝐿 𝑐𝑟 = the effective length of the column It can be seen that this relationship between the applied axial force and the buckling capacity of the column reflects the susceptibility for more slender members to experience a greater amount of deflection and hence a greater α value. This is due to the ratio of applied force and Euler buckling capacity increasing with an increase in slenderness. The derivation for the amplification of moments factor originates from a second order analysis of a centrally compressed beam or column and the simple relationship between moment and deflection curvature (Appendix A1). Hence from this it could be argued that the factor should apply to all columns irrespective of its material. Some codes do not appear to apply this factor in its current form when considering timber beam-columns however. This factor only stands accurate for members that are compressed along their axis alone to produce the second order
  9. 9. Harrison Wallen 9 deflections. For members that are compressed centrically as well as having point loads or end moments this factor is an approximation, refer to Chapter 2.2. Timoshenko and Gere (2009: 15) provide extensive calculations for a series of members induced by other load conditions, such as distributed and eccentric loading, as well as axial compression in order to derive an exact amplification factor for each case. They state that, “For values of P/Pcr less than 0.6, the error in the approximate expression is less than 2 per cent.” Therefore as the ratio of P/Pcr increases, so does the error - which stands true as increasing the applied load towards the critical buckling load will cause the deflection to tend towards infinity due to Pcr being derived from a sine curve. Annex I of British Standards Institution (2000: BS 5950-1:2000) shows the use of the amplification of moments factor for steel members acting as stocky beam-columns however it is not explicitly written as a factor but is instead written directly into the equations as a function of load applied and its compressional resistance. The Steel Construction Institute (2003: 515) confirms the importance of the amplification of moments factor in the design of beam-columns where an increase in slenderness causes an increase in the maximum moment acting on the member, Figure 1. Figure 1– Effect of the slenderness of a beam-column on the interaction equation resulting from combined axial compression and bending The Steel Construction Institute (2003: 515). Without the use of an amplification of moments factor in the design of steel beam-columns an unsafe design will result - increasingly with more slender columns.
  10. 10. Harrison Wallen 10 2 ELASTIC SECOND ORDER ANLYSIS OF BEAM-COLUMNS 2.1 The Gordon Rankine Method The Gordon Rankine method for the buckling of columns can be used within a second order analysis of beam-columns. The inclusion of the alpha factor takes into account the second order effects and the predictions obtained using this method are more accurate than the basic Euler critical buckling load. “Predictions of buckling loads by the Euler formula is only reasonable for very long and slender struts that have very small geometrical imperfections. In practice, however, most struts suffer plastic knockdown and the experimentally obtained buckling loads are much less than the Euler predictions. For struts in this category, a suitable formula is the Rankine Gordon formula which is a semi-empirical formula, and takes into account the crushing strength of the material, its Young's modulus and its slenderness ratio”.(Ross, Case, Chilver, 1999). The method uses the formula, 1 𝑁𝑏 = 1 𝑁𝑝𝑙 + 1 𝑁𝑐𝑟 which takes into account failures in both compression and bending. As the slenderness becomes increasingly small then Nb will tend towards Npl with failure in crushing and for increasingly large slenderness Nb will tend towards Ncr with failure in bending. This allows the equation to be a more robust solution for predicting the failure of columns with both low and high slenderness. The value of the buckling capacity is then used in the combination formula for the failure of beam-columns in uniaxial bending, 𝑁𝐸𝐷 𝑁𝑏 + 𝛼𝑀 𝑀𝑒𝑙 ≤ 1 This formula takes into consideration the combination of compressional and bending failure and can be used for members with an eccentric axial load causing uniaxial end moments on the strut. 2.2 The Secant Formula Through the use of the engineers bending formula and member equilibrium a formula can be derived defining an approximated curve of the deformed shape and a function to calculate the deflection at any point along the member for a column subject to eccentric rather than centric loads (Figure 2).
  11. 11. Harrison Wallen 11 Figure 2 - Deflection and interaction diagrams for an eccentrically loaded column. By considering moment equilibrium the engineers bending formula becomes, EI 𝛿2 𝑦 𝛿𝑥2 = −𝑃𝑦𝑜 − 𝑃𝑒 which is a second order inhomogeneous linear equation with complex roots. Setting P/EI = k2 the general solution becomes, 𝑦𝑔 = 𝑐1 𝑒 𝛼𝑥 cos(𝛽𝑥) + 𝑐2 𝑒 𝛼𝑥 sin(𝛽𝑥) where, 𝛼 = 0 and 𝛽 = 𝑘 giving, 𝑦𝑔 = 𝑐1 cos(𝑘𝑥) + 𝑐2sin(𝑘𝑥) with a particular solution, yp = - e 𝑦𝑜 = 𝑐1 cos(𝑘𝑥) + 𝑐2 sin(𝑘) − 𝑒 applying boundary conditions for the deflection of the column at any point x, yx, and rearranging gives, 𝑦𝑥 = 𝑒 [𝑐𝑜𝑠 (√ 𝑃 𝐸𝐼 𝑥) + 𝑡𝑎𝑛 (√ 𝑃 𝐸𝐼 𝐿 2 ) 𝑠𝑖𝑛 (√ 𝑃 𝐸𝐼 𝑥) − 1] Eq.2 The secant formula can find the maximum deflection, yo, by setting x=L/2 resulting in, 𝑦𝑜 = 𝑒 [𝑠𝑒𝑐 (√ 𝑃 𝐸𝐼 𝐿 2 ) − 1] Eq.3 (Beer, Johnston and DeWolf, 2004: 625-627)
  12. 12. Harrison Wallen 12 As this equation is a function of a sec curve, asymptotes at π/2, π, 3/2π etc. bound the solution below loads less than the Euler buckling load Pcrit as Figure 3 shows. Figure 3 - Load-deflection diagram formulated with the secant formula for columns with loads of varying eccentricity Gere (2001: 768). The secant formula is an accurate second order analysis for the elastic buckling of columns however it does not account for imperfections found in real situations. As timber has many in the form of material inhomogeneity, initial curvature and accidental load eccentricities, plotted graphs of experimental results may not fit directly with the secant curve. The Timber Engineering Company (1956: 542) suggest that the secant formula is the most accurate formula for calculating the critical load for long columns with eccentric loading and that any inaccuracies in other common design formula used to estimate such design loads are small enough to be acceptable in most engineering situations. This second order analysis, unlike the centrically loaded column formula (Appendix A), will not produce a simple amplification of moments factor to account for secondary moments as shown in Chapter 1.1; instead the amplification is modelled directly into the shape and nature of the sec curve and by rearranging Eq.2 the exact amplification of moments factor can be found (Eq.4), (Timoshenko and Gere, 2009: 13-14). 𝑦𝑥 = 𝑒 𝑐𝑜𝑠 (𝑘 𝐿 2) [𝑐𝑜𝑠 (𝑘 𝑙 2 − 𝑘𝑥) − 𝑐𝑜𝑠 (𝑘 𝑙 2 )] setting u=kl/2, 𝑦𝑥 = 𝑀𝑙2 8𝐸𝐼 2 𝑢2 cos(𝑢) [cos (𝑢 − 2𝑢𝑥 𝑙 ) − cos(𝑢)] Eq.4 or for central deflections, 𝑦0 = 𝑀𝑙2 8𝐸𝐼 2(1 − cos 𝑢) 𝑢2cos(𝑢) Eq.5
  13. 13. Harrison Wallen 13 Eq.4 has two parts; the first order analysis for a pin-ended beam with two equal end moments, and a function of x and u which describes the exact amplification of moments factor along any point of the beam-column, x (Timoshenko and Gere, 2009: 14-15). The first order part is also detailed by Calvert and Farrar (2008: 13-2) and derived in Appendix A2. Timoshenko and Gere (2009: 15) state that the exact amplification of moments equation can be replaced with the approximation, α as detailed in Chapter 1.1. The same mathematical analysis can be completed for beam-columns with different loading and end conditions resulting in varying functions for the exact amplification factor, all of which can be replaced by the approximation, α to a good degree of accuracy. Reece (1949: 180-181) explains that the maximum amplification factor for any type of loading occurs for a member with two equal end moments and that the factor is equal to sec(u) where similarly u=kl/2. This fits with Eq.3 and Eq.5 for the maximum deflection at mid-span. Reece (1949) produced a table of values for two factors K11 and K12, which describe the amplification factor. 2.2.1 Comparison of exact and approximate formulae for deflections Figure 4 – Graph showing the general comparison between exact and approximate formulae for the central deflections of columns under eccentric axial loading against a direct correlation guide line. Figure 4 was created using the formulae derived by Timoshenko and Gere (2009: 14-15): Eq.5 as the exact model and its corresponding approximation – a combination of the first order formula and α. It shows that for observable deflections that would generally form before complete failure of the member there is only a slight difference between the exact formula and any approximation
  14. 14. Harrison Wallen 14 using the amplification of moment’s factor. The data points were calculated using a value of elastic modulus of 7200 N/mm2 with values of slenderness ranging between 42.7 and 189.0 and eccentricities, e, of 16.67 % and 25 %. It is therefore justifiable to say that for these eccentricities and a large range of slenderness ratios the use of the amplification factor is a very good approximation. It also shows that for a column with end moments the exact mathematical model is more conservative than the approximation, but only mildly so. The values of the amplification of moments factor derived for Figure 4 ranged from 1 to 5.3. Deflections within the range of 100 mm and 140 mm are not unreasonable for large structures, and although they would have reached their limit for serviceability state the large values of α associated with these deflections show how influential it becomes in the combination equations as the member reaches its capacity. 2.3 British Standard Design Guidance The British Standards detail the design of timber columns in BS 5268-2:2002 in terms of permissible stress. For members subject to axial compression and bending the code states that the incident combinations should be proportioned such that, σm,a,|| σm,adm,|| (1- 1.5σc,a,|| σe ×K12) + σc,a,|| σc,adm,|| ≤1 where, σm,a,|| is the applied bending stress σm,adm,|| is the permissible bending stress σc,a,|| is the applied compression stress σc,adm,|| is the permissible compression stress (including K12) σe is the Euler critical stress K12 is the modification factor for compression members (British Standards Institution, 2002: BS 5268-2:2002: cl.2.11.6) If we consider the modification factor for compression members, K12, it can be seen that the contribution towards bending has been multiplied by an amplification of moments factor. K12 is derived from the Perry Robertson formula (Arya, 2009: 285) and takes into account the imperfections in the straightness of a column and the tendency to have accidentally applied eccentricity of the axial loads applied. The Perry Robertson formula is as follows including a British reduction factor N, Nσ= 𝜎 𝑦 + (𝜂 + 1)𝜎𝑒 2 − √[( 𝜎 𝑦 + (𝜂 + 1)𝜎𝑒 2 ) 2 − 𝜎 𝑦 𝜎𝑒] (Hearn, 1997: 29) by dividing through by the maximum compressive stress, σy, the equation becomes,
