Evaluation by a finite element method of the flexibility factor and fixity degree for the base plate connections commonly used
Using finite element approach, we have determined the flexibility factor and fixity degree of connections between a steel column and a reinforced concrete foundation. Two types of connections are studied. The first one consists of a base plate welded to the end of column and attached to the rei...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2007
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| Цитувати: | Evaluation by a finite element method of the flexibility factor and fixity degree for the base plate connections commonly used / M. Hamizi, N.E. Hannachi // Проблемы прочности. — 2007. — № 6. — С. 35-50. — Бібліогр.: 20 назв. — англ. |
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irk-123456789-481572013-08-15T18:56:43Z Evaluation by a finite element method of the flexibility factor and fixity degree for the base plate connections commonly used Hamizi, M. Hannachi, N.E. Научно-технический раздел Using finite element approach, we have determined the flexibility factor and fixity degree of connections between a steel column and a reinforced concrete foundation. Two types of connections are studied. The first one consists of a base plate welded to the end of column and attached to the reinforced concrete foundation by two anchor bolts. These bolts are placed on the major axis of the I-shaped section used as column. In the second configuration, the connection comprises a base plate and four anchor bolts placed out side the flanges of the I-shaped section. Two types of loadings are used, first connections were subjected to shear force and a bending moment, in the second case, the connections were subjected to shear force, a bending moment and an axial compressive force. To take into account the real behavior of these connections, an approach treating of the contact- friction problem between the base plate and the concrete foundation is retained. The method of approach is based on a unilateral contact law in which a Coulomb friction is added. The numerical resolution is ensured by the refined Lagrangian method. The moments-rotations curves, the flexibility factor according to the distance of the top of base plate curves are drawn. The fixity degrees of the connections are determinate and their influence on the loads and deformations are evaluated. Із використанням скінченноелементного підходу визначено значення коефіцієнта гнучкості та степеня стійкості вузлів з ’єднання стальної балки із залізобетонною основою. Досліджено два типи з ’єднань: у першому стальна опорна пластина з привареною до неї вертикальною балкою кріпиться до залізобетонної основи двома анкерними болтами, які знаходяться на осі симетрії двотаврової балки, у другому - чотирма болтами. Навантаження прийнято двох типів: перше з ’єднання зазнавало дії перерізувальної сили і згинального моменту, друге - перерізувальної сили, згинального моменту та осьової сили стиску. Реальну поведінку цих з ’єднань описували за допомогою підходу, який враховує умови контакту та тертя між опорною плитою і залізобетонною основою. Підхід базується на однобічності залежності для контактної задачі з кулонівським тертям. Для підвищення точності числових розрахунків застосовано модифікований метод Лагранжа. Отримано діаграми в координатах момент-кутове переміщення та коефіцієнт гнуч- кості-відстань від вершини вертикальної балки до опорної плити. Визначено вплив степеня стійкості з’єднань на допустимі навантаження і деформації. С использованием конечноэлементного подхода определены значения коэффициента гибкости и степени устойчивости узлов соединения стальной балки с железобетонным основанием. Исследуются два типа соединений: в первом стальная опорная пластина с приваренной к ней вертикальной балкой крепится к железобетонному основанию двумя анкерными болтами, которые расположены на оси симметрии двутавровой балки, во втором - четырьмя болтами. Задавались два типа нагружения: первое соединение подвергалось действию перерезывающей силы и изгибающего момента, второе - перерезывающей силы, изгибающего момента и осевой силы сжатия. Для описания реального поведения этих соединений использовался подход, учитывающий условия контакта и трения между опорной балкой и железобетонным основанием. Подход основан на односторонней зависимости для контактной задачи с кулоновским трением. Для повышения точности численных расчетов используется модифицированный метод Лагранжа. Получены диаграммы в координатах момент-угловое перемещение и коэффициент гибкости-расстояние от вершины вертикальной балки до опорной плиты. Определено влияние степени устойчивости соединений на допустимые нагрузки и деформации. 2007 Article Evaluation by a finite element method of the flexibility factor and fixity degree for the base plate connections commonly used / M. Hamizi, N.E. Hannachi // Проблемы прочности. — 2007. — № 6. — С. 35-50. — Бібліогр.: 20 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/48157 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Научно-технический раздел Научно-технический раздел |
| spellingShingle |
Научно-технический раздел Научно-технический раздел Hamizi, M. Hannachi, N.E. Evaluation by a finite element method of the flexibility factor and fixity degree for the base plate connections commonly used Проблемы прочности |
| description |
Using finite element approach, we have determined
the flexibility factor and fixity degree of
connections between a steel column and a reinforced concrete foundation. Two types of connections
are studied. The first one consists of a
base plate welded to the end of column and attached
to the reinforced concrete foundation by
two anchor bolts. These bolts are placed on the
major axis of the I-shaped section used as column.
