Referenced Literature

The publications with working links are available for download. The author of each publication is indicated.


Explanation of the embedding strength of particle board

T.A.C.M. van der Put

(Series 2007, nr 5)

By the earlier derived theory of the embedding strength, based on theequilibrium method of plasticity, it is possible to fully explain theempirical relations and test results of an investigation of theembedding strength of particle board leading to a new insight for theright design rules, (which should be in Eurocode 5).

The concluding results are given in:

“Derivation of the bearing strength perpendicular to the grain of locally loaded timber blocks ” in: “Holz als Roh- und Werkstoff”: “VOLUME 66, ISSUE 6 (2008), PAGE 409”.


 Comment on the thesis and Heron article of Raven et al.

T.A.C.M. van der Put (Dec. 2007) [download]

Because the theory is nowhere applied rightly in thesis and Heron 53-1article of Raven et al, an extension is given of the comment in Heron 53 (1980)- nr. 3 on the thesis and article to prove this and corrections, explanations and extensions are derived, as a necessary completion of the thesis.”.


The tensor-polynomial failure criterion for wood polymers

(Series 2005 nr.2)

The failure criterion of wood

(Series 2007, nr 4)

Derivation of the critical distortional energy principle for orthotropic composites like wood 

(Series 2007, nr 3)

These publications are replaced by the final concluding summary:

“A continuum failure criterion applicable to wood“ in: Journal of Wood Science volume 55 (2009) No.5. DOI 10.1007/s10086-009-1036-2

T.A.C.M. van der Put

As shown in the past, the tensor-polynomial equation can be regarded as a polynomial expansion of the real failure surface. The third-degree polynomial is shown to be identical to the real failure criterion while the second-degree part of the polynomial is identical to the orthotropic extension of the von Mises criterion for initial yield. The third degree polynomial hardening terms of the criterion are also shown to incorporate the earlier theoretical explained mixed-mode I-II fracture equation, showing hardening to be based on hindered micro-crack extension. For uniaxial loading, the failure criterion can be resolved in factors, leading to the derivation of extended Hankinson equations. This allows the relations between the constants of the total failure criterion to be elucidated as is necessary for data fitting of this criterion to provide a simple method to determine the constants by the simple uniaxial, oblique grain compression and tension tests. Based on this, the numerical failure criterion is given with the simple lower bound criterion for practice and for the codes.


Size effect in fracture mechanics of wide angle notched beams 

T.A.C.M. van der Put

(Series 2007 nr. 2)

A new explanation is given of the strength of wide angled notched timber beams by accounting for a Weibull type size effect in fracture mechanics. The strength of wood is described by the probability of critical initial small crack lengths. This effect is opposed by toughening by the probability of having a less critical crack tip curvature. The toughening effect dominates at the different wide angle notched beams showing different high stressed areas and thus different influences of the volume effect. This is shown to explain the other power of the depth in Eq.(18) and (19) than that of  the sharp notch value of 0.5 of Eq.(17). It further is shown to explain why for very small dimensions, also for sharp notches, the volume effect applies. The explanation by the Weibull effect implicates that the strength depends on small crack extension, in the neighbourhood of the macro crack tip. This initial crack population can be different for full scale members indicating that additional toughness tests have to be done on full scale (or semi full scale) test specimens.

An article on this subject is in review by Theoretical and applied fracture mechanics since 1 July 2006 and the last revision since 9 August 2009 and can be obtained after permission by the editor of that Journal.


Softening behaviour and correction of the fracture energy

T.A.C.M. van der Put

(Series 2007 nr. 1)

The area under the load-displacement softening curve gives the total external work on the test specimen and not the fracture energy. The fracture energy follows from half this area what is equal to the critical strain energy release rate at the first crack increment. For wood this correctly is applied for mode II. For mode I however, as for other materials, wrongly the total area is regarded, a factor 2 too high. In some applications, based on crack increment cycles, the error is even a multiple of this factor 2. On the other hand, the measurements at softening may show an apparent decrease of the specific fracture energy that can be explained by a small crack joining mechanism when the ultimate state of the ligament of the test-specimen is reached. Post fracture behaviour thus is not comparable with the behaviour of macro crack initiation.

It further is shown, by the kinetics of the process, that the irreversible work of an ultimate loading cycle is proportional to the activation energy of the fracture process and not to the driving force of the process. This explains why the crack velocity decreases with the increase of this irreversible work and increases with the stress intensity increase.