  15. 15. Harrison Wallen 15 Nσ σy = 1 2 + (η+1)σe 2σy -√[( 1 2 + (η+1)σe 2σy ) 2 - σe σy ] which can be written as, 𝑁𝜎 𝜎 𝑦 = 1 2 + (η+1)𝜋2 𝐸 2𝜆2σy -√[( 1 2 + (η+1)𝜋2𝐸 2𝜆2σy ) 2 - 𝜋2 𝐸 𝜆2σy ] with the British standards in mind we need to multiply the maximum compressive stresses, σy, by the modification factor, N, 𝜎 𝜎 𝑦 = 𝐾12= 1 2 + (η+1)𝜋2 𝐸 2𝑁𝜆2σy -√[( 1 2 + (η+1)𝜋2𝐸 2N𝜆2σy ) 2 - 𝜋2 𝐸 𝑁𝜆2σy ] Eq.6 giving K12 as shown in Annex B of British Standards Institution (2002: BS 5268-2:2002: Annex B), for members with a slenderness ratio greater than or equal to 5. It can be seen from Eq.6 that the actual applied stress, σ, is equal to the maximum applied stress, σy, multiplied by K12. The reduction factor N is taken as 1.5 (British Standards Institution, 2002: BS 5268-2:2002: Annex B). By multiplying the maximum applied compressive stress with this factor and the factor K12, to give actual applied compressive stresses, the original amplification factor of 1/(1-σc/σe) simply becomes 1/(1-[1.5K12σc]/σe) in the British Standard method for designing timber columns under combined axial and bending effects. It can be seen that the British Standard takes into account the amplification of moments experienced by the beam-column as well as taking into account any imperfections in the timber. The amplification factor has been derived directly through the second order analysis of a centrically loaded column (Appendix A) with additional modification factors N and K12. This modification factor is not the same as K12 described in Chapter 2.2 founded by Reece (1949). 2.4 Eurocode Design Guidance 2.4.1 Method 1 The European Committee for Standardisation (2004: BS EN 1995-1-1:2004+A1:2008) offers two alternative methods for the design of timber beam-columns. Cl.6.2.4 lays out the first of these where the following expressions should be satisfied, ( σc,0,d fc,0,d ) 2 + σm,y,d fm,y,d +km σm,z,d fm,z,d ≤1 ( σc,0,d fc,0,d ) 2 +km σm,y,d fm,y,d + σm,z,d fm,z,d ≤1 The codes require that the engineer consider the applied bending stress as a second order stress analysis derived from first principles, through the use of an amplification of moments factor, Theiler, Frangi and Steiger (2013: 1104) confirm this with the following expression,
  16. 16. Harrison Wallen 16 𝑀|| = 𝑀| 𝜇 where || and | denote first order and second order moment analyses respectively and μ is the amplification of moments factor denoted in this text as α. The code is not based on a permissible stress design calculation like the British Standard guidance but instead applies factors to combination effects rather than material properties. “The factor km makes allowance for re-distribution of stresses and the effect of inhomogeneity of the material in a cross-section.”, European Committee for Standardisation (2004: BS EN 1995-1- 1:2004+A1:2008: cl.6.1.6). Porteous and Kermani (2007: 165-166) explain that this factor km is applied to the ratio of moments about either the major or minor axis with Figure 5 showing the effect the factor km has on the failure criteria. Figure 5 - Axial force-moment interaction curve for bi-axial bending when either λrel,y or λrel,z > 0.3 , and with the factor km applied to the ratio of moments about the minor axis. (Porteous and Kermani, 2007: 166). As the diagram shows, the Eurocode has chosen to include a factor to increase the resistance against buckling. The failure criterion is increased when setting Mz/Muy to 1 giving N/Nu equal to (1-km) rather than equal to zero. 2.4.2 Method 2 Cl.6.3.2(3) - if the relative slenderness, λrel, is greater than 0.3 then the more stringent of the following conditions must be met in order to resist both axial compression and bending, otherwise it is required to refer to method one, σc,0,d kc,yfc,0,d + σm,y,d fm,y,d +km σm,z,d fm,z,d ≤ 1 σc,0,d kc,zfc,0,d +km σm,y,d fm,y,d + σm,z,d fm,z,d ≤ 1 These equations account for bending in the major, y, axis and the minor, z, axis whereas the British standards method only accounts for bending in one axis as is the case for most circumstances, i.e. bending will occur in the weakest axis. This consideration of bi-axial bending
  17. 17. Harrison Wallen 17 means the equations only remain safe if the column is restrained against lateral torsional buckling (Porteous and Kermani, 2007: 166). With a relative slenderness equal to or greater than 0.3 buckling can occur over pure compressive failure. Eurocode 5 accounts for secondary moments and deflections arising from combined axial compression and bending in cl.6.3.1.(1) which states, “The bending stresses due to initial curvature, eccentricities and induced deflection shall be taken into account, in addition to those due to any lateral load.” Therefore for method two the Eurocode does not expect the engineer to manually include the second order relationship as derived from first principles by modifying σm,y,d and σm,z,d before substituting the value into the design equation unlike with method one. The instability factors, kc,y and kc,z, account for instability within the strut and are calculated similarly to the lateral torsional buckling modification factors, χLT used in Eurocode 3. Theiler, Frangi and Steiger (2013: 1104) state that these factors consider “the P-delta effect, the variability of the strength and the stiffness properties within the timber members, the geometric imperfection of the timber members and the non-linear material behavior of timber when subjected to compression parallel to the grain and bending.” Where the P-delta effect refers to the secondary effects caused by the eccentric axial load. Investigations carried out by Prof. Hans Blaß derived this factor for use in the Eurocode and from this formulated the equations presented in the code (Blaß, 1987). This paper derived the second order analysis by plotting the approximations for moment-normal force-interaction of eccentrically loaded columns, as shown in Figure 6. From this the design method utilises a simple function only relying on the characteristic compressive strength parallel to the grain and the modulus of elasticity, Blaß (1987: 6-7), and also states that “With this method a more economic design of columns is possible’. Figure 6 - Approximations for eccentrically loaded columns as basis for a new column design proposal. (Blaß, 1987: 15).
  18. 18. Harrison Wallen 18 These investigations were based on stochastic and mechanical models in order to determine the ultimate loads for timber columns. TRADA (1991: 1.77) states that “To determine the characteristic ultimate loads of timber columns, a Monte Carlo simulation technique is used. For this purpose, a large number of columns is modelled … For different timber grades, slenderness ratios and end eccentricities, samples of columns are modelled.” This work conducted by H. J. Blaß introduced plastic theory into timber column design, which allowed for greater ultimate loads over simpler second order elastic theories. A statistical analysis was conducted on these simulations in order to determine the member resistance probability distribution, the work answered issues laid out by H. J. Blaß who stated that, “If the lower 5th -percentile of each of the parameters is used to calculate the characteristic strength of the member, the resulting value of the member strength will in general be lower than its 5th -percentile value.”, (TRADA, 1991: 1.75). 2.5 Conclusion There are numerous first and second order methods for designing columns ranging from simple theoretical equations derived from mechanics to semi-empirical methods taking into account the imperfections of timber through experimental observations of varying contributions to the final design equation as well as statistical analyses. Complex theoretical methods such as those detailed by Timoshenko and Gere (2009) can be simplified with the substitution of the amplification of moments factor. Semi-empirical methods such as the Gordon Rankine and British Standards methods also utilise this factor however it is not a simplification but instead a safe addition of taking into account second order effects within the main combination formulae. The Eurocode methods have taken a different stance by heavily relying on a Monte Carlo Simulation by Blaß (1987), this work has informed the complex equations used to determine the design loads meaning a more efficient result can be achieved. Unlike steel, timber is a very inhomogeneous material with many defects present and a microstructure that is observably very random and sometimes non-isotropic. This means that timber can be very unpredictable and design equations such as those from Calvert and Farrah (2008), and arguably the Gordon Rankine equation, do not have sufficient test data in-built within them to provide neither safe nor efficient results. Although both code methods are accurate to an extent, the material properties provided by the British Standards and the Eurocodes, along with how these are used and factored, are vital to providing safe and efficient designs. Whereas the British Standard’s design equation is a permissible stress method the Eurocode applies its modification factors to the final design load as well as each combination effect. This may allow for a more efficient design through applying the computational simulation observations to overall effects as opposed to material properties that can vary greatly. Two columns of the same grade may have very similar material properties but an imperfection in one will not affect the microscopic crushing or bending strength of each fiber, instead it will undermine the global
  19. 19. Harrison Wallen 19 stability of the member. It is apparent that reports on a statistical analysis for numerous timbers reacting to loading, as Blaß (1987) conducted, is required. This provided factors that directly account for changes in global stability as opposed to the material imperfections in permissible stress design, of which there will always be many. TRADA (1991: 1.79) concluded that, “(results) exceed the values based on elastic theory especially in the medium slenderness range between λ = 40 and λ = 100.” Comparisons with the Gordon Rankine and British Standard methods should confirm this when compared against experimental testing.
