S. BILYK, V. TONKACHEIEV: DETERMINING SLOPED-LOAD LIMITS INSIDE VON MISES TRUSS WITH ...
105–109
DETERMINING SLOPED-LOAD LIMITS INSIDE VON MISES
TRUSS WITH ELASTIC SUPPORT
DOLO^EVANJE MEJ NAGNJENE OBREMENITVE ZNOTRAJ VON
MISESOVEGA PALI^JA Z ELASTI^NO PODPORO
Sergiy Bilyk, Vitaliy Tonkacheiev
Kyiv National University of Civil Engineering and Architecture, Department of Steel and Wooden Structures,
Povitroflotskyi Ave. 31, 49600 Kyiv, Ukraine
vartist@ukr.net, thetvg@gmail.com
Prejem rokopisa – received: 2016-05-11; sprejem za objavo – accepted for publication: 2017-10-26
doi:10.17222/mit.2016.083
The scientific novelty of this work lies in the description of the methods of calculating and analyzing the symmetrical stability
loss of the von Mises truss in the case of sloped loads with an elastic support at the top of the truss. In this work, we consider a
deformed layout of the von Mises truss with an elastic support under a concentrated load at the apex joint about the vertical axis.
Numerical studies were carried out using the proposed methods. This work indicates the impact of the sloped load and elastic
support on the truss’ stability loss. We determined dependencies of critical external-load values on the rod’s starting pitch for
different situations. For each design situation, we obtained generalized expressions for determining critical force values for the
truss’ apex joint. We obtained dependencies and analytical expressions for determining the critical load on the dome structure,
based on the initial geometrical parameters, taking into account the reaction of the elastic support at the apex joint and the
direction of the external loads. Our studies show that increasing the support’s stiffness increases the sustainability of the von
Mises truss. Additionally, we obtained new results required for improving the stability of trusses under a sloped load with an
angle of 10–40° about the vertical axis.
Keywords: von Mises truss, elastic support, sloped load on joints, buckling, steel dome
Pri~ujo~i ~lanek predstavlja nov znanstveni prispevek za opis metode izra~una in analize izgube simetri~ne stabilnosti von
Misesovega pali~ja v primeru nagnjenega bremena z elasti~nimi podporami na vrhu pali~ja. V ~lanku avtorja obravnavata potek
deformacije von Misesovega pali~ja z elasti~no podporo pod koncentrirano obremenitvijo na vrhu vezi okoli vertikalne osi. S
predlaganimi metodami sta avtorja izvedla numeri~ne {tudije. V prispevku predstavita vpliv nagiba bremena in elasti~ne
podpore na izgubo stabilnosti pali~ja. Ugotovila sta, kak{en je vpliv kriti~ne zunanje obremenitve na za~etni polo`aj pali~ja pri
razli~nih pogojih izra~unanvanja. Za vsako izbrano situacijo sta avtorja dobila generaliziran izraz za dolo~itev vrednosti kriti~ne
sile v vrhu vezi pali~ja. Ugotovila sta, kak{ne so odvisnosti in dobila analiti~ne izraze za kriti~no obremenitev kupolaste
strukture na osnovi za~etnih geometrijskih parametrov, upo{tevajo~ reakcijo elasti~ne podpore na vrhu vezi in smer zunanje
obremenitve. [tudija je pokazala, da nara{~ajo~a togost podpore pove~uje nosilnost von Misesovega pali~ja. Dodatno sta avtorja
prispevka dobila nove rezultate za izbolj{anje stabilnosti pali~ja pod kotom od 10° do 40° nagnjeno obremenitvijo okoli verti-
kalne osi.
Klju~ne besede: von Misesovo pali~je, elasti~na podpora, nagnjena obremenitev vezi, uklon, jeklena kupola
1 INTRODUCTION
Dome structures are highly applicable due to a num-
ber of positive characteristics, including the following:
possibility to cover large amounts of space with a mini-
mum surface area, which in turn leads to reduced ex-
penses of the thermal energy for heating; a low material
consumption in comparison with other covering designs;
no need for intermediate supports; architectural express-
iveness; the ability to use small identical elements and
typical joint connections. Another positive feature of a
dome is a pretty good maintainability, achieved due to
the possibility to replace individual parts without the
dome structure losing its carrying capacity as a whole.1
Recent building practice pays most of the attention to
the use of efficient design solutions. One of the ways to
increase the efficiency of the building production is the
use of light spatial covering structures, including shells
and domes.
