Introduction-Plastic hinge concept-plastic section modulus-shape factor-redistribution of moments-collapse mechanism.
Theorems of plastic analysis - Static/lower bound theorem; Kinematic/upper bound theorem-Plastic analysis of beams and portal frames by equilibrium and mechanism methods.
Introduction-Plastic hinge concept-plastic section modulus-shape factor-redistribution of moments-collapse mechanism.
Theorems of plastic analysis - Static/lower bound theorem; Kinematic/upper bound theorem-Plastic analysis of beams and portal frames by equilibrium and mechanism methods.
Design of steel structure as per is 800(2007)ahsanrabbani
It does not offer resistance against rotation and also termed as a hinged or pinned connections.
It transfers only axial or shear forces and it is not designed for moment
It is generally connected by single bolt/rivet and therefore full rotation is allowed
Stiffness method of structural analysisKaran Patel
This method is a powerful tool for analyzing indeterminate structures. One of its advantages over the flexibility method is that it is conducive to computer programming.
Stiffness method the unknowns are the joint displacements in the structure, which are automatically specified.
Design of Reinforced Concrete Structure (IS 456:2000)MachenLink
This is the 1st Lecture Series on Design Reinforced Cement Concrete (IS 456 -2000).
In this video, you will learn about the objective of structural designing and then basic properties of concrete and steel.
Concrete properties like...
1. Grade of Concrete
2. Modulus of Elasticity
3. Characteristic Strength
4. Tensile Strength
5. Creep and Shrinkage
6. Durability
Reinforced Steel Properties....
1. Grade and types of steel
2. Yield Strength of Mild Steel and HYSD Bars
Design and Detailing of RC Deep beams as per IS 456-2000VVIETCIVIL
Visit : https://teacherinneed.wordpress.com/
1. DEEP BEAM DEFINITION - IS 456
2. DEEP BEAM APPLICATION
3. DEEP BEAM TYPES
4. BEHAVIOUR OF DEEP BEAMS
5. LEVER ARM
6. COMPRESSIVE FORCE PATH CONCEPT
7. ARCH AND TIE ACTION
8. DEEP BEAM BEHAVIOUR AT ULTIMATE LIMIT STATE
9. REBAR DETAILING
10. EXAMPLE 1 – SIMPLY SUPPORTED DEEP BEAM
11. EXAMPLE 2 – SIMPLY SUPPORTED DEEP BEAM; M20, FE415
12. EXAMPLE 3: FIXED ENDS AND CONTINUOUS DEEP BEAM
13. EXAMPLE 4 : FIXED ENDS AND CONTINUOUS DEEP BEAM
Design of steel structure as per is 800(2007)ahsanrabbani
It does not offer resistance against rotation and also termed as a hinged or pinned connections.
It transfers only axial or shear forces and it is not designed for moment
It is generally connected by single bolt/rivet and therefore full rotation is allowed
Stiffness method of structural analysisKaran Patel
This method is a powerful tool for analyzing indeterminate structures. One of its advantages over the flexibility method is that it is conducive to computer programming.
Stiffness method the unknowns are the joint displacements in the structure, which are automatically specified.
Design of Reinforced Concrete Structure (IS 456:2000)MachenLink
This is the 1st Lecture Series on Design Reinforced Cement Concrete (IS 456 -2000).
In this video, you will learn about the objective of structural designing and then basic properties of concrete and steel.
Concrete properties like...
1. Grade of Concrete
2. Modulus of Elasticity
3. Characteristic Strength
4. Tensile Strength
5. Creep and Shrinkage
6. Durability
Reinforced Steel Properties....