  20. 20. Harrison Wallen 20 3 DESIGN EQUATIONS COMPARISON – A PREREQUESIT TO THE EXPERIMENTS Using calculations detailed in Appendix B1 to B3, design loads for, BS 5268-2-2002, BS EN 1995-1-1:2004 and the Gordon Rankine method were determined in order to find the most efficient solution. The effects of changing the slenderness of the column and the degree of eccentricity of the axial load applied to it could also be compared and evaluated for compiling a test regime. The maximum axial loads for a square cross-section of 100 mm x 100 mm, grade C16 timber, were calculated for lengths of 1000 mm and 2400 mm with a variance of eccentricity from 0b to 1/4b. For the British Standard method values of K2, K3, K5 and K8 were taken as 1, for service class one, long term loading, no notching, and no member interactions respectively. The value of K7 was taken as 1.079 for lengths of 1000 mm and 2400 mm and values of K12 were calculated as 0.89 and 0.69 respectively. The change in K12 shows how much of an effect the slenderness of the member has on its value and the combination formula, dramatically reducing the effect of crushing on the overall failure criterion. Material properties were taken from British Standards Institution (2002: Table 8) for the British Standard method as well as the Gordon Rankine method. For the Eurocode method material properties were taken from the European Committee for Standardisation (2003: Table 1). The reduction factor, km was taken as 0.7 (cl.6.1.6(2)) and kmod was taken as 0.6 for permanent action loading (Table 3.1). A final load factor, assuming dead loading, of 1.35 was applied to the results. Table 1 – Comparison of design loads derived from three different design equations for varying eccentricity and slenderness. Method Design load (KN) Length = 1000mm Length = 2400 e = 0 e = b/6 e = b/4 e = 0 e = b/6 e = b/4 BS 5268-2-2002 135.9 64.9 51.7 105.8 54.3 44.4 BS EU 1995-1-1:2004 127.3 73.9 61 91.1 60 51.3 Gordon Rankine 143.9 66.9 53.0 112.1 55.9 45.5 Table 1 gives a good indication as to which method is the most efficient with the Eurocode calculations giving the greatest values for members with eccentric loading. This quickly indicates that the Eurocode can be the most efficient method for both low and high slenderness columns. Unlike the British Standards guidance for timber, the Eurocode does not design with a serviceability case, therefore any loads that are specified to be applied to the timber members require the addition of factors of safety such as those for dead and imposed loading and also for varying load combinations – it is important to note that in practice the maximum design loads could reduce even further under Eurocode guidance.
  21. 21. Harrison Wallen 21 It can also be seen that, as expected, increasing the eccentricity of axial load and the slenderness ratio reduces the design load. This reduction happens more dramatically with the British Standard and Gordon Rankine methods with a decrease of 58 % and 59 % respectively for lengths of 2400 mm from e = 0 to e = b/4. With respect to the Eurocode this decrease is only in the order of 44 % suggesting that the Eurocode is less affected by eccentricity than the other two methods. Also comparing against the two different lengths shows that both the British Standard and Gordon Rankine methods decrease by an average of 17 % whereas Eurocode values decrease by an average of 21 % across the different eccentricities. The Eurocode method is therefore more sensitive to length change than the other two. This would suggest that while the British Standard and Gordon Rankine methods are more diffident on increasing eccentricity the Eurocode design loads will reduce more with an increase in the slenderness of the column. 3.1 Effect of Eccentricity on the Maximum Design Load As a generalisation it can be seen that eccentric loading has a greater effect on less slender columns, with very slender columns having little extra effects being induced on them as eccentricity is increased. Comparing Table 2 and Table 3 it is shown that applying an eccentricity can have a greater effect on the design critical load calculated using the British Standards than the Eurocode with a greater percentage change from zero eccentricity to 50 % eccentricity. Table 2 – Percentage change on the critical load using BS 5268. Length (mm) Critical Load (KN) % Change e = 0 e = 1/2 1000 95.4 24.4 74.4 3000 29.3 15.1 48.4 6000 8.2 6.5 19.8 Table 3 – Percentage change on the critical load using EU 1995. Length (mm) Critical Load (KN) % Change e = 0 e = 1/2 1000 131.0 43.2 67.0 3000 37.2 23.6 36.6 6000 9.9 8.6 13.3 Figure B.1 and Figure B.2 in Appendix B4 illustrate plots describing the broad change in design loads for varying eccentricities and slenderness. In this case results were formulated using very short and instantaneous loading factors for the BS 5268-2:2002 and BS EN 1995-1- 1:2004+A1:2008 respectively - mimicking the loading combination to be applied to any members tested during the experiments.
  22. 22. Harrison Wallen 22 3.2 Conclusion The Eurocode method appears to be the most efficient way of designing timber columns for the lengths and eccentricities observed, except under zero eccentricity. The British Standards and Gordon Rankine methods share very similar equations and the difference in results obtained is most likely to come from the inclusion of the K12 modification factor in BS 5268-2-2002 taking into account any imperfections in straightness of the column. This causes the results from the British Standard to be consistently more conservative than the Gordon Rankine method results. As both slenderness and eccentricity are important factors for the stability of columns, and the fact that they alter the design code values in varying ways it is important to include a wide variety of values during the experiment column testing.
  23. 23. Harrison Wallen 23 4 EXPERIMENT METHODOLOGY This section describes the experimentation testing carried out to assess the accuracy of the code based methods for predicting the strength of timber props, subjected to pure axial compression and also axial compression and end moments due to eccentricity of end bearing. The design of the analysis required testing of a series of samples in order to determine and compare specific material properties and external action effects. C16 and C24 grades were selected to provide useful analyses for commonly used and widely available timbers. A high capacity Instron® column testing machine was used to apply a compressive force to members, for the loads induced the measuring apparatus will remain very accurate. Vertical displacement of the cross-head was measured digitally and lateral displacement was measured using a dial gauge, as shown in Figure 7. Figure 7 - Typical setup for the column test methodology including plate detail providing eccentricity. The props were tested with a range of three different eccentricities using steel plates as illustrated in Figure 8. These eccentricities were set as a ratio of the timber width, b. The eccentricities chosen were zero, b/6 and b/4. The prop lengths of 500mm, 900mm, 1900mm and 2400mm, with 3 different cross-sections. The 500mm length tests were carried out to establish the crushing strengths of the timber and therefore these tests were carried out with the eccentricity set at zero. Table 3 summarises the matrix of tests carried out. The lengths and cross-sectional dimensions of each member were checked and recorded with a tape measure prior to testing to account for tolerance variance during production.
  24. 24. Harrison Wallen 24 Table 3 - Testing matrix describing the number of tests carried out for each sample configuration. Eccentricity as a ratio of width, b Number of samples tested Section size (mm) and grade Section size (mm) and grade Section size (mm) and grade 44 x 94 C24 100 x 100 C16 150 x 150 C16 Length (mm) Length (mm) Length (mm) 2400 1900 900 500 2400 1900 900 500 2400 1900 900 500 0 1 1 1 1 1 1 1 2 1 1 1 2 b/6 1 1 0 0 4 1 0 0 2 1 0 0 b/4 1 1 0 0 2 1 0 0 1 1 0 0 Figure 8 - End plate arrangement. Each specimen was tested individually with displacement controlled loading. The vertical load was applied centrally down the column’s vertical axis by ensuring the column was located under the centre of the top pivot head. Figure 9 - Prop under applied compressive loading with eccentricity = b/4.
  25. 25. Harrison Wallen 25 The column was held in place with a minimal amount of pre-load and the dial gauge was placed in position and fixed. The initial load and vertical displacement was recorded at zero horizontal deformation and the vertical displacement was increased in increments of 0.5 mm or 1 mm depending on the rate to failure at which the member was deforming. At each displacement interval the vertical and horizontal displacements were recorded and the load was allowed to settle to a steady state and then recorded to ensure the true strength of the member was analysed. Figure 9 shows a typical specimen under applied axial load in the column-testing rig. The test was completed as the load carrying capacity of the column hit a plateau – the point at which the member had experienced minimal increase in load with a rapid increase in displacement; at this point the member had reached its structural capacity. 4.1 Measurement of Material Properties 4.1.2 Densities of Samples and their Moisture Content Samples were taken from the timber used during the experiment in order to calculate the average densities of the grade and section sizes used. These were roughly 500mm in length in order to get a good estimate. The samples were weighed on date of the experiment and then dried in warm machine room with measurements taken on two subsequent occasions. After 2 months of drying the samples were assumed to be dry and showed heavy cracking through the grain. From this data the amount of moisture lost can be calculated and hence the initial moisture content of the members can be estimated to a good degree of accuracy (Table 4). Table 4 – Determination of moisture content for samples taken from the timber used during the experiments. Sample Grade Section Size (mm) Density on date of experiment 27/11/14 (Kg/m3 ) Density on 3/12/14 (Kg/m3 ) Density on 2/2/15 (Kg/m3 ) Change in density (Kg/m3 ) Moisture content on date of experiment (Kg/m3 ) A C16 150 x 150 499 463 414 85 17.0 B C16 150 x 150 408 384 364 44 10.8 C C16 100 x 100 511 467 436 75 14.7 D C16 100 x 100 499 434 389 110 22.0 E C24 44 x 94 499 480 469 30 6.0 According to BS 5268-2:2002: cl.1.6.5 except in external uses, timber should be dried before grading and installation, with an allowable range of 12% to 18% (Service class 1 to 2) and be no greater than 20%. The NBS states that the strength of timber is, “affected by its moisture content - increasing as the moisture content reduces and vice versa. For example, the bending and the compression stresses for 'wet' timber, i.e. wood with a moisture content exceeding 20 per cent, are, 80 per cent and 60 per cent respectively of those for 'dry' timber (wood with a moisture content less than 20 per cent)."
