Influence of stiffness in bolted connections in wooden plane structure of truss type

Abstract

Trusses are structural systems commonly used in projects, being employed mainly in roof structures, present in most rural buildings. The design of trusses, as well as other structural systems, requires the determination of displacements, strains and stresses. However, the project is developed from an ideal model of calculation, considering free rotation between the elements of a connection. This paper presents a computer program for the analysis of bidimensional wooden trusses with connections formed with two screws per node. The formulation is based on the flexibility method, taking into account the influence of the effect of semi-rigid connections formed by two screws. An example of a structure is presented and analyzed by the program developed here, highlighting the importance of behavior analysis on semi-rigid connections.

Full text

TECHNICAL PAPER
____________________________________
1 Dr., Departamento de Engenharia Mecânica, Universidade Federal de São João del-Rei (UFSJ), Praça Frei Orlando 170, Centro,
São João del-Rei - MG, alchristoforo@ufsj.edu.br.
2 Departamento de Engenharia de Estruturas, Escola de Engenharia de São Carlos, Universidade de São Paulo, Av. Trabalhador
Sancarlense, 400, São Carlos - SP, giulianoromanholo@hotmail.com.
3 Dr., Departamento de Engenharia Mecânica, Universidade Federal de São João del-Rei (UFSJ), Praça Frei Orlando 170, Centro,
São João del-Rei - MG, tuliopanzera@ufsj.edu.br.
4 Dr., Departamento de Engenharia de Produção Civil, Centro Federal Tecnológico de Minas Gerais (CEFET-MG), Av. Amazonas,
7675, Nova Gameleira, Belo Horizonte - MG, pborges@civil.cefetmg.br.
5 Dr., Departamento de Engenharia de Estruturas, Laboratório de Estruturas de Madeira, Escola de Engenharia de São Carlos,
Universidade de São Paulo, Av. Trabalhador Sancarlense, 400, São Carlos - SP, frocco@sc.usp.br.
Recebido pelo Conselho Editorial em: 25-1-2011
Aprovado pelo Conselho Editorial em: 6-6-2011 Eng. Agríc., Jaboticabal, v.31, n.5, p.998-1006, set./out. 2011
INFLUENCE OF STIFFNESS IN BOLTED CONNECTIONS IN WOODEN PLANE
STRUCTURE OF TRUSS TYPE
ANDRÉ L. CHRISTOFORO1, GIULIANO A. ROMANHOLO2, TÚLIO H. PANZERA3,
PAULO H. R. BORGES4, FRANCISCO A. R. LAHR5
ABSTRACT: Trusses are structural systems commonly used in projects, being employed mainly in
roof structures, present in most rural buildings. The design of trusses, as well as other structural
systems, requires the determination of displacements, strains and stresses. However, the project is
developed from an ideal model of calculation, considering free rotation between the elements of a
connection. This paper presents a computer program for the analysis of bidimensional wooden
trusses with connections formed with two screws per node. The formulation is based on the
flexibility method, taking into account the influence of the effect of semi-rigid connections formed
by two screws. An example of a structure is presented and analyzed by the program developed here,
highlighting the importance of behavior analysis on semi-rigid connections.
KEYWORDS: trusses, semi-rigid connections, flexibility method, rural buildings.
INFLUÊNCIA DA RIGIDEZ DE LIGAÇÕES PARAFUSADAS EM ESTRUTURAS
PLANAS DE MADEIRA DO TIPO TRELIÇA
RESUMO: Treliças são sistemas estruturais comumente utilizados em projetos, empregadas
principalmente em estruturas de cobertura, presentes na maioria das construções rurais. O
dimensionamento de treliças, assim como o de outros sistemas estruturais, requer a determinação
dos deslocamentos, esforços, tensões e deformações atuantes em seus elementos constituintes. O
cálculo é desenvolvido com base em um modelo ideal, considerando-se o giro livre entre os
elementos componentes de uma ligação. Este trabalho objetiva apresentar um programa
computacional destinado à análise de treliças planas de madeira com ligações formadas com dois
parafusos por nó. A formulação é fundamentada no Método da Flexibilidade, levando-se em
consideração a influência do efeito semirrígido das ligações formadas por dois parafusos. Um
exemplo de estrutura auxiliar de cobertura é apresentado e analisado pelo programa desenvolvido,
evidenciando-se a importância da análise do comportamento semirrígido sobre as ligações.
PALAVRAS-CHAVE: estruturas de cobertura, ligações semirrígidas, método da flexibilidade,
construções rurais.

