CHAPTER II
LITERATURE REVIEW
2.1
Precast Concrete Construction in Malaysia
Precast concrete construction started in Malaysia with the erection of 7 blocks
of 17-storey flat, 4 blocks of 4-storey flat and 40 units of shop house opposite the
Kuala Lumpur General Hospital at the intersection of Jalan Pekeliling and Jalan
Pahang (Zakaria, 2002). This maiden project used the Larsen-Nielson system from
Denmark. This was the brainchild of the minister and several officers of the Housing
and Local Authority Ministry who visited several countries in Europe to learn about
precast concrete construction. The second construction project, which used precast
concrete, was the construction of 6 blocks of 17-storey flat, 3 blocks of 18-storey flat
and 66 units of shop house along Jalan Rifle Range, Penang. This project used the
French’s Estiot system (Tan, 2000). Since then, numerous precast concrete structures
such as high-rises, car parks, warehouses, factories, housings and retail units have
been built all over the country. Some latest examples are the Telekom Headquarter in
KL, townhouses in Cyberjaya, City Square in Johor Bahru, Flextronic Manufacturing
Plant in Senai, Putra Mosque in Putrajaya and Metal Pak Factory in Shah Alam
(Eastern, 2004).
7
2.2
Advantages of Precast Concrete Construction
Some of the advantages of using precast concrete construction are as follows:
2.2.1
Reduced Construction Time and Cost
In conventional construction method, time-consuming works such as
formworks, scaffoldings and curing are needed to produce a structural element. In
precast concrete construction method, structural elements are produced in
manufacturing plants while other activities at the construction site proceed. When the
structural elements are needed, they are immediately sent to the site and assembled
continuously, forming the structural frame and enclosing the building. In precast
concrete manufacturing plants, modern machineries are utilized with several
technicians attending to certain production process. This greatly reduced the number
of unskilled labor requirements. According to studies done by the “Prefabrication
Center Technology of Singapore”, labor requirement in precast concrete construction
is 46.5% less than in conventional construction methods (Tan, 2000).
2.2.2
Increased Quality of Structural Elements
Precast concrete elements are produced in plants using modern techniques
and machineries. Raw materials such as concrete, sand, reinforcing bars and
prestressing strands are under high level of quality control. Formworks used are of
higher quality than those used at construction sites. This allows truer shapes and
better finishes in precast components. Precast components have higher density and
8
better crack control, offering better protection from harsh weathers and sound
insulation. High density is achieved by using vibrating table or external vibrators
placed on formworks. Precast concrete also provide better fire resistance for
reinforcing bars. When compared to in situ concrete, this reduces the amount of longterm movement, which needs to be recognized in building design (Philips &
Sheppard, 1980).
2.2.3
Increased Durability and Load Capacity of Structural Elements
Prestressed precast concrete components have high structural strength and
rigidity, which are important to support heavy loads. This allows shallow
construction depth and long span in structural components. Fewer supporting
columns or walls result in larger floor space, which allow more flexibility in interior
design. Dense precast and prestressed concrete components are cast-in with smooth
steel, concrete or fiberglass. This result in components with smooth surfaces which
resist moisture penetration, fungus and corrosion. High density concrete reduces the
size and quantity of surface voids thus resisting accumulation of dirt and dust.
Precast concrete components are more durable to acid attack, friction, corrosion,
impact, abrasion and other environment effects. Precast concrete structures have
longer service years and require minimal repairs and maintenance.
9
2.3
Disadvantages of Precast Concrete
Some of the drawbacks of using precast concrete construction are as follows:
2.3.1
High Capital Cost
A large amount of resources must be invested initially to set up a precast
concrete plant. Sophisticated machineries are expensive and require heavy
investment. Precast concrete is mainly used in construction of high-rise buildings and
flats, which are at least five storeys high. Precast concrete is also utilized in
construction of housing estates where the design of houses is uniform. Other projects
where precast concrete is suitable are large stadiums, halls, factories, warehouses,
airports and hangars. The scale of the construction projects using precast concrete
must be large enough to ensure sufficient profit to offset the initial capital cost.
2.3.2
Sophisticated Connection Works
The behavior of connections determines the performance of precast concrete
structures. During erection of precast concrete structures, connections between
precast components must be supervised and done properly. This way, the intended
behavior of a connection (simple, semi-rigid or rigid) can be achieved. Apart from
that, a good sound insulation can be provided and water leakage problem can be
avoided. Skilled and well-trained labors are required to ensure proper connection is
produced during erection stages, which lead to additional cost.