1. Grade and types of steel
2. Yield Strength of Mild Steel and HYSD Bars
Design and Detailing of RC Deep beams as per IS 456-2000VVIETCIVIL
Visit : https://teacherinneed.wordpress.com/
1. DEEP BEAM DEFINITION - IS 456
2. DEEP BEAM APPLICATION
3. DEEP BEAM TYPES
4. BEHAVIOUR OF DEEP BEAMS
5. LEVER ARM
6. COMPRESSIVE FORCE PATH CONCEPT
7. ARCH AND TIE ACTION
8. DEEP BEAM BEHAVIOUR AT ULTIMATE LIMIT STATE
9. REBAR DETAILING
10. EXAMPLE 1 – SIMPLY SUPPORTED DEEP BEAM
11. EXAMPLE 2 – SIMPLY SUPPORTED DEEP BEAM; M20, FE415
12. EXAMPLE 3: FIXED ENDS AND CONTINUOUS DEEP BEAM
13. EXAMPLE 4 : FIXED ENDS AND CONTINUOUS DEEP BEAM
A review of constitutive models for plastic deformationSamir More
Materials like mild steel have defined yield point hence it is easy to distinguish between the elastic region and plastic region of deformation. But for materials that do not have specified yield point, it is hard to distinguish between elastic and plastic deformation region. In that case may be plastic deformation starts from beginning of the application of the load. For elastic region, stress and strain are in linear relationship with each other hence Hook’s law valid true. But for plastic region, the relation between stress and strain is nonlinear and complicated. So need for continuum plasticity model arises. The main aim of continuum plasticity model is to formulate mathematical model based on experimental results that can predict the plastic deformation of material under varying loading conditions and at different elevated temperature.
Design and analysis of Stress on Thick Walled Cylinder with and with out HolesIJERA Editor
The conventional elastic analysis of thick walled cylinders to final radial & hoop stresses is applicable for the internal pressures up to yield strength of material. The stress is directly proportional to strain up to yield point Beyond elastic point, particularly in thick walled cylinders. The operating pressures are reduced or the material properties are strengthened. There is no such existing theory for the stress distributions around radial holes under impact of varying internal pressure. Present work puts thrust on this area and relation between pressure and stress distribution is plotted graphically based on observations. Here focus is on pure mechanical analysis & hence thermal, effects are not considered. The thick walled cylinders with a radial cross-hole ANSYS Macro program employed to evaluate the fatigue life of vessel. Stresses that remain in material even after removing applied loads are known as residual stresses. These stresses occur only when material begins to yield plastically. Residual stresses can be present in any mechanical structure because of many causes. Residual stresses may be due to the technological process used to make the component. Manufacturing processes lead to plastic deformation. Elasto plastic analysis with bilinear kinematic hardening material is performed to know the effect of hole sizes. It is observed that there are several factors which influence stress intensity factors. The Finite element analysis is conducted using commercial solvers ANSYS & CATIA. Theoretical formulae based results are obtained from MATLAB programs. The results are presented in form of graphs and tables.
Similar to Module4 plastic theory- rajesh sir (20)
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Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
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Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
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Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
1. Structural Analysis - II
Plastic Analysis
Dr. Rajesh K. N.
Assistant Professor in Civil EngineeringAssistant Professor in Civil Engineering
Govt. College of Engineering, Kannur
Dept. of CE, GCE Kannur Dr.RajeshKN
2. Module IVModule IV
Plastic Theory
• Introduction-Plastic hinge concept-plastic section modulus-shape
factor-redistribution of moments-collapse mechanism-
• Theorems of plastic analysis - Static/lower bound theorem;
Kinematic/upper bound theorem-Plastic analysis of beams and
portal frames b equilibrium and mechanism methodsportal frames by equilibrium and mechanism methods.
Dept. of CE, GCE Kannur Dr.RajeshKN
2
3. Plastic Analysis Why? What?
• Behaviour beyond elastic limit?
Plastic Analysis - Why? What?
• Behaviour beyond elastic limit?
• Plastic deformation collapse load• Plastic deformation - collapse load
• Safe load load factor• Safe load – load factor
• Design based on collapse (ultimate) load limit• Design based on collapse (ultimate) load – limit
design
• Economic - Optimum use of material
Dept. of CE, GCE Kannur Dr.RajeshKN
3
5. Upper yield
point
A
point
Plasticess
B C
L i ld
range
stre
Lower yield
point
strainO strainO
Idealised stress-strain curve of mild steel
Dept. of CE, GCE Kannur Dr.RajeshKN
5
7. • Elastic analysisElastic analysis
- Material is in the elastic state
P f f d i l d- Performance of structures under service loads
- Deformation increases with increasing load
• Plastic analysis• Plastic analysis
– Material is in the plastic state
– Performance of structures under
ultimate/collapse loads/ p
– Deformation/Curvature increases without an
increase in load.
Dept. of CE, GCE Kannur Dr.RajeshKN
7
increase in load.