Influence of stiffness in bolted connections in wooden plane structure of truss type
Eng. Agríc., Jaboticabal, v.31, n.5, p.998-1006, set./out. 2011
999
INTRODUCTION
The use of wood as a structural element in Brazil has grown over the past few years because
of the research carried out in order to make it more competitive than other construction materials.
In the case of structures, wood can be used in temporary works, such as anchors and forms, or
as a structural element, such as beams, columns, poles, trusses and more. Wood is widely used in
structures of truss-type roofs, as those found in sheds and diverse rural buildings.
The flat design of roof structures is usually done through ideal calculation models, in which
the links are considered as perfectly flexible in rotation (trusses), i.e., transmitting only normal
forces among their structural components.
In practice, flat connections in wooden structures are designed in different ways, and
especially among them, the use of bolted connections. However, the existence of two or more
screws forms a semi-rigid effect of these connections.
The semi-rigid effect of connections lies between two ideal calculating models: the free
rotating (truss) and the perfectly rigid (porch) models.
According to RIBEIRO (1997), the study of connections began in England in the early
nineteenth century, with the study of the riveted beam-column type, which the moment-rotation
relationship was evaluated.
A work of great importance is that by JOHNSTON & MOUNT (1942), which analyzed
frames with semi-rigid connections. Later, SHOROCHNIKOFF (1950) reported the influence of
forces by the wind in semi-rigid connections for the same type of structure.
For metal structures, LOTHERS (1951) proposed equations to represent the elastic constraint
of semi-rigid connections. KRISHNAMURTHY et al. (1979) applied the Finite Element Method
(FEM) to obtain moment-rotation curves of connections with steel sheets, and JONES et al. (1983)
studied the influence of semi-rigid connections in steel columns.
In the case of studies of connections in wooden structures, the work by OLIVEIRA & DIAS
(2001) evaluated through the analysis of experimental results, the criterion for the design of the
metal pin connections, proposed by the technical standard NBR 7190/97 (Design of wooden
structures). Other relevant works involving the study of links and numerical methods applied to
wooden structures are the following: SANTANA & MASCIA (2009), SOUZA JUNIOR &
GESUALDO (2007), SOUZA JUNIOR & GESUALDO (2006), STAMATO & CALIL (2002),
CARVALHO (2002), GESUALDO (2001), SERAPHIM & FRANCO (2001), EMERSON &
FRIDLEY (1996), GROOM (1996) among others.
The present work aims at developing a computer program PS-R (Semi-Rigid Porch),
developed from the foundations of the flexibility method, to evaluate the effect of semi-rigid
connections formed by two screws in wooden plane structures of truss type. A roof structure is
evaluated by the PS-R program in order to verify the importance of this analysis.
METHODS
The modified stiffness matrix
0
M
[S ]
that accounts for the semi-rigid effect of the rotational
component for each element of the structure is developed according to the method of flexibility, as
expressed by Equation 1. Further details on obtaining the modified stiffness matrix can be found in
the work by WEAVER & GERE (1986).

André L. Christoforo, Giuliano A. Romanholo, Túlio H. Panzera et al.
Eng. Agríc., Jaboticabal, v.31, n.5, p.998-1006, set./out. 2011
1000
Rij Rj2 Rij Ri2
0
M
AE AE
0 0 - 0 0
LL
0 6Ce 3CLe 0 -6Ce 3CLe
0
[S ] =
22
Rj2 Rj3 Rj2
Rij Rj2
3CLe 2CL e 0 -3CLe CL
AE AE
- 0 0 0 0
LL
0 -6Ce -3CLe Rij Ri2
22
Ri2 Ri2 Ri3
0 6Ce -3CLe
0 3CLe CL 0 -3CLe 2CL e













((1
)
where,
A - cross-sectional area of the structural element;
E - longitudinal modulus of elasticity or Young's modulus;
L - length of the structural element;
I - moment of inertia of cross section;
C - coefficient of flexibility, obtained by:
 