10
2.3.3
Transportation, Handling Difficulties and Modification Limitation
Workers must be careful when handling precast concrete components to
avoid damage. Precast components are manufactured in plants, which are not always
situated in the vicinity of the construction sites. Precast components must be carted
from the plants to the sites using trailers. Usually, precast components are large and
heavy, creating difficulties in transportation. Upon arrival at the sites, portable or
tower cranes will lift the precast components into place for erection. Usually, to
increase the speed of construction, several cranes are used requiring large space.
Proper construction planning and site management is a must. Workers must be well
trained to ensure that precast components are positioned and connected properly to
avoid cases where the columns, beams, walls or slabs are not well aligned, dislocated
or out of plane. Precast concrete system is not flexible when future modification is
taken into account. For example, the walls of a flat built using load bearing precast
walls cannot be demolished for renovation purposes, as this will affect the stability of
the entire precast structure (Tan, 2000).
2.4
Design Criteria for Precast Concrete Connections
A connection is the total construction including the ends of the precast
components, which meet at it. A joint are the individual parts, which form the
connection. In the case of a beam-to-column connection, a bearing joint is made
between the beam and column, but when the assembly is completed with in situ
grout, the entire construction is a connection (Elliot, 1996). The performance of
precast concrete systems depends on the behavior of connections. The configuration
of connections affects the constructibility, stability, strength, flexibility and residual
forces in the structure. In addition, connections play a key role in the dissipation of
energy and redistribution of loads (Dolan, et al., 1987). Four rules to satisfactory
joint design are:
11
(a) Components are able to resist ultimate design loads in a ductile manner.
(b) Components can be manufactured economically and erected safely and
quickly.
(c) Tolerances of manufacturing and erection do not have adverse effect on the
intended behavior of connections.
(d) Final appearance of connections satisfy visual, fire and environmental
requirements.
The main criteria for the serviceability performance of connections are: strength,
ductility, durability and influence of volume change (Elliot, 1996).
2.4.1
Strength
Connection must be able to resist forces induced by dead and live gravity
loads, wind, earthquake and soil or water pressure. In addition, there are forces due
to restraint of volume changes in precast components. Apart from instability caused
by eccentricity of gravity and horizontal loads, unexpected loads from foundation
settlement, explosion and change of structure usage must be considered. Joints
strength is categorized by the type of induced stresses (compressive, tensile, flexural,
torsional and shear). A connection can have high resistance to one type of stress and
low resistance to another. Generally more than one type of joint are utilized to
achieve an overall result, rather than to provide resistance to a combination of
stresses.
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2.4.2
Ductility
Ductility is the ability of connections to take large deformations without
failure (between first yield and ultimate failure). Ductility is often related to moment
resistance in frame structures. In precast structures where connections are designed
using pinned joints between components, semi-rigid behavior is considered adequate
for moment resistance. In design, flexural yielding must take place at chosen plastic
hinge locations in order to control the post-elastic deformation capacity of the entire
precast structure.
2.4.3
Durability and Influence of Volume Changes
Durability of connections exposed to weather is affected by corrosion of
exposed steel elements or by cracking and spalling of concrete. Ideally, steel
elements must be covered with concrete or grout (no chlorides used). Otherwise it
must be painted, galvanized or using non-corrosive materials. Precast components
are of high quality, thus minimizing flexural cracking, provided tensile stresses are
within code limits. Improper detailing results in local cracking and spalling due to
restraint of movement or stress concentrations. Shortening effects of creep, shrinkage
and temperature reductions cause tensile stresses in precast components. Unless
some movement is allowed to take place, the stresses must be considered in design.
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2.5
Beam-to-Column Connections in Precast Construction
Beam-to-column connection is one of the basic features in the design of
multi-storey precast structures. In skeletal beam and column construction, the
vertical member is continuous (in design and construction term) and horizontal
components are framed into it at various levels. An alternative for low-rise unbraced
structures is to erect columns floor-by-floor and in situ connections are made
between precast components. There is an even split between the use of concrete
corbels with twin rib or rectangular beams and cast-in steel inserts forming the
invincible corbel. Beam-to-column connections can be categorized into two major
groups: simple connections and moment resisting connections (semi-rigid
connections and rigid connections).
2.6
Simply Supported Beam-to-Column Connections
The support of simply supported beams must meet the following
requirements:
(a) Allow for rotation of the beam above the support.
(b) Allow for horizontal movements of beam without the initiation of large
horizontal forces.
(c) During erection process, the supporting material must bridge the tolerances
between these elements and their support. This must be achieved without
considerably reducing the load-carrying capacity of the connection.
(d) Support able to transfer forces safely.
14
Pinned connections are designed to resist or transmit shear (gravity and uplifting)
and axial loads. In the first 3 requirements, bearing pads play a key role. In the last
requirements, detailing of the end zone of a beam is essential. Pinned connections
have simple detailing and can be erected in the simplest element-to-element bearing.