  26. 26. Harrison Wallen 26 All but one sample were measured to have moisture content below 20% and by averaging sample C and D a value of 18.35% is obtained. Therefore, for the purpose of this investigation, the timber tested shall be compared with values from service classes 1 and 2. From these results the factor K2, accounting for the moisture content of the timber, was chosen to represent service class 2 timbers for all samples, as recommend by BS 5268-2:2002: cl.1.6.5 for timbers with a moisture content of no more than 20%. K2 was therefore taken as 1 as recommend by BS 5268-2:2002: cl.1.6.5 for service class 2. The duration of loading factor K3 was taken as 1.75 for very short term loading according to BS 5268-2:2002: cl.2.8. Both K5 and K8 were taken as 1 as the props were not notched and acted as individual members. The depth factor K7 was calculated from BS 5268-2:2002 using dimensional parameters. The value of K12 was calculated using the material properties obtained experimentally for the test samples. These factors were used to calculate the design values of permissible bending stress and permissible compressive stress. 4.2 Measured Values of Bending Stress, Compressive Stress and Modulus of Elasticity The compressive strength of each cross-section was measured using the column test described above with length of 500mm and zero eccentricity. Table 5 displays the results. Table 5 – Experimental results and measurements of compressive stress. Dimensions (mm) Grade Length (mm) Eccentricity f(b) No. of repeat tests Nexp Avg. (KN) Area (mm2 ) Measured compressive Stress Avg. (N/mm2 ) Factored compressive Stress Avg. (N/mm2 ) 44 x 94 C24 500 0b 1 95 4136 22.9 13.1 100 x 100 C16 500 0b 2 108 10000 10.8 6.2 150 x 150 C16 500 0b 2 211 22500 9.4 5.4 The measured compressive stresses in Table 5 were calculated using the experimental maximum compressive load before failure and the area of the cross section. A K3 factor, taken from BS 5268-2:2002: cl.2.8, was included in the calculation with a value of 1.75 in order to take into account the very short load duration being applied to the members. This factor reduced the measured compressive stress as shown in Eq.7. The factor K12 was taken as 1 as values for the tested slenderness ratios ranged from 0.79 to 0.95 using standard values of compressive stress from BS 5268-2:2002. 𝑁𝑒𝑥𝑝 = 𝐾12 𝐾3 𝐴𝜎𝑐,𝑒𝑥𝑝 Eq.7 The bending strengths and moduli of elasticity for each cross section were determined experimentally. The beam was placed on pinned supports, with a central point load applied in the form of rectangular blocks, see Figure 10. Vertical displacement was recorded as the independent
  27. 27. Harrison Wallen 27 variable under displacement control and the load was recorded until failure. Figure 11 describes the test apparatus setup. Figure 10 – Beam under a central vertical point load. Figure 11 – Typical setup for the beam bending tests. The modulus of elasticity was calculated from the results using a graph of applied load against deflection. The gradient of each was measured and used to calculate E using the deflection formula for a pinned end member, see Table 6. Table 6 – Results of the beam tests to calculate the modulus of elasticity of each cross-section. Section size (mm) Grade Second moment of area (mm4 ) Span (mm) Gradient on graph Modulus of Elasticity (N/mm2 ) 44 x 94 C24 667275 1740 0.0443 7286 100 x 100 C16 8333333 755 2.0027 2154 150 x 150 C16 42187500 1735 1.8769 4840 The result obtained for a section size of 100 mm x 100 mm is well below the value recommended by both codes of standard used in this report. This could be due to a large defect in
  28. 28. Harrison Wallen 28 the member being tested that was not identified or the fact that the sample was considerably shorter than those used for the other section sizes. The bending strength was found by calculating the moment capacity of the member – see Table 7. A K3 factor, taken from BS 5268-2:2002: cl.2.8, was included in the calculation with a value of 1.75 in order to take into account the very short load duration being applied to the members. This factor reduced the measured bending stress as shown in Eq.8. 𝑁𝑒𝑥𝑝 𝐿 4 = 𝐾12 𝐾3 𝑍𝜎 𝑚,𝑒𝑥𝑝 Eq.8 Table 7 – Results of the beam test to calculate the bending strength of each cross-section. Section size (mm) Grade Z (mm3 ) Span (mm) P (KN) Measured bending strength (N/mm2 ) Factored bending strength (N/mm2 ) 44 x 94 C24 30331 1740 1.6 23 13.1 100 x 100 C16 166667 755 17.8 20 11.4 150 x 150 C16 562500 1735 32.3 25 14.3
  29. 29. Harrison Wallen 29 5 COMPARISON WITH EXPERIMENTAL RESULTS 5.1 Material Properties The material properties obtained from the experimentation on the timber samples are collated in Table 8. Values of density generally compare well against the tabulated values from the British Standard and Eurocode in Table 9 and Table 10 respectively. As the material properties were derived directly from experimental tests, in order to accurately compare them with the British Standard values, they have been factored in accordance with the code. Comparing Table 8 and Table 9 shows the experimental crushing and bending strengths to be much higher than those suggested by the British Standard. Timber is a very inhomogeneous material and it is expected that material properties will differ from member to member however this shows that the timber that was tested will be a lot stronger than the code would suggest. Comparisons with the Eurocode standard material properties show a good comparison with C24 graded samples but lower values of crushing strength were observed in the experimental data and bending strengths were observed higher than suggested by the code. Table 8 – Material properties obtained from experimental results. Section Size and grade (mm) Test Density Avg. (kg/m3 ) Dry Density Avg. (kg/m3 ) Test Moisture Content Avg. (%) Unfactored Crushing Strength Avg. (N/mm2 ) Unfactored bending Strength (N/mm2 ) Factored Crushing Strength Avg. (N/mm2 ) Factored bending Strength (N/mm2 ) Modulus of Elasticity (N/mm2 ) 44 x 94, C24 499 469 6 22.9 23 13.1 13.1 7286 100 x 100, C16 505 413 19 10.8 20 6.2 11.4 2154 150 x 150, C16 454 389 14 9.4 25 5.4 14.3 4840 Table 9 – Standard material properties from BS 5268-2:2002 (British Standards Institution, 2002: Table 8). Grade Average Density (kg/m3 ) Crushing Strength (N/mm2 ) Bending Strength (N/mm2 ) Modulus of Elasticity, Minimum (N/mm2 ) C24 420 7.9 7.5 7200 C16 370 6.8 5.3 5800 Table 10 – Standard material properties from BS EN 338:2009 (European Committee for Standardisation, 2003). Grade Average Density (kg/m3 ) Crushing Strength (N/mm2 ) Bending Strength (N/mm2 ) Modulus of Elasticity, 5% (N/mm2 ) C24 420 21 24 7400 C16 370 17 16 5400
  30. 30. Harrison Wallen 30 5.2 Code Comparisons with Results 5.2.1 British Standards Comparison The factors of safety achieved by the British Standard’s method against the experimental results, see Appendix C1: Table C.3, were plotted against slenderness in order to determine the accuracy of the equation. Material properties derived directly from the experimental investigations were used, detailed in Appendix C1: Table C.2. Figure 12 - Chart showing the relationship between slenderness and NExp/ NBS5268,m for varying eccentricity, e. Experimental results were also compared with the design load, NBS5268,s, calculated using BS 5268-2:2002 with values derived using the standard material properties and modification factors detailed in the specific clauses of the code, Appendix C1: Table C.1 and Figure C.1. The positive correlation tends to show that the British Standard’s method is increasingly conservative for more slender columns. This suggests that more slender members are not designed with such efficiency as stockier columns when using the British Standards. There is also a strong indication that as eccentricity is increased so is the efficiency of the design calculation. As investigated before in Chapter 3 it can be seen that increasing the eccentricity has a dramatic effect on the design load achievable, arguably greater than the effect of increasing slenderness with such a wide opening scatter correlation and the wide variance between data points of different eccentricity.
