Assessment of local bending stiffness in timber on basis of knot area ratios, resonance frequencies and measured strain fields

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Assessment of local bending stiffness in timber on basis of knot area ratios, resonance frequencies and measured

strain fields

Växjö September 2011 Ata Sina Farshid Shadram Peyman Karami Department of Civil Engineering

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Organisation/ Organization Författare/Author(s)

Linnaeus University Farshid Shadram, Ata Sina and Peyman Karami

Dokumenttyp/Type of document Handledare/tutor Examinator/examiner

Degree Project Anders Olsson Anders Olsson

Titel och undertitel/Title and subtitle

Assessment of local bending stiffness in timber on basis of knot area ratios, resonance frequencies and measured strain fields

Sammanfattning (på svenska)

I denna studie har 13 balkar av sydsvensk gran med tvärsnittstorlek av 45 x 145 mm och 3600 mm längd undersökts. Forskningen fokuserar på att bestämma den lokala böjstyvheten i balkar och utvärdera möjligheten att använda lokal böjstyvhet som en indikator för böjhållfastheten. För att bestämma balkarnas medelstyvhet mäts deras resonansfrekvens. Den lokala böjstyvheten kan därefter beräknas genom att storlek och position för kvistar mäts upp och genom att styvheten i kvistarna betraktas som försumbar. För verifiering av modellen belastades balkarna i fyrapunktsböjning och Aramis, ett optisk beröringsfritt system för att detektera töjningar på ytor användes. Eftersom töjningarna detekterades med Aramis vid kända belastningsnivåer kunde den lokala böjstyvheten bestämmas på basis av dessa mätningar. Resultatet visar att den lokala böjstyvheten som beräknas genom uppmätt resonansfrekvens, visuellt identifierade kvistar och antagande om nollstyvhet i kvistmaterialet överensstämmer ganska väl med den lokala böjstyvheten som beräknas från balkprovningen och de uppmätta töjningsfälten. Arbetet omfattade också att undersöka om uppmätta resonansfrekvenser som svarar mot högre böjmoder kunde beräknas med bättre överensstämmelse med en finit elementmodell som beaktade variationer i böjstyvheten jämfört med en modell som förutsatte konstant böjstyvhet. I detta avseende kunde dock ingen förbättring i överensstämmelse påvisas med modellen som tog hänsyn till variationer i böjstyvhet.

Nyckelord

Local bending stiffness, knot are ratio, resonance frequency, strain field measurement, wood Abstract (in English)

In this study, 13 boards of Norway spruce sized 45 x 145 mm in cross section with the length 3600 mm have been examined. This research focuses on determining the local bending stiffness of wooden boards and on evaluating the possibility of using it as an indicator property for bending strength. In order to assess local bending stiffness, resonance frequencies corresponding to bending modes are measured, and the influence of knots are considered using a visual knot measurement. Boards are also loaded in four-point bending and Aramis, an optical contact-free system for detecting strain fields on surfaces, is employed. From the strain fields detected at known load levels the local bending stiffness can be calculated. The result shows that the local bending stiffness calculated on basis of visual knot measurements agrees fairly well with the local bending stiffness calculated on basis of the strain fields detected using the Aramis system. It is also shown that the resonance frequencies corresponding to higher bending modes may be well captured computationally using a finite element beam model. But despite of what was expected the agreement between calculated resonance frequencies and measured resonance frequencies is better if the bending stiffness is assumed to be constant within the entire board than if the detected variation in bending stiffness is taken into account in the model.

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Abstract

In this study, 13 boards of Norway spruce sized 45 x 145 mm in cross section with the length 3600 mm have been examined. This research focuses on determining the local bending stiffness of wooden boards and on evaluating the possibility of using it as an indicator property for bending strength. In order to assess local bending stiffness, resonance frequencies corresponding to bending modes are measured, and the influence of knots are considered using a visual knot measurement. Boards are also loaded in four-point bending and Aramis, an optical contact-free system for detecting strain fields on surfaces, is employed. From the strain fields detected at known load levels the local bending stiffness can be calculated. The result shows that the local bending stiffness calculated on basis of visual knot measurements agrees fairly well with the local bending stiffness calculated on basis of the strain fields detected using the Aramis system. It is also shown that the resonance frequencies corresponding to higher bending modes may be well captured computationally using a finite element beam model. But despite of what was expected the agreement between calculated resonance frequencies and measured resonance frequencies is better if the bending stiffness is assumed to be constant within the entire board than if the detected variation in bending stiffness is taken into account in the model.

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Acknowledgement

This thesis is the final part of the Master Program in Structural Engineering at the School of Engineering, Linnaeus University. It is supported by Anders Olsson, Bertil Benquist, Marie Johansson and Jan Oscarsson. We are heartily thankful to our supervisor and specially Anders Olsson whose encouragements, guidance and supports from the initial to the final level enabled us to develop our project so far. Lastly, we offer our regards to all of those who supported us in any respect during the completion of the project.

