University Faculty of Engineering Lab Report: Plastic Behaviour

This laboratory report investigates the plastic behaviour of a continuous span beam and a fixed foot portal frame. The report details the experimental procedures, theoretical analysis, and results, including the determination of plastic moment capacity, collapse loads, and the formation of plastic hinges. The study employs the method of virtual work to calculate theoretical collapse loads and compares these values with experimental findings. The report covers various loading conditions and the use of interaction diagrams to define the permissible region for the frame's safety. The experimental setup involved loading the beam and frame with vertical and horizontal forces at specified ratios and monitoring the point at which plastic hinges formed. The results include testing images, measurement data, and comparisons between theoretical and experimental values, providing insights into the plastic behaviour of structural members. The conclusion highlights the good correlation between theoretical calculations and experimental results for different loading ratios. This report provides valuable information for understanding the plastic behaviour of structural elements.

University
Faculty of Engineering
Laboratory Report 1:
Plastic Behaviour of Beams & Frames
Written by:
Group 22:
Test Date:
Due Date:
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Table of Contents
1 Executive Summary..........................................................................................................3
2 Test 1.1 – Plastic moment capacity..................................................................................4
2.1.1 Plastic Section Modulus......................................................................................4
2.1.2 Elastic Section Modulus......................................................................................4
2.1.3 Shape factor....................................................................................................... 5
2.1.4 Plastic Moment...................................................................................................5
3 Test 1.2 - Plastic Behaviour of a Continuous Beam..........................................................6
3.1 Introduction...............................................................................................................6
3.2 Theoretical Analysis...................................................................................................6
3.2.1 Collapse Load......................................................................................................6
3.3 Experimental Procedure............................................................................................7
3.4 Results....................................................................................................................... 8
3.4.1 Testing images....................................................................................................8
3.4.2 Test Measurements............................................................................................8
3.4.3 Comparison........................................................................................................ 9
3.5 Conclusion............................................................................................................... 10
4 Test 1.3 - Plastic behaviour of a frame...........................................................................11
4.1 Introduction.............................................................................................................11
4.2 Theoretical Analysis.................................................................................................11
4.2.1 Method of virtual work.....................................................................................11
4.2.2 Beam Mechanism.............................................................................................12
4.2.3 Sway Mechanism..............................................................................................13
4.2.4 Combined Mechanism......................................................................................14
4.2.5 Interaction Diagram..........................................................................................15
4.3 Experimental Procedure..........................................................................................18
4.4 Experimental Results............................................................................................... 19
4.4.1 Testing Images..................................................................................................19
4.4.2 Test Measurement........................................................................................... 19
4.4.3 Comparison...................................................................................................... 20
4.5 Conclusion............................................................................................................... 21
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5 References......................................................................................................................22
1 Executive Summary
Knowing the plastic behaviour of a structure or structural member is important for
an engineer to consider during the design process. Modern design standards and
procedures are based on the design utilising the elastic range of a member, as this is
more conservative when compared to the plastic limit.
Ideally it would be preferred to carry out large scale in-situ testing of steel structures
to determine these characteristics, however this is not a feasible option for the
majority of sites & practicing engineers. This is where testing samples within the
laboratory is able to provide us with all the relevant data within a time & cost
effective manner which can be translated to the in-situ structure.
This report looks at the laboratory testing of a continuous span beam & a rigid fixed
foot portal frame to determine their plastic properties. The goal is to determine the
collapse loads required to form a plastic hinge.
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2 Test 1.1 – Plastic moment capacity
2.1.1 Plastic Section Modulus
The plastic section modulus is used to describe the characteristic of a member in
which it exhibits a plastic behaviour. The modulus is dependent on the position of
the plastic neutral axis. The plastic neutral axis divides the cross section into two
equal areas, in order for the area under compression to equal to the area under
tension
The plastic section modulus for a member with a circular cross section is
determined by the following equation;
S= d3
6
S=Plastic section modulus , ( mm3 ) d=Section diamter , ( mm )
S= d3
6
S= ( 3.20 mm )3
6
S=5.4613mm3
2.1.2 Elastic Section Modulus
The elastic section modulus is a geometric property of a member determined by the
first moment of area taken about the neutral axis.
