Analysis of a Semi-rigid Connection for Precast Concrete

Proceedings of the Institution ofCivil EngineersStructures and Buildings 163February 2010 Issue SB1Pages 41–51doi: 10.1680/stbu.2009.163.1.41

Paper 800067Accepted 09/09/2008Received 27/08/2009

Keywords: concrete structures/design methods & aids/buildings,structure & design

Mounir K. El DebsProfessor, Department ofStructural Engineering,University of Sao Paulo,Brazil

Anamaria M. MiottoFormer PhD student,Department of StructuralEngineering, University of SaoPaulo, Brazil

Ana Lucia H. C. El DebsAssociate Professor,Department of StructuralEngineering, University of SaoPaulo, Brazil

Analysis of a semi-rigid connection for precast concrete

M. K. El Debs BEng, MSc, PhD, A. M. Miotto BEng, MSc, PhD and A. L. H. C. El Debs BEng, MSc, PhD

This paper presents a study of a specific type of beam-to-

column connection for precast concrete structures.

Furthermore, an analytical model to determine the

strength and the stiffness of the connection, based on

test results of two prototypes, is proposed. To evaluate

the influence of the strength and stiffness of the

connection on the behaviour of the structure, the results

of numerical simulations of a typical multi-storey

building with semi-rigid connections are also presented

and compared with the results using pinned and rigid

connections. The main conclusions are: (a) the proposed

design model can reasonably evaluate the studied

connection strength; (b) the evaluation of strength is

more accurate than that of stiffness; (c) for a typical

structure, it is possible to increase the number of storeys

of the structure from two to four with lower horizontal

displacement at the top, and only a small increase of the

column base bending moment by replacing the pinned

connections with semi-rigid ones; and (d) although there

is significant uncertainty in the connection stiffness, the

results show that the displacements at the top of the

structure, and the column base moments present low

susceptibility deviations to this parameter.

1. INTRODUCTION

Precast concrete components are characterised by their ability

to be easily produced in factory plants. However, it is necessary

to connect these components to construct a building structure.

These connections are one of the main problems designers have

to face when using precast concrete structures.

The connections affect all stages of structures production, from

the adjacent parts of components manufacture to the assembly

of the structure. This in turn determines the behaviour of the

final structure. Consequently, they can be considered the most

important part of the design of precast concrete structures.

The simplest connections are pinned connections, which

produce structures subject to large bending moments. However,

connections that reproduce the behaviour of cast-in-place

reinforced concrete structures through the continuity of beams

and columns require more labour, reducing the inherent

advantages of precast concrete. The main problems of the

moment-resisting connections are the need for connecting

materials, steel and concrete, and tolerance adjustment, in

addition to the consequences of fragile concrete behaviour.

Thus, the beam-to-column connections of low-rise buildings

are generally designed to be pinned connections, owing to the

ease of production, instead of the better structural behaviour

resulting when the column bending moments are reduced. In

this case, the columns work as cantilevers and are fully

responsible for the structure stability. Consequently, when the

building height rises, the bending moments of the columns

rapidly increase, and the system becomes economically

impracticable. The alternatives to minimise this effect are the

use of special bracing elements or moment-resistant beam-to-

column connections. The adoption of a more difficult

connection is justified by the reduction of the column

bending moments when compared with pinned connection

systems.

The use of moment-resisting connections usually results in

rigid connections. The main objective is to obtain a final

structure that behaves as similarly as possible to the cast-in-

place concrete structures, whose structural analysis processes

are well known.

Between the pinned and the rigid connections, there are

intermediate behaviours, resulting in semi-rigid connections.

In steel structures and in concrete–steel composite structures,

the semi-rigid connection has been thoroughly studied. For

precast concrete structures, this subject was given some

attention in the 1980s (Bljuger, 1988). However, in the 1990s,

there was a massive mobilisation of researchers on behalf of

the COST C1 programme (control of the semi-rigid behaviour

of civil engineering structural connections). This programme

had seven working groups, one of which focused on the study

of connections in prestressed and reinforced concrete. Most of

the results obtained were related to precast concrete

connections (Chefdebien and Daldare, 1994; Elliott et al., 1998;

Lindberg and Keronen, 1992).

However, the use of semi-rigid connections, which has so far

been little explored in precast concrete structures, extends the

design possibilities and can bring significant benefits to the

design of multi-storey structures of small and medium-height

buildings.