The fracture energy is a function of the Griffith strength and thus is related to the effective width of the test specimen and not to the ligament length. This also has to be corrected. Based on the derivation of the softening curve, the reported fracture toughness of 720 kNm-15 of double-edge notched tests is corrected to 330 kNm-15 and the value of 467 kNm-15, based on the fracture energy, of the compact tension tests, also is corrected to the right value of 330 kNm-15. A revision of published mode I data, based on the fracture energy obtained by the area of the softening curve, thus is necessary.

The concluding final theory is published in:

“Softening behaviour and correction of the fracture energy” in:

Theoretical and applied fracture mechanics 48 (2007) 127-139, or: here



Derivation of the shear strength of continuous beams  

T.A.C.M. van der Put

(Series 2006 nr. 5)

As continuation on the theoretical explanation of the bearing strengths of locally loaded blocks, the bracing model is extended here and it is shown that, with the right dimensioning, always the shear strength is determining.

The elastic-plastic beam theory is extended for the influence of normal force and shear. Based on this extension the apparent contradictory test results of the shear- and bending strengths of beams and continuous beams is explained and also the shear and bracing action of beams loaded close to the supports is derived and verified by tests.

It appears that the theory of elasticity is not able to explain the data and to give the right stress distribution in two span beams, underestimating the bearing capacity by a factor 2/3, while the elastic-plastic beam theory gives a very precise description of the data and the determining shear- and bending strengths.

The derivations, confirmed by tests, lead to requirements for design rules of the Codes.

The concluding final theory will be published in 2010 in two articles as:


1. “Failure criterion for timber beams loaded in bending, compression and shear”, in:

Wood Material Science and Engineering.

where, based on the, for precise data explanation, necessary elastic-plastic strength calculation a derivation is given of the failure criterion for combined bending, compression and shear. This exact limit state criterion replaces the unacceptable unsafe criteria of the Eurocode 5, (EN 1995-1-1:2004)). It is shown that the thus far used principle of limited “flow” in axial compression as determining failure criterion, predicting e.g. no influence of a size effect, does not hold. In stead it is derived and confirmed by the data that bending tension failure is always determining showing the existence of a size effect and correction thus is necessary of the existing calculation method. Because the primary importance of the size effect for the strengths, also for combined bending- compression, a simple derivation of the size effect design equations is given and discussed.


2. “Derivation of the shear strength of continuous beams,” in:

Holz als Roh- und Werkstoff

The elastic- full plastic loading curve is for all materials sufficient to explain the strength of beams and beam columns loaded by bending and compression. This theory is extended for the influence of shear stress and it is shown to be the only way to explain the combined bending-shear strength from test results. Also the in the past derived bearing strength theory is extended here for bracing action. It will be shown for continuous beams as example, that besides moment redistribution by plastic flow in bending, a plastic shear flow mechanism exists that also is able to cause full moment redistribution. The derivations lead to requirements for the design rules and show how the shear stress may reduce the ultimate bending capacity.


A new fracture mechanics theory of orthotropic materials like wood

T.A.C.M. van der Put

(Series 2006, nr. 4)

A new, fracture mechanics theory is derived based on a new orthotropic isotropic transformation of the Airy stress function, making the derivation of the Wu-“mixed mode I – II” fracture criterion possible, based on the failure criterion of the flat elliptic crack. As a result of this derivation, the right fracture energy and theoretical relation between mode I and II stress intensities and energy release rates are obtained.

This is published in:

Engineering Fracture Mechanics 74 (2007) 771-781


Derivation of the bearing strength perpendicular to the grain of locally loaded timber blocks

T.A.C.M. van der Put

(Series 2006, nr. 3)

The theory and new developments of this project are now published by two articles:


1. “Derivation of the bearing strength perpendicular to the grain of locally loaded timber blocks” in Holz als Roh- und Werkstoff 66, (2008), issue 6 p. 409

Based on the equilibrium method of plasticity, the theoretical explanation of the bearing strengths of locally loaded timber blocks is given in the Appendices. The result of the numerical construction of the slip-lines can precisely be represented by an analytical function of a logarithmic spiral that can be shown to be the exact solution. This function in one variable can be given in the power law form leading to a theoretical and experimental value of that power of 0.5.

