Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 3, pp. 581-591, Warsaw 2015
DOI: 10.15632/jtam-pl.53.3.581

ELASTIC BUCKLING OF A TRIANGULAR FRAME SUBJECT TO

IN-PLANE TENSION

Krzysztof Magnucki

Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland

e-mail: krzysztof.magnucki@put.poznan.pl

Szymon Milecki

Institute of Rail Vehicles, TABOR, Poznań, Poland

The paper is devoted to elastic buckling of a symmetrical triangular frame under tensile
in-plane load. Threemathematical models of the triangular frame are formulated. The first
model deals with the pre-buckling state, the second one with the in-plane buckling state,
and the third onewith the lateral buckling state of the frame.The FEM-numericalmodel of
the frame is formulated and the critical loads are calculated. The comparison of the results
obtained in the analytical and numerical-FEManalysis is presented in tables and graphs in
figures.

Keywords: triangular frame, elastic buckling, critical load, mathematical model

1. Introduction

The theoretical basis of buckling problems of structures is elaborated in many papers and mo-
nographs. Horne and Merchant (1965) described the problem of stability of frames. Thompson
and Hunt (1973) presented a general theory of elastic stability of structures. Budiansky (1974)
presented the theory of buckling and post-buckling behaviour of elastic structures. Chen and
Liu (1987) described the theory of stability and implementation for structures. Bažant and Ce-
dolin (1991) presented an extensive review of stability problems of structures: columns, frames,
thin-walled beams, plates and shells. Simitses andHodges (2006) presented the fundamentals of
structural stability for columns, beams, rings and arches. Van der Heijden (2008) described the
elastic stability of solids and structures formulated and studied by professorW.T.Koiter. Kaveh
and Salimbahram (2007) presented a methodology for efficient calculation of buckling loads for
symmetric rectangular frame structures. Şakar et al. (2012) presented FEM study of dynamic
stability of a multi-span rectangular frame subjected to periodic loading.

The problem of triangular frame stability has been presented in literature only in several
publications. For example, Magnucki and Milecki (2012) presented flat elastic buckling of the
brake triangle in freight wagons, while Sobaś (2010) described the strength problems, especially
the fatigue strength of the brake triangles.

The subject of theoretical study presented in this paper is a symmetrical triangular frame
under tensile in-plane load (Fig. 1). The vertex C of the frame is fixed, whereas the vertexes A
andB are simply supported in the plane of the triangular frame.

The arms of length L1 are connected with a cross-beam of length L2. The cross-section of
the arms is rectangular, while the cross-section of the cross-beam is a circular ring (Fig. 2).



582 K.Magnucki, S. Milecki

Fig. 1. Scheme of the symmetrical triangular frame with the load

Fig. 2. Cross-section of the arms – 1 and cross-beam – 2

2. Mathematical models

2.1. Pre-buckling state

A half of the symmetrical triangular frame with the load F, internal normal force N2 and
bendingmomentM2 for the pre-buckling state is shown in Fig. 3.

Fig. 3. Scheme of a half of the frame for the pre-buckling state

The normal forces N(xi) and bendingmoments Mb(xi) in the elements of the frame:

• arm – 1

N(x1)=N1 =N2cosα+
1

2
F sinα Mb(x1)=

(

N2 sinα−
1

2
F cosα

)

x1+M2 (2.1)

• cross-beam – 2

N(x2)=N2 Mb(x2)=M2 (2.2)



Elastic buckling of a triangular frame subject to in-plane tension 583

The elastic strain energy of the frame is

Uε =
1

2EA1

L1
∫

0

[N(x1)]
2 dx1+

1

2EJ
(1)
z

L1
∫

0

[Mb(x1)]
2 dx1+

1

2EA2

L2/2
∫

0

[N(x2)]
2 dx2

+
1

2EJ
(2)
z

L2/2
∫

0

[Mb(x2)]
2 dx2

(2.3)

whereA1 = bc, A2 = π(d
2
1−d

2
0)/4 are the areas of the cross-sections of the arm and the cross-

beam, J
(1)
z = bc

3/12, J
(2)
z = π(d

4
1 −d

4
0)/64 – inertia moments of the cross-sections of the arm

and the cross-beam,E – Young’s modulus.
Taking into account the theorem ofMenabrea, two conditions are formulated