  31. 31. Harrison Wallen 31 5.2.2 Eurocode Comparison A similar approach was taken using the Eurocode guidance with tabulated data in Appendix C2. Figure 13 shows the Eurocode to be a slightly more efficient design approach for high slenderness columns except for a few outliers at zero eccentric data points. For the graph plotted using the standard code based material properties, Figure C.2 – Appendix C2, the design loads are even more efficient and the correlation is much more direct suggesting that some of the material properties derived experimentally could be slightly ambiguous. Figure 13 also shows that the code can be unsafe for instantaneous loading on low slenderness columns. This may be due to the fact that during the relatively fast crushing the stockier members would have become stronger, well above their usual crushing strength, and as shear was not fully induced the fibres would have compressed to form a stronger microstructure. This only occurred for one data point, at low slenderness. Again a pattern of high slenderness causing less efficiency arises with the Eurocode comparison however, ignoring the outliers, a less steep correlation is seen indicating that the Eurocode is more affected by slenderness as shown in Chapter 3. Figure 13 - Chart showing the relationship between slenderness and NExp/ NEU1995,m for varying eccentricity, e. 5.2.3 Conclusion Both design codes have been previously described to deal with the amplification of moments factor in different ways, from Figure 12 and Figure 13 it would seem that the Eurocodes offer a more efficient method of dealing with high slenderness columns. This may be due to its reliance on plastic analysis and a reduction on the combined effects of using multiple 5th -percentile
  32. 32. Harrison Wallen 32 material properties investigated by Blaβ (1987) and TRADA (1991). Slender columns behave in a different manner to stockier columns, although this is moderately dealt with by the British standards with the simplified formula for α, the factor K12 however appears to drastically reduce the design capacity of increasingly slender columns and therefore new experimental test data in the Eurocodes has observed and acted upon this overly conservative formula. Both graphs have very similar plots with Figure 13 just being shifted down in terms of the factors of safety. This indicates that the basic equations in each guidance material are almost identical when modification and loading factors are ignored. 5.3 Comparison of the Approximation Model and Experimental Data It has been shown in Chapter 2.2.1 that the use of the amplification of moments factor is a good determination for the second order effects of deformation and therefore will be used to compare against the experimental results in Figure 14. Figure 14 – Chart showing the approximate values of deflection for varying slenderness ratios with plotted experimental data for each. The plot shows that for columns with a low value of slenderness the experimental results lie close to the mathematical model but for very slender columns the model becomes increasingly conservative. As the exact mathematical model from Timoshenko and Gere (2009) is more conservative, in these cases, the use of this more complicated formula would not be necessary. λ = 189 λ = 83 λ = 64 λ = 43 λ = 55
  33. 33. Harrison Wallen 33 Both the Gordon Rankine method and the British Standard use this approximation, confirming that they are accurate enough to account for second order deflections however there is inefficiency with the amplification of moments approximation that causes very slender columns to be designed over conservatively. The general curve inherent of the secant curve is reflected well in the experimental data and the approximation formula is good at modelling this, the inefficiencies in high slenderness columns may only arise from material prediction inaccuracies. Figure 15 – Comparison between the design loads obtained using BS 5268-2:2002 and the Rankine method, with a 45 degree direct correlation line. The Gordon Rankine and British Standard methods are very similar mathematical models; only differing through the use of modification factors in the British Standard equation. Figure 15 shows how similar the two methods are. Data points that drop significantly below the direct correlation line arise from values with zero eccentricity as the British Standards produces a more conservative design load for these as shown in Chapter 3. As inaccuracies in both these equations arise from members with high slenderness it could be argued that the inefficiency arises from the bending model within the combination equations, more precisely the bending strengths being used. It may also be due to the values of the elastic modulus used. 5.4 Reasons for the Extra Strength Observed Chapter 2.2.1 showed how the amplification of moments factor increases with load, this is because the load applied is tending towards the Euler buckling load for that column causing α to converge to infinity. For this not to happen the Euler critical buckling load has to increase - the only material property associated with this formula is the modulus of elasticity; if this is increased
  34. 34. Harrison Wallen 34 the column would theoretically not be limited by its value of Pcr in the approximation formula used by the Gordon Rankine method and the British Standards. The values of elastic modulus used in Figure 14 were taken from British Standards Institution (2002) and may be too conservative especially as there is no guidance to apply modification factors to the standard material property in the beam-column equations. By increasing the values of E for C16 and C24 grade timbers to 7540 N/mm2 and 9360 N/mm2 respectively, an increase of 30 %, the approximation formula can predict the deflections more accurately, Figure C.3, Appendix C3. It tends to show that for the two highest slenderness ratios, 189 and 83, the approximation formula remains safe but produces a more efficient result than before using the standard values of elastic modulus. For less slender members the application of a 30 % increase in elastic modulus has caused some results to show an underestimate, and therefore unsafe result. Any modification of the value of elastic modulus may only have to be applied to very slender columns, or a rolling scale formula could be used minimising the effect on the current good efficiency for stocky columns but maximising it for very slender columns. Chapter 5.2 showed how similar the results from the British Standard and Eurocode became when the same measured material properties were used. Therefore if it is justifiable to increase the elastic modulus for the British Standard method using the amplification of moments factor then it would imply that doing the same for the Eurocode would also increase its’ efficiency. Currently both standards use very similar values for the elastic modulus for all grades and neither applies a modification factor directly to the value. As the Eurocode does not use the amplification of moments factor in the form that has been described in Chapter 1.1 the increase in the value of E would increase the factor kc,z or kc,y thus reducing the effect of crushing in the combination equation. Although crushing is not directly related to the elastic modulus as failure in bending is, having a greater value of E reduces the effect crushing has on the failure criterion allowing for a greater effect due to bending in method two of the Eurocode guidance. The value of E in the Gordon Rankine and Eurocode methods was substituted to produce a design load equal to that of the experimental results. It was found that the value of E had to be increased by a factor between 10 and 100. This shows that it is not purely the elastic modulus that is limiting the equations to produce very conservative design loads. A combination of increasing all material properties is required to match experimental results, this requires a more visual approach to matching the method result lines to experimental data points due to the number of material properties involved in the design equations.
  35. 35. Harrison Wallen 35 5.5 General Method Comparisons Using Design Equations and Standard Material Properties By plotting the experimental data against derived design loads, following the exact methods in the codes, observations of efficacy can be seen through trend lines. Figure 16 – Graph showing the relationship between the experimental results and varying design methods including trend lines for each. From Figure 16 it can be seen that the Euroce remains very efficient as a design method until it reaches high values of slenderness (low design loads) and begins to converge toward the British Standard and Gordon Rankine trend lines. Both the Gordon Rankine and British Standard trend lines are consistently relatively far away from the direct correlation line indicating their more conservative approaches. Figure C.4 in Appendix C4 has been created using the same values as Figure 16 but shows the relationship between the design load factor of safety and the slenderness of the column. It conforms to previous observations and shows clearly the effect of slenderness on the increase in the factor of safeties that are achieved. It indicates that the British Standard method is more influenced by slenderness than the Gordon Rankine method which would suggest that modification factors used in the British Standard are not well suited to very slender columns, it does however show that they produce a more efficient design at low slenderness with a cross over point between the British Standard being more conservative than the Gordon Rankine method at a
  36. 36. Harrison Wallen 36 slenderness value of 40. The Eurocode remains consistently more efficient than the other two methods however again it can be seen that it starts to become just as inefficient at a very high values of slenderness. A comparison can be plotted to show all three methods, the Euler buckling load, the crushing load, and experimental results for each section size and eccentricity applied. The plot for a section size of 150 mm x 150 mm with the zero eccentricity is shown in Figure 17, graphs for the two remaining section sizes and eccentricities can be found in Appendix C5, Figures C.5 to C.12. Figure 17 - Comparison between the effects of slenderness on the design load for different calculation methods at e = 0b for a section size of 150 x 150. It can be seen that for members with zero eccentricity the Eurocode guidance in fact becomes more efficient roughly between λ = 37 and λ = 57, this is also true for the other section sizes. This is a result of the plastic analysis carried out by TRADA (1991), which improves the estimation of effects due to compressive stresses, namely low slenderness columns. The figures with an eccentric loads applied, as shown in Appendix C5, also confirm the statement from Chapter 2.5 that the Eurocode has improved on simple elastic theory resulting in improvements in estimations corresponding to values between λ = 40 and λ = 100. Generally the experimental results mimic the general line of the Euler buckling load and the method lines however there is a pattern of high inefficiency shown with the wide gap between experimental results and prediction lines. This would suggest that the theory is predicting a good model for the effect that slenderness and eccentricity has on the ultimate design load of the columns but that they are being limited by material properties or factors being applied to them. A number of figures show experimental results lying outside the bounds of the Euler Critical load and the crushing strength for that particular grade of timber. This confirms that it is not just the elastic modulus that is too low when applying the code guidance to the experimental results.
  37. 37. Harrison Wallen 37 By analysing the method design curves on Figure 17 and the distance the experimental plots lie above them an estimation for the amount material properties should be increased by can be determined. An increase of 20 % for each material property was decided and the results are shown in Figure 18. Figure 18 - Comparison between the effects of slenderness on the design load for different calculation methods at e = 0b for a section size of 150 x 150 with an increase of 20 % for material properties. It can be seen how well the Eurocode method fits the line of experimental data points compared to the Gordon Rankine and British Standard design load lines. With an increase of 20 % on material property values both codes produce a much more efficient design load. The same modification can be applied to the other section sizes and eccentricities with only a few results producing unsafe estimations and the majority producing a much more efficient design load. Understandably timber, although graded, will always have varying properties across a selection of members and it is important to design to the weakest possible member. The use of the 5th -percentile accounts for this, hence why code guidance suggests such seemingly low values for the material properties. The new Eurocode guidance chose to keep the idea of grading timbers similar to that of the British Standard providing many advantages to trade and design however admit that, “There is however one disadvantage to the use of strength classes, which is that some grades and species may be assigned lower strength properties than their true characteristic values.” (TRADA, 1991: 1.123). This is a limitation to the use of timber in design, unlike steel which has very similar and predictable material properties across samples, unless manufactured, natural timber will always present challenges for designers to design to a safe and efficient standard.
