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Plastic
frame

 

 

 

 

 

 

 

 

Kingfan Ha

 

 

 

 

 

 

 

 

 

 

 

Submitted on 26/01/2017

 

 

 

 

 

 

BSc Level 6 2017-18

Directorate of Civil Engineering

School of Computing, Science and Engineering

 

Table of Contents
1.0        Introduction. 4
2.0        Apparatus. 5
3.0        Procedure. 8
4.0        Theory. 9
5.0        Results and calculations. 11
6.0        Discussion. 24
7.0        References. 25
 

 

 

 

 

 

 

 

 

 

 

Figure 1 –
Main apparatus without inserting the sample frame. 5

Figure 2 – Main apparatus
with after inserting sample frame. 5

Figure 3 – Steel frame
sample. 6

Figure 4 – Electronic
Calliper 6

Figure 5 – Ruler 7

Figure 6 – Data logger 7

Figure 7 – Frame example. 9

Figure 8 – Sway Failure
example. 9

Figure 9 – Beam failure
example. 10

Figure 10 – Combined
failure example. 10

Figure 11 – Tensile test
for group 1 frame. 11

Figure 12 – Height and
length of the frame. 12

Figure 13 – First group
breath section results. 12

Figure 14 – First group
depth section results. 13

Figure 15 – Frame and
section details. 14

Figure 16 – Beam failure. 14

Figure 17 – Sway failure. 15

Figure 18 – Combined
failure. 16

Figure 19 – Theoretical
Interaction diagram.. 17

Figure 20 – 18th Hole from
left – Beam failure. 17

Figure 21 – Vertical load
VS Vertical deflection (18th Hole) 18

Figure 22 – Horizontal
load VS Horizontal displacement (18th Hole) 18

Figure 23 – Middle hole –
Combine failure. 19

Figure 24 – Vertical load
VS Vertical displacement (Mid Hole) 19

Figure 25 – Horizontal
load VS Horizontal displacement (Mid Hole) 20

Figure 26 – 4th Hole from
left – Sway failure. 20

Figure 27 – Vertical load
VS Vertical displacement (4th Hole) 21

Figure 28 – Horizontal
load VS Horizontal displacement (4th Hole) 21

Figure 29 – Experimental
interaction diagram.. 22

 

1.0
        
Introduction

 

Plastic
theory was first introduced by the late Lord baker, Professors Horn, Heyman,
etc at Cambridge University. The design theory is simply as follow, when a
steel member has reach beyond its elastic limit from the apply load, the strain
increases as the stress stays the same. As the moment increases for a beam
section, a plastic hinge would form due to this behaviour at which the section
rotates at the plastic moment capacity.

Plastic
analysis is based on finding out the minimum load that causes the structure to
collapse. It occurs when enough hinges were formed that turns the structure
into a mechanism. The safe load equals to the collapse load divided by a load
factor.

Portal
frames are designed mainly using plastic design. The frame initially acts
elastically and then plastic hinges form as load continue to apply until it is
converted into a mechanism. A more accurate analyses takes the frame deflection
into consideration, though the secondary effects are only important to some
slender sway frames. (Nageim &
MacGinley)

It is
mentioned in plastic design codes specifications, that it requires the
construction material to have sufficient ductility for the plastic moment to be
able to be fully developed and sustained until it collapse due to failure. As for
steel material, the plastic hinge is require to be able to go through plastic
rotation without suffering from local buckling. (Wong)

Plastic
design make good use of the reserve strength for the structure with
consideration beyond the elastic state. The structure reserve strength allows
the structural member to continue to be loaded even after the maximum bending
capacity of the structure is reached, by utilized the elastic-plastic state
when the loading is still increasing.

Hence it
gives a more economical design which can use less material when using plastic
design method. In truth, theoretical plastic collapse load most of the time
will be less than the true load, which means it should always be a bit more
conservative when using the plastic design. (Wong)

 

 

 

 

 

 

 

 

 

 

 

 

2.0
        
Apparatus

The
apparatus used for the experiment are shown as below:

Figure 1 – Main apparatus without
inserting the sample frame

Figure 2 – Main apparatus with
after inserting sample frame

There
are 2 displacement gauge place on both top and side of the steel frame. The
vertical and horizontal pull are there to apply increment forces to the frame
manually until it fails. The steel member with 19 holes on it tells the load
ratio between the horizontal and vertical forces apply to the steel frame. Below
the steel member with 19 holes, there is a screw which is used to tighten the
pulls to induce an apply force to both top and side of the steel frame.