The present paper proposes a study of the strength and stiffness

of a specific type of beam-to-column connection for precast

concrete structures by considering tests in two prototypes. The

Structures and Buildings 163 Issue SB1 Analysis of a semi-rigid connection for precast concrete El Debs et al. 41

complete results of the experimental programme can be found

in Miotto (2002).

This paper presents an analytical model to determine both the

strength and stiffness of the connection. It also presents the

influence of these parameters on the structural behaviour by

comparing the numerical simulation results of a typical multi-

storey building with semi-rigid connections and the usual

pinned and rigid connection solutions.

2. STUDIED CONNECTION

The studied connection is a modified beam-to-column

connection currently used in Brazil for precast concrete

structures with heights up to 10 m. Details of the considered

connection are given in Figure 1. Compared with the current

connection, which is generally assumed to be a pinned

connection, it presents the following differences: (a) the

elastomer cushion is replaced by a special mortar cushion and

(b) the space between the beam end recess and the column was

filled with grout.

The cushion used was made of a mortar of cement and sand

with additions of soft aggregate, latex and short fibres to

produce a cement-based material with high deformability and

tenacity (El Debs et al., 2003). El Debs et al. (2006a) present

deformability values, obtained from stress–deformation curves.

This cushion was developed to avoid stress concentration and

also to allow small rotations of the beam.

The modifications do not alter the connection appearance and

the involved tolerances. However, concerning the manufacture,

there is some additional work to fill the upper part of the joint

with grout. Regarding the behaviour of the connection, a

significant transmission of negative bending moment is

expected, as well as a positive bending moment, but at a lower

level.

For the beam design, the negative moments introduced will

cause a reduction in the positive moments in the middle of the

span for the vertical load applied after the connection has been

accomplished.

For the column design, the bending moment transmission

produces a partial, but significant, reduction in the column

bending moments as well as the horizontal displacements at

the top level of the structure. As a consequence, the second-

order effects also tend to be reduced, in comparison to the

pinned connection structures. Further, there are lower bending

moments in the column base. The use of a structural system

with semi-rigid connections could therefore make it possible to

reduce the column cross-section or to increase the total height

of the buildings.

3. EXPERIMENTAL PROGRAMME

3.1. Characteristics of the prototypes

The experimental programme includes two prototypes:

prototype No. 1 represents an interior connection, and

prototype No. 2 represents an exterior connection. The main

geometric characteristics of the specimens are presented in

Figure 2. Owing to the loading procedure, the slab was

separated in such a way that prototype No. 2 would represent

two columns with reduced cross-sections. This reduction in the

column cross-section of the prototype represents a lightly

unfavourable situation. The reason for this option was the ease

of both the prototype manufacture and testing procedures.

The column and the corbel reinforcement, as well as the beam

reinforcement, were designed to represent a span modulation of

6 m 3 6 m. The continuity reinforcement was estimated for

these dimensions by considering the acting load after the

connection was effective. For the flexural continuity

reinforcement distribution, half of the required steel area was

adopted as bars passing through the column, while the rest of

the area was distributed bars in the structural concrete topping.

As a consequence of the available commercial bars, the

effective reinforcement consisted of 2˘16 mm (200 mm2)

passing through the column, and 6˘10 mm (470 mm2) on the

sides of the column. In prototype No. 2, the bars in the concrete

cover (6˘10 mm) were anchored on the edge of the slab and

the beam web bars (2˘16 mm) passed through the column, as

in prototype No. 1, to represent, in the actual situation, a

suitable anchorage in the column. More details of the prototype

construction can be seen in El

Debs et al. (2006b).

3.2. Test and loading

set-up

The test set-up and some

additional details are

presented in Figure 3. A

servo-controlled actuator of

500 kN capacity was used to

apply the loads. The loading

of prototype No. 1 was cyclic

and reversed, with two cycles

corresponding to 20% of the

predicted load, for each side,

two more cycles increasing

the load to 40% of the

predicted load, followed by

two more cycles of 60% of

the predicted load for the

negative moment and then

Fillingwith grout

Cushionof mortar

Bolt

Column

Beam

Hollow coreslab

Holes to passreinforcement rods

Cast-in-placeconcrete

Continuityreinforcement

Figure 1. The proposed connection with details of the alterations from current connection

42 Structures and Buildings 163 Issue SB1 Analysis of a semi-rigid connection for precast concrete El Debs et al.

raising the load monotonically until failure. In prototype No. 2,

the load was also cyclic and reversed, following the same

procedures, but aiming to obtain a failure provoked by the

positive moment.

