This document analyzes the buckling response of a frame system composed of a cross-laminated timber column connected to a steel beam. The system is modeled as having two degrees of freedom, with springs representing the bending stiffness of the column and axial stiffness of the beam. A Newton-Raphson algorithm is used to iteratively solve the nonlinear equilibrium equations as the load is incremented. The effects of varying the load magnitude, initial imperfections, and beam stiffness are examined. Increasing the load or imperfections reduces structural stiffness, while increasing beam stiffness enhances the overall strength of the system.

1. BUCKLING RESPONSE OF FRAME SYSTEM WITH 2
DEGREES OF FREEDOM.
Emmagnio Desir
Departmento de Ingeniería en Obras Civiles, Universidad de Santiago de Chile
emmagnio.desir@usach.cl
July 11, 2021
Abstract
In order to promote sustainable and ecological construction, wood is a very popular construction ma-
terial. In recent years, much progress has been made in the study of the physico-chemical characteristics
and mechanical properties of wood. In this paper we are interested in the non-linear response for a buck-
ling test of a 2 DOF system formed of a tubular metal beam and a Cross Laminate Timber (CLT) made
of Radiata pine[1]. The main work to be done is to evaluate the eect of the stiness of the beam on the
buckling response of the column, and then from a Newton Raphson algorithm we have to do the iterative
analysis of the system by step of loading. The main objective of this paper is to check the nonlinear
response of the system and the inuence of the beam on the column to nally be able to compare the
convergence which results from the analysis with the algorithm of Newton and the convergence which
derives from the nite element analysis produced with ANSYS APDL. The critical buckling load derived
from the equilibrium equations of the system makes it possible to calculate the displacement as function
of the initial imperfections of the system. Beam stiness has been shown to control column buckling
length. In conclusion, in a system with two DOFs the nonlinear response depends on the mechanical
properties of the materials which compose it and also on the overall stiness which is the sum of the
stinesses of the elements which compose the system. This study presents a broad perspective in the
analysis of structural systems composed of mixed materials with 2 DOF
1 Introduction
Slender structures deform or break when the maximum load they can support is exceeded, hence the notion
of critical load. According to Euler's buckling theory, the critical load is more or less a ratio between the
bending stiness and the length of the slender part. In the case of this present study, we have a system
composed of a CLT post of 3 layers of radiata pine wood should be considered with a longitudinal Young's
modulus equal to 11.29 GPa and Poisson's ratio equal to 0.18. The cross sectional area is composed of 3
layers of 150 mm x 30 mm (width x thickness) given a total thickness of 90 mm, the column is 3 meters high
and it is considered as bi-articulated at the upper end. The upper end of the column connects a tubular steel
beam with external diameter 70 mm and wall thickness 2.5 mm, the beam has a length of 2 m, one of the
ends of the beam is articulated with the column and embedded in the other end. For the steel strut consider
a Young's modulus of 210 GPa, and a Poisson's ratio of 0.3. See table1 shows the dierent mechanical
characteristics of CLT and steel.
Table 1: Properties of the pine wood
Elasticity modulus Poisson ratio Shear modulus
E11 : 11.29 GPa ν12 : 0.18 G12 : 0.71 GPa
E22 : 11.29 GPa ν13 : 0.18 G13 : 0.071 GPa
E33 : 0.38 GPa ν23 : 0.18 G23 : 0.071 GPa
Where:
E parallel =
1
30 ∗E longitudinal
G longitudinal =
1
16 ∗E parallel
G parallel =
1
10 ∗G longitudinal
1

2. 1.1 The structural system (Wood-Steel) with 2DOF
Figure 1: Column buckling problem, a) perfect structure, b) structure with imperfection at the centre of the column,
and c) structure with imperfection at the top of the column.
1.2 Geometrical and mechanical properties of the beam.
Figure 2: Steel beam cross section.
E = 210 GPa
Lb = 2 m
Area:
A =
π
4
× (D2
− (D − 2t)2
) = 530.14mm2
(1)
Axialstiffness : K2 =
A × E
Lb
= 55.66KN.mm (2)
1.3 Geometric and mechanical properties of the CLT
[2]
hi = 30mm
bi = 150mm
ai = 30mmm
L = 3000mm htot = 90mm
Ai = 4500mm2
Where i is the number of the layer i = 1, 2, 3
2