In the second configuration, the connection
comprises a base plate and four anchor
bolts placed out side the flanges of the I-shaped
section. Two types of loadings are used, first
connections were subjected to shear force and a
bending moment, in the second case, the connections
were subjected to shear force, a bending
moment and an axial compressive force. To
take into account the real behavior of these connections,
an approach treating of the contact-
friction problem between the base plate
and the concrete foundation is retained. The
method of approach is based on a unilateral contact
law in which a Coulomb friction is added.
The numerical resolution is ensured by the refined
Lagrangian method. The moments-rotations
curves, the flexibility factor according to
the distance of the top of base plate curves are
drawn. The fixity degrees of the connections
are determinate and their influence on the loads
and deformations are evaluated. |
| format |
Article |
| author |
Hamizi, M. Hannachi, N.E. |
| author_facet |
Hamizi, M. Hannachi, N.E. |
| author_sort |
Hamizi, M. |
| title |
Evaluation by a finite element method of the flexibility factor and fixity degree for the base plate connections commonly used |
| title_short |
Evaluation by a finite element method of the flexibility factor and fixity degree for the base plate connections commonly used |
| title_full |
Evaluation by a finite element method of the flexibility factor and fixity degree for the base plate connections commonly used |
| title_fullStr |
Evaluation by a finite element method of the flexibility factor and fixity degree for the base plate connections commonly used |
| title_full_unstemmed |
Evaluation by a finite element method of the flexibility factor and fixity degree for the base plate connections commonly used |
| title_sort |
evaluation by a finite element method of the flexibility factor and fixity degree for the base plate connections commonly used |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2007 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/48157 |
| citation_txt |
Evaluation by a finite element method of the flexibility factor and fixity degree for the base plate connections commonly used / M. Hamizi, N.E. Hannachi // Проблемы прочности. — 2007. — № 6. — С. 35-50. — Бібліогр.: 20 назв. — англ. |
| series |
Проблемы прочности |
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AT hamizim evaluationbyafiniteelementmethodoftheflexibilityfactorandfixitydegreeforthebaseplateconnectionscommonlyused AT hannachine evaluationbyafiniteelementmethodoftheflexibilityfactorandfixitydegreeforthebaseplateconnectionscommonlyused |
| first_indexed |
2025-07-04T08:26:29Z |
| last_indexed |
2025-07-04T08:26:29Z |
| _version_ |
1836704169583443968 |
| fulltext |
UDC 539.4
Evaluation by a Finite Element Method of the Flexibility Factor and
Fixity Degree for the Base Plate Connections Commonly Used
M. Hamizi and N. E. Hannachi
Université Mouloud Mammeri, Département de Génie Civil, Tizi-Ouzou, Algérie
УДК 539.4
Оценка методом конечных элементов коэффициента гибкости и
степени устойчивости типичных узлов соединения опорных
пластин с основанием
М. Хамизи, Н. Э. Ханнаши
Университет им. Мулуда Маммери, отделение гражданского строительства, Тизи-
Узу, Алжир
С использованием конечноэлементного подхода определены значения коэффициента гиб
кости и степени устойчивости узлов соединения стальной балки с железобетонным основа
нием. Исследуются два типа соединений: в первом стальная опорная пластина с прива
ренной к ней вертикальной балкой крепится к железобетонному основанию двумя анкер
ными болтами, которые расположены на оси симметрии двутавровой балки, во втором -
четырьмя болтами. Задавались два типа нагружения: первое соединение подвергалось
действию перерезывающей силы и изгибающего момента, второе - перерезывающей силы,
изгибающего момента и осевой силы сжатия. Для описания реального поведения этих
соединений использовался подход, учитывающий условия контакта и трения между опорной
балкой и железобетонным основанием. Подход основан на односторонней зависимости для
контактной задачи с кулоновским трением. Для повышения точности численных расчетов
используется модифицированный метод Лагранжа. Получены диаграммы в координатах
момент-угловое перемещение и коэффициент гибкости-расстояние от вершины верти
кальной балки до опорной плиты. Определено влияние степени устойчивости соединений на
допустимые нагрузки и деформации.
Ключевые слова: соединения колонн и оснований, коэффициент гибкости,
степень устойчивости, конечноэлементный подход, односторонний контакт,
кривые момент-угловое перемещение.
Introduction. Today, to study the steel connections between a steel column
and a reinforced concrete foundation as a perfectly pinned connection (R = 0) or
fully rigid connection (R = œ) is not quite realistic approach. Numerous and
well-documented studies in the past few decades have shown the nonlinear
behavior of the connections and their nonperfect rigidity or flexibility. The factor
of rigidity R expressed in kN-m/rad covers the whole range of the values
varying from zero to œ. The tools of design as well as the standards used in the
computer codes do not take into account this partial rigidity of the connections.
