Modern timber design approaches for traditional Japanese architecture: analytical, experimental, and numerical approaches for the Nuki joint

Abstract

This work fully investigates a specific timber joinery connection via experimental, analytical, and numerical methods. The selected joint is the Nuki joint: a mortised column with through-beam tenon. The experimental approach takes advantage of digital fabrication to reduce variations introduced by hand fabrication while the analytical approach builds on state-of-the-art embedment stress models. Material tests are used to calibrate the non-linear finite element model and analysis of the connection. Furthermore, the difference in behavior between prototypes of various beam thicknesses is examined across analysis approaches. This work not only sets out a workflow for digital fabrication, physical testing, and structural analysis for more complex joinery geometries, but also discusses the challenges and relevance of its application towards a reference library of joinery connections for modern timber construction.

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Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019
Form and Force
7 – 10 October 2019, Barcelona, Spain
C. Lázaro, K.-U. Bletzinger, E. Oñate (eds.)
Copyright © 2019 by FANG, MORADEI, LANDEZ, BRÜTTING, FISCHER, SHAO, SHERROW-GROVES,
FIVET, and MUELLER.
Published by the International Association for Shell and Spatial Structures (IASS) with permission.
Modern timber design approaches for traditional Japanese
architecture: analytical, experimental, and numerical approaches
for the Nuki joint
Demi FANG*a, Julieta MORADEIb, Jan BRÜTTINGc, Aliz FISCHERb, Daniel K. LANDEZa,
Benshun SHAOb, Nick SHERROW-GROVESb, Corentin FIVETc, Caitlin MUELLERa
a Massachusetts Institute of Technology (MIT), Cambridge, MA, USA
*Corresponding author: dfang@mit.edu
b Arup, San Francisco, CA, USA
c Swiss Federal Institute of Technology (EPFL), Fribourg, Switzerland
Abstract
This work fully investigates a specific timber joinery connection via experimental, analytical, and
numerical methods. The selected joint is the Nuki joint: a mortised column with through-beam tenon.
The experimental approach takes advantage of digital fabrication to reduce variations introduced by
hand fabrication while the analytical approach builds on state-of-the-art embedment stress models.
Material tests are used to calibrate the non-linear finite element model and analysis of the connection.
Furthermore, the difference in behavior between prototypes of various beam thicknesses is examined
across analysis approaches. This work not only sets out a workflow for digital fabrication, physical
testing, and structural analysis for more complex joinery geometries, but also discusses the challenges
and relevance of its application towards a reference library of joinery connections for modern timber
construction.
Keywords: timber, joinery, connections, Nuki joint, physical testing, finite element analysis, rotational stiffness
1. Introduction
Timber is one of the oldest known building materials in the world. In contrast to Western analogs, Asian
timber construction is characterized by its high degree of specialization in joinery and by connections
with interlocking geometries. The use of joinery connections largely faded with the introduction of metal
fasteners that continue to dominate post-industrialization timber connections.
Timber structures have seen a revival in recent decades due to growing awareness of timber’s reduced
environmental footprint compared to typical building materials such as steel or concrete. Buildings are
responsible for a third of global carbon emissions, and substituting wood for other materials could
prevent 14-31% of those emissions [1]. Current mass timber construction uses steel connections (e.g.
nail plates) which often results in a one-life span design of structures. The use of interlocking timber
joinery can allow for more sustainable, non-destructive disassembly of timber structures, as
demonstrated by historic precedents such as the Ise Shrines (Japan, 4 BC) [2].
The budding capabilities of today’s digital fabrication technology offer an opportunity to revitalize the
use of interlocking joinery in modern timber construction. Examples featuring 21st-century joinery
connections include the Yusuhara Bridge Museum (Japan, 2010) by Kengo Kuma and Associates, the
Tamedia Office Building (Switzerland, 2013) by Shigeru Ban Architects, and the Writers Theatre (USA,
2016) by Studio Gang Architects.

Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019
Form and Force
2
While there are benefits to reintroducing joinery connections into modern construction from the
perspectives of sustainability, constructability, and fabrication capabilities, the main obstacle is to
characterize and codify the mechanics of these connections. This paper is part of an ongoing
investigation into a workflow aimed to build up centralized knowledge on joint behavior. In this paper
the Nuki joint (Figure 1) is selected as an example to demonstrate the workflow. The simple geometry
of the Nuki – a beam piercing through a mortised column – enables a clearer understanding of its
mechanics, facilitating the calibration between analysis and experimental testing. This paper focuses on
predicting the rotational stiffness and behavior of the joint. The key steps of the workflow are 1) the
experimental testing of digitally fabricated specimens, 2) the analysis of the interlocking joint’s behavior
using principles of structural mechanics, and 3) the numerical simulation employing non-linear finite
element analysis (FEA). The future aim of these calibrations is to enable the characterization of joints
with more complex geometries and variations.
Figure 1. Nuki joint, a simple joinery connection studied in this work.
1.1 Literature review
Fascination with joinery connections from a historical and cultural point of view is evident from a
number of books and studies cataloguing their origins, evolutions, and geometries [3], [4]. The analyses
of timber joint mechanics range across analytical, numerical, and experimental methods. Some focus on
validation between experimental and numerical methods [5], [6], while others use a variety of analytical
models that are compared against physical tests [7]–[10]. A key behavior in the analysis of joinery
connections is embedment [11], which refers to the compression of one joint element (or part) into
another, a behavior that arises from the reduced stiffness of wood perpendicular to the grain. Building
upon embedment theory, Kitamori et al. [12] showed that additional embedment length outside the direct
contact region can significantly contribute to rotational stiffness. Komatsu et al. [12] considered an
elastoplastic material behavior where only a reduced material stiffness is taken into account when the
yield-point is reached, enabling an analytical description of the typical bi-linear moment-rotation
behavior of semi-rigid joinery connections. Recent works have applied his theory to different joinery
connections [13], [14]. As for the Nuki joint in particular, Chang et al. [15] compared analytical and
experimental models but do not discuss plastic effects of the joint behavior. Guan et al. [6] compared
experimental and numerical models for Nuki joints with wedges.
In summary, the state of the art for analyzing interlocking joinery through experimental testing,
mechanical analysis, and numerical simulation have yet to be synthesized and calibrated. This work
particularly builds on own previous work by Fang and Mueller [7], Moradei et al. [16], and Fang et al.
[17]. This work presents new results on digitally fabricated specimens made of glue-laminated timber
as well as numerical results to compare with the analytical and experimental approaches.

Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019
Form and Force
3
2. Methodology
The Nuki joint was here parameterized by beam depth Bd, beam width Bw, column width Cw, column
depth Cd, which are notated in Figure 1. For experimental testing, joint specimens were manufactured
with dimensions Bd = Cw = Cd = 8.25 cm. Two different beam widths Bw, 2.54 cm and 3.81 cm were
studied. A visual overview of the experimental and numerical testing setup is shown in Figure 2.
Figure 2. Setup, support and loading conditions of the (a) experimental testing and (b) numerical simulation.
2.1 Material properties
Glue-laminated timber elements were used for the experimental prototypes. Material testing was
conducted first to determine the material properties and to apply them in the analytical and numerical
models. Most relevant mechanical properties are summarized in Table 1.
Table 1. Mechanical properties determined for the material used in wood specimens.
Ec,0
Mean
(standard dev.)
Ec,90
Mean
(standard dev.)

c,y,90
Mean
(standard dev.)
PR90
[MPa]
[MPa]
[m/m]
[%]
13,500 (1340)
385 (89)
0.025 (0.001)
11.8%
Ec,0 compressive stiffness (modulus of elasticity) parallel to grain
Ec,90 compressive stiffness (modulus of elasticity) perpendicular to grain

c,y,90 yield compressive strain perpendicular to grain
PR90 reduced stiffness of Ec,90 after yielding
2.2 Experimental setup
Six glulam specimens for each joint geometry were digitally fabricated with a robotic CNC router. The
opening in the columns has rounded corners equal to the size of router ball end, i.e. 9.5 mm. In the test
setup, which is shown in Figure 2(a), the column element of the joint was fixed to the table, while the
beam element was attached to a hand-screw-actuated load cell of 4400 N capacity. The beam-column
connection was not press-fit, and varied in tightness. The total gap between the beam and column was
in the 0-2 mm range. A linear displacement was applied at the beam end using the load cell, and the
resulting rotation at the joint was measured by two string potentiometers.
2.3 Analytical model
The analytical model employed to prescribe the moment-rotation behavior of the joint is based on
embedment theory and uses an elastic-plastic material model. In the model, the bending moment M is

Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019
Form and Force
4
expressed in terms of the rotation angle θ using trigonometric functions and structural mechanics. For
details on the analytical model, the reader is referred to Fang et al. [17].
2.4 Numerical model
To numerically model the joint behavior, explicit finite element analysis was employed using the LS-
Dyna v10.0 software package. Figure 2(b) illustrates the numerical simulation setup, designed to capture
key aspects of the physical testing setup. The top and the bottom of the column are restrained from
displacement and rotation. The surfaces where the loading clamp is installed are rigidly connected
through their nodes. Those nodes are then coupled to the loading steel rod with a hinge. The loading is
applied to the rod via a prescribed displacement that allows capture of the failure mechanism.
Beam and column are discretized with 45,000 fully integrated solid elements in total. A transversely
anisotropic material model (LS-Dyna MAT 143) is employed for both components in the FEA analysis
[18], [19]. This model allows to capture physically nonlinear effects (e.g. plasticity and damage) as well
as the anisotropy between grain parallel and grain perpendicular behavior of the wood.
3. Results
The moment-rotation behavior obtained from all three models are summarized in Figure 3. The
following subsections give detailed descriptions of each method’s results.
Figure 3. Summary of results from experimental, analytical, and numerical approaches. (a) Moment-rotation
behavior for specimens of Bw = 2.54 cm; (b) moment-rotation behavior for specimens of Bw = 3.81 cm.
3.1 Experimental Results
Figure 4. Rotational stiffness testing of prototypes. Progressive video frames from an example specimen.
0
400
800
1200
0 0.05 0.10 0.15 0.20
Moment [Nm]
Rotation [rad]
Experimental Numerical Analytical
Moment [Nm]
Rotation [rad]
00 0.05 0.10 0.15 0.20
400
800
1200
(a) (b)
Bw = 2.54 cm (1.00 in.) Bw = 3.81 cm (1.50 in.)

Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019
Form and Force
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Figure 4 shows a sequence of images taken during the experimental testing of one joint specimen. From
the measured test data (Figure 3, grey) it is apparent that in general Nuki joints show a two-phase
moment-rotation behavior: a first initial (linear elastic) stiffness and a second stiffness post-yielding.
The change between the two phases happens gradually. Tests ended when the load cell reached its
capacity. For nearly all specimens of Bw = 3.81 cm, this limitation prevented the estimate of secondary
stiffness.
Variability in testing results is partially due to wood being a heterogeneous material. It can be further
influenced by changes in temperature and humidity between fabrication and testing of the prototypes,
as that affects the tightness of the fit between the beam and the column. Results could be more consistent
with a much larger number of specimens.
3.2 Analytical results
The analytical model employs an elastic-plastic material behavior which is appropriate for describing
the two-phase behavior demonstrated by the physical prototype. After calibration of the analytical model
to the experimental test for both beam widths, one unique decay coefficient α was selected as 4.5/Bd.
This parameter relates to shape of the embedment outside the direct contact between beam and column.
For α and all other parameters the reader is referred to Fang et al. [17]. The analytical model does not
include failure prediction; the calculations were manually ended at 100 and 150 mrad for Bw = 2.54 cm
and Bw = 3.81 cm, respectively. In general, after calibration the modeled rotation behavior matches the
experimental data (Figure 3).
3.3 Numerical results
The main parameters of the material model, such as Young’s moduli and strengths, were calibrated
based on the physical material testing (see values in section 2.1). Other values, such as ν = 0.39 Poisson
ratio, were estimated from literature [20]. Contact between beam and column is defined by numerical
parameters representing the contact stiffness, the friction between surfaces, and the initial gap size. In
the model, the gap size between the beam and the column was set up based on observation of the
fabrication, with 0.5 mm gap each side of the beam. At the top and bottom the gap was smaller, 0.025
mm, to ensure numerical stability. The analysis assumed no initial stress from joining the two pieces.
Figure 5. Stress in the beam perpendicular to grain (MPa) throughout the numerical simulation for the
Bw = 2.54 cm specimen.
Selected images of the progression of the numerical simulation are shown in Figure 5. The contour plot
shows the stresses perpendicular to the grain of the beam. The simulation was stopped when the analysis
showed numerical instabilities as the damage at the contact became excessive. The stress distribution is
in a good agreement with the assumption of the analytical model, and with the observations of the local

Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019
Form and Force
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behavior at the physical testing. A close-up of the connection behavior for all three methodologies is
shown in Figure 6.
Figure 6. Close-up of the joint for (a) experimental testing, (b) analytical model, and (c) numerical simulation.
3.4 Comparison of stiffness values across models
In this section, based on the two-phase moment-rotation behavior, absolute and relative stiffness values
are compared across all three models. ‘Stiffness 1’ refers to the initial stiffness, and ‘Stiffness 2’ refers
to the post-yielding stiffness. Table 2 compares the absolute stiffness values. Table 3 compares the
relative increase in stiffness with respect to the two tested beam widths. Note that Stiffness 2 values for
the experimental testing of specimen with Bw = 3.81 cm are not recorded because experimental testing
had to be halted before a post-yielding stiffness could be observed.
Table 2. Comparison of absolute stiffnesses across models.
Bw = 2.54 cm
Bw = 3.81 cm
Stiffness 1
[kN-m/rad]
Stiffness 2
[kN-m/rad]
Stiffness 1
[kN-m/rad]
Stiffness 2
[kN-m/rad]
Experimental mean
16.0
5.46
25.1
--
Analytical model
18.7
4.29
28.0
6.44
Numerical model
18.9
5.10
28.1
6.74
% difference between
analytical and
experimental*
+17%
-21%
+10%
--
% difference between
numerical and
experimental*
+18%
-7%
+11%
--
* signed percent error assuming experimental mean as the ‘expected value’
The results in Table 2 demonstrate that the analytical and numerical models overestimate the initial
stiffnesses (Stiffness 1) by about 18% and 11% for the two different beam widths respectively.
Table 3. Comparing effects of beam width variation across models.
% increase in stiffness between Bw = 2.54 cm and Bw = 3.81 cm
Stiffness 1
Stiffness 2
Experimental
57%
--
Numerical
49%
32%
Analytical
50%
50%

Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019
Form and Force
7
The results in Table 3 demonstrate that in the experiments there is a slightly larger relative increase in
initial stiffness from increasing beam width than numerically and analytically predicted.
4. Conclusion and future work
The rotational stiffness of Nuki joints was tested with digitally fabricated specimens made of glue-
laminated timber. The rotational behavior was modeled both analytically and numerically, and these
models were calibrated with the experimental results through material testing. While neither model
captures wood behavior at a micro level, calibrating key parameters allowed analytical and numerical
models to capture the overall behavior of the joints. For behavior at failure, the models were more
limited: the analytical model does not capture failure by definition, while the numerical model resulted
in numerical instabilities as the local failure at the contact surface became excessive. However, the
accuracy of failure prediction is not crucial because for design recommendations, the region of initial
stiffness will be of primary interest.
While digital fabrication allowed quick and precision woodwork, the variability of the wood due to
change in humidity or temperature over time did not allow the connection fit to be precise. In practice,
majority of the typical connections are press-fit, eliminating this issue. For non-press-fit connections,
more care needs to be taken to the consistency of the fabrication environment for precise fit. Design
values also need to be defined considering this variability. In the future, further experimental testing
could increase the number of specimens and improve the consistency of results against both models to
verify that all parameters result in model predictions close to expected behavior.
This paper demonstrated analytical and numerical models effectively predicting rotational stiffness
behavior and absolute values within 20% of experimental results after material calibration. With the
now calibrated model parameters it is possible to analyze other joint types and geometries. MAT 143
material also proved to be suitable to numerically simulate other joint geometries, allowing the
numerical design of more complex connections where analytical models are not available. Further, the
obtained semi rigid moment-rotation behavior could be integrated into a global FEA model via rotational
springs.
Overall, the potential for both models to be used for other joinery geometries was demonstrated. This
workflow can be applied to further joinery connections in an effort to build up a centralized knowledge
base of joinery connections, empowering designers to apply these historically-inspired, sustainable
connections more widely in timber construction.
Acknowledgements
The authors would like to acknowledge Nordic Structures and Bensonwood for providing the prototype
material, Autodesk BUILD Space for the fabrication facilities, Arup’s internal research grant for funding
JM, AF, BS, and NSG, and Stephen Rudolph at MIT CEE for assisting the physical tests.
Author Contributions
DF, JB, and CM developed the analytical model. DKL and DF carried out digital fabrication and
physical testing. AF, BS, and JB built and analyzed the numerical model. DF and JM provided overall
coordination and management. NSG, CF, and CM supervised.
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