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Table of contents

1. Introduction ... 1

1.1 Background ... 1

1.2 Aim and scope: ... 2

2. Theory ... 2

2.1. Aramis system: ... 2

2.2. Differential equation of strain and bending moment in order to calculate bending stiffness . ... 4

2.3. Knot measurement ... 6

2.4. Calculation of natural frequency ... 9

2.5. Fast Fourier Transforming (FFT) ... 10

3. Expected results... 10

4. Methods and measurements ... 11

4.1. Dynamic excitation of the boards and natural frequency measurement ... 11

4.2. Density Measurement ... 12

4.3. Knot measurement ... 13

4.4. Finite element beam modeling and natural frequency calculation ... 14

4.5. Comparing the calculated frequencies basis on variable and constant EI with measured frequencies ... 15

4.6. Measurement of strain field of the boards by using Armis system in combination with a four-point loading: ... 15

4.7. Calculation of the local bending stiffness on basis of measured strain fields: ... 17

5. Results and analysis: ... 18

5.1. Local bending stiffness assessed from strain field measurement: ... 18

5.2. Result and Analysis of knot measurement and frequency measurement: ... 20

6. Discussion... 37

6.1. Comparison of bending stiffness obtained from strain field measurement and knot and frequency measurement ... 37

6.2. Comparison of calculated frequencies according variable and constant EI with measured frequencies. ... 40

7. Conclusions ... 40

8. References ... 41

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1. Introduction

A long term goal of the research within timber engineering is to find an accurate and effective method for strength grading of wooden boards. This master thesis is concerned with the assessment of local bending stiffness of wooden boards of Norway spruce which is the most common species in use for construction purposes at present. The following describes a brief background to the entire field of research, the aim and scope of the present study.

1.1 Background

In order to use timber as load bearing material in structures, it is necessary to ensure that the timber has the capacity to carry the loading. To achieve a relation between the properties of timber and the strength value which can be used in the design, systems of strength classes have been adopted. In Europe, the requirements of twelve strength classes of sawn softwood timber have been defined, they are denoted as: C14, C16, C18, C20, C22, C24, C27, C30, C35, C40, C45 and C50, where the character “C” refers to coniferous species and the number after it refers to the characteristic value of bending strength (in MPa) of timber pieces graded to that particular class. The characteristic value means that 5% of the pieces graded in the class may have a lower strength value than indicated by the strength class characteristic value and at least 95% exceed it. To ensure that even the few pieces with strength below the characteristic value will not fail during service, an additional material safety factor is used, which is often 1.3 for structural timber, another safety factor is used to account for the uncertainty of loads. This is a reason, why it is necessary to find a better method for strength grading of timber boards [8].

In addition to bending strength, the European system of strength classes (EN 338, CEN 2003a) sets requirements for two other properties, namely density and bending stiffness (mean value of modulus of elasticity [MOE]). These three properties of timber can be named as the grade determining properties of the timber.

One way to determine the true strength of wooden boards is using the destructive test. In this method, the board is loaded until failure. Afterwards, of course, it is not usable for its intended purpose as a load carrying component. The other way of prediction of strength is to measure the other properties of timber pieces like bending strength, bending stiffness and density as mentioned above. These measurements are made by some suitable nondestructive testing (NDT) methods.

Obviously, predicting strength by using indirect methods always include some uncertainty, because the capability of an indirect measurement to predict strength can never be perfect and always includes measurement errors.

The properties that are obtained by NDT-methods and used as predictors of grade determining properties are called grade indicating properties. The effectiveness of a system depends on the prediction capability of the grade indicating properties and the accuracy by which they can be measured.

This research focuses on different methods for assessment of stiffness properties within boards on a local level.

The occurrence of knots is one of the most important characteristics which influence the strength of a board and therefore the first method consists of a non-destructive test for measuring of the local bending stiffness by using visual knot area ratio. In this method some weak sections with the length of 150 mm are determined so that every weak section includes one or more knots with minimum diameter of 13 mm in one end of the knot.

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The second method consists of measuring the bending stiffness according to two dimensional measuring strain fields of loaded boards by using non-contact optical equipment (Aramis System). The test was performed on the section with the length of 400 mm located around the middle of the board and it varies from board to board depending on which section includes more knots.

All tests are performed on 13 beam boards of dimension 45*145*3900 mm (Norway spruce).

By taking the result of two mentioned methods on 13 Norway board timbers and using some mathematics calculation, the local bending stiffness of the boards, as an indicator property of bending strength can be calculated from two different method.

Both of the tests are performed in the laboratory of Linnaeus University in Växjö, Sweden.

Also a finite element model of the beams has been established to calculate the natural frequencies of the boards by considering the variable and constant bending stiffness along the boards and to evaluate how well is the first method, the calculated frequencies will be compared to measured natural frequencies related to bending modes which is obtained from a dynamic excitation method. How much the calculated frequencies fit to measured frequencies, the calculated bending stiffness from first method are more reliable.

1.2 Aim and scope:

The aim of this thesis is:

 to calculate, on basis of measurement results, the bending stiffness of weak sections in boards

 to compare the local bending stiffness, calculated on basis of knot area ratios and resonance frequency, with local bending stiffness calculated on basis of strain field detected by the use of an optical equipment (Aramis System) for the beams loaded in four points bending.

The research has been done just on 13 boards of Norway spruce since examining more specimens could conclude more certain results whereas just one cross section dimension has been used (45x145 mm). Also a proper continuation of the work should include a larger project so that larger samples of several timber boards to be investigated.

On the other hand there are some limitations to determine the weakest sections which should be examined to measure the strain field. In some cases choosing the cracked weak sections was unavoidable and it caused deviation in the results.

