Plastic Behaviour of Beams and Frames

This report documents an experiment investigating the plastic behavior of beams and frames made of ductile material. The study focuses on the formation of plastic hinges due to bending under loading. Three tests were conducted: Test 1.1 determined the plastic moment of a circular rod under tension; Test 1.2 analyzed a continuous two-span beam under incremental loading; and Test 1.3 examined a rigid portal frame under incremental loading. Each test involved setting up the structure, applying loads until failure, recording results, and analyzing the data using both theoretical and experimental methods. The report compares theoretical and experimental collapse loads, discussing potential sources of error and concluding that for continuous beams, loading position doesn't affect collapse load, while for portal frames, load ratios slightly influence the results. The report highlights the relationship between plastic moment, yield stress, and geometry.

Plastic Behaviour of Beams and Frames

Contents
Introduction...........................................................................................................................................1
Background Information........................................................................................................................1
Test 1.1..................................................................................................................................................2
1.1.0 Laboratory Test Set-Up.............................................................................................................2
1.1.1 Test Results...............................................................................................................................2
1.1.2 Analysis.....................................................................................................................................2
1.1.3 Discussion.................................................................................................................................3
Test 1.2..................................................................................................................................................3
1.2.0 Laboratory Test Set-Up.............................................................................................................3
1.2.1 Results......................................................................................................................................4
1.2.2 Analysis.....................................................................................................................................4
1.2.3 Discussion.................................................................................................................................6
Test 1.3..................................................................................................................................................6
1.3.0 Laboratory Test Set-Up.............................................................................................................6
1.3.1 Results......................................................................................................................................7
1.3.2 Analysis.....................................................................................................................................8
1.3.3 Discussion...............................................................................................................................10
Conclusion...........................................................................................................................................11

Introduction
The following report entails an experimentation of the behaviour of ductile material. The
following report will specifically observe the enabling of plastic hinges due to bending
caused by loadings. The plastic hinges that from are regions of localised plasticity where an
increase in rotations occurs due to plastic moment of the beam section being equal to a
constant bending moment. This report aims to observe the development of plastic collapse
mechanisms and compare experimentally determined plastic collapse values against
theoretical ones.
Background Information
Plastic collapse methods are used in obtaining failure loads of structures. The method
involves analysis of structure mechanics after the onset of plasticity at certain points along
the structure that are unable to withhold loads any longer. An elastic analysis is used mainly
for practical design however a plastic analysis is used for considering a wide spectrum of
responses along the structure when it is performing in the inelastic region and when
designing for the ultimate loading.
For simple beam structures, the moment curve relationship for a rectangular cross section is
shown above. For M<My, the beam behaves elastically however at A; the moment M = the
yield moment My and the relationship loses its linear behaviour. The moment then further
moves towards the fully plastic moment Mp and the curve relationship tends to infinity. The
ratio of Mp to the yield moment is known as the shape factor Sf. When a load is applied to
the structure, it is in its elastic state and will resist deformation; however, when it reaches
the point of zero resistance, plastic hinges form. When the plastic hinges in the structure
equal n+1 the structure will collapse (n = degree of indeterminacy of the structure).

Test 1.1
1.1.0 Laboratory Test Set-Up
1. Metal rod placed under tension machine
2. Rod is stretched from both sides until failure occurs
1.1.1 Test Results
Diameter (d) = 3.2mm
Yield Stress (σy) = 245MPa
1.1.2 Analysis
The plastic collapse method aims to determine the failure load of a structure by finding the
onset of plasticity at certain points along the structure.
The Plastic Moment of the circular rod in this experiment is found by utilizing the following
calculation:
M p=S σ y
Where Mp = Plastic Moment
S = Plastic Section Modulus
σy = Yield Stress
Circular Rod:
Ac= AT= Asemicircle
Asemicircle= π d2
4
S=( A¿¿ c × h)+ ( AT ×h ) =2 Ah ¿
y= 2d
3 π
h= y
2
Where h = distance to the plastic neutral axis of semicircle; y = distance to centroid
∴ S= π d2
4 × 2 d
3 π