Steel inserts such as plates or rolled sections are anchored into connecting elements
to increase bearing capacities. Steel inserts are later surrounded with grout for
durability.
2.6.1
Beams Supported by Steel Corbels
Figure 2.1 shows this type of connection. In ultimate limit state, the steel
corbel is loaded by a vertical load Fvu and a horizontal tensile force Fhu. The steel
corbel is an I-beam or two vertically placed U-beams. The vertical force Fvu is
obtained from the equilibrium of vertical forces, as shown in Figure 2.2.
Fvu = f c bl e /{3.7 + 4a / l e }
The magnitude of the vertical force exerted at the end of the corbel on the concrete
face can be calculated by the formula below:
N cu = Fvu (0.22 + 1.33a / l e )
This force must be resisted by the reinforced part of the column below the
steel corbel. At the end of the steel corbel, a vertical load Ncu acts in an upward
direction on the concrete. This vertical load must be balanced either by the dead load
acting in the upper part in the column, or by reinforcement in this part of the column
if the dead load is not present. The transfer of the vertical load Ncu from the corbel to
the column should be design appropriately. The compressive forces in the column
accumulated from upper stories must be considered as well.
15
Figure 2.1: Beams supported by steel corbels (Bruggeling & Huyghe, 1991).
Figure 2.2: Engineering model for the connection of beams supported by
steel corbels (Bruggeling & Huyghe, 1991).
2.6.2
Welded Connection of a Beam to the Support Restrained by Horizontal
Displacements Only
Figure 2.3 shows this type of connection. The tensile force in the connection
should be transferred to the column and the beam safely. The steel parts of the
connections should be anchored into the concrete using bars welded to these steel
parts. Figure 2.4 illustrates a welded connection of a slab to the support, restrained by
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horizontal displacements only. Figure 2.5 depicts a welded connection of a TT-slab
to the support, restrained by horizontal displacements only. In this connection, large
tensile forces, with a mainly shortening effect, can be exerted on the connection.
Brittle failure can occur under service loads if the forces are not transferred to the
structural components.
Figure 2.3: Welded connection of a beam to the support
(Bruggeling & Huyghe, 1991).
Figure 2.4: Welded connection of a slab to the support
(Bruggeling & Huyghe, 1991).
Figure 2.5: Welded connection of a TT-slab to the support
(Bruggeling & Huyghe, 1991).
17
2.7
Moment Resisting Beam-to-Column Connections
Moment resisting connections are used to assure sufficient stability of portal
frames. They are used to limit the depth of beams and other structural components
without reducing their stiffness. They are mainly found between beams and columns
and between slabs and walls. Progressive collapse of multistorey structures can be
avoided by utilizing these connections (Bruggeling & Huyghe, 1991). Moment
resisting connections are essential to develop frame action in precast structures. The
connections must be strong enough to resist loadings and stiff enough to resist
horizontal sway of the precast building (Dolan, et al., 1987).
Moment resistance of at least 50 to 75 kNm is required in relieving sagging
moments in beams or increasing the stiffness of frames. A couple force of 300 kN
and a lever arm of 150 to 250 mm is required to generate the moment, using concrete
area of 20 000 mm2 (grade C40 assumed) and 16 to 20 mm diameter high tensile bars
(grade 460 assumed). Ductile failure must be allowed to happen and the limiting
strength should not be governed by shear friction, insufficient length of weld, plates
embedded in thin sections or other details which may lead to brittleness (Elliot,
1996).
2.7.1
Beams on Top of a Column
Figure 2.6 shows a beam-to-column connection, which can only resist
negative bending moments. At failure, the vertical component of the bending
moment and the load from the beam is concentrated in the beam-bearing pad (mortar
joint)-column area. This area is known as the support zone. To avoid premature
failure of column or the beam, this zone must be detailed carefully. During
construction, the beam is placed onto the column. The beam and column are rigidly
18
connected by reinforcement. The height of the beam must not exceed the width of the
column. Rigidity is achieved by overlapping bars behind a curve in one of the bars,
as shown in Detail IIIb in Figure 2.6. The overlap must be made possible in the
vertical part of the bars behind the curve. The sum of the length of overlap and the
radius of the loop must be smaller than the height of the beam. As a result of this, the
diameter of the bars as well as the moment capacity of the connection is limited.
This type of connection cannot take positive bending moment. The length of
the support zone is the key to the safety of this connection. Small corbels can be
provided on top of the column to provide sufficient support length. These
connections are only used in exceptional case where there are no other alternatives.
The bars protruding from the beam is causing difficulties in manufacturing,
transporting and erection process.
Figure 2.6: Rigid beam-to-column connection (Bruggeling & Huyghe, 1991).