There are the following types of steel dome structures
characterized by special design features: reticulated
structures, ribbed, ribbed-ring and ribbed-ring structures
with connections. Despite the minimal material con-
sumption, the characteristic of reticulated domes – the
complexity of the joint connections – is an important
factor, which affects the carrying capacity and the cost.
For small (12–30 m) spans, it is cost-effective to use
plain ribbed domes that require a similar material con-
sumption as the reticulated ones under the same condi-
tions.
Hinge joints are traditionally used for the ribbed-
dome designs. The biggest problem of the hinge-joint
design is the buckling of the apex joint and the elements
adjacent to it. It is considered conventional to use a
three-hinged arch as the design model for a ribbed
dome.2 The dependence between the dome load and
displacement of the arch’s apex joint is nonlinear.3,4,5
Materiali in tehnologije / Materials and technology 52 (2018) 2, 105–109 105
UDK 519.6:620.172:624.014.2 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 52(2)105(2018)
Deformation of such a system is usually analyzed using
three-hinged rod-system models.
The first study of a three-hinged rod system was
conducted by R. V. Mises6 and the systems themselves
are called von Mises trusses. We conducted a wide range
of studies, taking into account the elastic reactions of the
system and elastic supports at the joints,7,8 getting close
to that of Zeman’s machines.9 There are researches of
Stoker’s column buckling.10 There are thorough
researches of the von Mises truss buckling based on
dynamic criteria for the elastic supports.7,11 The basic
research of the von Mises truss buckling under a vertical
symmetrical load and a possible symmetric or
asymmetric stability loss is reported in references.9,12,13
Also, there are researches using the finite-element
analysis for non-linear structures.14,15,16
Earlier studies are insufficient in terms of their use
when determining the buckling of more complex systems
such as ribbed domes with rings. The top of a dome
model can be considered as a model consisting of the
von Mises truss combination. However, we must con-
sider the impact of the adjacent elements. The impact of
the adjacent elements on the truss’ apex joint ought to be
modeled as an additional elastic support, and exactly that
was done in this study. At the same time, the structure of
a dome can have a dedicated element of the von Mises
truss under a sloped load. The impact of circular dome
elements can be modeled in the form of additional elastic
supports at dome’s joints.
These studies are based on generalized characteristics
of the von Mises truss buckling under a sloped load,
obtained with the research from reference.17
2 STATEMENT OF THE PROBLEM
The purpose of this study is to determine the depen-
dencies of the maximum values of the relative force at an
apex joint, at which the joint starts loosing its stability,
on the initial pitch of the rods in the system with diffe-
rent input parameters of a design scheme. A deformed
scheme of the von Mises truss with an elastic support
and a sloped load at the apex joint of the truss is pre-
sented on Figure 1.
The methods of research are described in reference17
and implemented with the algorithm from Figure 2.
For convenience of the implementation of numerical
studies, a generalized stability criterion17 is presented in
the parametric form:
P
E A
A B C D
⋅
⋅
= + + +
cos p
cal
(1)
where A
k
E A
a
a
=
⋅
⋅
⋅
 p
cal
0
0
B
a
a a
=
−
− ⋅
⎛
⎝
⎜
⎞
⎠
⎟ + −
⎛
⎝
⎜
⎞
1
1
1
0
0
2
0
tan
tan
tan






0l
p
p
p
0l
p
⎠
⎟
2
C
a
a a
=
−
+ ⋅
⎛
⎝
⎜
⎞
⎠
⎟ + −
⎛
⎝
⎜
⎞
1
1
1
0
0
2
0
tan
tan
tan






0l
p
p
p
0l
p
⎠
⎟
2
D
a
= − ⋅ −
⎛
⎝
⎜
⎞
⎠
⎟2
1
0
sin
tan



0l
0l
p
– parameters of Equation (1).
The maximum values for the relative joint load of the
apex joint, at which the truss starts to lose stability, are
defined for the value range of the initial rod pitch about
the vertical axis of 45–89° using the "golden section"
method.18
We selected a segment from p/a0 = 0 to the relative
displacement value, at which the first derivative of
generalized stability criterion function (1) changes its
sign to the opposite as the criterion for determining the
initial range for searching extremes.
3 RESULTS AND DISCUSSION
Some results of the numerical studies are presented
in Figure 3. When increasing the force pitch about the
vertical axis, the maximum relative force increases, and
its procession slows down somewhat, but upon reaching
the maximum relative force, the loss of the joint stability
takes place faster than at the low pitch (Figure 3a).