S= π ×3.22
4 × 2 ×3.2
3 × π =5.46
M p=5.46 × 245
¿ 1.34 N . m
1.1.3 Discussion
The circular rod was expected to deform plastically once the plastic moment had reached a
capacity of 1.34Nm. The plastic moment obtained in this test will be further used as a
constant value for tests 1.2 and 1.3. Since the metal rod is of similar material and
dimensions to the beams used in tests 1.2 and 1.3; it can be assumed that the plastic
moment will remain constant. From this test it can be concluded that plastic deformation
will occur once the beam reaches the plastic moment calculated above.
Test 1.2
1.2.0 Laboratory Test Set-Up
1. Continuous two-span beam is incrementally loaded with weights at a steady pace
2. Load is placed with respective to ratio given by lab handout per group
3. Beam loaded till failure
4. The load is weighed for calculations
Figure 1.2.2 Post failure and
formation of plastic hinge.

1.2.1 Results
Groups Ratio Loading Dist
(x1) (mm)
Loading Dist
(x2) (mm)
Load (kg) Load (x1) (N) Load (x2) (N) Total Load (N)
1 1.50 180.00 120.00 4.58 26.98 17.99 44.97
2 2.00 200.00 100.00 4.05 26.49 13.24 39.73
3 2.50 214.28 85.72 4.02 28.19 11.28 39.47
4 3.00 225.00 75.00 3.36 24.70 8.23 32.93
5 3.50 233.33 66.67 3.87 29.53 8.44 37.96
6 4.00 240.00 60.00 3.70 29.05 7.26 36.32
1.2.2 Analysis (for Group 1)
Σ M =0
Figure 1.2.3 Diagram of continuous
beam
Figure 1.2.4 Diagram of plastic hinge
formation
45N
27N F N?
180mm 120mm
X1 X2

Σ M A = (−45× 180 ) + ( F ×300 )=0
F= 8100
300 =27 N
The above calculation can be used to calculate the theoretical forces at the supports for
each different ratio. (F represents collapse load)
Detereming the experimental forces using plastic analysis;
External virtual work=InternalVirtual Work
Σ F ×δ =Σ M p ×θ
δ=0.15 θ
1.5 F ×0.15 θ=¿
¿( M ¿¿ p ×θ)+( M ¿ ¿ p ×θ)+(M ¿¿ p × θ)¿ ¿ ¿
1.5 F ×0.15 θ=3 M p θ
F= 3 × 1.34
1.5× 0.15
¿ 17.87 N
44.67−17.87=Fcollaspe=26.8 N
Experimental Calculation for Groups 1-6.
Groups Ratio Loading Dist
(x1) (mm)
Loading Dist
(x2) (mm)
Load (x1) (N) Load (x2) (N) Total Load (N)
1 1.50 180.00 120.00 17.87 26.80 44.67
2 2.00 200.00 100.00 13.40 26.80 40.20
3 2.50 214.28 85.72 10.72 26.80 37.52
4 3.00 225.00 75.00 8.93 26.80 35.73
5 3.50 233.33 66.67 7.66 26.80 34.46
6 4.00 240.00 60.00 6.70 26.80 33.50

1 . 00 1 . 5 0 2 . 0 0 2 . 5 0 3 . 0 0 3 . 5 0 4 . 0 0 4 . 5 0
22.00
23.00
24.00
25.00
26.00
27.00
28.00
29.00
30.00
Collapse Load vs Ratio
Ratio
Collapse Load (N)
1.2.3 Discussion
From Test 1.2 the relationship between plastic moment and collapse load was observed.
Theoretically determined values of collapse load were compared and contrasted against
experimentally determined values for collaspe load using the plastic moment (Mp) value
obtained from Test 1.1. The theoretical values for collapse load were obtained through
simple force analysis whilst the experimental value were determined using the plastic
analysis method approach. From the results it was observed that the ratio given per
calculation did not affect the collapse load of the metal rod. The graph in figure 1.2.5 shows
how the collapse load remains constant even when the ratio is altering (comparison of
theoretical collapse load to experimental collapse load). From this test it can be concluded
that the location of loading along a structure plays no effect on collapse load of the
structure.
Test 1.3
1.3.0 Laboratory Test Set-Up
1. A rigid portal frame is set up using the same rod material used in Test 1.1 and 1.2
2. The portal frame is loaded incrementally in accordance with the ratio given to group
3. Portal frame is loaded till failure occurs
4. The load at which failure occurred is recorded on scales
Figure 1.3.1 Rigid portal frame set-
up
Figure 1.2.5 Theoretical vs Experimental