3.3. Materials

The mechanical properties of

the concrete and the grout

are presented in Table 1. The

properties were determined

using tests on cylindrical

samples of 150 mm diameter

and 300 mm height for the

concrete and 50 mm diameter

and 100 mm height for the

grout on the same day of the

prototype testing. The

compressive strength (fc) and

tensile strength (f t) were

calculated based on an

average of six specimens and

the modulus of elasticity (E)

was calculated as an average

of three samples.

The steel used to produce the

dowels for the prototypes

was SAE 1020 with a

25.4 mm diameter and

nominal yielding stress (fy)

of 250 MPa. The reinforcement steel was Brazilian standard

CA-50 with a nominal yielding stress of 500 MPa. The values

for the yielding stress (fy) and ultimate strength (fu) of the

steel are given in Table 2 and are the average of four tested

specimens. The hollow core slab with an average compressive

strength of 45 MPa was furnished by a precast concrete

company.

3.4. Bending moments–rotation curves for connections

The main results concerning semi-rigid connections are the

bending moment–rotation curves. The bending moment was

calculated considering the reaction to half of the load

introduced by the actuator at a distance of 1.40 m (on the

column face). The rotation of the connection was obtained by

averaging the relative displacements given by the transducers

placed on the upper part of the prototype and on the upper

face of the column corbel and dividing by 450 mm, which is

the distance between these positions.

Figure 4 shows the bending moment–rotation curves for

prototypes No. 1 and No. 2. These curves correspond to the

envelope curves in the cyclic loading stages. As can be

observed, the failure load for prototype No. 1 occurred in the

direction of the negative moment, while the opposite occurred

for prototype No. 2.

190

400

200

50

150

300

250 70

600

200

1050

800

Column Column

20

Bearingcushion

Hollowcore slabCast-in-place

concrete

Bolt (Ø 25 mm)�

Continuityreinforcement

600

600

400 200 1290

100

200

20

(a) (b)

Upper view

Figure 2. Geometry and details of the tested prototypes: (a) prototype No. 1; (b) prototypeNo. 2 (dimensions in mm)

1400 mm

1600 mm

20 mm

Transducers

450 mm

Transducers

Support

Support

Reaction structure

Actuator

Device toapply the load

Figure 3. Test set-up and some details of the prototype tests(dimensions in mm)

Beam and column concrete Concrete topping Grout

Prot. No. 1 Prot. No. 2 Prot. No. 1 Prot. No. 2 Prot. No. 1 Prot. No. 2

fc : MPa 49.0 49.5 33.2 28.5 52.6 59.4f t: MPa 3.2 3.9 2.9 2.5 4.7 4.5E: GPa 32.8 36.3 31.6 24.8 13.4 14.9

Table 1. Mechanical properties of concrete and grout


4. PROPOSED DESIGN MODEL

4.1. Ideal bending moment–rotation curves for

connection

For negative moments, three stages can be observed from the

results

(a) a very short and stiff initial stage

(b) a second stage, with less stiffness

(c) a final stage of yielding.

For positive moments, it is also possible to observe an initial

stage of high stiffness and then a very long stage where the

moment increases slowly, indicating that the material is

yielding.

For negative and positive moments, the initial stiffer stages of

the connections tend to lose stiffness with repeated loading;

therefore, the use of a bi-linear diagram is proposed for both

cases, as shown in Figure 5. In this way, the connection

behaviour can be described by only two parameters, for each

bending moment direction, the yielding moment My (Myp for

positive moment and Myn for negative moment) and the

stiffness k� (k�p for positive moment and k�n for negative

moment).

4.2. Negative moments

The yielding moment can be determined based on Figure 6,

where a rectangular distribution of stresses is assumed for the

grout and the cushion, and the dowel contribution is neglected.