Assuming an idealized behavior for column base connections (perfectly pinned or
fully rigid) does not reflect the true behavior of these connections and even less
© M. HAMIZI, N. E. HANNACHI, 2007
ISSN 0556-I7IX. Проблемы прочности, 2007, N 6 35
M. Hamizi and N. E. Hannachi
the consequences for the results o f analyses, m ainly the internal force distribution
between members and structural deformations. The partial flexibility (X = 1/R ) of
column bases and its consequences, like a fixity degree y, on the analysis o f steel
structures have drawn less attention from research community than beam-to-
column connections. Nevertheless, results from all studies tend to confirm that
column base connections exhibit semi-rigid behavior. Galambos [1] was the first
to demonstrate the effect o f column base fixity on strength o f column. He
concluded that the buckling strength o f rigid frames could benefit positively from
the partial rigidity o f column base connections. Two decades later, N ixon [2]
picked up the Galambos theoretical equations and demonstrated that the increased
strength o f column could lead to non-negligible savings in light industrial
buildings. In 1970, an experimental research was undertaken at Laval University
o f Quebec (Canada). The results obtained by Lizotte and Beaulieu [3] showed
that the degree o f base plate fixity o f a simple two-anchor-bolt column base,
nom inally assumed to be pinned, was such that it could be considered as a fixed
base connection until buckling occurred. Also, the mom ent developed at the base
before buckling did not induce rotation o f the base plate. Later, Picard and Dion
[4], Samson and Beaulieu [5], and Perruse and Beaulieu [6 ], showed that the
presence o f axial load significantly increases the degree o f column base fixity,
and i f considered in column analysis, it reduces the effective length o f the
column, reduces second order (P — A) effects, and leads to m ore realistic bending
m om ent distribution in the column. The base restraint coefficient (G L) then
recom m ended by Canadian Standard S16 (CSA 1978 and CSA 2001) [7, 8 ] for
assumed pinned or fixed column base connection appeared conservative and was
consequently not appropriate. Also Eurocode 3 [9] treats these connections either
as pinned or fixed. Knowing the true degree o f fixity, N ixon’s equations led to
m ore accurate evaluations o f column buckling loads. A few years later, Beaulieu
and Picard [10] showed from the results o f the experimental program that column
buckling seems to occur in the elastic behavior zone o f the colum n-foundation
connection, that is, the linear portion at the beginning o f the M — 6 curve. Also,
they showed that the num ber o f anchors and their size do not influence the
buckling resistance, but they believed that eventually substantial material economy
could be gained i f the true rigidity o f column base connections was taken into
consideration; for instance, in the design o f industrial structures. Experimental
research by M elchers [11] also demonstrated the m om ent resistance capacity of
assumed pinned column base connections and identified parameters that influence
column base rigidity, such as base plate thickness; anchor bolt size and column
size. These results were confirmed by Pensirini and Colson [12]. They show that
the initial stiffness and ultimate capacity o f the connection are significantly
dependent on the column axial load. In 1996, Ermopoulos and Stamatopoulos
[13] reached the conclusion that increased axial loading confers higher rigidity to
column base connection. They identified parameters that influence column base
rigidity, such as a base plate thickness, anchor bolt size, the concrete stress, the
nonlinear contact between base plate and concrete foundation. One year later, the
same authors [14] proposed an analytical m odeling o f column base plates under
cyclic loading based on mathem atical model. Following these results, another
study undertaken by Kootolen and Baniotopoulos [15] showed the effect o f axial
36 ISSN 0556-171X. Проблемы прочности, 2007, № 6
Evaluation by a Finite Element Method
load on the displacements o f base plate. They simulated the nonlinear contact
between base plate and concrete foundation. The last study has been carried out
by Dumas, Beaulieu, and Picard [16]. Results obtained from finite element model
show that consideration o f the semi-rigidity o f column base connections increases
the accuracy o f the analysis results and leads to a decrease in structure weight and
deformation.
1. D evelopm ent o f th e M odel by F in ite E lem ent. Taking into account
studies enumerated above and the various recommendations m ade by the authors,
we built a two-dimension finite element model based on the nonlinear analysis o f
the structure to simulate the behavior o f column base plate connection. The model
takes into account the nonlinearity o f m aterials and the nonlinearity o f contact
between the foundation and the base plate, where it simulates the rising o f the
base plate and where friction at the interface concrete foundation-base plate is
ensured by four nodes quadratic elements [17]. The m odel is established in
CASTEM 3M computer code.