∂Uε
∂N2
=
1

A1

L1
∫

0

N(x1)cosαdx1+
1

J
(1)
z

L1
∫

0

Mb(x1)x1 sinαdx1+
1

A2

L2/2
∫

0

N2 dx2 =0

∂Uε
∂M2

=
1

J
(1)
z

L1
∫

0

Mb(x1) dx1+
1

J
(2)
z

L2/2
∫

0

M2 dx2 =0

(2.4)

Thus, after integration, two equations are obtained

α11
M2
L2
+α12N2 =

1

2
β1F α21

M2
L2
+α22N2 =

1

8
F (2.5)

fromwhich

N2 =
1

2

α11−4β1α21α11
α22−α12α21

F M2 =
1

8

4β1α22−α12
α11α22−α12α21

FL2 (2.6)

where

α11 =
A2L

2
2

4J
(1)
z

sinα

cos2α
α12 =1+

A2
A1
cosα+

A2L
2
2

12J
(1)
z

sin2α

cos3α

β1 =
( A1L

2
2

12J
(1)
z cos2α

−1
)A2
A1
sinα α21 =1+

J
(1)
z

J
(2)
z

cosα α22 =
1

4
tanα

Thus, the normal tension force of arm (2.1) is as follows

N1 =
1

2

( α11−4β1α21
α11α22−α12α21

cosα+sinα
)

F (2.7)

Thenormal forceN2 (2.6)1 is a compressive force acting on the cross-beamwhich causes buckling
of the frame.

2.2. Flat buckling state – critical load

The critical state for the flat buckling of the symmetrical triangular frame is demonstrated
by symmetrical flexure of the cross-beam and the angles of rotation of the vertexes A andB in
the xy plane. A scheme of the flat buckling mode for one half of the frame is shown in Fig. 4.
Short discussion of this buckling problemwas presented by Bažant and Cedolin (1991).
The detailed scheme of the load and displacements for the arm and the cross-beam of the

frame is presented in Fig. 5.



584 K.Magnucki, S. Milecki

Fig. 4. Scheme of the flat buckling mode

Fig. 5. Scheme of the load and displacements for the flat buckling state

The bendingmoment in the arm

Mb(x1)=MA−N1v1(x1)−RAyx1 (2.8)

The differential equation of the bending line of the arm

EJ(1)z
d2v1
dx21
−N1v1(x1)=−MA+RAyx1 (2.9)

or

d2v1
dx21
−k21yv1(x1)=−

MA

EJ
(1)
z

+
RAyx1

EJ
(1)
z

(2.10)

where k1y =
√

N1/(EJ
(1)
z ) is a coefficient.

The solution of this equation is in the form

v1(x1)=C
(1)
1 sinh(k1yx1)+C

(1)
2 cosh(k1yx1)+

MA
N1
−
RAy
N1
x1 (2.11)

whereC
(1)
1 ,C

(1)
2 are the integration constants.



Elastic buckling of a triangular frame subject to in-plane tension 585

The boundary conditions are as follows

v1(0)= 0 v1(L1)=0
dv1
dx1

∣

∣

∣

∣

∣

L1

=0 (2.12)

fromwhich

C
(1)
1 =

MA
N1

1+k1yL1 sinh(k1yL1)− cosh(k1yL1)

k1yL1cosh(k1yL1)− sinh(k1yL1)
C
(1)
2 =−

MA
N1

(2.13)

and the reaction

RAy = k1yMA
[1+k1yL1 sinh(k1yL1)− cosh(k1yL1)

k1yL1cosh(k1yL1)− sinh(k1yL1)
cosh(k1yL1)− sinh(k1yL1)

]

(2.14)