Figure 3 – Steel frame sample

There
are total 3 groups in the session and each group is given a steel frame as
shown on Figure3. The side of the frame sample are treated as column, and the
top member are treated as beam.

Figure 4 – Electronic Calliper

The
calliper is used to measure the breadth and depth of the frame. Three readings
were taken on both columns and the beam to get an average value for the breadth
and the depth.

Figure 5 – Ruler

The
ruler was used to measure the length of the beam and the height of the column
after placing the frame into the main apparatus.

Figure 6 – Data logger

The data
logger is used to record the horizontal/ vertical displacement made by the
steel frame. And then combing all the results to make 2 graphs, including the
vertical displacement VS vertical applied load and the horizontal displacement
vs horizontal load applied.

 

 

 

 

 

 

 

3.0
        
Procedure

The
procedures are mentioned as below according to the laboratory session:

1.      Three groups were each given a
steel frame, a calliper and a ruler

2.      Measured the breadth and depth of
the steel frame section in three different parts along the member to obtain an
average value, and the length of the beam. The section details were then
written on a piece of paper

3.      Untighten the fix support and put
the steel frame into the main apparatus. Aligned to make it stand horizontally
with a height of 206mm for both column

4.      Adjusted to a specific hole to
get a load ratio for one of the failure mechanism

5.      Load was then applied by
tightening the screws below the hole, results were then recorded into the data
logger

6.      Load kept on applying till the
steel frame was out of the elastic region as observed on the computer

7.      Unloaded the steel frame

8.      The steel frame was then properly
removed

9.      The test was then repeated for
two more other failure mechanisms

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4.0
        
Theory

The
theory will be illustrated with an example.

When it
comes to plastic frames using plastic analysis, the collapse mechanism will not
be looking obvious straightaway, and it needs to consider several
possibilities. It is generally most advantageous to use work equation method
for simple rectangular frames.

The
frame below (Figure 7) will be given a uniform full plastic moment of 20kNm.
With an un-factored load to find the load factor

 against collapse.

Figure 7 – Frame example

All the
possible locations to form plastic hinges are at A, B, C, D, and E. at which
there is a change of slope in the bending moment diagram. Between the hinges
are called “Critical sections”, and it is impossible to form plastic hinge in
the “Critical sections”. Then it follows three valid mechanisms which have
plastic hinges at the critical sections.

1.      Sway Mechanism

Figure 8 – Sway Failure example

In this
case, there is no downward movement of the 10kN load for a rigid-link, small,
movement of the mechanism. The work equation is then:

In
plastic moment mechanisms, the bending moment at any plastic hinge is always
associated to the direction of rotation of the hinge, which means plastic
hinges will always do positive virtual work.

2.      Beam Mechanism

Figure 9 – Beam failure example

The
stanchions remain vertical resulting only a small movement of the mechanism.
And no movement for the 5kN load. Equation as shown below:

3.      Combined mechanism

Figure 10 – Combined failure example

This is
a combination of 1 and 2. It is necessary to combine because 2 hinges at
location B will be eliminated and replaced by a rigid joint instead. Without
it, the mechanism will have 2 degrees of freedom, at which it wouldn’t be
possible to relate all the movements in the virtual mechanism to variable

. The work equation is
presented as:

After calculations
for all possible mechanism, the load factor against collapse will be minimum

 value which is 1.92. (Davies &
Brown, 1996)

 

5.0
        
Results and calculations

Theoretical

Figure 11 – Tensile test for group 1
frame

The
tensile strength test gives the yield strength of the steel member to calculate
the plastic moment for the steel member. Yield strength occurs in the end of
the linear section, at which the steel will start behaving plastically.

Reading
from the table of results provided in the excel file and the graph, it reaches
a maximum stress of 363.57

 for the linear section of the line.

 

Figure 12 – Height and length of the
frame

Length
of the beam

=
301mm,

=
298mm,

=
298mm

=
299mm = 300mm

Breath
of the section

First
group

Figure 13 – First group breath
section results

=
13.36mm

Second
group

=
13.34mm,

=
13.32mm,

=
13.19mm

=
13.28mm

Third
group

=
13.50mm,

=
13.50mm

=
13.50mm

Total

=
(13.31 + 13.28 + 13.50)/3 = 13.36mm

 

Depth of
the section

First
group

Figure 14 – First group depth
section results

=
3.25mm

Second
group

=
3.05mm,

=
3.10mm

=
3.08mm

Third
group

=
3.13mm,

=
3.18mm

=
3.16mm

Total

=
(3.25 +3.08 + 3.16)/3 = 3.14mm

 

Figure 15 – Frame and section
details

Plastic
moment

Beam
failure

Figure 16 – Beam failure

External
work done =

Internal
work done =

External
= Internal

 =

 

Sway
failure

Figure 17 – Sway failure

External
work done =

Internal
work done =

External
= Internal

 =

 

 

 

 

Combined
failure

Figure 18 – Combined failure

External
work done =

Internal
work done =

External
= Internal

 =

 =

 =

Figure 19 – Theoretical Interaction
diagram

The
theoretical interaction diagram is made with a collapse load of 320N in beam
failure, 233N in sway failure and

 =

 as a function for combine failure. The graph
will be used to compare to the experimental interaction diagram.