  38. 38. Harrison Wallen 38 6 CONCLUSIONS AND RECOMMENDATIONS 6.1 Conclusions The amplification of moments factor is required in some form in order to account for the very prominent second order effects experienced by beam-columns. The secant formula provides a very good approximation for the deflection curve of a column subject to combined axial compression and equal end moments however it does not account for material imperfections of which there are many for timber. A formula derived as the α factor is a good approximation for the exact method obtained purely through mechanics. The approximation model derived by Timoshenko and Gere (2009: 15) is a bad estimation for high slenderness columns. Semi-empirical methods such as the Gordon Rankine and British Standard methods improve on the mechanical methods by introducing modification factors and material properties however they are also weak estimations for high slenderness columns. The Eurocode method provides more efficient design loads for columns in most cases but also become much less efficient at very high slenderness. On the face of it the British Standard and Gordon Rankine methods are very similar however they produce very different results with the Gordon Rankine method producing less conservative results. The inclusions of modification factors in the British Standard for a permissible stress calculation have caused this reduction in efficiency. The Eurocode on the other hand uses characteristic strengths applying factors to overall combination effects derived from statistical analysis, allowing for an improvement on the efficiency of design. The British Standard method is more affected by eccentricity whereas the Eurocode is more affected by slenderness, however high slenderness values pose a problem for the efficiency of both codes indicating that the Eurocode has resolved the problem of eccentricity, and hence second order moments, to a further extent than the British Standard. Experimental results are best matched in terms of how they change with slenderness by the Eurocode with design load results producing curves that match even better than those of the theoretical exact model derived by Timoshenko and Gere (2009). Investigations conducted by TRADA (1991) for the Eurocode have greatly improved the efficiency of columns within the range of λ = 40 and λ = 100 due to the implementation of more efficient mathematical models that take into account plastic analysis and characteristic strength. Therefore the problem with these design equations lies with the material properties and modification factors used on them rather than the basic equations producing the characteristic curves for each section size and eccentricitys. A rough estimation of an increase in 20 % for every material property produced design load lines that were much closer, and therefore efficient, than using the standard guidance given in the codes for material property values. As a number of members across two different grades were tested it would be reasonable to conclude that in most
  39. 39. Harrison Wallen 39 cases the material properties can be increased for that particular grade of timber. It is hard to tell whether this will be accurate for all members and a much wider sample of members will need to be tested to reach a valid conclusion. It can be concluded however that the advances in analysis methods and modelling have allowed the Eurocode to produce relatively simple yet accurate equations to account for secondary effects on beam-columns, over and above those derived through theoretical assumptions or by the British Standards. The use of manufactured timbers such as CLT and Glulam provide members that are, on a macroscopic level, more homogeneous and therefore should display more coherent material properties across grades. As these are utilised more and more throughout the industry as a sustainable and sometimes more efficient material, especially on relatively high-rise buildings, second order effects should be predicted well through the use of the Eurocode guidance and design loads should match those observed in reality even more accurately. 6.2 Recommendations for Future Work Results displayed in this thesis should be backed up by a wider range of samples in order to validate the results and determine whether samples are consistently stronger that set out in the codes of practice. Results based on longer load durations, and creep should also be considered. Experimentation on manufactured timbers could be carried out to provide a more reliable analysis on the effects of second order moments induced in beam-columns, as material properties should be standardised more efficiently and realistically. Eventually new factors could be brought into the Eurocode to provide more efficient designs for slender columns of particularly high grades through further computational and statistical analysis that is backed up by experimental observations.
  40. 40. Harrison Wallen 40 APPENDIX A A1: Derivation of the Moment Amplification Factor (Adapted from, Timoshenko and Gere, 2009: 31-32), Figure A.1 - Deflection diagram for a beam subject to centric axial compression. First order deflection yI: The moment is given by, M = PyI substituting into the Engineer’s bending formula, EI 𝛿2 𝑦 𝛿𝑥2 = −𝑃𝑦𝐼 Setting P/EI = k2 𝛿2 𝑦 𝛿𝑥2 + 𝑘2 𝑦𝐼 = 0 writing in the form ay”+by’+cy = 0 then b2 – 4ac < 0 so the solution has complex roots with the general solution in the form, 𝑦𝑔 = 𝑐1 𝑒 𝛼𝑥 cos(𝛽𝑥) + 𝑐2 𝑒 𝛼𝑥 sin(𝛽𝑥) where 𝛼 = −𝑏 2𝑎 = 0 and 𝛽 = √4𝑎𝑐−𝑏2 2𝑎 = 𝑘 giving, 𝑦𝑔 = 𝑐1 cos(𝑘𝑥) + 𝑐2sin(𝑘𝑥) applying the boundary c nonditions of yI = 0 when x = 0 gives c1 = 0 𝑦𝑔 = 𝑐2sin(𝑘𝑥) applying the boundary conditions of yI = 0 when x = L results in either c2 = 0 or sin(kL) = 0 for a real solution sin(kx) = 0 and therefore kL = π or k = π/L giving, 𝑦𝐼 = 𝑐2sin( 𝜋 𝐿 𝑥) applying initial conditions yI = ɑ when x = L/2 gives, ɑ = 𝑐2sin ( 𝜋 2 ) or c2 = ɑ therefore a first order solution for the deflection at any point x along the beam is, 𝑦𝐼 = ɑsin ( 𝜋 𝐿 𝑥) Second order deflection yII; The second order deflection and moments can be added directly to the first order values via the principle of super positioning,
  41. 41. Harrison Wallen 41 y = yI + yII and M = -P(yI + yII) following a similar process as before yields, EI 𝛿2 𝑦 𝛿𝑥2 = −𝑃(𝑦𝐼 + 𝑦𝐼𝐼) 𝛿2 𝑦 𝛿𝑥2 + 𝑘2 𝑦𝐼𝐼 = −𝑘2 (ɑ𝑠𝑖𝑛 ( 𝜋𝑥 𝐿 )) taking the homogeneous part of the differential equation will give the same result as the first order analysis, 𝑦𝐼𝐼 = 𝑐2sin ( 𝜋 𝐿 𝑥) and therefore, 𝑦𝐼𝐼 " = − 𝜋2 𝐿2 𝑐2sin ( 𝜋 𝐿 𝑥) by substituting these equations into the original differential equations it follows that, − 𝜋2 𝐿2 𝑐2sin ( 𝜋 𝐿 𝑥) + 𝑘2 𝑐2 𝑠𝑖𝑛 ( 𝜋𝑥 𝐿 ) = −𝑘2 ɑ𝑠𝑖𝑛 ( 𝜋𝑥 𝐿 ) replacing k2 and rearranging to find c2 gives, 𝑐2 = − 𝑃ɑ 𝐸𝐼 𝑃 𝐸𝐼 − 𝜋2 𝐿2 = ɑ 𝜋2 𝐸𝐼 𝑃𝐿2 − 1 = ɑ 𝑃𝑐𝑟 𝑃 − 1 resulting in the second order deflection being, 𝑦𝐼𝐼 = 1 𝑃𝑐𝑟 𝑃 − 1 ɑsin( 𝜋 𝐿 𝑥) or 𝑦𝐼𝐼 = 1 𝑃𝑐𝑟 𝑃 − 1 𝑦𝐼 substituting these equations into the equation for moment, 𝑀 = 𝑃(𝑦𝐼 + 𝑦𝐼𝐼) = 𝑃 (ɑsin ( 𝜋 𝐿 𝑥) + 1 𝑃𝑐𝑟 𝑃 − 1 ɑsin ( 𝜋 𝐿 𝑥)) 𝑀 = 𝑃 (ɑsin( 𝜋 𝐿 𝑥)) ( 1 1 − 𝑃 𝑃𝑐𝑟 ) = 𝑀 𝑥 ( 1 1 − 𝑃 𝑃𝑐𝑟 ) = 𝑀 𝑥 𝛼 showing the applied moment to be amplfied by a factor α of 1/(1-P/Pcr). A2: Derivation of the first order maximum deflection for a beam with two equal end moments Using the Engineer’s bending formula,