Thus, the angle of theA vertex rotation in the xy plane is

θ
(1)
A =

dv1
dx1

∣

∣

∣

∣

∣

0

=
MAL1

EJ
(1)
z

f1y(k1yL1) (2.15)

where

f1y(k1yL1)=
1

k1yL1

{

sinh(k1yL1)−
1+k1yL1 sinh(k1yL1)− cosh(k1yL1)

k1yL1cosh(k1yL1)− sinh(k1yL1)
[cosh(k1yL1)−1]

}

(2.16)

Similarly, the bendingmoment in the cross-beam is

Mb(x2)=−MA+N2v2(x2) (2.17)

The differential equation of the bending line

d2v2
dx22
+k22v2(x2)=

MA

EJ
(2)
z

(2.18)

where k2 =
√

N2/(EJ
(2)
z ) is a coefficient.

The solution of this equation is

v2(x2)=C
(2)
1 sinh(k2x2)+C

(2)
2 cosh(k2x2)+

MA
N2

(2.19)

The boundary conditions

v2(0)= 0 v2(L2)=0 (2.20)

fromwhich the integration constants are

C
(2)
1 =−

MA
N2

1− cos(k2L2)

sin(k2L2)
C
(2)
2 =−

MA
N2

(2.21)

Thus, the angle of theA vertex rotation in the xy plane is

θ
(2)
A =

dv2
dx2

∣

∣

∣

∣

∣

0

=−
MAL2

EJ
(2)
z

1− cos(k2L2)

k2L2 sin(k2L2)
(2.22)



586 K.Magnucki, S. Milecki

The consistency condition for the angles of in-plane rotation of the vertex A for the arm and
cross-beam

θ
(1)
A = θ

(2)
A (2.23)

fromwhich the nonlinear algebraic equation is in the following form

kJzzf1y(k1yL1)+2
1−cos(k2L2)

k2L2 sin(k2L2)
= 0 (2.24)

where kJzz = J
(2)
z /(J

(1)
z cosα) is a dimensionless parameter.

Taking into account the normal forces N2 (2.6)1 and N1 (2.7), the critical load from this
equation is determined.

2.3. Lateral buckling state – critical load

The critical state for the lateral buckling of the triangular frame is demonstrated by sym-
metrical flexure of the cross-beam in the xz plane and the corresponding angles of rotation of
the vertexes A andB. A scheme of the lateral bucklingmode for the half of the frame is shown
in (Fig. 6).

Fig. 6. Scheme of the lateral buckling mode

The flexure of the cross-beam in the xz plane causes bending and torsion of the arms. A
scheme of the moments in the vertex A is shown in Fig. 7.

Fig. 7. Scheme of the moments in the vertexA



Elastic buckling of a triangular frame subject to in-plane tension 587

The bendingmomentM
(1)
Ay and torsional momentM

(1)
At in the vertex A for the arm are

M
(1)
Ay =M

(2)
Ay cosα M

(1)
At =M

(2)
Ay sinα (2.25)

whereM
(2)
Ay is the bendingmoment in the vertex A for the cross-beam.

Analogically, the bending angle θ
(1)
Ab and torsion angle θ

(1)
At in the vertex A for the arm are

θ
(1)
Ab = θ

(2)
A cosα θ

(1)
At = θ

(2)
A sinα (2.26)

where θ
(2)
A is the bending angle in the vertexA for the cross-beam.

The detailed scheme of the load and displacements for the arm and the cross-beam of the
frame is presented in Fig. 8.

Fig. 8. Scheme of the load and displacements for lateral buckling state

The bendingmoment in the arm is

Mb(x1)=M
(1)
Ay −N1w1(x1)−RAzx1 (2.27)

This bending problem is analogous to the bending of the arm for flat buckling state (10), thus
the bending angle of the vertex A is

θ
(1)
Ab =

dw1
dx1

∣

∣

∣

∣

∣

0

=
M
(1)
AyL1

EJ
(1)
y

f1z(k1zL1) (2.28)

where

f1z(k1zL1)=
1

k1zL1

{

sinh(k1zL1)−
1+k1zL1 sinh(k1zL1)− cosh(k1zL1)

k1zL1cosh(k1zL1)− sinh(k1zL1)
[cosh(k1zL1)−1]