The elastic section modulus for a member with a circular cross section is
determined by the following equation;
Z= π × d3
32
Z=Elastic section modulus , ( mm3 )d=Section diamter , ( mm )
Z= π × d3
32
Z= π × ( 3.20 mm )3
32
Z=3.2170 mm3
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2.1.3 Shape factor
The shape factor is a numerical means of relating the plastic section modulus to the
elastic section modulus for a particular cross sectional shape of a member. The
typical value for a circular cross-section is 1.7. The shape factor for the member
tested is calculated below;
Shape Factor= S
Z
Shape Factor= Plastic section modulus , ( mm3 )
Elastic section modulus , ( mm3 )
S=Plastic section modulus , ( mm3 ) Z=Elastic section modulus , ( mm3 )
Shape Factor= 5.4613
3.2170
Shape Factor=1.6976
Shape Factor ≈1.70
2.1.4 Plastic Moment
The plastic moment ( M p ) of a structural member is defined as the point in which the
entire cross section has reached its yield stress. Theoretically, this is the maximum
bending moment in which the section can resist before a plastic hinge forms. This
hinge will continue to deform under additional loading until such point in which the
material fails.
The Plastic moment of a member is determined by the following equation;
M p=S × σ y
M p=Plastic moment , ( Nm )
S=Plastic section modulus , ( m3 )σ y=Y ield stress , ( MPa )
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3 Test 1.2 - Plastic Behaviour of a Continuous Beam
3.1 Introduction
The aim of this experimental testing is to obtain an understanding on the behaviour
of a continuous span beam subjected to vertical forces. The loading action of each
span will be based on a specified ratio. Key items to watch for will be the location
and severity of plastic hinges during the gradual loading process.
3.2 Theoretical Analysis
3.2.1 Collapse Load
To determine the theoretical load at which a plastic hinge will form in the
member the virtual work method is used. The method involves balancing the
external work with the internal work and solving for the unknown collapse load.
Data;
L=0.30m M p=Plastic moment , ( Nm )
∑ External Work =∑ Internal Work∑ External Work =V c × ( L
2 )×θ
∑ Internal Work=3 × M p ×θ
V c × ( L
2 ) ×θ=3 × M p × θ V c × ( 0.30 m
2 ) ×θ=3 ×1.36 Nm ×θ
V c × 0.15 m× θ=4.08 Nm × θV c × 0.15 m=4.08 NmV c= 4.08 Nm
0.15 m V c=27.20 N
W c= V c
2 W c= 27.20 N
2 W c=13.60 N
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3.3 Experimental Procedure
The experimental testing conducted at the University of Wollongong High Bay Lab
was to investigate the plastic behaviour of a continuous span beam containing two
spans under particular loading conditions. The beam was constructed from a circular
3.2mm diameter steel rod with a yield stress of249 MPa.
The test apparatus shown in Figure 3.2.1-1 enables the spans of the beam to be
loaded with different vertical forces at a specified ratio. The forces are applied by a
spreader bar positioned below the frame.
Figure 3.2.1-1 - Beam testing diagram
A suspended bowl is positioned along the spreader bar at a particular distance to
vary the loading of vertical & horizontal forces. The ratio for this experiment was the
vertical force was twice that of the horizontal force.
The bowl attached to the spreader bar was gradually loaded with led pellets to
increase the vertical force on both spans simultaneously. The beam was visually
monitored to determine the point at which a plastic hinge was forming, at which
point no additional load was applied.
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Once the beam was seen to have developed a plastic hinge it was deemed to have
failed. The loaded bowl was weighed to determine the amount of additional mass
the beam was able to support at the point of failure.
The tabulated results & theoretical comparison are shown in the following section.
3.4 Results
3.4.1 Testing images
Figure 3.4.1.1 shows the test beam after plastic hinges have formed at both the
point of loading & internal support.