3Rij Ri Rj
2EI
CL 4e 3 4e e 1



, and
eRij ; eRi; eRj - dimensionless parameters of flexibility.
The coefficients of flexibility contained in the modified stiffness matrix element of the beam,
allow assessing the structural behavior through three different forms of analysis: the first form
accounts for the effect of semi-rigid connection, the second one considers the structure as a porch
and the third one considers it as a truss. To analyze the behavior of semi-rigid connections via the
software developed in FORTRAN language, it is necessary to know the coefficients of flexibility
that make the modified stiffness matrix (equation 1) in a stiffness matrix for porch beam and in a
stiffness matrix for truss beam, respectively.
The determination (calibration) of the coefficient of flexibility responsible for transforming
the stiffness matrix in the modified stiffness matrix for a porch beam is performed according to the
structural model of a beam embedded in their edges with a concentrated force of intensity F, applied
in mid-span, as illustrated in Figure 1.
For the structural variables were assigned the following numeric values: F = -5 kN ; A = 50 cm2 ; I = 467.67 cm4 ; E = 2000
kN/cm2 ; L = 200 cm
FIGURE 1. Bi-clamped beam.
Several attempts were made to find the rotational coefficient of semi-stiffness, which aims to
transform the stiffness matrix into the modified stiffness matrix of a beam porch element. It was
found that for a coefficient of 1.0 × 109, the values of the transverse displacements at the point of

Influence of stiffness in bolted connections in wooden plane structure of truss type
Eng. Agríc., Jaboticabal, v.31, n.5, p.998-1006, set./out. 2011
1001
application of force as well as the relative rotations at the ends of the beam converged to the results
provided by commercial software SAP 2000. For values above 1.0 × 109, the results for the
displacements with the use of these programs, PS-R and SAP 2000, remained constant.
To determine the coefficient of rotational semi-stiffness that represents the structure as bolted,
the same tests (variation of rotational stiffness) were carried out; however, the structure considered
in SAP 2000 was bi-clamped (hinged). The coefficient of semi-rigid rotation found for this case was
1.0 × 10-9.
It is important to note that the rotational coefficients of semi-rigidity herein determined are
independent of the geometrical and physical parameters used in numerical simulations and that the
axial stiffness was kept constant over the whole data analysis, varying only the rotational stiffness.
In addition to analyzing the effect of semi-rigid connections, the PS-R program also allows
evaluating the structure as a porch or a truss.
In the case of the connection formed by two screws, the standards adopted for the calculation
and the respective provisions were established by the normative NBR 8800/1986 (Project and
execution of steel structures in buildings), replaced by the current version NBR 8800/2008, to
quantify the forces acting on the screws of the connection. The removal of the screws (Figure 2)
confers the presence of a moment of resistance in the connection.
FIGURE 2. Detail of bolted connection.
The shear resistance force of the screws FR and the resistance moment MR of the connection
are expressed by Equations 2 and 3, respectively:
R V NV
F=ΦR
,
uPNV fAR 42,0
(2)
pRR eFM 
(3)
where,
p
e
- spacing between screws;
P
A
- gross area, based on the nominal diameter “dp” of the screw;
u
f
- tensile strength of the screw material;
RNV - nominal shear strength, and
V
Φ
- weighting coefficient of the shear resistant strength.