}

(2.29)

and k1z =
√

N1/(EJ
(1)
y ) is a coefficient, J

(1)
y = b

3c/12 – inertia moment of the cross-sections of
the arm.
The torsion angle θ

(1)
At in the vertex A

θ
(1)
At =

M
(1)
AtL1

GJ
(1)
t

=2(1+ν)
M
(1)
AtL1

EJ
(1)
t

(2.30)

where:G=E/[2(1+ν)] is the shearmodulus of elasticity, ν – Poisson’s ratio, and the torsional

constant J
(1)
t of the rectangular cross-section

• for b¬ c

J
(1)
t =µb

3c µ=
1

3
−

[

0.178+0.153
b

c
−0.138

(b

c

)2]

(2.31)

• for c¬ b

J
(1)
t =µbc

3 µ=
1

3
−

[

0.178+0.153
c

b
−0.138

(c

b

)2]

(2.32)



588 K.Magnucki, S. Milecki

The bendingmoment in the cross-beam is

Mb(x2)=−M
2
Ay +N2w2(x2) (2.33)

This bending problem is analogous to the bending of the cross-beam for flat buckling state
(2.17), thus the bending angle of the vertexA is

θ
(2)
A =

dw2
dx2

∣

∣

∣

∣

∣

0

=−
M
(2)
AyL2

EJ
(2)
z

1− cos(k2L2)

k2L2 sin(k2L2)
(2.34)

Taking into account expressions (2.26) for the bending angle θ
(1)
Ab and torsion angle θ

(1)
At , the

consistency condition for the angles of the vertex A is as follows

θ
(1)
Ab cosα+θ

(1)
At sinα= θ

(2)
A (2.35)

Substituting expressions (2.28) and (2.30) into equation (2.35) and making simple transforma-
tion, the nonlinear algebraic equation is obtained in the following form

kJzyf1z(k1zL1)cos
2α+2(1+ν)kJzt sin

2α+2
1− cos(k2L2)

k2L2 sin(k2L2)
= 0 (2.36)

where: kJzy = J
(2)
z /(J

(1)
y cosα), kJzt = J

(2)
z /(J

(1)
t cosα) are dimensionless parameters.

Taking into account the normal forces N2 (2.6)1 and N1 (2.7), the critical load from this
equation is determined.

2.4. Example of the triangular frame – values of the critical load

An examplary calculation of the critical load is carried out for a symmetrical triangular
frame: length of the cross-beam L2 = 1352mm, angle between the arms and the cross-beam
α = π/9, diameters of the cross-section of the cross-beam d0 = 50mm, d1 = 60mm, area of
the cross section of the arms A1 =1000mm

2 and material constants E =2 ·105MPa, ν =0.3.
Models with different b parameter have been investigated. Values of other parameters are the
same as the parameters of the construction which is used in railway industry. The values of

critical loads F
(Anal)
CR,Flat and F

(Anal)
CR,Lat have been calculated on the basis of non-linear algebraic

equations (2.24) and (2.36), respectively. The results of calculations are specified in Table 1.

Table 1.Values of critical loads – analytical solution

b [mm] 15 20 25 31.623 35 40 50 60

F
(Anal)
CR,Flat [kN] 360.6 403.0 448.2 510.0 541.5 587.1 672.5 747.8

F
(Anal)
CR,Lat [kN] 363.9 394.6 403.4 399.2 388.7 377.1 357.1 342.8

Flat buckling occurs when the width b of the arm is small (b ¬ 18mm), and the lateral
buckling occurs for greater width (18mm¬ b).

3. Numerical FEM model

3.1. FEM model of the triangular frame

The symmetrical triangular frame is abeamstructure (Fig. 9), hence it ismodeledwithbeam
elements – system SolidWorks Simulation 2013 (Fig. 10). The model of the frame is defined in
the rectangular coordinate system. The loading force is applied to the vertexes of the frame
(Fig. 11).