1.3.1 Results
Groups Ratio Loading Dist
(x1) (mm)
Loading Dist
(x2) (mm)
Load (kg) Vertical (N) Horizontal (N) Total Load (N)
1.00 1.50 180.00 120.00 3.66 21.52 14.35 35.87
2.00 2.00 200.00 100.00 3.71 24.23 12.12 36.35
3.00 2.50 214.28 85.72 3.53 24.70 9.88 34.58
4.00 3.00 225.00 75.00 3.74 27.53 9.18 36.71
5.00 3.50 233.33 66.67 3.74 28.56 8.16 36.72
6.00 4.00 240.00 60.00 3.79 29.76 7.44 37.20
Figure 1.3.2 Load being increased
underneath the portal frame
V
H
150mm
300mm
Figure 1.3.3 Portal beam setup
for Group 1
150mm
V
H
150mm
300mm
` ``

1.3.2 Analysis (for Group 1)
Σ M =0
V =1.5 H
Σ F=1.5 H +H −F =0
2.5 H=F
H= 36 N
2.5 =14.4 N
∴ V =1.5 ×14.4 N =21.6 N
The calculation above demonstrates the theoretical calculation of the horizontal and vertical
forces for Group 1 with
a ratio of 1.5 H:V.
Determining horizontal
and vertical forces using
plastic analysis.
150mm
F
X2 X1
`
F = 36N
V H
120mm 180mm
θθ
θ θ
V
H
Figure 1.3.4 Portal frame setup with
loading bar underneath

The (figure) and (figure) above show two elementary plastic collapse mechanisms; beam collapse
and sway collapse respectively.
The number of hinges (n) required to form a plastic collapse mechanism in a structure having r
degrees f redundancy is n = r + 1. In the case of the portal frame above, r = 3 and n = 4.
The calculation below shows the analytical method of finding forces using plastic analysis of frames.
In beam collapse mechanism calulcation, the distance is half the length of the top of the portalẟ
frame since the vertical force is acting midway ( = 0.15). In sway collapse calculation, the distanceẟ ẟ
is the full length of the side of the portal frame since the horizontal force is acting at the top ( =0.3).ẟ
Beam Collapse Mechanism:
External work done=InternalWork Done
ΣW × δ=Σ M p ×θ
θ
θ θ
θ
∆ ∆
H
V
Figure 1.3.5 Beam Collapse
Figure 1.3.6 Sway Collapse

W vertical ×δ= ( M p ×θ ) + ( M p × θ ) + ( M p ×θ ) + ( M p × θ )
W vertical ×δ=4 M p θ
W vertical ×0.15 θ=4 M p θ
W vertical= 4 ×1.34
0.15
¿ 35.7 N
Sway Collapse Mechansim:
External work done=Internal Work Done
ΣW × δ=Σ M p ×θ
W horizontal ×δ = ( M p ×θ ) + ( M p × θ ) + ( M p ×θ ) + ( M p ×θ )
W horizontal ×0.3 tanθ=4 M p θ
W horizontal= 4 ×1.34
0.3 =17.9 N
Combined Beam Sway Collapse Mechanism:
(W ¿¿ horizontal × 0.3 tanθ)+ ( W vertical ×0.15 θ ) = ( M p × θ ) + ( M p ×θ ) + ( M p × θ ) + ( M p ×θ ) + ( M p ×θ ) + ( M p × θ ) ¿
(0.3 H +0.15 V )θ=6 M p θ
0.3 H +0.15 V =6 × 1.34
0.3 H +0.15 V =8.04 N
0 5 1 0 1 5 2 0 2 5 3 0 3 5
-10
0
10
20
30
40
50
60
Experimental vs Theoretical Loading
Horizontal Load (N)
Vertical Load (N)
Figure 1.3.6 Experimental Theoretical