8. AssumptionsAssumptions
• Plane sections remain plane in plastic
condition
S l d l b h• Stress-strain relation is identical both in
compression and tensionp
Dept. of CE, GCE Kannur Dr.RajeshKN
8
9. Process of yielding of a section
• Let M at a cross-section increases gradually.
• Within elastic limit, M = σ.Z
• Z is section modulus, I/y
• Elastic limit – yield stresses reached
M ZMy = σy.Z
• When moment is increased, yield spreads into innerWhen moment is increased, yield spreads into inner
fibres. Remaining portion still elastic
Fi ll th ti ti i ld• Finally, the entire cross-section yields
Dept. of CE, GCE Kannur Dr.RajeshKN
9
11. Change in stress distribution duringChange in stress distribution during
yielding
yσ yσ yσσ
yσ yσ yσσ
y
Rectangular cross section
Dept. of CE, GCE Kannur Dr.RajeshKN
11
13. Plastic hingePlastic hinge
h h l l ld d h• When the section is completely yielded, the
section is fully plastic
A f ll l b h l k h• A fully plastic section behaves like a hinge –
Plastic hinge
Plastic hinge is defined as an yielded zone duePlastic hinge is defined as an yielded zone due
to bending in a structural member, at which
large rotations can occur at a section atlarge rotations can occur at a section at
constant plastic moment, MP
Dept. of CE, GCE Kannur Dr.RajeshKN
13
14. Mechanical hinge Plastic hinge
Reality ConceptReality Concept
Resists zero Resists a constant
t Mmoment moment MP
Mechanical Hinge Plastic Hinge with M = 0Mechanical Hinge Plastic Hinge with MP= 0
Dept. of CE, GCE Kannur Dr.RajeshKN
14
15. • M – Moment corresponding to working load• M – Moment corresponding to working load
• My – Moment at which the section yieldsy
• MP – Moment at which entire section is under yield stress
yσ
CC
T
yσ
MP
Dept. of CE, GCE Kannur Dr.RajeshKN
15
16. Plastic momentPlastic moment
• Moment at which the entire section is
under yield stress
C T
c y t y
C T
A Aσ σ
=
=
2
c t
A
A A⇒ = =
•NA divides cross-section into 2 equal parts
2
A
2
y
A
C T σ= =
Dept. of CE, GCE Kannur Dr.RajeshKN
1616
17. yσ
C
yc
2
y
A
σ=
yt
yc
A
T
ZSi il
σ
2
yT σ=
•Couple due to
A
σ
yZσSimilar toyσ
( )
A
y y Zσ σ⇒ + =•Couple due to
2
yσ ( )
2
y c t y py y Zσ σ⇒ + =
Plastic modulus
is the shape factor( )1pZ
>
Dept. of CE, GCE Kannur Dr.RajeshKN
17
p( )
Z
18. Shape factor for various cross-sections
b
Shape factor for various cross sections
Rectangular cross-section:
d
Section modulus
( )3 212bdI bd( )
( )
12
2 6
bdI bd
Z
y d
= = =
2
A bd d d bd⎛ ⎞
( )
2
2 2 4 4 4
p c t
A bd d d bd
Z y y
⎛ ⎞
= + = + =⎜ ⎟
⎝ ⎠
Plastic modulus
Z
Shape factor = 1.5pZ
Z
Dept. of CE, GCE Kannur Dr.RajeshKN
1818
19. Circular sectionCircular section
A
d
( )
2
p c t
A
Z y y= +
d 2 3
2 2
8 3 3 6
d d d dπ
π π
⎛ ⎞⎛ ⎞
= + =⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠8 3 3 6π π⎝ ⎠⎝ ⎠
1 7pZ
S = =
( )4 364d d
Z
π π
= = 1.7S
Z
= =
2 32
Z
d
Dept. of CE, GCE Kannur Dr.RajeshKN
19
20. Triangular section
3
bh⎛ ⎞
⎜ ⎟
Triangular section
2
36
2 24
bh
bh
Z
h
⎛ ⎞
⎜ ⎟
⎝ ⎠= = 2
3
h
2 24
3
h
A
3
CG axis h
( )
2
p c t
A
Z y y= + b
E l i
cy
S = 2.346
Equal area axis
ty
Dept. of CE, GCE Kannur Dr.RajeshKN
20
b
22. Load factor
collapse load M Zσ
Load factor
P y Pcollapse load M
Load factor
working load M Z
Zσ
σ
= = =
Rectangular cross-section:
2
4
P y P y
bd
M Zσ σ= =
2 2
6 1 6
ybd bd
M Z
σ
σ σ= = =4
y y
2 2
2 25yPM bd bd
LF
σ
σ
⎛ ⎞ ⎛ ⎞
∴ = = ÷ =⎜ ⎟ ⎜ ⎟
6 1.5 6