Elastic buckling of a triangular frame subject to in-plane tension 589

Fig. 9. Numerical model of triangular frame

Fig. 10. Discretization of numerical model with the beam elements

Fig. 11. Boundary conditions and loads of the numerical model

3.2. Example of the triangular frame – values of the critical load

An examplary calculation of the critical load is carried out for the symmetrical triangular
frame: length of the cross-beamL2 =1352mm,angle betweenarmsand the cross-beamα=π/9,
diameters of the cross-section of the cross-beam d0 =50mm, d1 =60mm, the area of the cross
section of the arms A1 = 1000mm

2 and material constants E = 2 · 105MPa, ν = 0.3. The

values of critical loads F
(FEM)
CR,Flat and F

(FEM)
CR,Lat are calculated on the basis of the finite element

method. The results of calculations are specified in Table 2. An examplary FEM calculation of
the triangular frame with dimension b=25mm is shown in Fig. 12.

Flat buckling, similarly to the analytical model, occurs when the width b of the arm is small
(b¬ 18mm), and the lateral buckling occurs for greater width (18mm¬ b).



590 K.Magnucki, S. Milecki

Table 2.The values of critical loads – FEM solution

b [mm] 15 20 25 31.623 35 40 50 60

F
(FEM)
CR,Flat [kN] 353.7 390.7 435.6 492.4 520.9 562.1 638.2 704.7

F
(FEM)
CR,Lat [kN] 361.6 393.5 400.5 393.2 385.5 373.4 352.7 338.6

Fig. 12. Scheme of the FEMmodel of the triangular frame

4. Comparison analysis

The values of critical loads calculated analytically and numerically (FEM) are similar. The
difference between them is less than 3%. Comparison of these values is graphically shown for
flat buckling (Fig. 13) and for lateral buckling (Fig. 14).

Fig. 13. Comparison of the critical values obtained from the analytical and numerical (FEM)
methods for flat buckling

Fig. 14. Comparison of the critical values obtained from the analytical and numerical (FEM)
methods for lateral buckling



Elastic buckling of a triangular frame subject to in-plane tension 591

5. Conclusions

Theoretical studies of the buckling problem of the symmetrical triangular frame carried out on
the basis of the analytical and numerical FEMmodels allow one to conclude that:

• flat buckling of the frame occurs when the width b of the arm is small (b¬ 18mm),

• lateral buckling of the frame occurs when the width b of the arm is greater (18mm¬ b),

• maximum of the critical load exists for the width b=25mm,

• the analytical model of the frame accurately describes the flat and lateral buckling phe-
nomena. The analytical and FEM results comply with each other.

The presented study, considering the lateral buckling, has not been undertaken before.

References

1. Budiansky B., 1974, Theory of buckling and post-buckling behaviour of elastic structures, Ad-
vances in Applied Mechanics, 14, 1-65

2. Bažant Z.P., Cedolin L., 1991,Stability of Structures. Elastic, Inelastic, Fracture, and Damage
Theories, Oxford University Press, NewYork, Oxford

3. ChenW.F., LiuE.M., 1987,Structural Stability: Theory and Implementation, Elsevier,NewYork

4. Horne M.Z., Merchant W., 1965,The Stability of Frames, PergamonPress, NewYork

5. Kaveh A., Salimbahrami B., 2007, Buckling load of symmetric plane frames using canonical
forms,Computers and Structures, 85, 1420-1430

6. Magnucki K., Milecki S., 2012, Elastic stability of a brake triangle (in Polish), Modelowanie
Inżynierskie, 44, 199-208

7. Simitses G.J., Hodges D.H., 2006,Fundamentals Structural Stability, Butterworth-Heinemann,
an imprint of Elsevier

8. Şakar G., Öztürk, Sabuncu M., 2012, Dynamic stability of multi-span frames subjected to
periodic loading, Journal of Constructional Steel Research, 70, 65-70

9. Sobaś M., 2010, New generation of brake triangles for freight care (in Polish),Pojazdy Szynowe,
3, 21-30

10. ThompsonJ.M.T.,HuntG.W., 1973,AGeneral Theory of Elastic Stability, JohnWiley&Sons,
London, NewYork, Sydney, Toronto

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Manuscript received August 26, 2014; accepted for print December 19, 2014