Figure 2.7 shows connections BC28 and BC29, which are dowel connections.
They are designed to eliminate field welding during erection of precast structures.
The bending moment capacity of these connections is low with a design moment of
33 kNm. A large deformation is required to develop this moment. Therefore, these
connections are seldom chosen to provide moment resistance for frame action.
However, they do provide some resistance and may be useful in some cases (Dolan,
et al., 1987).
19
Continuous beam-tocolumn connection
Figure 2.7: Connection BC28 and BC29 (Dolan, et al., 1987, Elliot, 1996).
A continuous beam-to-column connection is also shown in Figure 2.7. The
beam is seated and dowelled on to the top of column. Bearing plates are provided
between the components for various reasons: to ensure a uniform bearing pressure, to
ensure beam reaction is transferred to column in the intended position, to avoid
eccentricity of load, to prevent local spalling and to accommodate tolerances (Elliot,
1996). The size of bearing pad should be at least 75 × 75 mm or h/3 in larger
columns. The connection can transfer vertical forces by means of confinement links
and also transfer horizontal forces by means of steel reinforcement placed on top of
the column.
20
2.7.2
Connection between Beams and a Column
Figure 2.8(a) shows connection BC16A, which is a beam-to-column
connection. This connection used continuous reinforcement cast into a composite
topping to provide moment resistance. The design moment resistance of this
connection is about 161 kNm (Dolan, et al., 1987). Although this connection is
strong and very ductile, the energy dissipation during cyclic loading is low. The
prying action of the beam rotating about the corbel greatly reduced the strength of
this connection.
Figure 2.8(b) shows connection BC25, which is another type of beam-tocolumn connection. The connection consists of bolts between end plates cast into the
column and beam elements. Connection BC25 is used for extending columns through
large beams. Energy dissipation of the connection under cyclic loading is low and its
design moment is 179 kNm (Dolan, et al., 1987). The strength of the connection is
limited by the bolt capacity while bolt yielding accounted for most of the energy
dissipation. Full bolt capacity can be achieved if additional tie confinement is used.
However, the sudden buckling failures of the bars and the possibility of frame
collapse suggest that connection BC25 cannot be chosen as the main moment
resisting component.
Connection BC26 is a cast in-place connection, where a precast beam is
constructed into a cast in-place column, as shown in Figure 2.8(c). This connection is
somewhat similar to cast-in-place construction. To increase the capacity and ductility
of the connection as well as to avoid shear ties being pulled out during failure, stirrup
confinement using 135° anchorage provisions can be utilized. The strain hardening
of the reinforcement contributes to the strength of this connection. The design
moment of this connection is 161 kNm (Dolan, et al., 1987).
21
Figure 2.8(a): Connection BC16A (Dolan, et al., 1987)
Connection BC27, which is a fully post-tensioned connection, is shown in
Figure 2.8(d). This connection display good ductility and its design moment is about
236 kNm. Failure occurred by fracture of one of the post-tensioned strands. High
initial stiffness is obtained by post-tensioning, which is equivalent to the uncracked
concrete with a modulus of elasticity of 51 000 MPa (Dolan, et al., 1987). This
stiffness is effective until the initial post-tensioning forces have been overcome.
Figure 2.8(b): Connection BC25 (Dolan, et al., 1987)
Figure 2.8(c): Connection BC26 (Dolan, et al., 1987)
Figure 2.8(d): Connection BC27 (Dolan, et al., 1987)
22
2.7.3
Connection of a Beam to the Shaft of a Column
Figure 2.9 shows a connection of a beam to the shaft of a column. This
connection is utilized when the column is not interrupted at each level of a multistory
building. The beam is connected to the shaft of the column by means of a welded
connection. This type of connection can also be obtained by prestressing although it
is more expensive and time-consuming in construction.
Figure 2.9: Connection of a beam to the shaft of the
column (Bruggeling & Huyghe, 1991).
A classical welded plate connection, connection BC15 is shown in Figure
2.10. A steel plate is welded to the longitudinal reinforcement and a companion plate
is cast into the adjacent precast members, which is a column in this case. A closure
plate is welded between the two embedded plates to complete the connection. When
the connection failed under negative moment, either one of the bars fractured at the
end of the weld or the field welds have failed. Failure is usually a result of direct
tension and the prying action of beam rotating about the corbel of the column. The
design moment of this connection is 161 kNm and can be increased by reducing the
eccentricity of the load path through the connection and minimizing the prying action
of the beam (Dolan, et al., 1987). Under positive moment, the moment capacity of
the connection is comparable to its negative moment capacity. However, the
connection displayed limited energy limitation under cyclic loading. The tension
loadpath through this connection is intricate (unpredictable). This caused
23
deformations which are normally not considered in design stages to occur. To avoid
this problem, the eccentricity of the load path must be minimized.