1.1. U n ila te ra l C o n ta c t (the S ig n o r in i P ro b le m ). In numerous simulations,
the law o f unilateral contact used is illustrated by the problem o f Signorini. Let us
consider a deformable body in contact with a rigid body (Fig. 1), the conditions o f
unilateral contact o f Signorini having to be respected in all points o f the
deformable bodies located in the contact T c are [1 8 ] :
h < 0 ,
R n < 0 ,
h ■ R n — 0 ,
( 1 a)
( 1b)
( 1 c)
where h is the interstice or the displacement o f a point o f contact in the normal
direction to the contact n, and R n is the component o f the norm al effort.
Equation (1a) translates the condition o f impenetrability, equation (1b) - the fact
that the normal force o f contact is compression, and equation ( 1c) represents the
condition o f complementarily ( if the point is in contact then h — 0 and R n ^ 0 , if
the point leaves the contact then h < 0 and Ryt = 0) [18].
Imposed forces f s
г с n : external normal
with the contact
Fig. 1. Contact between a deformable body and rigid body (the Signorini problem).
ISSN 0556-171X. Проблемы прочности, 2007, № 6 37
M. Hamizi and N. E. Hannachi
1.2. C o u lo m b s L aw . The force at the point o f contact can be broken up into a
normal force R n and a tangential force R 7 (R = R n • n + R 7 • t ). The model of
Coulomb is written in the following w ay [18]:
\R t \ ^ f * \ R n \,
R 7 < /u\ R n \ ^ v 7 = 0 (adherence),
R t = - P \ R ~r (slip),
(2 a)
(2 b)
(2 c)
where v i is the tangential relative speed between the two bodies and u is the
friction coefficient o f Coulomb (Fig. 2) which includes all the local parameters,
such as roughness between the two bodies.
Fig. 2. The Coulomb law.
v
v t
1.3. E q u ilib r iu m w ith o u t F r ic tio n . A deformable body Q 1 and a rigid body
Q 2 are considered, we note by Q = Q i U Q 2 the total o f the two bodies. The
deformable body is subjected to the imposed displacements u d on the zone T u,
to applied loads f s on the zone , and to forces o f volumes f V acting on the
field (Fig. 1)
d iv Ö + f v = 0 in Q, (3)
u = u d on r u , (4)
[ a ] - n = f s on T a , (5)
[a ]• n = R on Tc . (6 )
1.4. E q u ilib r iu m w ith F r ic tio n . Equilibrium o f deformable body Q 1 with
frictional contact is described by
W body = 2 f [O' ]{£}dV ~ f f v • u d V ~ f f v • u d S - W cont. (7)
2 v v rc
38 ISSN 0556-171X. npoôneMbi nponnocmu, 2007, № 6
Evaluation by a Finite Element Method
The work o f the actions o f contact on the deformable body is written as
W cont = f f (R nn • U n + R t • u t ) d S (8)
rc
with
u t = ( U2 — U1) — u n • n.
The actions o f rigid body Q 2 on body Q 1 are described by
fc o n t = f f (R n • n + R t ) d S . (9)
rc
1.5. F in ite E le m e n t M o d e lin g .
a = C /£ (behavior law),
£ = [B ]u (interpolation o f deformations),
u = [N ]uk (interpolation o f displacements).
In matrix form the Eq. (7) is written
Wbody = 2 u T [K ]u — u T { F } ( 1 0 )
with rigidity matrix
[K ] = f B T [ C ] B d V ,
V
and vector o f the external efforts
{ F } = f [N T ] f v d V + f [N T ] f s d S + f [N T ] f (xmtd S .
The equilibrium o f the system with frictional contact amounts to m inimize
the energy equation under the following constraint:
\d iv o = 6 W body = 0,
̂ - - - - T ( 1 1 )
[h = (( u 2 — U1 ) • n ) • n = [G ] u n = 0.
1.6. M e th o d o f R e so lu tio n (A u g m e n te d L a g ra n g ia n P ro b le m ). The m ethod of
resolution is based on combination o f the penalization and Lagrangian methods.
We have to insert a large term a (penalization coefficient) and X (Lagrangian
multiplier) into the energy equation (10) [19]:
Гс Гс
ISSN 0556-171X. Проблемы прочности, 2007, № 6 39
M. Hamizi and N. E. Hannachi
1 ^
W bodv( u , A) = - u T [K ]u + AT [G]T u + - u T [G ][G ]T u = 0 ,
^Wbody ( u , A ) = 0
d W
du
d W
„ dA
_ 0’ f[K + a G G T ]u K + [G ]Ak = F ,
= 0, " | a k + 1 = A K + a[G ]T u K .
( 1 2 )
(13)
2. B ehav io r Law s. For the finite elements model, we adopted for the column
and the base plate, and anchors bolts the behavior laws illustrated by Figs. 3 and 4,
respectively.