Imposing force equilibrium in the vertical and horizontal

directions, the moment equilibrium related to point C, which is

aligned with Rcu, the solution can be determined. Suppose

Rs ¼ As fy and Rg ¼ ycn fcgbw, the moment at C is

Myn ¼ As fyzn1

with

zn ¼ he � d9e �ycn2

2

and

ycn ¼ As fyfcgbw

3

where As is the cross-section area for the continuity

reinforcement, fy is the yielding stress for the continuity

reinforcement, he is the height of the beam end recess, d9e is the

distance from the reinforcement to the beam face, ycn is the

height of compression stress block at beam end recess, fcg is

Diameter: mm fy:MPa fu: MPa

10 576 68312.5 611 70516 589 764

Table 2. Steel strength

�250

�200

�150

�100

�50

0

50

�4 �2 0 2 4 6 8 10 12

Rotation 10� 3

Ben

ding

mom

ent:

kN m

Prototype No. 1

Prototype No. 2

Figure 4. Envelope of bending moment–rotation curve forconnection

M

Myp

kφn

Myn

φ

Figure 5. Proposed bending moment–rotation relationship

d�e

zn

le

ycn

Rcu

xcu

xcu/2

Vhe

Rq

Rs

M

C

Col

umn

face

Figure 6. Stress diagram in the connection for negativemoments


the compressive strength of the grout, bw is the beam end

recess width and xcu is the length of the compression stress

block under the cushion.

The value of xcu varies with the cushion stiffness. If the

cushion is very deformable, its value tends towards the length

of the beam end recess. If a triangular stress diagram at the

beam end recess is adopted, the value of xcu will be 2‘e=3.

Based on the test results, the beam end recess is supposed to be

fully rigid. The deformability of the connections is therefore

due to: (a) a strain concentration of a main crack in the face of

the column on the tensioned side; (b) the joint deformation on

the compressed side; and (c) cushion deformation. Figure 7

shows the deformed position of the beam end recess. It can be

seen that the following components with their respective

stiffnesses were considered

(a) reinforcement (ks)

(b) grout of the joint (kg) and cushion (kcu).

The position of the centre of rotation CR, which coincides with

point C used to obtain the yielding moment, can be calculated

from

ycr ¼ks(he � d9e)þ kg ycn=2ð Þ

ks þ kg4

The stiffness of the continuity reinforcement is

ks ¼�sAs

wy5

where wy is the opening of the pronounced crack, close to the

column face, and its value can be estimated through the

following expression used by Miotto (2002), where the first

part is given in Federation Internationale du Beton (FIB, 1999)

and the second part is given in Engstrom (1992a)

wy ¼ 2(1þ Æw)s1

Æw�

8(1þ Æers,ef )�s

2

�maxEs

" #1=1þÆw

þ �s

Es4�6

with

Æe ¼Es

Ec,top7

�max ¼ 2:5ffiffiffiffiffiffiffiffiffiffifc,top

q8

rs,ef ¼As

Ac,ef9

where As is the area of the continuity reinforcement, Ac,ef is

the involved area of the continuity reinforcement, �s is the

stress in the continuity reinforcement, � is the average

diameter of the continuity reinforcement, Es is the steel

modulus of elasticity, Ec,top is the concrete topping modulus of

elasticity, fc,top is the compressive strength of concrete topping,

Æw ¼ 0:4 and s1 ¼ 1.

In this case, as the reinforcement is placed in the concrete

topping, Ac,ef can be calculated from the product of the

thickness of the concrete topping and the distance of the

influenced area from the tensioned bars.

Since the objective is to obtain the stiffness of the

reinforcement when yielding, the stress in the continuity

reinforcement �s is the yielding stress (fy).

The stiffness corresponding to the joint filled with grout is

kg ¼ycnbwDgj

10

where bw is the beam width and Dgj is the deformability of the

grout joint, in terms of stress.

The deformability values for the mortar joints can be found in

the technical literature (Barboza, 2002; Bljuger, 1988).

Because the moment equilibrium is related to the rotation

centre CR, the connection stiffness can be calculated using the

expression

k�n ¼ ks he � ycr � d9eð Þ2 þ kg ycr �ycn2

� �2" #

11

4.3. Positive moments

The yielding moment can be determined based on Figure 8,

where a rectangular block of stresses was considered in the

concrete topping. The eventual reaction of the cushion close to

the column is neglected.

φ

φ

φ

φks

hekg

kcu

ycr

xcu/2

ycn/2

VMCR

Figure 7. Deformed position of the beam and deformationcomponents for the negative moment


Imposing the force equilibrium in the vertical and horizontal

directions and the moment equilibrium referring to point C,

which corresponds to the dowel position, the bending moment

in C can be obtained from

Myp ¼ Fsdzp12

with

zp ¼ he �ycp2

13

and

Fsd ¼ c�d2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifyd fcc,max

q14

where ycp is the height of the compression stress block at the

top of the beam end recess and is calculated by

ycp ¼ Fsd= fcc,maxbf (bf is composite beam width); c is a

coefficient which can be found in the technical literature

(CEB–FIP, 1991); �d is the dowel diameter; fyb is the yielding

stress of dowel steel; and fcc,max is the higher value between

the compressive strength concrete and compressive strength

grout in contact with the dowel.