Figure 2.10: Connection BC15 (Dolan, et al., 1987).
2.7.4
Beam-to-Column Connection Using Bolts in Sleeves
A beam-to-column connection using bolts in sleeves is shown in Figure 2.11.
This type of connection uses steel or hard plastic sleeves, which are cast in the
column and bounded by the usual confinement links and main steel. Seating cleats or
similar attachments are received by steel bolts or threaded rods that are placed
through the sleeves. Structurally, this connection is identical to cast-in sockets. The
difference is that the bolt is not a tight fit in the sleeve, causing the resulting pressure
inside the sleeve to be transferred directly to the concrete. In 1992, Ahmad, S.A.M.
from University of Southampton did a full scale testing using M24 grade 8:8
threaded bolts supporting angle cleats to the face of 300 × 300 mm precast columns
to investigate the effects of the concentration of bolts on connector capacity in his
PhD Thesis, “Behavior of Sleeved Bolt Connections in Precast Concrete Building
Frames”. The number of bolts acting as connector varied from 1 to 4 (i.e. 2 rows with
2 bolts in each row). The strength of concrete varied from 61.9 N/mm2 to 30.5
N/mm2. R8 links with 50 mm diameter were used as confinement reinforcement
beneath the bolts. Table 2.1 shows the ratio of failure load for the multi-bolt
connection to that of a single bolt.
24
Table 2.1: Ratio of failure load for multi-bolt to single bolt
No. of bolts
Load capacity/Single bolt capacity
2
1.7
3
2.6
4
2.8
(Source: Elliot, 1996)
In the tests conducted by Ahmad, specimens using high strength concrete
failed by flexural (due to the small connector eccentricity) and by shear yielding of
the bolts. Large “ovulation” of the sleeves and concrete splitting beneath the
connector were reported as the failure mode of specimens using lower strength
concrete.
Figure 2.11: Bolts in cast-in sleeves in column (Elliot, 1996).
25
2.7.5
Column Insert
Column insert is defined as steel section, which is embedded in a precast
column to transfer axial and shear forces. In some cases, it can also transfer bending
and torsion moments to the column. This type of connection has been widely
researched. Therefore, there is much confidence in the design of column insert
connections compared to other types of connections in precast construction. Some
examples of column inserts are as follows:
(a) universal column or beam (UC, UB)
(b) rolled channel, angle or bent plate
(c) rolled rectangular or square hollow section (RHS, SHS)
(d) narrow plate
(e) threaded dowels or bolts in steel or plastic tubes
(f) bolts in cast-in steel sockets
Figure 2.12 gives several types of cast-in steel inserts at beam-to-column
connections. Figure 2.12(a), (b) and (c) are either solid or tubular while Figure 2.12
(d) (e) and (f) are cast-in sections. Column inserts can be grouped into:
(a) ‘wide sections’, when the breadth of the bearing surface bp is in the range of
75mm to 0.4b
(b) ‘thin plates’, which includes thin walled rolled sections with wall thickness
less than 0.1b or 50 mm. Usually additional bearing surfaces are required in
thin section connectors
(c) ‘broad sections’, when the distance from the cover to the sides of insert is
small enough to cause shear cracking. As a result, the allowable stresses of
the connection are reduced
26
SHS and RHS are the most widely used shapes for the insert. The reasons are;
wide range of sizes (general range is 100 × 50 mm to 250 × 150 mm), good torsional
properties (in non-symmetrical cases), uniform shape (easier to weld additional
plates and rebars) and high shear to flexural capacity (greater than UBs).
Figure 2.12(a): Cast-in steel inserts using solid or hollow billet with top
steel reinforcing bars (Elliot, 1996).
Figure 2.12(b): Cast-in steel inserts using solid billet with welded
plate in beam (Elliot, 1996).
27
Figure 2.12(c): Cast-in steel inserts using solid or hollow sections with
threaded dowel and top angle fixing (Elliot, 1996).
Figure 2.12(d): Cast-in steel inserts using open box and notched plate in
beam (Elliot, 1996).
Figure 2.12(e): Cast-in steel inserts using rolled H-section and bolted
on cleat (Elliot, 1996).
28
Steel plate
Steel angle
UB section
Shear stud to
prevent tension
pull out
Section view
Elevation view
Additional plate
to increase
bearing width
Figure 2.12(f): Cast-in steel inserts using universal beam section with steel angle and
plate (CIDB, 1999).