Fig. 3 Fig. 4
Fig. 3. Adopted stress-strain relations for the steel column and base plate connection.
Fig. 4. Adopted stress-strain relations for the anchor bolts.
For the foundation concrete, the material is considered to operate in the
elastoplastic field with the Young modulus E C = 29 GPa, Poisson ratio y C = 0.18,
tensile strength f t = 3 MPa, and compressive strength f c = 38 MPa.
3. N um erical Exam ples. In this study, two types o f connections are
analyzed. The first one consists o f a base plate welded to the end o f column and
attached to the reinforced concrete foundation with two anchor bolts. These bolts
are placed on the m ajor axis o f the I-shaped column cross section, one anchor bolt
on each side o f the web (Fig. 5). In the second configuration, the connection
comprises a base plate and four anchor bolts placed outside the flanges o f the
I-shaped section (Fig. 6 ). Two loading types are used. First, the connections were
subjected to shear force and a bending moment only, then the connections were
put under shear force, a bending m om ent and an axial compressive force (Fig. 7).
In this case a bending moment is caused by the offset compressive load.
Different eccentricities and variable axial loadings (P = 100 to 600 kN) are
chosen, in order to show the influence o f these parameters o f the degree o f fixity
o f the column base connections.
The following measures have been in order to perform correctly this study:
• An interaction between the holes in the base plate and the anchor bolts is
ensured by considering a unilateral contact between these two bodies.
• In order to simplify the mesh, the anchor bolts that are o f circular sections
are simulated in this study by bolts o f square sections o f equivalent surface.
40 ISSN 0556-171X. npoôneMu npounocmu, 2007, № 6
Evaluation by a Finite Element Method
70 mm 70 mm
Fig. 5
) mm
Fig. 6
Fig. 5. Detail of two anchors bolts connection FT (HE100 B: A = 26-102 mm2, Ix = 449.5-103
mm4, Sx = 89.91-103 mm3, h = 100 mm, b = 100 mm, t f = 10 mm, tw = 6 mm).
Fig. 6 . Detail of four anchor bolts’ connection CFT (HE160 B: A = 54.3-102 mm2, Ix =
= 2492-103 mm4, Sx = 311.5-103 mm3, h = 160 mm, b = 160 mm, f = 13 mm, tw = 8 mm).
Fig. 7. Finite element mesh of the 3D model.
• The simulation o f the anchor bolts is made so that the nodes coincide with
the nodes o f the holes o f the base plate.
• To take into account the problem o f contact friction between the base plate
and the foundation, the nodes as well as the degrees o f freedom o f the two bodies
are selected so that they coincide.
• The same precaution is also taken w ith regard to the nodes and the degrees
o f freedom o f the anchor bolts and the concrete foundation.
• Traction in the concrete develops only in the higher part o f bolt (on the
third o f L P ).
ISSN 0556-171X. npoôneMU npounocmu, 2007, № 6 41
M. Hamizi and N. E. Hannachi
• The loadings are introduced in the forms o f increments (ensured by
CASTEM 3M code).
• Precautions are also taken with regard to the m easurem ent o f rotations in
levels which coincide with the experimental study [4] and this rotation 0 =
= ( a — b ) /x in which a and b are the displacement measured on the flanges o f
the column at the im posed distances x [16] (Fig. 7).
• The deform ation o f the soil under the concrete foundation is neglected
owing to the fact that the bending m om ent developed at the column base plate
seems weak to force the foundation.
• The results o f our model were com pared with the experimental results
obtained in [4].
The details o f connections are shown in Table 1.
T a b l e 1
Studied Parameters Values
Connection FT1 FT4 CFT1 CFT3 CFT6 CFT6 CFT6 CFT6 CFT6
Column length L, mm 1220 1220 1220 1220 1220 1220 1220 1220 1220
Eccentricity e, mm 150 300 300 300 300 300 300
Axial load P, kN 300 200 100 200 300 400 600
Distance x with the top
of the base plate (mm)
0
170
350
0
170
350
430
0
170
350
430
0
170
350
430
0
170
350
430
0
170
350
430
0
170
350
430
0
170
350
430
0
170
350
430
4. Results. The following m om ent-rotation curves were obtained (Figs. 8-12).
4.1. M o m e n t-R o ta tio n C urves a n d C o m p a riso n w ith L a v a l U n iversity
C urves.
Rotation (rad)
Fig. 8 . Moment-rotation curve connection HE100 B with 2 anchors (“Level 1”, axial load P = 0):
(A) FT1 experimental; (■) FT1 model.
42 ISSN 0556-171X. npoôneMU nponnocmu, 2007, № 6
Evaluation by a Finite Element Method .