 

Experimental

Figure 20 – 18th Hole from left –
Beam failure

Figure 7
shows a beam failure where the top steel member deflects but the side remains
the same. Beam failure occur at 223.76N vertically and 24.71N horizontally. With
a load ratio of

 

Figure 21 – Vertical load VS
Vertical deflection (18th Hole)

 

Figure 22 – Horizontal load VS
Horizontal displacement (18th Hole)

 

 

 

 

 

Figure 23 – Middle hole – Combine
failure

As shown
on Figure 8, the frame is experiencing both sway and beam failure by observing
the deflected shape due to applied load.

The
combine failure occurred at an apply load of 162.24N vertically and 165.4N
horizontally. With a load ratio of

 

Figure 24 – Vertical load VS
Vertical displacement (Mid Hole)

 

Figure 25 – Horizontal load VS
Horizontal displacement (Mid Hole)

 

Figure 26 – 4th Hole from left –
Sway failure

Figure 9
shows the frame having a sway failure. The failure occurred at an apply load of
181.94N horizontally and 46.9N vertically. With a load ratio of

Figure 27 – Vertical load VS
Vertical displacement (4th Hole)

 

Figure 28 – Horizontal load VS
Horizontal displacement (4th Hole)

 

Figure 29 – Experimental interaction
diagram

The experimental
interaction diagram is made with a vertical load of 223.8N, horizontal load of
181.9N and a combine failure of vertical force 162.2 and horizontal force of
165.4N.

Hazard

Persons
at Risk

Raw
Risk

Comments
or Control Measures Specified by the Assessor

Residual
Risk

Probability
of Occurrence

Severity
of Occurrence

Risk

Probability
of Occurrence

Severity
of Occurrence

Risk

 
Sharp edges of material
 
 
 
Slippery floor
 
 
 
 
 
 

 
Students, teachers
 
 
 
Students, teachers
 
 
 
 
 

 
M
 
 
 
 
L
 
 
 
 

 
M
 
 
 
 
H
 
 
 

 
M
 
 
 
 
M
 
 
 
 

 
Self-aware
of the dangerous objects present in the laboratory, careful handling
 
 
 
Keep floor dry, or wear
non-slippery shoes. Keep out of dangerous object
 

 
L
 
 
 
 
L
 
 
 
 

 
M
 
 
 
 
M
 
 
 

 
M
 
 
 
 
M
 
 
 
 
 

6.0
        
Discussion

From the
interaction diagram, it is showing the theoretical results has a bigger
permissible region in terms of both vertical and horizontal forces. Which
experimentally, it is a slight less permissible in terms of vertical and
horizontal forces. This is obvious true because there are a lot more
considerations to be made at which it is neglected in the plastic analysis.

The temperature
and pressure of the environment, might change slightly how the material behaves
due to slight expansion or contraction, even though it has a small effect.

Relating
to plastic design specifications, there could be a second order effects within
the frame itself, which is small enough to be neglected in most design.

Frame
could have imperfections, lack of verticality. Also, member imperfections in
terms of straightness and residual stress.

There
are advantages when comparing plastic design method over elastic design method,
but there are also limitations to its use. For plastic design, it is assumed
that all cross section can sustain the plastic moment

 without undergoing local buckling. To achieve this,
the sections must be compact or it needs to be either class 1 and class 2 cross
sections in Eurocode 3. (Wong)

The
consideration of effects of lateral-torsional buckling on plastic behaviour of
the structure was also ignored. Hence, a sufficient lateral restraint should be
applied to all members that were designed by the plastic method to prevent
lateral buckling. The steel material also needs to be ductile enough to perform
plastic rotation without failure. (Wong)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7.0
        
References 

 

Davies, J. M.,
& Brown, B. A. (1996). Plastic design
to BS 5950. London: Blackwell Science.

Nageim, H. A. a. L. J. M. U. U. K., & MacGinley, T. J. a. Steel Structures : Practical Design Studies,
Third Edition (3rd ed.).

Wong, B. Plastic analysis and
design of steel structures.

 

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