3. Figure 3: Calcutation properties of the column, using 3 layers CLT section.
Atotal =13500mm2
1.4 Mechanical properties of the CLT [2]
Inertia
I =
bi × htot3
12
= 9112500mm4
(3)
Eective rgidity by shear analogy method
EIeff =
n
X
i=1
Ei · bi ·
hi3
12
+
n
X
i=1
Ei · Ai · ai2
= 99198000KN · mm2
(4)
Eective shear stiness
GAeff =
a2
( h1
2·G1·b ) +
n−1
X
i=2
hi
Gi · bi
!
+ ( hn
2·Gn·b )
# = 1165KN (5)
Where : i is the number of the layer and a = htot − h1
2 − hn
2 ; n = 3 layers thus a = 60mm
Stiness of the CLT
K1 = (π2
)
EIeff
4 ∗ L
= 81.58KN.mm (6)
2 Methods
To balance the forces acting on the structure we used the energy method[3] Πep = U − W, where U is the
set of internal forces and W is the set of forces caused by external forces. The resistance of the structure
to the forces is represented by a system with a torsional spring (K1) in the middle of the column (bending
stiness) and a longitudinal spring (K2) at the top of the column which represents the rigidity of the steel
beam (axial stiness).
For each spring we have found an equilibrium equation from which at the end we have a system of
nonlinear equations with initial imperfection angles α1 and α2, angles of rotation after the deformation θ1
and θ2 and lateral displacement ∆U. These components constitute the unknowns of the set of equations
formed. Newton Raphson's algorithm was implemented to solve the system of equations and predict the
lateral displacement of the column using an iterative analysis of 50 steps. Finally we did a nite element
analysis simulation with ANSYS APDL[4], we compared the results of the 2 processes, more information in
the results and discussions section.
3

4. 2.1 Hypotheses
[2]
Figure 4: Real behaviour of the structure where θ1=θ2
Figure 5: A) The frame B)The system stiness with one torsional spring and one linear spring C) bar 2 D) bar 1.
Internal energy
U(θ1; θ2) =
1
2
K1((θ2 − α2) − (θ1 − α1))2
+
1
2
K2((
L
2
sin θ1 −
L
2
sin α1) + (
L
2
sin θ2 −
L
2
sin α2))2
(7)
External energy
W(θ1; θ2) = P((
L
2
−
L
2
cos θ1) − (
L
2
−
L
2
cos α1) + (
L
2
−
L
2
cos θ2) − (
L
2
−
L
2
cos α2)) (8)
Potencial energy of the system [3, postnote]
Π(ep) = U − W (9)
2.2 Out of balance forces (Residuals)
∂Π
∂θ1
= −K1(θ2 − θ1 + α1 − α2) + K2
L
2
(sin(θ2) + sin(θ1) − sin(α1) − sin(α2)) · cos(θ1) − p
L
2
sin(θ1) (10)
4

5. ∂Π
∂θ2
= −K1(θ2 − θ1 + α1 − α2) + K2
L
2
(sin(θ2) + sin(θ1) − sin(α1) − sin(α2)) · cos(θ2) − p
L
2
sin(θ2) (11)
After some mathematical manipulation:
∂Π
∂θ1
= R1and
∂Π
∂θ2
= R2 (12)
R1 = −K1(θ2 −θ1 +α1 −α2)+K2(
L2
8
·sin(2 sin θ1))+K2 ·
L2
4
·cos θ1 ·sin θ2 −sin α1 −sin α2 −
PL
2
·sin(θ1) (13)
R2 = K1(θ2−θ1+α1−α2)+K2·
L2
8
sin(2·θ2)+K2·
L2
4
·cos(θ2)·(sin(θ1)−sin(α1)−sin(α2))−
PL
2
·sin(θ2) (14)
The Jacobian of the system:
K1 + K2
L2
4 − P L
2 −K1 + K2
L2
4
−K1 + K2
L2
4 K1 + K2
L2
4 − P L
2
!
·
θ1
θ2
=
0
0
(15)
det[M] = ((K1 + K2
L2
4
) × (K1 + K2
L2
4
− P
L
2
) − (−K1 + K2
L2
4
) × (−K1 + K2
L2
4
)) (16)
Buckling critical load
Pcr =
4 · K1
L
(17)
3 Buckling analysis of the CLT column
[2]
3.1 Case A
Elastic buckling load estimation using Euler's buckling formula[5] and considering a solid isotropic homo-
geneous timber column with Elong = 11.29 GPa and the inertia of the CLT secrion, considering both ends
hinged K = 1.
Pcr =
π2
EI
(KL)2
= 112.82KN (18)
3.2 Case B
Elastic buckling load estimation using Euler's buckling formula and considering the corrected (EI)e, as
proposed in the CLT Handbook using the gamma or shear analogy method.
Pcr =
π2
EIeff
(KL)2
= 108.78KN (19)
5