André L. Christoforo, Giuliano A. Romanholo, Túlio H. Panzera et al.
Eng. Agríc., Jaboticabal, v.31, n.5, p.998-1006, set./out. 2011
1002
According to the version of the normative document NBR 8800/1986, the weighting values
for bolts ASTM A325 and ASTM A490 equals to 0.65 and the minimum distance between their
centers should not be less than
p
3d
(constructive disposition).
Table 1 indicates, according to the specifications of the material, the shear resistance of the
screw according to their diameter.
TABLE 1. Materials used on screws.
Specification
Resistance (kN cm-2)
Nominal Diameter “dp” (mm)
ASTM A325
82.50
12,7 dp 25,4
72.50
25,40 dp 38,10
ASTM A490
103.50
12,7 dp 38,10
The analysis of the behavior of semi-rigid connections formed by two screws through the
program PS-R is carried out according to the correction of rigid connections, which is due to an
iterative process. This correction is made only when the resistance moment of the connection
becomes lower than that of the applicant. To determine in advance the strains on the structure, the
connections are considered to be perfectly rigid, and soon after this consideration, the bending
stresses acting on the connection are compared in module with the resistance that it presents.
The calculation of the resistance moment of the connection by the PS-R program is conducted
as a function of the spacing and diameter of the screws, being equal for all nodes of the structure
and constant throughout the analysis (equation 3). Attention should be paid for the minimum
dimension of the screws provided by the normative, to not impair the reliability of the results.
After calculating the bolted moment and the resistance moment of the connections, the
program compares these two values. If the bolted moment is less than or equal to the resistance
moment, the connections are considered rigid and the value of the moment acting on the connection
is the bolted moment. If the bolted moment in a connection is greater than the resistance moment,
the program recalculates the whole structure so that it finds a new equilibrium configuration. This
procedure is incremental and iterative and, for the bolted connections beyond their resistance, the
value of the coefficient of semi-rigid rotational connection was successively decreased in 1% and
all strains in the structure (redistribution of the surplus moment) were recalculated.
EXAMPLE OF APPLICATION AND RESULTS
Figure 3 illustrates the structure analyzed by the PS-R program.
FIGURE 3. Type-A structure.
The identification of nodes and elements of the components of type A structure are illustrated
in Figure 4.

Influence of stiffness in bolted connections in wooden plane structure of truss type
Eng. Agríc., Jaboticabal, v.31, n.5, p.998-1006, set./out. 2011
1003
In the analysis of the type-A structure, the following values of structural variables are used: F = 50
kN ; A = 90 cm2 ; I = 1,687.50 cm4 ; E = 2000 kN/cm2 ; L = 200 cm; specification of screw material
= ASTM A325; dp= 1.3 cm ; ep= 4 cm.
FIGURE 4. Discretization of the type-A structure.
The type-A structure is evaluated by three different forms of analysis: first, the displacements
of nodes 2 and 3 (Figure 4) are limited to values less than L/200, according to the NBR 7190/1997
specifications, so to ensure that the structure to be designed is within the linear elastic regime (as
required in the project), in order to verify the influence of the semi-rigid effect under small
displacements; in the second analysis, all structural variables are maintained, with the exception of
force, which is gradually increased in order to verify the intensity responsible for applying two
connections beyond their resistance; the third form of analysis aims to determine the amount of
force necessary to apply four connections of the structure beyond its resistance, allowing to
compare the values of bending moments for both, rigid and semi-rigid connections.
It is important to make clear that the whole structure is designed to withstand the strain and
suffer small displacements, ensuring linear elastic behavior of materials, the designer's task is to
find the best dimensions and arrangements of structural elements as well as choose the most
suitable material. Thus, the second and third forms of analysis aforementioned are presented merely
to verify the stiffness loss of the connections calculated by the program.
Table 2 shows the values of nodal displacements of the type-A structure, expressed in
centimeters, obtained under the three forms of analysis that the PS-R program performs, with a
intensity force F = 50 KN.
TABLE 2. Values of nodal displacements for the type-A structure.
Node
Flexible (truss)
Semi- Rigid
Rigid (porch)
Displ. (x)
(cm)
Displ. (y)
(cm)
Displ. (x)
(cm)
Displ. (y)
(cm)
Displ. (x)
(cm)
Displ. (y)
(cm)
1
0
0
0
0
0
0
2
0.00235
-0.36612
0.00248
-0.36150
0.00248
-0.36150
3
-0.00235
-0.36612
-0.00248
-0.36150
-0.00248
-0.36150
4
0
0
0
0
0
0
5
-0.02484
-0.36025
-0.02456
-0.35602
-0.02456
-0.35602
6
0.0248
-0.36025
0.02456
-0.35602
0.02456
-0.35602
where: Displ. (x) - nodal displacement of the element in direction of x-axis; Displ. (y) – nodal
displacement of the element in direction of y-axis.
According to the PS-R program, no connection was required beyond their capacity limits, i.e.,
the calculated acting moments were all less active to the resistance moment, of intensity