2.25
4 1.5 6
yLF
M
σ∴ = = ÷ =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
Dept. of CE, GCE Kannur Dr.RajeshKN
2222
23. Factor of safetyFactor of safety
Yield Load
Factor of Safety=
WorkingLoad
yW
W
=
YieldStress
WorkingStress
yσ
σ
= =
g
( )
= 1.5
/1 5
yσ
σ
=
( )/1.5yσ
El ti A l i F t f S f tElastic Analysis - Factor of Safety
Plastic Analysis - Load Factor
Dept. of CE, GCE Kannur Dr.RajeshKN
23
24. Mechanisms of failure
• A statically determinate beam will collapse if one plastic
hinge is developedhinge is developed
• Consider a simply supported beam with constant cross• Consider a simply supported beam with constant cross
section loaded with a point load P at midspan
• If P is increased until a plastic hinge is developed at the point
of maximum moment (just underneath P) an unstable
t t ill b t dstructure will be created.
• Any further increase in load will cause collapse• Any further increase in load will cause collapse
Dept. of CE, GCE Kannur Dr.RajeshKN
24
25. • For a statically indeterminate beam to collapse, more than oney p
plastic hinge should be developed
• The plastic hinge will act as real hinge for further increase of
load (until sufficient plastic hinges are developed for
collapse )collapse.)
• As the load is increased, there is a redistribution of moment,, ,
as the plastic hinge cannot carry any additional moment.
Dept. of CE, GCE Kannur Dr.RajeshKN
25
26. Beam mechanismsBeam mechanisms
D i bDeterminate beams
& frames: Collapse
f fi l i Simple beam
after first plastic
hinge
Dept. of CE, GCE Kannur Dr.RajeshKN
2626
27. Indeterminate beams &
frames: More than onea es: Mo e t a o e
plastic hinge
to develop mechanismp
Fixed beam
l h d l h d fPlastic hinges develop at the ends first
Beam becomes a simple beamBeam becomes a simple beam
Plastic hinge develops at the centreg p
Beam collapses
Dept. of CE, GCE Kannur Dr.RajeshKN
27
28. Indeterminate beam:
More than one plasticMo e t a o e p ast c
hinge to develop
mechanism
Propped cantilever
l h d l h f d fPlastic hinge develops at the fixed support first
Beam becomes a simple beamBeam becomes a simple beam
Plastic hinge develops at the centreg p
Beam collapses
Dept. of CE, GCE Kannur Dr.RajeshKN
31. Methods of Plastic Analysisy
• Static method or Equilibrium method
- Lower bound: A load computed on the basis of an assumed- Lower bound: A load computed on the basis of an assumed
equilibrium BM diagram in which the moments are not greater than
MP is always less than (or at the worst equal to) the true ultimate
l dload.
• Kinematic method or Mechanism method or Virtual work• Kinematic method or Mechanism method or Virtual work
method
- Work performed by the external loads is equated to the internal
work absorbed by plastic hinges
Upper bound: A load computed on the basis of an assumed- Upper bound: A load computed on the basis of an assumed
mechanism is always greater than (or at the best equal to) the true
ultimate load.
Dept. of CE, GCE Kannur Dr.RajeshKN
31
32. • Collapse load (Wc): Minimum load at whichp c)
collapse will occur – Least value
• Fully plastic moment (MP): Maximum moment
capacity for design – Highest valuecapacity for design – Highest value
Dept. of CE, GCE Kannur Dr.RajeshKN
32
33. Determination of collapse load
1. Simple beam
Determination of collapse load
1. Simple beam
Equilibrium method:
W l
Equilibrium method:
.