Considering the size of the recesses in the beam, smaller and thicker section
is more economical. The minimum breadth of insert is 6 mm for rolled sections and 4
mm for box sections with the condition that the insert is sealed to the passage from
air and moisture. The insert is set in the column by casting it directly into the
column. Another way is by placing the insert into a preformed hole in the column
using grout or resin-mortar. In the former method, the insert must penetrate the sides
of the mould by means of platting. Forming a hole in the column and casting the
insert into it is a good method to avoid the uneven surfaces problem in column.
However, one problem is that cracking may propagate around the corners of the hole
when it is lifted from the mould. This problem is prone to occur when the breath of
the hole is greater than 1/3 of the breadth of the column. Epoxy mortar should be
used to replace grout in cases where the connection is subjected to heavy loads. For
the connection to be classified as heavily loaded, the working stress below the insert
is greater than 0.4fcu. of the grout (for linear elastic analysis only). Epoxy mortar is a
quick setting material with considerable compressive and tensile strength. Epoxy
mortar can also be poured into holes with much smaller diameters. Smaller holes in
the column greatly enhanced crack resistance (Elliot, 1996).
29
2.8 Nonlinear Static Analysis
2.8.1
Materially Nonlinear Analysis
Material nonlinear analysis is utilized if the material stress-strain relationship
is significantly nonlinear. Figure 2.13 considers for example the idealized stressstrain relationship for a steel bar. This is linear in the elastic range so that an elastic
analysis would predict the correct deformed configuration provided the yield stress is
not exceeded. If yielding occurs, the stiffness of the bar decreases resulting in a
nonlinear stress-strain law. Therefore, incremental loading is required to trace the
complete material response. This is illustrated for a simple two bar arrangement as in
Figure 2.15. LUSAS has a number of different material models, which permit
modelling of a variety of physical materials including ductile metals and concrete.
2.8.2
Geometrically Nonlinear Analysis
In geometrically nonlinear analysis the effect of structural deformation on the
structural stiffness and on the position of applied loads is considered. A simply
supported beam with uniformly distributed loading as shown in Figure 2.14
illustrates this effect. Linear solution would predict the familiar simply supported
bending moment and zero axial force. However, in reality, as the beam deforms, the
angle of inclination of the beam at the supports introduces an axial component of
force. This force may become significant if the deformations and consequently the
angle of inclination become large.
30
2.8.3
Deformation Dependent Boundary Conditions
For this type of analysis, the boundary conditions are modified depending on
the deformed shape of the structure. The mass is subjected to a pressure, P and is
initially supported by a single spring. As the load is increased, contact is established
with a second spring, which alters the load-deformation response of the structure.
Nonlinear boundary conditions are imposed by using joint elements or through the
slide line facility. A joint element with an initial gap and zero stiffness is used to
connect the mass to the spring. Once the gap has been closed the element is given a
stiffness so that it forms a rigid link.
Figure 2.13: Typical idealized uniaxial stress-strain relationship for steel.
(a) Problem Definition Axial forces
(b) Axial Forces Induced By Large
Figure 2.14: Geometrically nonlinear response of a
simply supported beam.
31
(a) Problem definition.
(b) Stress-strain relationship for Bar 1.
(c) Stress-strain relationship for Bar 2.
(d) Combined response of system.
Figure 2.15: Nonlinear response of a two bar system.
32
2.9
Nonlinear Solution Procedures
In nonlinear analysis, it is not possible to directly obtain a stress distribution,
which equilibrates a given set of external loads. A solution procedure in which the
total required load is applied in a number of increments is applied. In each increment,
a linear prediction of the nonlinear response is made and subsequent iterative
corrections are performed in order to restore equilibrium by the elimination of the
residual forces.
Iterative corrections depend on some form of convergence criteria, which
indicates to what extent an equilibrium has been achieved. This kind of solution
procedure is therefore commonly referred to as an incremental iterative method, as
illustrated in Figure 2.16. Nonlinear solution in LUSAS is based on the NewtonRaphson procedure. Specifications of the solution procedure are defined using the
Nonlinear Control properties assigned to load case. For the analysis of nonlinear
problems, the solution procedure adopted greatly influences the results. In order to
reduce this dependence, wherever possible, nonlinear control properties incorporate a
series of generally applicable default settings and automatically activated facilities.
Figure 2.16: Incremental iterative method.
33
2.9.1
Iterative Procedures
In Newton-Raphson procedure, an initial incremental solution is predicted
based on the tangent stiffness from which the incremental displacements and its
iterative corrections are derived. In the standard Newton-Raphson procedure, each
iterative calculation is always based on the current tangent stiffness. For finite
element analysis, this involves the formation and factorization of the tangent stiffness
matrix at the start of each iteration. Although the standard Newton-Raphson method
usually converges rapidly, continuous manipulation of stiffness matrix is expensive.
The need for a robust yet inexpensive procedure leads to the development of
modified Newton-Raphson methods.