The deformed position of the beam end recess, which is

assumed to be rigid, can be obtained from Figure 9. It can be

observed that the following deformation components with the

associated stiffnesses are considered

(a) compressed concrete (kc)

(b) tension in the dowel (ktd)

(c) shear in the dowel (ksd).

As kc approaches infinity, the rotation centre CR is aligned,

which automatically defines its position. In this case, CR also

coincides with the point C used to calculate the yielding

moment.

The stiffness of the connection can be calculated by

considering the equilibrium of moments referred to the centre

of rotation CR, which produces the following expression

k�p ¼ ksd hc �ycp2

� �2

15

The stiffness associated with the shear deformation of the

dowel is

ksd ¼Fsd

avy16

where avy is the transverse displacement of the dowel when the

maximum force is attained.

More information for the calculation of the displacement can

be found in the technical literature (Engstrom, 1992b). This

displacement can also be obtained from CEB model code 1990

(CEB–FIP, 1991), which indicates 0:10�d.

4.4. Comparison with experimental results

Based on the presented equations, the characteristic values of

the moment–rotation curves for prototypes No. 1 and No. 2

can be obtained. The geometry values of the prototypes are

given in Figures 2 and 3. Most of the material mechanical

properties were previously presented. The compressive strength

of the mortar cushion, determined by cylinder samples of

50 mm diameter and 100 mm height, is 25 MPa. Table 3 shows

other values used to obtain the theoretical curves.

The following parameters were also considered

(a) joint deformability of 0.1 3 10�4 m/MPa (Bljuger, 1988)

(b) c coefficient of 1.2 in Equation 14

(c) displacement avy of 0:10�d for Equation 16.

Col

umn

face

le/2

le

Ftd

Fsd

zp

Rc C

ycp V

M

Figure 8. Stress diagram in the connection for positivemoments

le/2

ktdksd

φ

φ

φ

φ

kC � 00

CRV

M

Figure 9. Deformed position of the beam and deformationcomponents for the positive moment


Table 4 presents the connection stiffness and the yielding

moments calculated with the proposed procedures, regarding

points CR ¼ C. The difference between the interior and external

columns is attributable to the material properties.

Figure 10 shows the experimental moment–rotation curves

compared with the theoretical values given in Table 4. For this

comparison, the experimental moments were calculated with

respect to points CR ¼ C, which means lever-arms of 1.246 m

for prototype No. 1 and 1.280 m for prototype No. 2. For

prototype No. 2, which was intended to fail with the positive

moment, the strength and stiffness were well predicted.

Regarding prototype No. 1, the theoretical model is in good

agreement with the connection strength. However, for the

stiffness estimation, the design model furnished values which

are not very close to the experimental ones for both cases. On

the other hand, if the displacement analysis is considered, the

model will also furnish safer values.

With these results, it can be observed that the strength is

relatively well evaluated, but the stiffness evaluation is much

less precise. This is a consequence of the uncertainties present

in the calculation: the crack opening, the deformability of the

joint filled by grout and the displacement of the dowel until

the maximum force is reached.

5. NUMERICAL MODELLING OF A TYPICAL

STRUCTURE

In order to evaluate the influence of the connection stiffness on

the structure behaviour, numerical simulations of a typical

multi-storey building with connections of different grades of

stiffness were performed.

The analysis was restricted to the global stability and the

increase of bending moments in the columns owing to the

structural deformation. The purpose was to show that some

Prototype No. 1Interior column

Prototype No. 2Exterior column

Distance from the reinforcement to the beam top (d9e) 25 mmContinuity reinforcement 870 mm2

2˘16 + 6˘10400 mm2

2˘16Average diameter of the continuity reinforcement 12.5 mm 16 mm

Note: the continuity reinforcement outside the beam web of prototype No. 2 was not takeninto account as it is not anchored appropriately on the slab border vicinity.