Є
g,
Ü Sо
2
Rotation (rad)
Fig. 9. Moment-rotation curve connection HE160 B with 4 anchors (“Level 1”, P = 0): (■) FT4
experimental; (A) FT4 model.
Rotation (rad)
Fig. 10. Moment-rotation curve connection HE160 B with 4 anchors (“Level 1”, P = 600 kN):
(■) CFT6 experimental; (A) CFT6 model.
4.2. M o m e n t-R o ta tio n C u rves o f O th er C o n n ectio n s.
Rotation (rad)
Fig. 11. Moment-rotation curve HE100 B with 2 anchors under various axial loadings: (■) FT1,
P = 0; (A) CFT3, P = 200 kN; (♦) CFT1, P = 300 kN.
ISSN 0556-171X. Проблемы прочности, 2007, № 6 43
M. Hamizi and N. E. Hannachi
Rotation (rad)
Fig. 12. Moment-rotation curve connection HE160 B with 4 anchors (FT4) for various axial load:
( • ) P = 0; (A) P = 100 kN; (■) P = 200 kN; (♦ ) P = 300 kN; (+) P = 400kN; ( x ) P = 600 kN.
4.3. F le x ib il ity F a c to r C u rves were obtained (Figs. 13-15).
Fig. 13. Flexibility factor curves of FT1 and FT4 connections according to the distance to the top of
the base plate: (■), (A) theoretical curves of FT1 and FT4, respectively; (□), (A) model curves of
FT1 and FT4, respectively.
Distance to the top of the base plate (mm)
Fig. 14. Flexibility factor curves of CFT1 and CFT3 connections according to the distance to the top
of the base plate: (■) theoretical curve of pinned column; (A) theoretical curve of fixed column;
(*) CFT1 model; (O) CFT3 model.
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Evaluation by a Finite Element Method
Distance to the top of the base plate (mm)
Fig. 15. Flexibility factor curves of CFT6 connection according to the distance to the top of the base
plate for various axial load: (+) theoretical curve of pinned column; (■) CFT6 model, P = 100 kN;
(▲) CFT6 model, P = 200 kN; ( • ) CFT6 model, P = 300 kN; (X) CFT6 model, P = 400 kN; (*)
CFT6 model, P = 600 kN; dashed line corresponds theoretical curve of fixed column.
4.4. C a lcu la tio n o f th e F ix ity D eg re e . Once flexibility factor is calculated,
we carried out the fixity degree o f the various connections. For FT connections
subjected only to shear force and bending moment, their fixity degrees are
calculated by using the Eq. (14) suggested by Brun and Picard [18] and we
obtained the results o f Table 2,
Y = -----(14)
1 +
3EIX,
L
For the CFT connections subjected to the axial load in addition to shear force
and bending moment, the fixity degrees are calculated by using the linear portion
o f the m om ent-rotation curve obtained by the model. For each position o f the
column where rotations are evaluated, a factor is calculated by the Eq. (15) and
the fixity degree o f connection is given by the Eq. (16) [4]:
X mod x — X fx
X rx X fx
(15)
with
2xL — 3x
X f x = — — (fixed connection),
4E IL
(16)
and
_ L2 - 3x 2
rx _ 6E IL
(pinned connection), (17)
y _ 1 — (1 8 )
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M. Hamizi and N. E. Hannachi
T a b l e 2
Fixity Degrees of the FT Connections
Connection 2 0, (kN- mm)- 1
FT1 13-10'-7 0.258
FT4 0.8 -10- 7 0.505
T a b l e 3
The Fixity Degrees of CFT Connections
Connections FT1 CFT3 CFT6 ,
P = 100 kN
CFT6 ,
P = 200 kN
CFT6 ,
P = 300 kN
CFT6 ,
P = 400 kN
CFT6 ,
P = 600 kN
Fixity degree 0.715 0.855 0.690 0.710 0.790 0.910 0.990
For CFT1, CFT3 and CFT6 connections, the values o f fixity degree are
summarized in Table 3.
4.5. In flu e n c e o f the F ix ity D e g re e on the E ffo r ts a n d D e fo rm a tio n s . To
illustrate the influence o f the rigidity o f column base connection in structure
analysis, a study on a simple frame (Fig. 16) subjected to axial loads P = 350 kN
and lateral load F = 44.5 kN is carried out by considering different cases o f fixity
degree (y = 0 , 0.5, and 0.7, and X = 1).
Fig. 16. Frame used to evaluate the efforts and displacements for various fixity degrees [(A, D)
joints with variable rigidity; (B, C) joints with fixed rigidity; E = 2 -105 MPa].