2. Theory

The major theory of this research is:

1. A good conception of the local bending stiffness can be achieved on basis of knot area ratios and resonance frequency

2. A good resolution of the bending stiffness will be reflected in calculated resonance frequencies.

Also the following theoretical backgrounds are used in this research:

2.1. Aramis system:

Aramis system is an optical 3D measuring system. This special device accomplishes all the measurements without any contact with the object. Aramis calculates the deformation of decisive items and provides a specific knowledge about the behavior of the measured object.

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Nowadays Aramis system is used in the most industry estimations, here is the major usage field of Aramis system:

 Testing of materials

 Evaluation of strength grading

 Dimensioning of components

 Examination of non-linear behavior

 Characterization of creep and aging processes

 Determination of Forming Limit Curves (FLC)

 Assessment of FE models

 Determining of the material characteristics

 Analysis of the behavior of homogeneous and inhomogeneous materials during deformation

 Strain calculation

The main hardware and software components in Aramis system can describe as:

 Sensor with two cameras (only for 3D setup)

 Stand for secure and steady hold of the sensor

 Sensor controller for power supply of the cameras and to control image recording

 High-performance PC system

 ARAMIS application software v6.1 and GOM Linux 10 system software or higher

Figure 2.1: A rectangular cross section

Aramis system recognizes the surface structure of the investigated object in digital camera images and benefits all the coordinates to the image pixels. The first image represents the undeformed object.

During deformation of the measuring object, Aramis records all the images and later calculates deformation and displacement of the object regarding to the first image or undeformed object (Aramis system saves these images as different stages).

If the measuring object has homogeneous surface (or few object characteristics) then it is needed to prepare this surface (for example with applying a stochastic color spray on the surface of the object).

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Aramis is especially suited to three dimensional deformation measurements with combination of static or dynamic loads, for analyzing and calculating of both deformation and strain in the object.

2.2. Differential equation of strain and bending moment in order to calculate bending stiffness [9].

To understand the bending stress in an arbitrary loaded beam, consider a small element cut from the beam as shown in Figure 4.3 The beam type or actual loads does not affect the derivation of bending strain equation.

ε = ΔL/L (Eq. 2.1)

Figure 2.3: Small beam element in bending

Using the line segment, AB, the length before and after deformation can be used to calculated the strain as

(Eq. 2.2)

The line length AB is the same for all locations before bending. However, the length A'B' becomes shorter above the neutral axis (for positive moment) and longer below. The line AB and A'B' can be described using the radius of curvature, ρ, and the differential angle, dθ.

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(Eq. 2.3)

(Eq. 2.4)

Notice that the y coordinate is assumed upward from the neutral axis, where there is no strain.

Putting these together gives,

(Eq. 2.5)

This relationship gives the bending strain at any location as a function of the beam curvature and the distance from the neutral axis.

When a moment acts on a beam, the beam rotates and deflects. The relationship between the radius of curvature, ρ, and the moment, M, at any given point on a beam is:

(Eq. 2.6)

Figure 2.4: Small relation between bending moment ‘M’ and radius of curvature ‘ρ’

This relationship was used to develop the bending stress equation but it can also be used to find the bending stiffness, EI, due to moment and radius of curvature.

Recall from calculus, the radius of curvature for any point of a function, y = f(x), can be calculated according to Eq. 2.7.

(Eq. 2.7)

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For beams, it is convenient to note the deflection as v (upward is positive) instead of y. Also, for small deflections, the first derivative or slope, dy/dx, is small. If it is squared, then it is very small and can be assumed to be zero. In other words, (dv/dx)² ≈ 0. This simplifies the radius of curvature equation to

(Eq. 2.8)

Figure 2.5: Geometry of Beam Bending and Deflection

Combining this with the bending stress equation gives standard moment-curvature equation,

(Eq. 2.9)

By combination of Eq. 2.5, 2.8 and 2.9 the bending stiffness can be calculated related to bending moment and strain value (Eq. 2.10).

(Eq. 2.10)

2.3. Knot measurement

The theory which is used to measure the knot area ratios in this research is that the knots are supposed a non-material part of the board. To calculate the bending stiffness (EI) along the board according to first method (knot are ratios measurement) it is necessary to evaluate the second moment of inertia of all parts of the boards including weak sections and homogenous sections.

The second moment of inertia of a beam cross-sectional area measures the beam ability to resist bending. The larger the moment of inertia the less the beams will bend. The moment of inertia is a geometrical property of a beam and depends on a reference axis. The smallest moment of inertia about any axis passes through the centroid.

The following are the mathematical equations to calculate the moment of inertia:

(Eq. 2.11)

(Eq. 2.12)

Which:

y is the distance from the x axis to infinitesimal area dA.

x is the distance from the y axis to infinitesimal area dA.

dx is the width of infinitesimal area dA.

dy is the height of infinitesimal area dA.

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For a rectangular cross section (Figure 2.6) the second moment of inertia relative to the x axis can be calculated as:

(Eq. 2.13)

Which:

b is the width of the rectangular section h is the length of the rectangular section

In this experiment, for all sections which are supposed homogenous Eq. 2.13 is used to calculate the second moment of inertia by considering:

b=45 mm h=145 mm

Knots are considered as the non-material parts of weak sections and it has been also assumed that properties of knots are extended along length of weak section. As a result, to calculate the second moment of inertia of weak section, the projected knot areas located within the window, illustrated in Figure 2.7, should be excluded.