  42. 42. Harrison Wallen 42 −𝑀 𝐸𝐼 = 𝛿2 𝑦 𝛿𝑥2 𝐸𝐼 ∫ 𝛿2 𝑦𝛿𝑥 = ∫ −𝑀𝛿𝑥 𝐸𝐼𝛿𝑦 = −𝑀𝑥 + 𝑐1 where, 𝛿y = 0 at x = L/2 , 0 = − 𝑀𝐿 2 + 𝑐1 giving , 𝑐1 = − 𝑀𝐿 2 𝐸𝐼 ∫ 𝛿𝑦𝛿𝑥 = ∫ −𝑀𝑥 + 𝑀𝐿 2 𝐸𝐼𝑦 = − 𝑀𝑥2 2 + 𝑀𝐿 2 𝑥 + 𝑐2 where, y = 0 at x = 0 , giving, c2 = 0 for any point along the beam, x , 𝑦 = − 𝑀𝑥2 2𝐸𝐼 + 𝑀𝐿𝑥 2𝐸𝐼 and setting x = L/2 for the maximum deflection which occurs at the centre of the beam . 𝑦 = − 𝑀𝐿2 8𝐸𝐼 + 𝑀𝐿2 4𝐸𝐼 = 𝑀𝐿2 8𝐸𝐼
  43. 43. Harrison Wallen 43 APPENDIX B B1 - Derivation of N for the Gordon Rankine Method 𝑁 𝑁𝑏 + 𝛼𝑀 𝑀𝑒𝑙 = 1 Where Mel = Zσm,adm,|| 𝑁 𝑁𝑏 + 1 1 − 𝑁 𝑁𝑐𝑟 𝑁𝑒 𝑀𝑒𝑙 = 1 𝑁 𝑁𝑏 + 𝑁𝑐𝑟 𝑁𝑐𝑟 − 𝑁 𝑁𝑒 𝑀𝑒𝑙 = 1 𝑁 𝑁𝑏 + 𝑁𝑐𝑟 𝑁𝑒 𝑀𝑒𝑙(𝑁𝑐𝑟 − 𝑁) = 1 𝑁𝑀𝑒𝑙(𝑁𝑐𝑟 − 𝑁) 𝑁𝑏 + 𝑁𝑐𝑟 𝑁𝑒 = 𝑀𝑒𝑙(𝑁𝑐𝑟 − 𝑁) 𝑁𝑀𝑒𝑙 𝑁𝑐𝑟 − 𝑁2 𝑀𝑒𝑙 + 𝑁𝑐𝑟 𝑁𝑏 𝑁𝑒 − 𝑀𝑒𝑙 𝑁𝑏 𝑁𝑐𝑟 + 𝑁𝑀𝑒𝑙 𝑁𝑏 = 0 −𝑁2 𝑀𝑒𝑙 + 𝑁(𝑁𝑐𝑟 𝑁𝑏 𝑒 + 𝑀𝑒𝑙 𝑁𝑐𝑟 + 𝑀𝑒𝑙 𝑁𝑏) − 𝑀𝑒𝑙 𝑁𝑏 𝑁𝑐𝑟 = 0 which has the solution for N of, 𝑁 = −𝑏 ± √𝑏2 − 4𝑎𝑐 2𝑎 where, 𝑎 = 𝑀𝑒𝑙 𝑏 = 𝑁𝑐𝑟 𝑁𝑏 𝑒 + 𝑀𝑒𝑙 𝑁𝑐𝑟 + 𝑀𝑒𝑙 𝑁𝑏 𝑐 = 𝑀𝑒𝑙 𝑁𝑏 𝑁𝑐𝑟 B2 - Derivation of N for the British Standards Method σm,a,|| σm,adm,|| (1- 1.5σc,a,|| σe ×K12) + σc,a,|| σc,adm,|| =1 𝑁𝑒 𝑍 σm,adm,|| (1- 1.5 𝑁 𝐴 σe ×K12) + 𝑁 𝐴 σc,adm,|| =1 ( 𝑁𝑒 Zσm,adm,|| × 𝑁 (N- 1.5𝑁2 Aσe ×K12) ) + 𝑁 Aσc,adm,|| =1
  44. 44. Harrison Wallen 44 𝑁2 𝑒 NZσm,adm,|| − 1.5𝑁2Zσm,adm,|| Aσe ×K12 + 𝑁 Aσc,adm,|| =1 𝑁𝑒 a − bN + 𝑁 𝑐 = 1 which has the solution for N of, 𝑁 = −√(−𝑎 − 𝑏𝑐 − 𝑒𝐶)2 − 4𝑎𝑏𝑐 + 𝑎 + 𝑏𝑐 + 𝑒𝑐 2𝑏 where, a = Zσm,adm,|| 𝑏 = 1.5Zσm,adm,|| Aσe ×K12 𝑐 = Aσc,adm,|| B3 - Derivation of N for the Eurocode Method For results, only method 2 is required, σc,0,d kc,yfc,0,d +km σm,y,d fm,y,d + σm,z,d fm,z,d = 1 𝑁 Akc,yfc,0,d +km 𝑁𝑒 Zfm,y,d = 1 N ( 1 Akc,yfc,0,d +km 𝑒 Zfm,y,d ) = 1 giving, N = 1 ( 1 Akc,yfc,0,d +km 𝑒 Zfm,y,d ) Dividing N by 1.35 then applied a factor of safety for dead loads to the final result.
  45. 45. Harrison Wallen 45 B4 – Eccentricity Effects on Different Lengths Figure B.1 – Effect of eccentricity on the critical load advised by BS5268 at varying lengths, L. Figure B.2 - Effect of eccentricity on the critical load advised by EU 1995 at varying lengths, L.
  46. 46. Harrison Wallen 46 APPENDIX C C1 - British Standard Results Comparison Table C.1 – Material properties and factors used for NBS 5268,s: with modification factors and standard material properties, for Table C.3. Section size (mm) Grade Crushing Strength (N/mm2 ) Bending Strength (N/mm2 ) Modulus of Elasticity, Minimum (N/mm2 ) K2 K3 K5 K7 K8 K12 44 x 94 C24 7.9 7.5 7200 1 1.75 1 1.136159421 1 Varies 100 x 100 C16 6.8 5.3 5800 1 1.75 1 1.128452643 1 Varies 150 x 150 C16 6.8 5.3 5800 1 1.75 1 1.079228237 1 Varies Table C.2 – Material properties and factors used for NBS 5268,m: without factors, with measured material properties, for Table C.3. Section size (mm) Grade Crushing Strength (N/mm2 ) Bending Strength (N/mm2 ) Modulus of Elasticity (N/mm2 ) K2 K3 K5 K7 K8 K12 44 x 94 C24 22.9 23 7286 1 1 1 1 1 Varies 100 x 100 C16 10.8 20 2154 1 1 1 1 1 Varies 150 x 150 C16 9.4 25 4840 1 1 1 1 1 Varies Table C.3 – Comparison of results with BS 5268. Section size (mm) Grade Length (mm) Eccentricity f(b) Nexp (KN) NBS 5268,m without factors, with measured material properties (KN) NExp/ NBS5268,m NBS 5268,s with factors and standard material propertie s (KN) NExp/ NBS5268,s 44 x 94 C24 900 0b 80 32.3 2.48 26.8 2.98 1/6b / 22.7 / 17.5 / 1/4b / 20.0 / 15.1 / 1900 0b 65 8.2 7.90 7.7 8.41 1/6b 23 7.5 3.06 6.8 3.38 1/4b 17 7.2 2.35 6.4 2.64 2400 0b 36 5.2 6.86 5.0 7.21 1/6b 22 5.0 4.44 4.6 4.78 1/4b 14 4.8 2.90 4.4 3.16 100 x 100 C16 900 0b 132 80.2 1.22 98.5 1.34 1/6b / 51.9 / 48.1 / 1/4b / 45.0 / 38.8 / 1900 0b 130 28.8 1.20 59.3 2.19 1/6b 120 24.4 4.47 34.8 3.45
  47. 47. Harrison Wallen 47 Section size (mm) Grade Length (mm) Eccentricity f(b) Nexp (KN) NBS 5268,m without factors, with measured material properties (KN) NExp/ NBS5268,m NBS 5268,s with factors and standard material propertie s (KN) NExp/ NBS5268,s 1/4b 58 22.8 2.34 29.4 1.97 2400 0b 170 18.7 1.57 42.5 4.00 1/6b 90 16.9 4.93 28.4 3.17 1/4b 60 16.1 3.46 24.6 2.44 150 x 150 C16 900 0b 292 189.3 1.38 239.5 1.22 1/6b / 138.8 / 112.9 / 1/4b / 122.9 / 90.0 / 1900 0b 250 153.9 1.18 193.1 1.29 1/6b 227 113.8 1.70 95.8 2.37 1/4b 194 101.8 1.67 78.1 2.48 2400 0b 220 130.0 1.04 161.6 1.36 1/6b 150 99.1 1.26 85.9 1.75 1/4b 100 89.5 0.96 71.1 1.41 Figure C.1 – Chart showing the relationship between slenderness and NExp/ NBS5268,s for varying eccentricity, e
  48. 48. Harrison Wallen 48 C2 - Eurocode Results Comparison Table C.4 – Material properties and factors used for NEU 1995,s: with factors and standard material properties, for Table C.6 Section size (mm) Grade Crushing Strength (N/mm2 ) Bending Strength (N/mm2 ) Modulus of Elasticity, 5% (N/mm2 ) Kmod Km 44 x 94 C24 21 24 7400 1.1 0.7 100 x 100 C16 17 16 5400 1.1 0.7 150 x 150 C16 17 16 5400 1.1 0.7 Table C.5 – Material properties and factors used for NEU 1995,m: without factors, with measured material properties, for Table C.6. Section size (mm) Grade Crushing Strength (N/mm2 ) Bending Strength (N/mm2 ) Modulus of Elasticity, 5% (N/mm2 ) Kmod Km 44 x 94 C24 22.9 23 7286 1 1 100 x 100 C16 10.8 20 2154 1 1 150 x 150 C16 9.4 25 4840 1 1 Table C.6 – Comparison of results with EU 1995. Section size (mm) Grade Length (mm) Eccentricity f(b) Nexp (KN) NEU 1995,m without factors, with measured material properties (KN) NExp/ NEU1995,m NEU 1995,s with factors and standard material propertie s (KN) NExp/ NEU1995,s 44 x 94 C24 900 0b 80 36.7 2.18 40.0 2.00 1/6b / 24.4 / 30.0 / 1/4b / 20.9 / 26.7 / 1900 0b 65 9.5 6.84 10.6 6.15 1/6b 23 8.4 2.74 9.7 2.37 1/4b 17 8.0 2.14 9.3 1.82 2400 0b 36 6.1 5.95 6.7 5.34 1/6b 22 5.6 3.94 6.4 3.45 1/4b 14 5.4 2.60 6.2 2.25 100 x 100 C16 900 0b 132 72.8 1.81 134.2 0.98 1/6b / 49.4 / 79.2 / 1/4b / 42.6 / 65.7 / 1900 0b 130 32.1 4.05 80.8 1.61 1/6b 120 26.6 4.52 57.0 2.11 1/4b 58 24.5 2.37 49.7 1.17
  49. 49. Harrison Wallen 49 Section size (mm) Grade Length (mm) Eccentricity f(b) Nexp (KN) NEU 1995,m without factors, with measured material properties (KN) NExp/ NEU1995,m NEU 1995,s with factors and standard material propertie s (KN) NExp/ NEU1995,s 2400 0b 170 21.1 8.05 55.5 3.06 1/6b 90 18.6 4.85 43.1 2.09 1/4b 60 17.5 3.43 38.8 1.55 150 x 150 C16 900 0b 292 163.0 1.79 318.4 0.92 1/6b / 118.4 / 183.9 / 1/4b / 104.1 / 151.8 / 1900 0b 250 148.4 1.68 270.1 0.93 1/6b 227 110.5 2.05 166.6 1.36 1/4b 194 98.0 1.98 139.9 1.39 2400 0b 220 136.3 1.61 225.5 0.98 1/6b 150 103.7 1.45 148.5 1.01 1/4b 100 92.6 1.08 126.9 0.79 Figure C.2 - Chart showing the relationship between slenderness and NExp/ NEU1995,s for varying eccentricity, e.