The analysis o f the frame has been carried out using a matrix method o f
structural analysis w ith pure linear deformation joints [20]. This method consists
in modifying the rigidity matrix o f the frame elements to take into account the
jo in t rigidity using a fixity degree, which can vary from 0 to 1. The (P — A)
effects are included in calculations. The bending moments and lateral
displacements obtained o f first order analysis are m ultiplied by a factor o f
amplification U 0 [4]
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U о =
S p А i
i =1
F h
(19)
where F is the lateral load, P t the axial load in the column, A t the lateral
displacement o f the column, and h the height o f column.
Table 4 gives the moments as well as the side displacement at the head and
the base o f column for each degree o f fixity.
T a b l e 4
Efforts and Deformations in the Column for the Various Degrees of Fixity
Fixity degree Case 1
у = О
Case 2
у = 0.5
Case З
у = 0.7
Case 4
у = 1
Moment at the head
of column (kN • m)
56.2о З2.9 28.0 24.6
Moment at the base
of column (kN • m)
О 28.1 З1.9 З4.З
Side displacement
of the column (mm)
З1.15 12.2 9.6 6.7
4.6. C o m p a riso n o f the R esu lts . We compared the results obtained for
various fixity degrees used in the study (Table 5).
T a b l e 5
Comparison of the Moments and Displacements at the Head of Column
According to the Degrees of Fixity
Case 1/Case 2 Case 1/Case З Case 1/Case 4
Reduction of the moment
at the head of column
40% 50% 56%
Reduction of the side
displacement
60% 69% 78%
C onclusions. The m odel gives good results; the various comparisons reflect
it well (see Figs. 8-10).
The assumptions that we adopted are not far from reality; the results
obtained by the m odel fit well with the experimental results for the first steps of
loadings and then under the estimated rotations. That is certainly with the fact
that in reality the rotation of the foundation is not negligible. The results obtained
are the following:
(i) for the connections w ith two anchors bolts w ithout axial load, the model
sticks very well the experimentation for 0<O.O1 rad, after the m odel gives
rotations lower by 10% o f the experimental ones (Fig. 8 );
1
1
ISSN 0556-171X. Проблемы прочности, 2007, № 6 47
M. Hamizi and N. E. Hannachi
(ii) for the connections w ith four anchors bolts w ithout axial load, the model
stick very well w ith the experiment for 0 < 0 .0 0 1 rad, after the m odel gives
rotations lower by 20% o f those o f the experiment (Fig. 9);
(iii) for the connections w ith four anchors bolts with axial load, the model
stick very well w ith the experiment for 0 < 0.005 rad, after the m odel gives
rotations lower by 25% o f those o f the experiment (Fig. 10).
According to these results, presence o f m ore anchor bolts in the connection
prevents rotations o f the column base plate to foundation. On the other hand,
presence o f axial load on the level o f connection does not eliminate the rotation o f
the foundation to soil as reported in [3]. Even without axial load applied to the
column, the connections FT1 and FT4 have a flexional rigidity (quite significant
resistance to rotation to be considered in calculations). The connections with four
anchor bolts have a higher rigidity than the connections w ith two anchor bolts.
Presence o f axial load on the top o f column produces a significant increase in
fixity degree o f the connections compared to that obtained when no axial load is
applied. I f we take into account the rigidity o f the column base connections as a
beam-colum n behavior; the principal advantages are: effective reduction length of
the column, reduction o f the m om ent at the head o f column, increase in the
m om ent at the base o f column, reduction o f side displacement at the head of
column, and reduction o f the second order ( P — Д ) effects.
I f the minim um o f fixity degree o f the studied connections is equal to 0.5 in
the presence o f an axial load o f compression, real displacement will be equal only
to 40% of the displacement taken in calculations. We also see that the columns
bear only about 60% o f their capacity. The decrease is due not only to the
(P — Д) effects, but also because the column operaty in double curve and that
maximum bending m om ent in the column decreases when a m om ent is developed
in the jo in t at the base o f columns column. We propose to take the fixity degree of
column base connections equal to 0.5 (y = 0.5) in calculation o f steel frames, since
beyond this value, the reduction o f the moments and displacements at the head o f
column is o f less importance. I f the rigidity o f the jo in t at the base o f the column
is considered, one can make the choice o f a m ore economic section for the
columns.