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Figure 2.7: Definition of window width [5].

For the mentioned complex object, the second moment of inertia can be calculated approximately using equation 2.14, due to parallel axis theorem. The parallel axis theorem compensates for each smaller object’s centroid’s distance from the actual centroid.

Figure 2.8: Projection of knots on weak section.

(Eq. 2.14)

Where

A is the area of projection of knots.

d is the distance between centre of mass of knots and x axis.

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2.4. Calculation of natural frequency [6]

As we know from structural dynamics, if we have an undamped 2-DOF (2 Degree of Freedom) vibrating system (Eq. 2.15).

(Eq. 2.15)

The solution to Eq. 2.15 will consist of a complementary solution plus a particular solution.

The complementary solution, obtained by setting the right hand side of Eq. 2.15 to zero, involves the dynamical properties of the system called natural frequency.

0

0 (Eq. 2.16)

To obtain the solution for free vibration the following steps maybe followed.

Assume that the system undergoes harmonic motion of the form:

cos (Eq. 2.17)

cos (Eq. 2.18)

Where and are signed constants that determine the amplitudes of the two respective sinusoidal motions. Substitute this assumed solution into the equations of motion to obtain the following algebraic eigenvalue problem:

0

0 (Eq. 2.19)

Since this is a set of homogeneous linear algebraic equations, the only nontrivial solutions of Eq. 2.9 Correspond to values of that satisfy the characteristic equation

0 (Eq. 2.20)

That is, values of for which the determinant of the coefficients of Eq. 2.19 is equal to zero.

This is a polynomial equation of order 2 in .

Solve for the roots of the characteristic equation. Label these roots and with . (In mathematical terminology and are called eigenvalues.)

The parameters and are called the circular natural frequencies (circular, since they are expressed in the units rad/sec; natural, since there is no excitation of the system; and frequencies, since they are the rate of oscillation of the harmonic motion in Eq. 2.17, 2.18).

Although and are sometimes called the natural frequencies of the system, that name is usually reserved for the natural frequencies in hertz (=cycle/s),

,

By extending this technique, the natural frequencies of free vibration undamped MDOF (Multi Degree of Freedom) system can be obtained considering Eq. 2.21., 2.22.

0 (Eq. 2.21)

det 0 (Eq. 2.22)

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2.5. Fast Fourier Transforming (FFT) [7]

FFT analyzer is a powerful tool to derive natural frequencies corresponding to axial and bending modes of vibration.

The general frequency domain is the method to express the applied loading in term of harmonic components, the response to each component evaluating, and then superposing the harmonic responses to obtain the total response. To do so Fourier series represented in form of the frequency-domain-analysis procedure.

Thus, formulate this procedure in terms of a numerical-analysis to make it practical.

Because limited cases are available for which the Fourier integral transform of the applying load functions, and the evaluation of these integrals are tedious .To evaluate these forms in efficient numerical techniques, direct Fourier transformation equations can be used, which are a more effective form for the numerical solution procedures. It is used to gain a sequence of discrete data in the frequency domain. It should be clear that Discrete Fourier Transform (DFT) is the procedure for analysis digital data, but again DFT is not recommended, because practically in real time applications and for large values of samples in the sequence.

A Fast Fourier Transform (FFT) is an efficient algorithm to compute the Discrete Fourier Transform (DFT). It is an ingenious way of achieving rather than the DFT.

Frequency-domain data are obtained by converting time-domain data using a mathematical technique referred to as a Fast Fourier Transform (FFT). This allows each

vibration component of a complex machine-train spectrum to be shown as a discrete frequency peak. Cooley and Turkey published Fast Fourier Transform (FFT) algorithm. This developing in the field of data by degreasing number of calculations and mathematic

operations required for the discrete Fourier transformation. Because of the fact of ’a realistic signal could have 1024 samples which requires over 20,000,000 complex multiplications and additions’ (Anon 2009c). Frequency components in various frequency bands of interest are extracted by using narrow-band analog filters.

The first tow modes of extracted frequencies from these algorithm operations (FFT) are interested in our research to be used in the calculation of the modulus of elasticity (MOE) that is required to assign the properties of the board of our interest. Therefore FFT is economical method to perform the analysis by using software with microprocessor technology through dedicated analyzers as well as desktop computers (Thorby 2008) and (Clough et al.2003).

3. Expected results

This research focuses on determining the local bending stiffness of wooden boards. The expected results in this thesis is that the bending stiffness calculated on basis of knot area ratios and measured and resonance frequency is in good agreement with the bending stiffness obtained from strain field measurement by using non-contact optical equipment and also the agreement between calculated and measured resonance frequencies will be improved if instead of constant value of bending stiffness, the detected variable is considered in the model and within the entire board.

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4. Methods and measurements

As described above in the Introduction, the objective of the work was to study how well some different NDT-methods and their combinations can predict more accurate bending stiffness of the boards as an indicator property of strength grading. The methods are used in this research include some different experiment has been done in laboratory and related calculations. In the following, a description is given about how each measurement was used in the thesis.