Table 3. Values used to obtain the theoretical curves

Interior column Exterior column

Negative moments Positive moments Negative moments Positive moments

Stiffness: MN/rad 60.9 6.4 25.1 6.3Yielding moment: kN m 202.0 36.4 96.3 36.3

Table 4. Theoretical values of connection stiffness and yielding moments

�250

�200

�150

�100

�50

0

50

�10 �8 �6 �4 �2 0 2 4 6 8 10

Rotation 10� 3

Ben

ding

mom

ent:

kN m

Prototype No. 1

Theoretical for internal connection

Prototype No. 2

Theoretical for external connection

Figure 10. Comparison of experimental and theoretical bending moment–rotation curves forconnection


advantages can be obtained when the column-to-beam

connections are considered to be semi-rigid.

Figure 11 represents a typical two-storey frame structure of

three spans. The loads indicated in Figure 11 are already

magnified by the safety factors. Figure 12 shows the structural

models: (a) pinned before the connections between beams and

columns were effective and (b) semi-rigid after the connections

between beams and columns were effective. The dead-load (g),

which corresponds to the weight of the structure itself, is

working in the pinned connections model, while the live-load

(q) and the wind load (W) are working in the semi-rigid

connections model.

The ªz coefficient method proposed by Franco and Vasconcelos

(1991) was chosen for the performed analysis. The analysis also

verifies the need to consider second-order effects by

demonstrating a simplified estimate of these effects. Concisely,

it consists in calculating the ªz coefficient, which evaluates the

deformability of the structure and, multiplied by the horizontal

loads, can also take into account non-linear effects. The ªz

coefficient is given by

ªz ¼1

1� ˜Md=M1dð Þ17

where M1d is the first-order moment at the bottom of the

structure owing to the lateral loads and ˜Md is the first

evaluation of the second-order moments, calculated from the

structure deformations due to the first-order moments.

For design purposes, if ªz is less than 1.1, then there is no need

to consider the overall second-order effects, and if ªz value is

less than 1.2 and greater than 1.1, the moments obtained in the

first-order analysis must be multiplied by ªz . In this study, the

ªz coefficient is applied even if it is lower than 1.1.

The displacements of the structure can be obtained using the

reduced values of flexural stiffness in order to consider the

non-linear behaviour of the materials. The usual values are

(EI )red ¼ 0.4EI for beams and (EI )red ¼ 0.8EI for columns, in a

framed structure, and (EI )red ¼ 0.4EI for fixed-end columns

(cantilever action) and pinned beams (El Debs, 2000). In the

absence of data to establish the stiffness reduction for semi-

rigid connections, the mean value of 0.60 is considered.

The finite-element-based program ANSYS 8.0 was used to

process the calculation. The semi-rigid connections were

simulated by spring elements, COMBIN39, available in the

software library. The COMBIN39 elements are capable of

representing a spring with non-linear behaviour, so the bilinear

moment–rotation curve defined for the connections was used.

In order to analyse the effect of the stiffness of the

connections, the following alternatives were considered

(a) pinned connections

(b) semi-rigid connections with the values present in Table 5

(c) rigid connections (fully restrained connections).

The values in Table 5 were calculated considering the material

properties

(a) precast concrete compressive strength of 35 MPa

(b) cast-in-place concrete compressive strength of 25 MPa

g: kN/m q: kN/m W: kN

Top storey

Intermediatestorey

26

Beam sections: 300 mm 650 mmColumn sections: 300 mm 400 mm

�

�6·0 m 6·0 m 6·0 m

3·75

m3·

75 m

20 10 10

2014

g qt t�

g qi i�

Wt 10 kN�

Wi 10 kN�

Figure 11. Analysed structure and considered loads

(b)

(a)

Figure 12. Structural model before (a) and after (b) theconnections become effective: (a) pinned connections;(b) semi-rigid connections


(c) continuity reinforcement and dowel of same diameters and

strength of the prototype

(d ) elasticity modulus of concrete of 30 GPa, which is

approximately the mean value between the precast and the

cast-in-situ concrete.

To calculate the negative and positive moments, safety factors

for the materials (1.4 for concrete and 1.15 for steel) were

introduced.

Table 6 presents the main results obtained for the analysed

situations. Several observations can be made.

(a) The displacement at the top of the structure for semi-rigid

connection is 13.7% of the value considering pinned

connection.

(b) The ªz coefficient is also significantly reduced.

(c) The bending moment in the column base for semi-rigid

connection is 41.9% of the value considering pinned

connection for the load combination G + Q +W.