Figure 3.4.1-2 - Collapse of test beam
3.4.2 Test Measurements
The following data was recorded for the test conducted;
Mass of Rings ∧ ¯+ Bucket +Load=3777.20 g
Mass of Rings ∧¯¿ 543.90 g
Mass of Bucket=403.30 g
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Mass of Load= ( Mass of Bucket +Load )− ( Mass of Bucket+ Rod∧Rings )
Mass of Load= ( 3777.20 g )− ( 403.30 g+543.90 g )
Mass of Load=2830.00 g
Convert to weight to force
W c=Collapse load
W c=2830.00 g x 9.81m/ s2
W c=27.76 N
3.4.3 Comparison
Percentage of Error
Error= Actual force ( N )
Theoretical force ( N ) −100 %
Error= 27.76 N
27.20 N −100 %
Error=2.06 %
Group Ratio Mp (Nm) Load on span 1 Load on span 2
No. V : H Nm N N
21 1.5 : 1 1.30 25.94 17.29
22 2.0 : 1 1.36 27.20 13.60
18 2.5 : 1 1.30 25.92 10.38
19 3.0 : 1 1.31 26.23 8.72
20 3.5 : 1 1.36 27.20 7.81
Table 3.4.3-1 - Comparison of theoretical with different loading ratios
Group Ratio Mp (Nm) Load on span 1 Load on span 2
No. V : H Nm N N
21 1.5 : 1 1.30 27.54 19.15
22 2.0 : 1 1.36 27.76 13.88
18 2.5 : 1 1.30 27.81 12.84
19 3.0 : 1 1.31 27.27 10.87
20 3.5 : 1 1.36 29.27 10.47
Table 3.4.3-2 - Comparison of experimental with different loading ratios
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3.5 Conclusion
The comparison of theoretical calculations to the results obtained from the
experimental testing showed a good correlation for all ratio’s of vertical force to the
historical force.
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4 Test 1.3 - Plastic behaviour of a frame
4.1 Introduction
The aim of this experimental testing is to obtain an understanding on the behaviour
of a portal frame under a combined loading action. The loading action will be based
on a specified ratio of vertical & horizontal forces. Key items to watch for will be the
location and severity of plastic hinges during the gradual loading process.
4.2 Theoretical Analysis
4.2.1 Method of virtual work
The method of virtual work which is the form of the principle of least action is used
to determine the forces & movements within a rigid body.
The virtual work method will be utilized by assuming the external work is equal to
the internal work, allowing us to solve for the required unknown values. This is
represented by the following equation;
∑ External Work =∑ Internal Work
Two principles which the method is based on are;
1) The deformations within a structure after it has collapsed occur by rotation at
the plastic hinges, the simple supports & internal pin connections.
2) The virtual work principle can be applied to these deformations listed in 1.
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4.2.2 Beam Mechanism
A beam mechanism occurs when a load is applied to a span that is restrained on
either end causing the formation of a plastic hinge. The top member of the frame in
which the experiment was conducted can be represented by a beam with fixed ends
carrying a point load at mid span. To find the collapse load associated with the beam
mechanism the method of virtual work was employed and tabulated below;
∑ External Work =∑ InternalWork
∑ External Work =V c × δ
∑ Internal Work= ( M p ×θ ) + ( M p × θ ) + ( M p ×α ) + ( M p ×α )
Figure 4.2.2-3 - Beam mechanism
M p=1.360 Nm
δ= ( 0.15 m ×θ ) = ( 0.15 m× α )
V c × δ = ( M p ×θ ) + ( M p ×θ ) + ( M p × α ) + ( M p × α )
V c × δ = ( 1.360 Nm ×θ ) + ( 1.360 Nm ×θ )+ (1.360 Nm × α ) + ( 1.360 Nm ×α )
V c × ( 0.15 m× θ )= ( 2.720 Nm ×θ )+ ( 2.720 Nm × θ )
V c= ( 2.720 Nm× θ ) + ( 2.720 Nm ×θ )
( 0.15 m×θ )
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