4.1. Dynamic excitation of the boards and natural frequency measurement

The idea is to hit the board with suitable impact load and measure the natural frequency of the board. The method has the advantage that forces needed and deflections induced are fairly small. To prepare a free-free boundary conditions of each board, the boards are suspended in rubber bands (see figure 4.1) and the dynamic excitation is conducted by hitting with an impulse hammer in the end section and on the narrow edge for excitation of axial modes and edgewise bending modes, respectively. The vibrations are detected using an accelerometer which is fastened on the end section and on the narrow section when measuring acceleration in the longitudinal direction and in the transversal direction, respectively (see figure 4.1). The signal from the accelerometer is transformed by a FFT-analyzer then fast Fourier transformation is used for calculation of resonance frequencies (also called eigenfrequency) corresponding to axial and bending modes of vibration. The precision in measurements depends on the frequency range defined, which for measurements in the axial and transversal direction was set to 0-5000 Hz and 0-1000 Hz, respectively. The received precision of the detected eigenfrequencies were in both cases better than ± 0.25 %. [2].

Figure 4.1: The board is suspended by two rubber bands to reassemble free-free conditions [2].

Also the dynamic modulus of elasticity ( ) can be calculated based on the measured resonance frequency in axial mode ( ), the length (L) and the measured density (ρ) of a beam.

4. ρ. L . (Eq. 4.1)

The resonance frequencies corresponding to axial and bending modes of vibration are measured. The measured result for four boards is shown in Figure 4.2 and Figure 4.3. In the figures 4.2 & 4.3, peaks determine the resonance frequencies which are marked with small cross lines. The six lowest resonance frequencies for bending modes and axial modes for all thirteen boards is recorded and stored.

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Figure 4.2: Measured transversal vibration content for board number one to four

Figure 4.3: Measured axial vibration content for board number one to four 4.2. Density Measurement

The global density of the boards is measured by determination of the volume and weight of the boards. The global density of the boards was measured by weighing and measuring the volume of the specimens in laboratory. The volume of the boards is calculated based on length, width and thickness of the boards. The board density, ρ, was simply calculated as the mass of the board divided by its volume.

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4.3. Knot measurement

One of the non-destructive measurements is evaluation of size, type and position of the knots.

Before applying the bending test, the knot geometry of the weakest sections of the boards is recorded. In order to facilitate the calculation of different knot measures, the knots are divided into eight types, A to H (See Figure 4.4)

Figure 4.4: Different knot types for classification [5].

Also a co-ordinate system is determined as it is shown below (Figure 4.5):

Figure 4.5: The system of co-ordinate used for the boards.

For each board, some weak sections with the length of 150mm is determined visually and every weak section include one or more knots which grown in it since failure usually

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originates at or near a knot. In this evaluation, just the knots with the diameter more than 13mm (at one end of them) are considered. The Data file including the size, type and position of the knots of 13 boards is documented. One example is shown below (Figure 4.6):

Figure 4.6: A sample of projected knot area section.

By writing some codes in MATLAB software, the area of projected knot (hatch area) is calculated by using the coordination of the knot as illustrated in Figure 4.6. Then as explained in Theory the obtained values is used to calculate the second moment of inertia (I) for weak and sound sections of the boards.

4.4. Finite element beam modeling and natural frequency calculation

A finite element beam model is established via MATLAB to determine the bending stiffness of the boards. For this purpose, every beam is divided into many elements and the length of each element is approximately about 5 mm. the as mentioned in theory the global stiffness matrix and mass matrix is computed for a two dimensional element by using CALFEM software. Then by assuming a free-free boundary condition and also assuming an arbitrary value of module of elasticity, the natural frequencies of the beam model, based on what explained in theory, can be calculated by writing some codes of CALFEM. The program is run for different modules of elasticity values till the first natural frequency get fit to first measured natural frequency corresponding to bending mode of vibration. Then the module of elasticity and the second moment of inertia for every finite element is calculated and distribution of the bending stiffness along the board can be obtained.

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By using the same way the resonance frequency of homogenous beam can be calculated by assuming the certain second moment of inertia along the board which could be compared to measured resonance frequency.

4.5. Comparing the calculated frequencies basis on variable and constant EI with measured frequencies

It’s been proven that the occurrence of the knots in timber boards decreases the natural frequencies [7] furthermore how much the knots is detected more precisely the calculated natural frequencies get more fit to the measured natural frequencies.

As described in 4.3, the deviation of calculated natural frequencies rather than measured frequencies can be calculated in order to calculate the error of calculated natural frequencies (Eq. 4.2).

, , / _,

. . .

, , / _,

(Eq. 4.2)

Which:

, is the calculated eigenfrequency of the boards.

, is the measured eigenfrequency of the boards.

Then error values can be calculation by multiplying the vector (Eq. 4.2) with its transposed form (Eq. 4.3).

(Eq. 4.3)

This error can be assessed for calculated natural frequency assuming variable and constant second moment of inertia respectively along the boards and it is expected that the error obtained using variable EI for boards (i.e when the presence of knots is taken into account) is lower than when constant EI values are used for the boards.

4.6. Measurement of strain field of the boards by using Armis system in combination with a four-point loading:

The strain field is measured by consecutive load tests in which one side of the specimen was studied during each test. As it is described in the theory part the main system in this method is Aramis. Below is a complete description of this method, location of loading machine, location of Aramis cameras and the output.