(d ) For the analysed parameters, the positive moment at the

connection occurs only for load combination G + G and its

value is low.

Based on the results presented, other similar framed structures

were simulated, with an increasing number of storeys. Vertical

horizontal loads were repeated for the intermediate storeys,

keeping the same load for the top level. Table 7 shows the

results.

Based on the results in Table 7, two primary conclusions can

be drawn. First, it is possible to progress from a two-storey

frame with pinned connections to a four-storey one with semi-

rigid connections. The displacement at the top would be lower,

and the bending moment in the column base would increase

slightly, from 44.65 to 49.29 kNm. Second, even for a five-

storey frame, the positive moments at the connection would be

lower than the yielding moments, which indicate the possibility

of another increase in the height; however, in this case, there

would be a large increase in the column base moments.

Interior column Exterior column

Negative moments Positive moments Negative moments Positive moments

Stiffness: MN/rad 63.5 5.4 26.4 5.4Yielding moment: kN m 147.5 24.2 70.2 24.2

Table 5. Design values of connection stiffness and yielding moments

Loads G + Q + W Loads G + W

Connections a*: mm ªz Mb.ªz†:

kN mMv.ªz

‡:kN m

a*: mm ªz Mb.ªz†:

kN mMv.ªz

‡:kN m

Pinned 29.77 1.19 44.65 0 29.77 1.12 42.02 0Semi-rigid§ 4.07 1.03 18.72 — 4.07 1.02 18.54 3.99Rigid§ 1.99 1.01 15.27 — 1.99 1.01 15.04 15.00

* a – displacement at the structure top level (average of four columns)† Mb – moment at the column bottom (average of four columns)‡ Mv – positive moment at beam–column connection, (—) means that there is only negative moment§ Dead load actuates on pinned connections

Table 6. Main results for the analysed alternatives

Load G + Q + W Load G + W

Connection n* a:mm

ªz Mb.ªz:kN m

Mv.ªz:kN m

a: mm ªz Mb.ªz:kN m

Mv.ªz:kN m

Pinned 2 29.77 1.19 44.65 0 29.77 1.12 42.02 0Semi-rigid 2 4.07 1.03 18.73 — 4.07 1.02 18.54 3.99

3 11.30 1.05 33.94 — 11.30 1.03 33.29 8.52

4 21.81 1.07 49.29 0.06 21.81 1.05 48.37 13.13

5 36.30 1.10 66.26 4.75 36.30 1.06 63.85 17.00

* n – number of storeys

Table 7. Results when the number of storeys is increased for semi-rigid connections


Because of the uncertainty in the calculation of connection

stiffness, a numerical simulation was performed to analyse the

effect of this value on the studied parameters. Therefore, three

different stiffnesses were considered: 0.5, 1.0 and 2.0 times the

calculated values presented in Table 5, while maintaining the

same resistance moments. Table 8 presents the results of this

analysis for the load combination G + Q +W.

The results indicate that the variation of connection stiffness

becomes more important when the structure turns more

deformable as the storey number increases. It is also possible to

observe that the connection stiffness affects the displacements

more than the column base bending moments. For the case of

the four-storey structure, when the connection stiffness

decreases 50%, the displacement at the structure top level

increases 36.8%, while the bending moment in the column base

only increases 14.4%. The results show that the displacements

at the top of the structure and the column base moments

present low susceptibility to deviations of this parameter.

6. CONCLUSIONS

Based on the results obtained, some conclusions can be drawn.

(a) The proposed design models can reasonably evaluate the

studied connection strength for both negative and positive

moments.

(b) The evaluation of strength is more accurate than that of

stiffness.

(c) In the example of a typical structure, it is possible to

increase the number of storeys of the structure from two to

four with lower horizontal displacement at the top and

only a small increase of the column base bending moment

by using semi-rigid connections.

(d ) Although there is significant uncertainty in the connection

stiffness, the results show that the displacements at the top

of the structure and the column base moments present low

susceptibility to deviations of this parameter. In the

example of a typical four-storey structure, the moment in

the column base increases only 14.4% when the connection

stiffness is reduced to half the calculated stiffness.

ACKNOWLEDGEMENTS

The authors would like to acknowledge the Brazilian

government agencies FAPESP and CNPq for the scholarship

and financial support given to this research.

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Analysis of a Semi-rigid Connection for Precast Concrete

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