A slow convergence rate can be significantly improved by adopting an
iterative acceleration technique. In cases of severe and localized nonlinearity,
encountered typically in materially nonlinear or contact problems, some form of
acceleration may be a prerequisite to convergence. In LUSAS, iterative acceleration
is performed by applying line searches. Basically, the line search procedure involves
extra optimization iterations in which the potential energy associated with the
residual forces at each iterative step is minimized. Selection of line search parameters
depends largely on the type of problem and experience. However, a maximum of 3 to
5 line search iterations with a tolerance of 0.3 to 0.8 is usually sufficient (the closer
the tolerance is to unity, the more slack the minimum energy requirement).
In cases where both material and contact nonlinearities are encountered,
convergence difficulties arise when evaluating material nonlinearities in
configurations where the contact conditions are invalid because the solution is not in
equilibrium. To avoid this situation, contact equilibrium is established using elastic
properties from the previous load increment before the material nonlinearity is
resolved.
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2.9.2
Incremental Procedures
In Newton-Raphson solution procedures, it is assumed that a displacement
solution is obtained for a given load increment and within each load increment, the
load level remains constant. Such methods are referred to as constant load level
incrementation procedures. However, where limit points in the structural response
are encountered (for example in the geometrically nonlinear case of snap-through
failure) constant load level methods will, at best, fail to identify the load shedding
portion of the curve and, at worst, fail to converge at all past the limit point. The
solution of limit point problems therefore leads to the development of alternative
methods, including displacement incrementation and constrained solution methods.
Constrained methods differ from constant level methods in that the load level
is not required to be constant within an increment. The load and displacement levels
are constrained to follow some pre-defined path. In LUSAS, 2 forms of arc-length
method have been applied. First is the Crisfield’s modified arc-length procedure in
which the solution is constrained to lie on a spherical surface defined in displacement
space. For the one degree of freedom case this becomes a circular arc. Second is the
Rheinboldt’s arc-length algorithm, which constrains the largest displacement
increment (as defined by the predictor) to remain constant for that particular
increment. The use of the arc-length method over constant load level methods can
improve convergence characteristic and has the ability to detect and negotiate limit
points. In LUSAS, control of arc-length solution procedures is through the
Incrementation section of the Nonlinear Control properties. If required, the solution
may be started under constant load control and automatically switched to arc-length
control based on a specified value of the current stiffness parameter. The required
stiffness parameter for automatic conversion to arc-length control is input in the
Incrementation section of Nonlinear Control properties.
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2.9.3
Incremental Loading
The selection and level of incrementation depend on the problem that needs
to be solved. Incremental loading for nonlinear problems can be specified in four
methods, which are:
(a) Manual Incrementation, where loading data in each load increment is
specified separately.
(b) Automatic Incrementation, where a specified load case is factored using fixed
or variable increments.
(c) Mixed Incrementation, where manual and automatic incrementations are
combined.
(d) Load Curves, where the variation of one or more sets of loading data is
specified as a load factor vs. load increment or time step load curve.
There are 2 methods of automatic incrementation. In the first method,
uniform incrementation is applied. For each increment, the current load factor is
multiplied by the specified load components to generate the applied load. The second
method is the Variable Incrementation Alternatively, where the variable
incrementation is requested. In this case, the current load factor is automatically
varied according to the iterative performance of the solution. The variation is a
function of the required number of iterations and a specified desired iterative
performance. Therefore, when the number of iterations taken is less than the desired
value, the incremented load factor is subsequently increased and conversely, if the
number of iterations is greater than the desired value, it is decreased. Variable
incrementation is used in conjunction with either constant load level or arc-length
solution methods and is an effective way of automatically adapting the performance
of the solution procedure to the degree of nonlinearity encountered. The overall
effect is to increase and decrease the numerical effort in the areas of most and least
nonlinearity respectively.
36
When necessary, manual and automatic incrementation procedures are mixed.
When mixing manual and automatic incrementation, certain rules apply. Load cases
are respecified as often as required. If the automatic procedure is specified, it will
continue until one of the termination criteria is satisfied. In switching from manual to
automatic control, any loading input under the manual control is remembered and
held constant, while the automatic procedure is operating. In switching from
automatic back to manual control, any loading accumulated under automatic control
is forgotten and must be input as a manual load case if required. If prescribed
displacements are being used, then in any switching from one type of control to
another, the effect of prescribed displacements will be remembered and will not need
to be input again.
Where an increment has failed to converge within the specified maximum
number of iterations, it is automatically reduced and reapplied. This process is
repeated according to values specified in the step reduction section until the
maximum number of reductions has been tried. In a final attempt to achieve a
solution, the load increment is increased to try to step over a difficult point in the
analysis. If the solution still failed to converge, the solution is terminated.