1. Two flexible supports with 2610mm distance between (A in figure 4.7)

2. A ruler hanging above the supports which will allow the beam to move horizontally.

3. Two point loads (establish by two compressors) acting on the beam (B in figure 4.7).

Each point load is located 870mm (h is height of the beams and 6h=870mm) from the support. Furthermore the distance between point loads is 870mm too.

4. Aramis cameras (which will be use to measure displacement of the beam in order to a four-point loading) are located in front of the color sprayed surface (C in figure 4.7)

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Figure 4.7: Aramis system and bending machine

As it is described in the theory part in order to have a homogeneous surface of the boards, it is needed to apply a color spray (first applied a white spray and later black spray) on the surface of the beam. It makes it possible for Aramis to recognize points on the beam and follow the change in displacements between different positions when the beam is loaded. Effect of the point loads on the beam is tension on upper side and compression on the bottom side (i.e.

bending along the beam). During bending, Aramis cameras take images from the beam in the fixed distance of 400mm (furthermore this length should conclude the weakest section of the board). In the same time when the beam is loaded, Aramis cameras start taking pictures, each image saves as a stage in Aramis software. The first stage is the undeformed object. In each stage Aramis software records both the image of deformed beam and the load quantity which affected on the beam at the meantime.

Aramis gives the strain values of all the points in each stage (Aramis calculate the strain values from a comparison between deformed point from each stage in compare with same point in the stage zero (undeformed stage)).

In the Aramis, software will define more than 70 stages in most of the boards but the last stages show the final collapse of the beam. In this thesis and method, some stages are defined before the occurring of first crack on the beam and will compare with a stage, over stage 0 (for example comparison between stage 10 and 30). The reason for taking stage 10 as first stage is because at the beginning of loading, beam is going to lift up to the supports i.e. there isn’t any bending happening along the beam until stage 10.

Later it defines 2 sections on the beam in Aramis software for measuring of the strain fields (for both stages 10 and 30). These two sections are located a little bit far from the edge of the beam in order to be far from the cracks which are located in the edge of the beams.

Figure 4.8 shows distribution of the strain rates in these two sections with color variation which is took from Aramis system, in this picture even shows the load quantity which affected on the beam at the meantime.

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Figure 4.8: the distribution of the strain rates with color variation 4.7. Calculation of the local bending stiffness on basis of measured strain fields:

As it is described in the theory part, Aramis gives the result of strain values in two different sections, close to the upper and lower edge of the beam respectively. After the beam is loaded in four-point bending a constant bending moment acts on an examined cross section of the beam. By calculating of the bending moment corresponding to a certain load, and knowing the strain corresponding to this load, it is possible to calculate the bending stiffness EI using formula below:

(Eq. 4.4)

(Eq. 4.5)

(Eq. 4.6)

Where:

1

x is the strain value of each point in the upper section

2

x is the strain value of each point in the bottom section

y is the vertical distance between the same points in both sections.

2 2 x v

d

d is the curve scope.

x is the total strain value of each point in both sections.

M is the bending moment which is established by the loads effect on the beam.

Therefore EI values can be calculated for all the boards in the parts of the boards where the strain fields are detected. Calculations are performed using the software MATLAB. Also the distribution of the local bending stiffness along the boards can be obtained. For this purpose the examined section with the length of 400 mm is divided to 20 mm sections and by

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calculating the average of strain in every section, the related bending stiffness can be obtained.

5. Results and analysis:

After performing different experiments and calculations on 13 boards in laboratory the results are presented as below. The results include two parts. First part illustrates and interprets the measured strain field and calculated local bending stiffness by using bending machine and Aramis system. Second part illustrates obtained bending stiffness along the boards on basis of knot area ratios. Finally, third part represent the lack of fit between measured, calculated and expected (assuming homogeneity) eigenfrequencies of the boards.

5.1. Local bending stiffness assessed from strain field measurement:

Figure 5.1 shows a board examined in bending machine so that the strain field is measured by Aramis.

Figure 5.1: existence of the knots in the weak section examined in bending machine in order to measure by Aramis.

The board shown in Figure 5.1 has more than 3 knots in the weakest part of it. These knots are located both in the edge and surface of this section. When the beam is loaded, the weak section is subjected to a constant bending moment which causes compression and tension in lower and upper part of the beam, respectively. For all load stages, the existence of the knots influences the strain rates and it causes a sudden change in strain value in the place of the knots and it is due to the behavior of knots in wood members exposed to loading.

Figure 5.2 shows the strain rates of the two sections (as it is described on theory part) in the x direction (the strain rates graph is designed by Excel program obtained from Aramis outputs in combination with a four-point loading).

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Figure 5.2: Longitudinal surface strains in the x direction, top row: strain rates in upper section (tension side), bottom row: strain rates in lower section (compression side) In this graph, the strain rate of the upper section on the beam is shown by a blue line (tension side) and bottom side is shown by a red line (compression side).

As it is visible, in the position that board includes knots, the strain distribution fluctuates.

Regarding to the strain graph (Figure 5.2), the strain rate changes suddenly between 160 to 220 mm. Whenever the strain increases then bending stiffness decrease and vice versa.