André L. Christoforo, Giuliano A. Romanholo, Túlio H. Panzera et al.
Eng. Agríc., Jaboticabal, v.31, n.5, p.998-1006, set./out. 2011
1004
105.60 kN cm, noting that this value was calculated by the program and was the same for three
analysis presented. This result is observed in Table 2, where displacement values for the structure
with rigid and semi-rigid connections are exactly the same.
For the second form of analysis, the intensity of the force responsible for applying the first
two connections beyond their resistance limits equals to 75 kN. Table 3 shows values of bending
moments acting on the structure nodes, expressed in kN cm, considering semi-rigid and rigid
connections.
TABLE 3. Values of bending moments acting on type-A structure by a force of 75 kN.
Rigid (porch)
Semi-Rigid
Connectivity
Bending moment
(kN cm)
Bending moment
(kN.cm)
Element
Node (i)
Node (j)
Node (i)
Node (j)
Node (i)
Node (j)
1
1
2
-24.01698
-121.53668
-23.6591
-120.17243
2
2
3
36.96272
-36.962762
36.64768
-36.64786
3
3
4
121.53668
24.01690
120.17243
23.65691
4
1
5
24.01690
-63.12227
23.65691
-63.27271
5
2
5
28.97094
29.16207
28.39852
28.96128
6
2
6
55.60302
3.19366
55.12605
2.89919
7
3
5
-55.60302
-3.19366
-55.12605
-2.89919
8
3
6
-28.97094
-29.16277
-28.39852
-28.96128
9
6
4
63.12227
-24.01690
63.27271
-23.65691
10
5
6
37.15385
-37.15385
37.21062
-37.21062
According to the program, nodes 2 and 3 (Figure 4) were applied beyond their resistance
capacities and, for the redistribution of the residual bending moment to the entire structure, 465
iterations were necessary.
For the third form of analysis, the intensity of the force responsible for applying the first four
connections beyond their limits of resistance is equal to 372 kN. Table 4 shows the values of
bending moments acting on the nodes of the structure, expressed in kN cm, considering the semi-
rigid and rigid connections.
TABLE 4. Values of bending moments acting on the type-A structure by a force of 372 kN.
Rigid (porch)
Semi-Rigid
Connectivity
Bending moment
(kN.cm)
Bending moment
(kN.cm)
Element
Node (i)
Node (j)
Node (i)
Element
Node (i)
Node (j)
1
1
2
-119.12384
-602.82193
-56.83838
-120.08168
2
2
3
183.33508
-183.33508
68.54579
-68.54579
3
3
4
602.82193
119.12384
120.08168
56.83838
4
1
5
119.12384
-313.08645
56.83838
-120.13901
5
2
5
143.69586
144.64384
-31.55450
-7.31987
6
2
6
275.79100
15.84055
83.09039
-32.70217
7
3
5
-275.79100
-15.84055
-83.09039
32.70217
8
3
6
-143.69586
-144.64389
31.55450
7.31987
9
6
4
313.08645
-119.12384
120.13901
-56.83838
10
5
6
184.28311
-184.28311
94.75671
-94.75671

Influence of stiffness in bolted connections in wooden plane structure of truss type
Eng. Agríc., Jaboticabal, v.31, n.5, p.998-1006, set./out. 2011
1005
According to the program, nodes 2, 3, 5 and 6 (Figure 4) were applied beyond their resistance
capabilities and, for the redistribution of residual bending moments to the entire structure 1,135
iterations were necessary.
CONCLUSIONS
The force value of 50 kN responsible for causing small displacements (L/200) in the structure
originated applying moments on the connections for both, rigid and semi-rigid calculation models,
lower than the resistance moment (105.60 kN cm). The displacements values shown in Table 2
indicate great similarity in terms of displacement between the flexible (truss), the rigid and the
semi-rigid models. For design conditions in which the applicant moment is lower than the resistance
moment, the truss model represents as a good calculation alternative, since it does not require the
use of iterative processes, contrary to what happens with the PS-R program, which does consider
the effect of semi-rigid connections.
The use of the PS-R program allowed finding values of the forces responsible for applying
some connections beyond their respective resistances, which represents an alternative calculation
tool for the analysis of truss displacements and stresses. However, it should be noted that the
rigidity of the connection is usually greater than that of the beam elements, resulting in failure of the
wood before the occurrence of maximum application to the connection. Studies involving the
consideration of non-linear physics to the wood allow a more precise analysis of this effect, being
the subject of interest for the development of future work.
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