The presence of the elastic support at the apex joint
in the absence of a sloped load leads to an increase in the
maximum value of the relative force. By increasing the
stiffness as well, as for a very small truss’ rod pitch about
the horizontal axis (5–10°), it is possible to get no nega-
tive values of the relative force on the graph, as indicated
S. BILYK, V. TONKACHEIEV: DETERMINING SLOPED-LOAD LIMITS INSIDE VON MISES TRUSS WITH ...
106 Materiali in tehnologije / Materials and technology 52 (2018) 2, 105–109
Figure 1: Deformed scheme of the von Mises truss with an inclined
load and elastic support at the apex joint of the truss where: 0l –
initial rod slope about the vertical axis; a0 – half part of truss span; p
– force pitch at the apex joint about the vertical axis; k – elastic
support stiffness parameter for the apex joint; P – sloped load
in Figures 3b and 3c. With a further increase in elasti-
city, the dependence becomes closer to linear (Fig-
ure 3c).
In the case of the classic Mises truss, dependence
between the initial truss’ rod pitch and the maximum
value of the relative force at the apex joint assumes the
shape of a downward curve. In the case of the sloped
load of the apex joint, we considered a force pitch of
10–80° about the vertical axis, provided that it is always
smaller than the initial rod pitch.
With the algorithm (Figure 2), we obtained the
dependencies between the maximum force value at the
apex joint and the initial rod’s slope (Figure 4).
It is possible to describe the obtained dependence
(Figure 4a) with the following polynomial of the third
degree:
P
P
E A
K K K Kl l lmax
cos
rel p
cal
=
⋅
⋅
= ⋅ + ⋅ + ⋅ +

  1 0
3
2 0
2
3 0 4 (2)
where 0l is the initial rod pitch, while coefficients K1,
K2, K3, K4 for different force-pitch p values are given in
Table 1.
Table 1: Coefficients from Equation (2)
p K1 K2 K3 K4
0 -2.09e-06 5.62e-04 -5.04e-02 1.51
10 -2.14e-06 5.76e-04 -5.16e-02 1.54
20 -2.38e-06 6.34e-04 -5.65e-02 1.68
30 -3.15e-06 8.20e-04 -7.13e-02 2.08
40 -6.69e-06 1.63e-03 -1.33e-01 3.63
Dependence of the relative displacement values, at
which the system starts losing the stability of the initial
rod pitch, is shown in Figure 5.
It is possible to describe the obtained dependence
(Figure 5) with the following polynomial of the third
degree:


  rel
p
0
= = ⋅ + ⋅ + ⋅ +
a
K K K Kl l l5 0
3
6 0
2
7 0 8 (3)
where 0l – the initial rod pitch; coefficients K1 … K5
for different force pitch p values are shown in Table 2.
Table 2: Coefficients of Equation (3)
p K5 K6 K7 K8
0 -2.79e-06 6.88e-04 -6.42e-02 2.24
10 -2.92e-06 7.19e-04 -6.68e-02 2.31
20 -3.45e-06 8.45e-04 -7.71e-02 2.61
30 -4.92e-06 1.19e-03 -1.04e-01 3.35
40 -8.38e-06 1.98e-03 -1.65e-01 4.96
S. BILYK, V. TONKACHEIEV: DETERMINING SLOPED-LOAD LIMITS INSIDE VON MISES TRUSS WITH ...
Materiali in tehnologije / Materials and technology 52 (2018) 2, 105–109 107
Figure 3: Dependence of the relative force at the apex joint P*cosp/EAcal on the relative vertical displacement of joint p/0l; a) 0l=80°; p=0;
k=0; b) p=0; k in a range of (0…0.02); c) 0l=85°; p=0; k in a range of (0…0.02) where 0l – initial rod pitch about the vertical axis; a0 –
half of truss span; p – force pitch at the apex joint about the vertical axis, k – elastic support stiffness parameter for the apex joint
Figure 2: Algorithm of numerical research of joint buckling of von
Mises trusses where 0l – initial rod pitch about the vertical axis; a0 –
half of truss span; p – force pitch at the apex joint about the vertical
axis; EAcal – rod’s compressive stiffness; k – elastic support stiffness
parameter for the apex joint; i – number of iterations during the study
inserted in options A, B, C, D (1) instead of vertical joint dis-
placement value p/a0; m – iteration step (dependence graph scale);
P*cosp/EAcal – relative force at the apex joint or active load value;
p/a0 – relative vertical displacement or vertical displacement indi-
cator
In the case when the apex joint had elements pro-
viding a certain joint stiffness (elastic support), we
considered the cases, in which the stiffness value was
within (0.001 ... 0.2) for a truss span of 18 m.