  50. 50. Harrison Wallen 50 C3 – Effect of Increasing the Elastic Modulus Figure C.3 - Chart showing the approximate values of deflection using a greater E for varying slenderness ratios with plotted experimental data for each. C4 – Method Comparison using Standard Material Properties and Factors Figure C.4 - Graph showing the relationship between the Design load factor of safety and slenderness for varying design methods including trend lines for each λ = 189 λ = 83 λ = 64 λ = 43 λ = 55
  51. 51. Harrison Wallen 51 C5 – Comparison of Results for Different Section Sizes and Eccentricity Figure C.5 - Comparison between the effects of slenderness on the design load for different calculation methods at e = 0b for a section size of 44 x 94. Figure C.6 - Comparison between the effects of slenderness on the design load for different calculation methods at e = 1/6b for a section size of 44 x 94
  52. 52. Harrison Wallen 52 Figure C.7 - Comparison between the effects of slenderness on the design load for different calculation methods at e = 1/4b for a section size of 44 x 94 Figure C.8 - Comparison between the effects of slenderness on the design load for different calculation methods at e = 0b for a section size of 100 x 100
  53. 53. Harrison Wallen 53 Figure C.9 - Comparison between the effects of slenderness on the design load for different calculation methods at e = 1/6b for a section size of 100 x 100 Figure C.10 - Comparison between the effects of slenderness on the design load for different calculation methods at e = 1/4b for a section size of 100 x 100
  54. 54. Harrison Wallen 54 Figure C.11 - Comparison between the effects of slenderness on the design load for different calculation methods at e = 1/6b for a section size of 150 x 150 Figure C.12 - Comparison between the effects of slenderness on the design load for different calculation methods at e = 1/4b for a section size of 150 x 150 C6 – Determination of E for the Gordon Rankine Method 𝑁 𝑁𝑏 + 𝛼𝑀 𝑀𝑒𝑙 = 1 Where Mel = Zσm,adm,|| 𝑁 ( 1 𝑁𝑐𝑟 + 1 𝑁𝑝𝑙 ) + 1 1 − 𝑁 𝑁𝑐𝑟 𝑁𝑒 𝑀𝑒𝑙 = 1 𝑀𝑒𝑙 𝑒 ( 1 𝑁𝑐𝑟 + 1 𝑁𝑝𝑙 ) + 1 1 − 𝑁 𝑁𝑐𝑟 = 𝑀𝑒𝑙 𝑁𝑒
  55. 55. Harrison Wallen 55 ( 𝑀𝑒𝑙 𝑒 1 𝑁𝑐𝑟 ) + ( 𝑀𝑒𝑙 𝑒 1 𝑁𝑝𝑙 ) + 1 1 − 𝑁 𝑁𝑐𝑟 = 𝑀𝑒𝑙 𝑁𝑒 𝑀𝑒𝑙 𝑒 𝑁𝑐𝑟 + 𝑁𝑐𝑟 𝑁𝑐𝑟 − 𝑁 = 𝑀𝑒𝑙 𝑁𝑒 − 𝑀𝑒𝑙 𝑁𝑝𝑙 𝑒 which can be written as, 𝐴 𝑁𝑐𝑟 + 𝑁𝑐𝑟 𝑁𝑐𝑟 − 𝐵 = 𝐶 where, 𝐴 = 𝑀𝑒𝑙 𝑒 𝐵 = 𝑁 𝐶 = 𝑀𝑒𝑙 𝑁𝑒 − 𝑀𝑒𝑙 𝑁𝑝𝑙 𝑒 which can be rearranged to, 𝑁𝑐𝑟 2 (1 − 𝐶) + 𝑁𝑐𝑟(𝐴 + 𝐵𝐶) − 𝐴𝐵 = 0 which has the solution, 𝑁𝑐𝑟 = −𝑏 ± √𝑏2 − 4𝑎𝑐 2𝑎 where, 𝑎 = 1 − 𝐶 = 1 − ( 𝑀𝑒𝑙 𝑁𝑒 − 𝑀𝑒𝑙 𝑁𝑝𝑙 𝑒 ) 𝑏 = 𝐴 + 𝐵𝐶 = 𝑀𝑒𝑙 𝑒 + (𝑁 ( 𝑀𝑒𝑙 𝑁𝑒 − 𝑀𝑒𝑙 𝑁𝑝𝑙 𝑒 )) 𝑐 = −𝐴𝐵 = − 𝑀𝑒𝑙 𝑒 𝑁
  56. 56. Harrison Wallen 56 APPENDIX D D1 – Risk Assessment
  57. 57. Harrison Wallen 57 D2 – Method Statement Method Statement: Crushing of timber and blockwork columns to failure in the heavy structures lab Dr Mike Byfield
  58. 58. Harrison Wallen 58 10/11/2014 The timber props and dry laid blockwork columns must be tested in accordance with the following procedure: 1. The area must be coned off appropriately to prevent trip hazards and to isolate the area. 2. The blockwork columns or timber props must be placed in position manually. 3. The scaffold bars must be installed to prevent the ejection of the test pieces during the test. 4. The timber columns must be manually held vertical whilst the cross-head is lowered until the prop is held in place by the reaction with the top plate of the column test machine. 5. The displacement gauges to measure horizontal movement must be placed on scaffold bars. 6. The machine must be set to displacement control during tests. Increments of displacement must not exceed 5mm. 7. When the test is finished the test specimens must be safely disposed of so as to prevent them becoming trip hazards.
  59. 59. Harrison Wallen 59 REFERENCES Arya, C., 2009. Design of Structural Elements: Concrete, Steelwork, Masonry and Timber Designs to British Standards and Eurocodes. 3rd ed. Oxon: Taylor & Francis. Beer, F. P., Johnston, E., R., Jr., DeWolf, J., T., 2004. Mechanics of Materials (In SI Units). 3rd ed. [e-book]. Available through: Google Books <books.google.co.uk> [Accessed 25 November 2014]. Blaß, H., J., 1987. International council for Building research Studies and Documentation: Working Commission W18A – Timber Structures. Design of Timber Structures. CIB-W18A/20- 2-2, pp.1-17. British Standards Institution, 2000. BS 5950-1:2000 Structural use of steelwork in building – Part 1: Code of practice for design – Rolled and welded sections. London: BSI. British Standards Institution, 2002. BS 5268-2:2002 Structural use of timber – Part 2: Code of practice for permissible stress design, materials and workmanship. London: BSI. Calvert, J., R., Farrar, R., A., 2008. An Engineering Data Book. 3rd ed. Basingstoke: Palgrave Macmillan. European Committee for Standardisation, 2003. BS EN 338:2003 Structural timber – Strength classes. Brussels: CEN. European Committee for Standardisation, 2004. BS EN 1995-1-1:2004+A1:2008 Eurocode 5: Design of timber structures – Part 1-1: General – Common rules and rules for buildings. Brussels: CEN. Gere, J. M., 2001. Mechanics of Materials. 5th ed. Brooks/Cole. Hearn, E. J., 1997. Mechanics of Materials 2 – The Mechanics of Elestic and Plastic Deformation of Solids and Structural Materials. 3rd ed. [e-book]. Available through: Google Books <books.google.co.uk> [Accessed 25 November 2014]. Porteous, J., Kermani, A., 2007. Structural Timber Design – to Eurocode 5. Blackwell Publishing. Reece, P., O., 1949. An Introduction to the Design of Timber Structures. London: Spon.
  60. 60. Harrison Wallen 60 Ross, C., T., F., Case, J., Chilver, A., 1999. Strength of Materials and Structures, [e-book]. Available through: Elsevier <www.elsevier.com/books> [Accessed 12 December 2014]. Royal Institute of British Architects: NBS, Wooden performance: moisture content of timber. [online] Available at: <http://www.thenbs.com/topics/constructionproducts/articles/woodenPerformance.asp> [Accessed 10 February 2015]. The Steel Construction Institute, 2003. Steel Designers’ Manual. 6th ed. Blackwell Publishing. Theiler, M., Frangi, A., Steiger, R., 2013. Engineering Structures. Strain-based calculation model for centrically and eccentrically loaded timber columns, [e-journal] 56, 1103-1116. Available through: Elsevier <www.elsevier.com/locate/engstruct> [Accessed 1 November 2014]. Timber Engineering Company, 1956. Timber Design and Construction Handbook. New York: F. W. Dodge Corporation. Timoshenko, S. P., Gere, J. M., 2009. Theory of elastic stability. 2nd ed. New York: Dover Publications, Inc. TRADA, 1991. Proceedings of the 1991 International Timber Engineering Conference - Volume 1. Design of Columns. pp. 1.75 – 1.81 TRADA, 1991. Proceedings of the 1991 International Timber Engineering Conference - Volume 1. CEN Standards for Strength Classes and the determination of Characteristic Values. pp. 1.122 – 1.127

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