Р е з ю м е
Із використанням скінченноелементного підходу визначено значення коефі
цієнта гнучкості та степеня стійкості вузлів з ’єднання стальної балки із
залізобетонною основою. Досліджено два типи з ’єднань: у першому стальна
опорна пластина з привареною до неї вертикальною балкою кріпиться до
залізобетонної основи двома анкерними болтами, які знаходяться на осі
симетрії двотаврової балки, у другому - чотирма болтами. Навантаження
прийнято двох типів: перше з ’єднання зазнавало дії перерізувальної сили і
згинального моменту, друге - перерізувальної сили, згинального моменту та
осьової сили стиску. Реальну поведінку цих з ’єднань описували за допо
могою підходу, який враховує умови контакту та тертя між опорною пли
тою і залізобетонною основою. Підхід базується на однобічності залежності
48 ISSN 0556-171X. Проблеми прочности, 2007, N 6
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для контактної задачі з кулонівським тертям. Для підвищення точності
числових розрахунків застосовано модифікований метод Лагранжа. Отрима
но діаграми в координатах момент-кутове переміщення та коефіцієнт гнуч-
кості-відстань від вершини вертикальної балки до опорної плити. Визна
чено вплив степеня стійкості з’єднань на допустимі навантаження і дефор
мації.
1. T. V. Galambos, “Influence of partial base fixity on frame stability,” ASCE J.
Struct. Div., 86 (ST-5), 85-108 (1960).
2. C. D. Nixon, The Design o f Light Industrial Buldings, Ph.D. Thesis,
Department of Civil Engineering University of Alberta, Alta, Edmonton
(1979).
3. J. Lizotte et D. Beaulieu, Influence de la Fondation sur la Stabilité des
Poteaux, Rapport GCT-81-05, Département de Génie Civil, Université Laval,
Quebec (1981).
4. A. Picard et J. Dion, Étude Expérimentale des Assemblages Poteau -
Fondation dans les Charpentes d ’Acier, Rapport GCT-81-04, Département
de Génie Civil, Université Laval, Quebec (1981).
5. G. Samson and D. Beaulieu, Étude de la Stabilité d ’un Poteau avec Attache
Semi-Rigide a la Fondation, Rapport GCT-83-01, Département de Génie
Civil, Université Laval, Quebec (1982).
6. B. Pérusse et D. Beaulieu, Étude Expérimentale de la Rigidité d ’un
Assemblage Poteau - Fondation de Type Standard, Rapport GCT-85-07,
Département de Génie Civil, Université Laval, Quebec (1985).
7. Standard CSA S16.1-78. Limit States Design o f Steel Structures, Canadian
Standards Association, Toronto (1978).
8. Standard CSA S16-01. Limit States Design o f Steel Structures, Canadian
Standards Association, Toronto (2001).
9. Eurocode 3. Design o f Steel Structures. 1.1. General Rules and Rules fo r
Building (1992).
10. D. Beaulieu et A. Picard, “Contribution des assemblages avec plaque
d’assise a la stabilité des poteaux,” Constr. Métall., 2, 3-19 (1985).
11. R. E. Melchers, “Modeling of column-base behavior,” in: R. Bjorhovde, J.
Brozzetti, and A. Colson (Eds.), Connections in Steel Structures: Behavior,
Strength, and Design, Elsevier Science Publ. Co., New York (1988), pp. 151
157.
12. P. Penserini et A. Colson, “Caractérisation des liaisons structure métallique -
fondation: application en flambement des poteaux,” Constr. Métall., 2,
43-52 (1992).
13. J. C. Ermopoulos and G. N. Stamatopoulos, “Mathematical modeling of
column base plate,” J. Constr. Steel Res., 36, No. 2, 79-100 (1996).
14. G. N. Stamatopoulos and J. C. Ermopoulos, “Interaction curves for column
base plate connections,” J. Constr. Steel Res., 44, No. 1-2, 69-89 (1997).
ISSN 0556-Î7ÎX. Проблемы прочности, 200?, № б 49
M. Hamizi and N. E. Hannachi
15. M. J. Konotoleon and C. C. Baniotopoulos, “Computational aspects on the
frictional unilateral contact problem arising on steel base plate connections,”
Comp. Struct., 78, 303-309 (2000).
16. M. Dumas, D. Beaulieu, and A. Picard, “Characterization equations for steel
column base connections,” Canadian J. Civil Eng., 33, No. 4, 409-420
(2006).
17. M. Hamizi et D. Beaulieu, Études Expérimentale et Numérique des Pontages
Métalliques, Rapport GCT-87-05, Département de Génie Civil, Université
Laval, Quebec (1987).
18. L. Champaney, “Contact unilatéral entre solides élastiques,” Notes de cours
‘Eléments Finis’ du DESS Dynamique des Structures Modernes dans leur
Environnement.
19. J. C. Simo and T. A. Laursen, “An augmented Lagrangien treatment of
contact problems involving friction,” Comp. Struct., 42, No. 1, 97-116
(1992).
20. P. Brun et A. Picard, Étude d'un Assemblage Imparfaitement Rigide et des
Effets de son Utilisation dans un M ultiétagé, Rapport GCT-76-03,
Département de Génie Civil, Université Laval, Quebec (1976).
R e c e iv e d 14. 05 . 2007
50 ISSN 0556-171X. Проблемы прочности, 2007, № 6
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