In the case of an elastic-support inception into the
apex joint at a large initial rod pitch about the vertical
axis, we do not see the maximum relative-force values.
This is due to the lack of a pronounced extremum on the
graph of the relative-force dependency at the apex joint
on the relative apex-joint displacement (Figure 3c).
In the mixed case of a sloped load in the presence of
an elastic support at the apex joint, we considered a
system with elastic-support stiffness parameters within
(0.001: 0.02) for the force pitch at the apex joint in a
range of 10–40° about the vertical axis for the truss span
of 18 m.
Averaging the maximum relative force in the pre-
sence of the elastic support at the apex joint becomes
more complicated due to the need to consider not only
the stiffness of the elastic support, but also the truss span
and rod’s stiffness for each case of the calculations.
According to the above numerical-algorithm studies
(Figure 2), the maximum relative force at the apex joint
in the presence of the elastic support can be defined as
the sum of the maximum relative-force values without
the elastic support, considering only components B, C,
S. BILYK, V. TONKACHEIEV: DETERMINING SLOPED-LOAD LIMITS INSIDE VON MISES TRUSS WITH ...
108 Materiali in tehnologije / Materials and technology 52 (2018) 2, 105–109
Figure 6: Dependence of the maximum relative force at the apex joint Pmax*cosp/EAcal on the initial rod slope 0l, with different values of force
pitch p, and a0=9m; a) k=0.001; b) k=0.02 where a0 – half of the truss span; p – the force pitch at the apex joint about the vertical axis, k –
elastic support stiffness parameter for the apex joint
Figure 5: Dependencies of the relative vertical displacements of the
truss’ apex joint p/a0, at which the joint buckling appears, on the
initial truss’ rod pitch 0l for different force pitch p values where 0l
is the initial rod pitch about the vertical axis; a0 is the half of truss
span; p is the force pitch at the apex joint about the vertical axis
Figure 4: Dependence of the maximum relative force at the apex joint Pmax*cosp/EAcal on the initial rod pitch 0l; a) p < 0l; k=0; b) p=0;
a0=9m; k in a range of (0.001…0.2) where 0l – initial rod pitch about the vertical axis; a0 – half of truss span; p – force pitch at the apex joint
about the vertical axis, k – elastic support stiffness parameter for the apex joint
D, while considering the A component separately, see
Equation (1). Hence, we get a generalized equation to
determine the maximum relative force:
[ ]P
P
E A
P
a k
E Akmax
max
max
cos
rel p
cal
rel
rel
cal



=
⋅
⋅
= +
⋅
⋅
0
(4)
Where Pmax
rel – the maximum relative force obtained
from Equation (2); rel – relative vertical displacement of
the apex joint, at which the system starts losing its
stability, calculated with Equation (3); a0 – half of the
von Mises truss span; k – parameter of the rigidity of the
elastic support at the ridge node; E – the elasticity
modulus; Acal – the estimated cross-section of truss’ rods;
p – the force pitch about the vertical axis.
Below is the search algorithm of the maximum rela-
tive force in the presence of an elastic support at the apex
joint:
1. We can use Table 1 to determine the maximum rela-
tive force at the apex joint;
2. We can use Table 2 to determine the appropriate
relative vertical displacement of the apex joint;
3. The values found, as well as the truss’ span and rods’
stiffness should be inserted into Equation (4), from
which we get the maximum relative force considering
the presence of the elastic support at the apex joint.
The results of this algorithm are shown in Figure 6.
4 CONCLUSIONS
1. We found dependencies of the maximum joint-load
values, at which the von Mises truss starts losing its
stability, on the initial truss’ rod pinch, which in turn
allows the designer to evaluate the carrying capacity
of dome structures with known geometrical parame-
ters.
2. We obtained general expressions to determine the
maximum relative-force values for different design
cases, which make it possible to evaluate the carrying
capacity of a dome with an elastic support at the apex
joint and under a sloped load.
3. The proposed method allows us to check the stiffness
of a complex system at the earlier stages of design.
4. The proposed study is acceptable for use when consi-
dering the elastic buckling of von Mises trusses with
the elements with a pitch characteristic for dome
coverings, provided that the local stiffness is ensured
for each element.
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S. BILYK, V. TONKACHEIEV: DETERMINING SLOPED-LOAD LIMITS INSIDE VON MISES TRUSS WITH ...
Materiali in tehnologije / Materials and technology 52 (2018) 2, 105–109 109