2.9.4
Solution Termination
In manual incrementation, the solution is automatically terminated following
the execution of one increment. In automatic incrementation, the solution progresses
one Nonlinear Control chapter at a time. The accomplishment of each Nonlinear
Control chapter is controlled by its Termination parameters. Termination may be
specified in 3 ways, which is by limiting either the maximum applied load factor,
maximum number of applied increments or maximum value of a named freedom.
37
In cases where more than one criteria is specified, termination occur when the
first criteria is achieved. Failure to converge within the specified maximum number
of iterations will either result in a diagnostic message and termination of the solution
or, if automatic incrementation is being used, a reduction of the applied load
increment. When necessary, the solution is continued from an unconverged
increment although the consequences must be considered. In addition, the solution
will be terminated if, at the beginning of an increment, more than two negative pivots
are encountered during the frontal elimination phase.
2.9.5
Cracking Concrete with Crushing Model
The cracking concrete with crushing material model is based on a multisurface plasticity approach to represent the nonlinear behavior of concrete in both
tension and compression. This model simulates directional softening and crushing in
compression using the same yield functions. Cracks in tension are assumed to form
when the major principal stress reaches the tensile strength, which is when a
permanent crack plane is formed. Multiple cracks can form at non-orthogonal
directions to one another. This model also simulates nonlinear behavior in
compression with hardening and softening functions applied to the local yield
surfaces. The local yield surface is calibrated to provide a close fit to the HoekBrown strength envelope and it includes the effects of strength increase with triaxial
confinement.
In tension zones, permanent crack planes result in directional loss of strength.
In compression zones, the planes are not permanent but may rotate and result in an
isotropic loss of strength. In both tension and compression, unloading from the yield
surface is assumed to be elastic. For materials that are dominated by cracking and
crushing is not important, the Concrete Cracking Model is recommended. Table 2.2
shows the material properties for the numerical models. Figure 2.17(a) and (b) shows
38
the tensile and compressive behavior of concrete. Figure 2.18(a) shows the tensile
behavior of steel reinforcement bar, threaded rod and shear link. Figure 2.18(b)
shows the tensile and compressive behavior of steel plate and angle.
Table 2.2: Material properties for numerical models.
Items
Concrete:
Elastic
Mass density, ρc
Young’s Modulus, Ec
Poisson’s ratio
Plastic
(Cracking & Crushing model)
Compressive strength, fc
Tensile strength, ft
Strain at peak compressive stress, εcp
Strain at end of compressive softening curve, εco
Strain at end of tensile softening curve, εto
Steel (reinforcement bar, threaded rod & shear link):
Elastic
Mass density, ρrs
Young’s Modulus, Ers
Poisson’s ratio
Plastic
(Stress potential model)
Initial uniaxial yield stress, fy
Ultimate yield stress, fy max
Hardening gradient, Erp
Strain at end of hardening curve, εro
Steel (plate and angle):
Elastic
Mass density, ρss
Young’s Modulus, Ess
Poisson’s ratio
Plastic
(Stress potential model)
Initial uniaxial yield stress, fsy
Ultimate yield stress, fsy max
Hardening gradient, Esp
Strain at end of yield curve, εso
Strain at end of hardening curve, εst
(Source: BS 8110, 1985 & Dzulkarnian, 2002)
Values
-5
3
2.4 × 10 N/mm
2
26,000 N/mm
0.2
50 N/mm2
3.158 N/mm2
0.002
0.003
0.004
7.85 × 10-6 kg/mm3
200,000 N/mm2
0.3
460 N/mm2
560 N/mm2
2121 N/mm2
0.02
7.85 × 10-6 kg/mm3
200,000 N/mm2
0.3
460 N/mm2
640 N/mm2
2121 N/mm2
0.02
0.06
39
σn (Ten, N/mm2)
3.158
Softening
0.004
ft
Ec = 26000
εto
ε (Ten)
Figure 2.17(a): Tensile behavior of concrete.
σn (Comp, N/mm2)
0.003
50
0.002
fc
Softening
Ec = 26000
εcp
εco
ε (Comp)
Figure 2.17(b): Compressive behavior of concrete.
40
σ (Ten, N/mm2)
560
fy max
Hardening
Erp = 2121
460
0.02
fy
Ers = 200000
εro
ε (Ten)
Figure 2.18(a): Tensile behavior of steel reinforcement bar, threaded rod
and shear link.
σ (Ten & Comp, N/mm2)
fsy
460
Hardening
Esp = 2121
0.06
640
0.02
fsy max
εso
εst
Ess = 200000
ε
Figure 2.18(b): Tensile and compressive behaviour of steel angle and plate.