Assuming a modulus of elasticity of 11GPa the bending stiffness for a homogeneous board is calculated as:

3 2

6 125.7

12 145 . 0 045 . 10 0

11 KNm

EI  





This means, between 160mm and 220mm the bending stiffness should be less than 125.7.

Figure 5.3 shows distribution of the local bending stiffness along the mentioned board. This graph is designed by writing a function in MATLAB program and using the average strain values in each 20 mm of the board (in order to eq.4-4, 4-5, 4-6 the local bending stiffness formula which is presented in the method part)

‐3.00E+00

‐2.50E+00

‐2.00E+00

‐1.50E+00

‐1.00E+00

‐5.00E‐01 0.00E+00 5.00E‐01 1.00E+00

0.00E+005.00E+011.00E+021.50E+022.00E+022.50E+023.00E+023.50E+024.00E+024.50E+02

Strain rates [%]

Longitudinal direction of the beam [mm]

Stain rates of board number 08 for tension and compression sections in the x direction

Ten com

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Figure 5.3: The distribution of bending stiffness along the board on basis of strain field measurement

The result is as expected in the section between 160 to 220mm and the local bending stiffness decrease due to existence of knots.

5.2. Result and Analysis of knot measurement and frequency measurement:

As mentioned in section 4.4 the distribution of bending stiffness along the boards on basis of knot measurement can be obtained. As shown in Figure 5.4 for sections assumed as a homogenous sections the EI value is constant and for the weak sections with the length of 150 mm a reduction in EI values can be observed.

Figure 5.4: The distribution of bending stiffness along the board on basis of knot measurement and frequency measurement

The results of bending stiffness assessment related to strain field measurement and knot and frequency measurement on all thirteen boards are presented as follows. Red line in third graph

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‐2.00E+00

‐1.00E+00 0.00E+00 1.00E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00 7.00E+00

0.00E+001.00E+022.00E+023.00E+024.00E+025.00E+02

Strain rates [%]

Stain rates of board number 01 for tension and compression sections in the x direction

ten co

of each board shows the local average of each board in order to make it comparable to third graph.

Figure 5.1.a: Board number 01

Figure 5.1.b: Stain rates of board number01 for both sections in the x direction

Figure 5.1.c: Distribution of the local bending Figure 5.1.d: Distribution of the local bending stiffness on basis of knot and frequency stiffness basis on strain field measurement

measurement in board 01 in board 01

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‐1.00E+00 0.00E+00 1.00E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00

0.00E+001.00E+022.00E+023.00E+024.00E+025.00E+02

Strain rates [%]

ten com

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‐8.00E‐01

‐6.00E‐01

‐4.00E‐01

‐2.00E‐01 0.00E+00 2.00E‐01 4.00E‐01

0.00E+001.00E+022.00E+023.00E+024.00E+025.00E+02

Strain rates [%]

Stain rates of board number 03 for tension and compresion sections in the x direction

Ten com

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‐4.00E‐01

‐2.00E‐01 0.00E+00 2.00E‐01 4.00E‐01 6.00E‐01 8.00E‐01 1.00E+00

0.00E+001.00E+022.00E+023.00E+024.00E+025.00E+02

Strain rates [%]

Ten com

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‐1.00E+00

‐5.00E‐01 0.00E+00 5.00E‐01 1.00E+00 1.50E+00 2.00E+00

0.00E+001.00E+022.00E+023.00E+024.00E+025.00E+02

Strain rates [%]

Ten com

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‐2.00E+00

‐1.00E+00 0.00E+00 1.00E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00

0.00E+001.00E+022.00E+023.00E+024.00E+025.00E+02

Strain rates [%]

Ten com

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‐8.00E‐01

‐6.00E‐01

‐4.00E‐01

‐2.00E‐01 0.00E+00 2.00E‐01 4.00E‐01 6.00E‐01

0.00E+001.00E+022.00E+023.00E+024.00E+025.00E+02

Strain rates (%)

Ten com

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‐3.00E+00

‐2.50E+00

‐2.00E+00

‐1.50E+00

‐1.00E+00

‐5.00E‐01 0.00E+00 5.00E‐01 1.00E+00

0.00E+001.00E+022.00E+023.00E+024.00E+025.00E+02

Strain rates [%]

Ten com

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‐3.00E+00

‐2.50E+00

‐2.00E+00

‐1.50E+00

‐1.00E+00

‐5.00E‐01 0.00E+00

0.00E+001.00E+022.00E+023.00E+024.00E+025.00E+02

Strain rates [%]

Stain rates of board number 09 for tension and compression sections in the x direction

Ten com

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‐1.50E+00

‐1.00E+00

‐5.00E‐01 0.00E+00 5.00E‐01 1.00E+00 1.50E+00 2.00E+00

0.00E+001.00E+022.00E+023.00E+024.00E+025.00E+02

Strain rates [%]

Ten com

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‐3.00E+00

‐2.00E+00

‐1.00E+00 0.00E+00 1.00E+00

0.00E+001.00E+022.00E+023.00E+024.00E+025.00E+02

Strain rates [%]

Ten com

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‐3.00E+00

‐2.00E+00

‐1.00E+00 0.00E+00 1.00E+00 2.00E+00 3.00E+00

0.00E+001.00E+022.00E+023.00E+024.00E+025.00E+02

Strain rates [%]

Ten com

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‐1.00E+00 0.00E+00 1.00E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00

0.00E+001.00E+022.00E+023.00E+024.00E+025.00E+02

Strain rates [%]

Ten com

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The first six measured resonance frequencies related to bending modes and calculated resonance using constant EI and fluctuating EI respectively are presented in the chart 7.1.