6. 3.3 Case C
Elastic buckling load estimation using Euler's buckling formula and considering the corrected (EI)ef f , as
proposed in the CLT Handbook using the gamma or shear analogy method, and the eective shear coecient.
Pe, ν =
Pe
1 + K·P e
GAeff
= 81.85KN (20)
Where: Pe =
π2
·E05·Ieff
(Ke·L)2
E05 = 0.82 · E = 9.26GPa and Ieff = EIeff
Emean = 8786359.6mm4
4 Numerical solution
The solutions of the nonlinear quadratic equation system were found from an incremental solving method
developed with Newton Raphson's algorithm in MATLAB software. This process consists of incrementing i
+ 1 loading step the critical load derived from the equilibrium equations. To do this we dened a tolerance
which is close to zero which we compared with the determinant of the Jacobian matrix[6], this comparison
shows us the limit when the stiness tends towards zero to stop the iterative process. From the vector of
residual moments and the vector of external forces we have to determine the convergence of the structure.
The non-linear response of the structure is in accordance with the deformation shown in gure Figure 1b.
Considering a maximum deformation at the height of the column (typical case of a bi-articulated column), if
we divide the height into two equal sections we will see that the response forces vs displacement are the same
if not that the direction of displacements has changed . The displacement of the lower sections is strictly
positive and that of the upper section is strictly negative. This is in accordance with our forecasts see gure
Figure 4 because the lateral displacement according to the rst mode of buckling of the column is done at
mid-height and creates two angles of rotation θ1 and θ2 which are respectively positive and negative. We
can write ||θ1|| = ||θ2||.
4.1 Eect of load smaller than the critical load with varios imperfection.
To study the nonlinear characteristic of the structure we have applied a load which is a little smaller than
the critical load it can withstand is a factor of 0.98 and we apply dierent levels of imperfections. The
curves show that after passing through the non-linear zone, the lateral displacement of the structure remains
rectilinear until it reaches the critical point.
Figure 6: Force Vs Displacement
when F = 0.98 ∗ Pcr and α = 0.1.
Figure 7: Force Vs Displacement
when F = 0.98 ∗ Pcr and α = 0.01.
Figure 8: Force Vs Displacement
when 0.98 ∗ Pcr and α = 0.001.
6

7. 4.2 Solution convergence study
Figure 9: Convergence of the system
4.3 External load sensitivity analysis
In this section we analyze the inuence of the variation of the load applied on the structure. To do this we
applied an initial imperfection of 0.001 rad and we varied the load with factors: 1.5, 2 and 4. The non-linear
analysis shows that when the critical load increases the rigidity of the system decreases and leads to the ruin
of the system as shown by the curves below. Mathematically we can explain this phenomenon by calculating
the determinant of the Jacobian matrix, if the result is zero and the rigidity of the system is also zero.
Figure 10: Force Vs Displacement
when F = 1.5 ∗ Pcr and α = 0.001.
Figure 11: Force Vs Displacement
when F = 2 ∗ Pcr and α = 0.001.
Figure 12: Force Vs Displacement
when 4 ∗ Pcr and α = 0.001.
4.4 Imperfection sensitivity analysis
In the previous section we have already talked about the eect of increasing the critical load, now we will
increase the load by a factor of 1.5 and vary the initial imperfection from the following values: 0.1, 0.01,
0.001, 0.0001, 0.00001 and 0.000001. By varying the initial imperfection of the structure we can see that the
non-linear behavior also changes so that with a greater imperfection the structure behaves as being totally
nonlinear, moreover when we decrease the imperfection the linearity of the structure increases considerably.
7

8. Figure 13: Force Vs Displacement
when F = 1.5 ∗ Pcr and α = 0.1.
Figure 14: Force Vs Displacement
when F = 1.5 ∗ Pcr and α = 0.01.
Figure 15: Force Vs Displacement
when 1.5 ∗ Pcr and α = 0.001.
Figure 16: Force Vs Displacement
when F = 1.5 ∗ Pcr and α = 0.0001.
Figure 17: Force Vs Displacement
when F = 1.5 ∗ Pcr and α = 0.00001.
Figure 18: Force Vs Displacement
when 1.5 ∗ Pcr and α = 0.000001.
4.5 Steel strut stiness sensitivity analysis
In the idea of studying the inuence of the steel beam on the CLT column we varied the rigidity of the beam,
to do this we multiplied the stiness of the beam by factors such: 2, 5 and 10. This variation shows that
if stiness of the beam is increased, the bearing capacity of the structure also increases (global stiness).
Mathematically that's mean the determinant of the Jacobian matrix increases.
Figure 19: Force Vs Displacement
when F = 1.5 ∗ Pcr, α = 0.00001 and
K2 = 2 ∗ K2.
Figure 20: Force Vs Displacement
when F = 1.5 ∗ Pcr, α = 0.00001 and
K2 = 5 ∗ K2.
Figure 21: Force Vs Displacement
when F = 1.5 ∗ Pcr, α = 0.00001 and
K2 = 10 ∗ K2.
5 Finite element analysis
The simulation of the structural behavior of the system was carried out from ANSYS APDL[4], several
analyzes such as linear statics, eigen-buckling and non-linear statics were carried out with dierent level of
geometric imperfection in order to explores dierents response possibilities of the system. We considered a
system with 6 degrees of freedom including three of displacement and three of rotations.
8