This power representation of the stress spreading model of the strength increase, by confined dilatation, provides simple rules for the code and a simple design method that precisely matches to the data in all circumstances and loading cases and explains the apparent contradictory test results of Suenson, the Eurocode, the French rules, Graf, Korin and Augustin et al. and, as shown before, explains other comparable loading cases as e.g. by pin dowel connections


2. “Correction of the Eurocode 5 design rules for partial compression perpendicular to the grain” in: Wood Science and Technology (2010).

where a derivation is given of the empirical Madsen equation which is the basis of the proposed design rules for locally loaded beams and blocks of Eurocode 5.  The explanation of the meaning of the constants leads to an extended Madsen equation, which is shown to apply approximately for thin bearing blocks only.  Based on the plasticity theory the theoretical explanation of the strength data is given. It is shown when the design rules are up to a factor 6 too conservative and when too unreliable. It also is shown why the conclusion of no influence of the dimensions and depth of the test specimen on the strength is opposed by other investigations and only applies for the special test specimen dimensions of the Eurocode investigation. It further is demonstrated that the theory precisely explains and fits to all known data and provides a very simple reliable design rule for the Eurocode (given in Appendix B). The analysis further shows that design rules for bearing blocks don’t apply unconditionally for support stresses on beams. This design has to be based on the failure criterion for combined stresses showing that (except for unwanted early local failure due to under dimensioned bearing plates) the shear strength is determining for failure as also is shown to be the cause of failure of the Eurocode test beams.


Discussion and extension of CIB-W18/39-12-6 on bearing strength

T.A.C.M. van der Put

(Series 2006 nr. 2)

The theoretical explanation of the results of the CIB-W18/39-12-6 publication is lacking and is given and discussed here. The theory was published long ago and is applied in many reports of the Stevin Laboratory as e.g. in [3], where it is shown to be the only possible theory to explain the very high embedding strengths of particle board in compression.

The theory also is published in CIB-papers e.g. in [1] and in a report for the CIB-Stability Committee and more recently in [2], where it a was shown that the theory fully and precisely explains the data of Ballerini of [4] and the Karlsruhe data of joints with one dowel.

This publication is replaced by the publications mentioned at Series 2006 nr. 3


The energy approach of fracture of beams by joints loaded perpendicular to the grain

T.A.C.M. van der Put

(Series 2006 nr. 1) [download]

In the CIB-W18 publication: CIB-W18/33-7-7, the full derivation of the √n/n c    -factor of the Compliance method was not given and also the derivation of the general compliance equation of joints in the middle of the beam, including the influence of the change of the normal force at a crack increment, was not given but only was send afterwards to the interested people.

Both derivations are given here now as supplement on the CIB-W18/33-7-7 publication and it is important to know that the theory explains precisely all known data at that time as for instance of the investigations of the Karlsruhe University.

See also the publications mentioned at Series 2006 nr. 3


Rheology and transformations of polymers like wood

T.A.C.M. van der Put

(Series 2005 nr. 3)

A new theory of:

Solidification, Nucleation, Glass transition, Annealing, Diffusion, Rouse/Zimm spectra, Power law, Reaction order, etc., and determination of aging and decomposition processes in wood.

Parts of this theory are (or will be) published in separated articles. The first two are:


1. “Theoretical derivation of the WLF- and annealing equations” in:

Journal of Non-Crystalline Solids 356 (2010) 394–399

where, based on the deformation kinetics approach, the theoretical derivation is given of the empirical WLF-equation of the time–temperature equivalence. The same is done for annealing at glass transition. The derivation provides a general theory for any loading history and replaces the inconsistent free volume model.


2. A consistent theory of nucleation and growth in: Journal of crystal growth

where the classical nucleation and growth model is shown to be inconsistent. Therefore a new theory is derived, showing that nucleation is just a common example of the kinetic theory of transport processes, with a special property of the activation volume parameter.


A new consistent theory of fracture mechanics of wood

T.A.C.M. van der Put

(Series 2005, nr. 1.1)

Overview of the derivation of a theory of fracture mechanics; the softening curve with the explanation of the measurements; the power law; the mixed “mode I – II” interaction equation; the energy method of notched beams and of joints loaded perpendicular to the grain; a new Dugdale model; the right crack growth theory; the corrected fictitious crack model with the right cohesive stress and the right fracture energy; the volume effect of wide angle notches and of the stress intensity and fracture energy; the orthotropic-isotropic transformation of the Airy function; the relation between  the mode I and mode II stress intensities and energy release rates in all directions

Parts of the theory are published as given at Series 2006-4, 2007-1 and 2.

The concluding final theory will be published in 2010 as chapter of a book on Fracture Mechanics by Nova Science Publishers


Evaluation of Embedment Strength

A.J.M. Leijten

(2004) [download]

Data for:

Reliability Analyses of connections with dowel type fasteners. This study focuses on the main influencing independent parameters of the embedment strength being; the timber density and diameter of the fastener. This information can be used to be incorporated in model design codes and to feed probabilistic design models of timber connections.