The deviation of measured natural frequencies of bending modes and calculated natural frequencies with average in EI for thirteen boards is shown and compared.

Measured frequency

Calculated frequency with fluctuating EI

Calculated frequency with constant EI

Frequency (Hz)

Deviation from measured frequency

Error Frequency

(Hz)

Error

Board 1

60 60 0.00

0.127

60 0.00

0.1178

159.5 166.1 0.04 165.5 0.04

287 324.1 0.13 324.4 0.13

438 534.9 0.22 536.2 0.22

593.5 737.9 0.24 724.9 0.22

762 799.1 0.05 801 0.05

Board 2

60 60 0.00

0.1215

60 0.00

0.0987

158.5 166.7 0.05 165.4 0.04

291.5 325 0.11 324.3 0.11

443.5 540.9 0.22 536.1 0.21

601.5 745.1 0.24 724.8 0.20

775.5 798.3 0.03 800.9 0.03

Board 3

56 56 0.00

0.0863

56 0.00

0.0795

149.5 155.4 0.04 154.4 0.03

275.5 301.1 0.09 302.8 0.10

420.5 496.8 0.18 500.5 0.19

575.5 695 0.21 676.6 0.18

739.5 731.8 -0.01 747.6 0.01

Board 4

60.5 60.5 0.00

0.0973

60.5 0.00

0.0888

157 165.3 0.05 166.8 0.06

294.5 329.9 0.12 327 0.11

448.5 536.6 0.20 540.6 0.21

627 754.4 0.20 730.9 0.17

800 812.2 0.02 807.6 0.01

Board 5

56.5 56.5 0.00

0.2566

56.5 0.00

0.2338

152 156.7 0.03 155.7 0.02

278 306.2 0.10 305.3 0.10

437 509 0.16 504.7 0.15

577.5 694 0.20 682.3 0.18

611 754.7 0.24 754 0.23

785 1060.6 0.35 1053 0.34

Table5.1: Values of frequency obtained from measurements and calculations based on variable and certain bending stiffness.

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Calculated frequency with fluctuating EI

Calculated frequency with constant EI

(Hz)

Error

Board 6

60.5 60.5 0.00

0.2334

60.5 0.00

0.1893

154 168.3 0.09 166.8 0.08

278 327.9 0.18 327.1 0.18

426 545.4 0.28 540.6 0.27

581.5 766.7 0.32 730.9 0.26

737.5 817.2 0.11 807.6 0.10

Board 7

59.5 59.5 0.00

0.1075

59.5 0.00

0.1023

156.5 162.7 0.04 164.1 0.05

286 316.4 0.11 321.6 0.12

437 524.5 0.20 531.7 0.22

606.5 746.8 0.23 718.8 0.19

773.5 797 0.03 794.2 0.03

Board 8

49 49.0 0.00

0.0704

49.0 0.00

0.0466

125 133.1 0.06 135.1 0.08

242 264.8 0.09 264.9 0.09

384.5 450.4 0.17 437.9 0.14

538.5 632.3 0.17 592.1 0.10

692 668.1 -0.03 654.2 -0.05

Board 9

58 58 0.00

0.1177

58 0.00

0.1227

151.5 157.9 0.04 160 0.06

277 310.1 0.12 313.7 0.13

420 506.9 0.21 518.5 0.23

574 713.8 0.24 701 0.22

751.5 751.4 0.00 774.5 0.03

Board 10

58 58.3 0.01

0.136

58 0.00

0.1148

152 161.3 0.06 159.9 0.05

278.5 314.3 0.13 313.5 0.13

422.5 523.3 0.24 518.2 0.23

583 717.3 0.23 700.6 0.20

739 787.1 0.07 774.2 0.05

Contd. table 5.1: Values of frequency obtained from measurements and calculations based on fluctuating and constant bending stiffness.

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Calculated frequency with variable in EI

Calculated frequency with average in EI

(Hz)

Error

Board 11

50 50.0 0.00

0.0576

50.0 0.00

0.0630

130.5 136.9 0.05 137.8 0.06

242.5 265.5 0.09 270.2 0.11

378 435.5 0.15 446.7 0.18

550.5 625.2 0.14 603.9 0.10

717.5 665.2 -0.07 667.2 -0.07

Board 12

50 50.0 0.00

0.0418

50.0 -0.01

0.0369

136.5 138.4 0.01 137.8 0.01

254 268.0 0.06 270.2 0.06

397 442.6 0.11 446.7 0.13

556.5 632.9 0.14 603.8 0.09

736.5 674.3 -0.08 667.2 -0.09

Board 13

52.5 52.5 0.00

0.6632

52.5 0.00

0.4762

142.5 147.6 0.04 144.7 0.02

266.5 287.8 0.08 283.8 0.06

391.5 477.5 0.22 469.1 0.20

416.5 681.8 0.64 634.1 0.52

570 720.4 0.26 700.7 0.23

Contd. table 5.1: Values of frequency obtained from measurements and calculations based on fluctuating and constant bending stiffness.