9. The mechanical characteristics of the structure which are dened in section 1 have been inserted into
structural elements such as: for the CLT we used the shell element 181 and for the steel beam we used the
beam element 188. A connection contact was used to materialize the articulation between the beam and the
column.
The predictions made by during the numerical analysis are veried, the dierent no-linear responses of the
structure according to the assigned imperfections are veried. The deformation of the column is consistent
with the Force VS Displacement curves produced in MATLAB, it has been shown that despite the stiness
of the steel beam which inuences the structural behavior of the column, the maximum displacement is at
mid-height. of the latter.
5.1 Eigenvalue buckling load
The linear static nite element analysis of the structure allows to nd the critical load and the lateral
displacement of the structure quoted Figure 19. Without considering the geometric imperfections, the
behavior of the structure and completely linear.
Figure 22: The elastic buckling loads of the system.
5.2 Non-linear geometric analyses
By applying geometric imperfections to the structure (geometric non-linearity), it can be seen that the force
vs displacement response curves of the structure show that with a greater imperfection the structure passes
rapidly from the elastic domain to the non-elastic domain contrary to reduction of imperfection. These
analyzes verify the predictions made in the numerical analysis section.
Figure 23: Force Vs Displacement
when initial imperfection = L/100
Figure 24: Force Vs Displacement
when initial imperfection = L/1000
Figure 25: Force Vs Displace-
ment when initial imperfection =
L/10000
9

10. 6 Comparative study
In this section we will compare the dierent results derived from the dierent analyzes performed on the
system to nally bring conclusions.
6.1 Elastic buckling
Critical load that can carry he CLT column
Pe, ν =
Pe
1 + K·P e
GAeff
= 81.85KN (21)
Critical load that can carry the system
Peigenbuckling = 100.56KN (22)
Theoretical ones and discuss your observations
K =
s
π2 × EI(eff)
Pcr × L2
= 1.08 (23)
The critical load derived from the single column derived from equation 21 is 81.85 KN, while that found
from linear static nite element analysis is 100.56 KN. In the rst case, the column being calculated alone
there is only the stiness of the CLT that acts, in the second case the structure has been calculated in its
entirety so the axial stiness of the beam has been taken into account. By adding the rigidity of the CLT
column and the axial stiness of the beam if we compare the sum with the critical loads that the structure
can withstand in both cases, we can conclude that the column alone oers 80 percent of the load-bearing
capacity of the structure.
By comparing the critical load obtained from the linear static analysis at the nite element and the
formula of EULER using the eective stinesses of the CLT, we derived an eective length factor of 1.08.
since the CLT column is considered to be bi-articulated, the theoretical eective length factor is 1, hence
the dierence between these two factors is trivial. 1 = 1.08.
6.2 Non-linear buckling analyses
Comparing the nite element analysis gure 26 and the numerical analysis gure 27 dier in the displace-
ment of the structure which is approximately 30 percent greater in the numerical results. The nonlinear
behavior of the structure is the same case. The variation of the rigidity of the metal beam has taught us that
by increasing the rigidity of the beam we also increase that of the system. The variation of the geometric
imperfection shows us that the greater the imperfection, the greater the nal deformation.
Figure 26: ANSYS response Figure 27: MATLAB response
10

11. 7 Conclusion
In this paper we have investigated the eect of the stiness of a steel beam on the buckling behavior of a CLT
by means of a 2 DOFs system. The characteristics of the wood which constitutes the column of CLT were
determined from the shear analogy method, we determined the eective rigidity of the CLT from which we
determined the critical load of the column alone which was subsequently compared with the critical load of
the entire system. The eects of the stiness of the steel beam on the overall stiness of the system against
buckling were analyzed.
To analyze the system we used the energy method to derive the residual equations from the balance of
internal and external forces. The equations derived from the analysis were solved using Newton Raphson's
algorithm with 50 load steps dened at the start and at the end we have 101 iterations. The predictions of
the numerical analyzes made with MATLAB were validated with the nite element analysis carried out with
the ANSYS software, an element size of 50 mm was chosen for a structure of 3m x 0.15mx 0.9 m (Length,
width and thickness). Static linear and static nonlinear analyzes with the arc length method were carried
out to determine the critical load of the structure and the nonlinear behavior of the structure.
In addition, we found the convergence of the structure and explored dierent possibilities of variation
of the characteristics. We varied the critical load, geometric imperfection and stiness of the steel beam.
These dierent changes have allowed us to explore the nonlinear behavior of the system on all possible forms.
We believe that the present modelling strategy represents a robust platform for further investigation on
the structural response of CLT structures with a particular view to elaborate more accurate guidelines for
structural design. This will be the subject of future works.
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