4
u
P
W l
M =
MP
M 4 P
u
M
W
l
∴ =u
l
Dept. of CE, GCE Kannur Dr.RajeshKN
3333
34. Virtual work method:
E IW W=
.2
2
u P
l
W Mθ θ
⎛ ⎞
=⎜ ⎟
⎝ ⎠
uW
4 P
u
M
W
l
∴ =
2θl
θ
2
θ
Dept. of CE, GCE Kannur Dr.RajeshKN
34
35. 2. Fixed beam with UDL 2
l
2
2
.
,
24
CENTRE
w l
M =
2
.
12
ENDS CENTRE
w l
M M>=
Hence plastic hinges will develop at the ends first.
MP
M
MC1
MB1
MC2
MMP
MB1
MP
Dept. of CE, GCE Kannur Dr.RajeshKN
35
36. 2
.
2 uw l
M =
uw
Equilibrium:
2
8
PM =
16 PM
∴
2θ
θ θ
2
P
uw
l
∴ =
Virtual work: E IW W=
( )
0
22 2
l
l
M
θ
θ θ θ
⎛ ⎞
+⎜ ⎟⎛ ⎞
⎜ ⎟⎜ ⎟
16 PM
w∴ =( )22 2
2 2
u Pw M θ θ θ⎛ ⎞
= + +⎜ ⎟⎜ ⎟
⎝ ⎠⎜ ⎟
⎝ ⎠
2uw
l
∴ =
Dept. of CE, GCE Kannur Dr.RajeshKN
36
⎝ ⎠
37. 3. Fixed beam with point load3. Fixed beam with point load
uW
2θ
θ θ
u
MP
MP
2θ
Virtual work:
( )2
l
W Mθ θ θ θ
⎛ ⎞
= + +⎜ ⎟
Equilibrium:
Virtual work:
( )2
2
u PW Mθ θ θ θ= + +⎜ ⎟
⎝ ⎠
2
4
P u
l
M W=
8 P
u
M
W
l
∴ =
8 P
u
M
W
l
∴ =
Dept. of CE, GCE Kannur Dr.RajeshKN
37
38. 4. Fixed beam with eccentric point load4. Fixed beam with eccentric point load
uW
Equilibrium:
u
a b
2 P u
ab
M W
l
=
q
MP
l
2 PM l
W∴ =
MP
uW
ab
∴ =
Dept. of CE, GCE Kannur Dr.RajeshKN
38
39. Virtual work:
1 2a bθ θ=uW
Virtual work:
θ θ+
1θ
a b
2θ
1 2
b
a
θ θ⇒ =
1 2θ θ+ a
⎡ ⎤( ) ( )1 1 1 2 2u PW a Mθ θ θ θ θ= + + +⎡ ⎤⎣ ⎦
b⎡ ⎤
( )2 2 22 2u P
b
W b M
a
θ θ θ
⎡ ⎤= +⎢ ⎥⎣ ⎦
( )
2 2
2
2
2 2P P
u
M Mb a bW
b aba
θ θ
θ
+⎡ ⎤∴ = =+⎢ ⎥⎣ ⎦
2 P
u
M l
W
ab
=
Dept. of CE, GCE Kannur Dr.RajeshKN
2
40. 5. Propped cantilever with point load at
midspanmidspan
MMC2
MP
MP
MP
MC1
MB1
Dept. of CE, GCE Kannur Dr.RajeshKN
40
41. uW
2θ
Vi t l kVirtual work:
E IW W=
Equilibrium:
E IW W
( ) ( )2
l
W Mθ θ θ
⎛ ⎞
= +⎜ ⎟
.
0.5
4
u
P P
W l
M M+ =
( ) ( )2
2
u PW Mθ θ θ= +⎜ ⎟
⎝ ⎠
4
6 PM
6 P
u
M
W
l
∴ =
6 P
u
M
W
l
∴ =
Dept. of CE, GCE Kannur Dr.RajeshKN
41
42. 6. Propped cantilever with UDL
2
wl Maximum positive BM
8
wl p
x1
MMP
MP
At collapse
x2
E
Required to locate E
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42
q
43. 2
l ⎛ ⎞
2
2 2 2
2 2
u u
E P P
w lx w x x
M M M
l
⎛ ⎞= − − =⎜ ⎟
⎝ ⎠
( )1
For maximum,
0EdM
=
2
0
dx
=
2 0
2
u P
u
w l M
w x
l
− − = ( )2
2 0.414x l=From (1) and (2),
2
11.656 P
u
M
w
l
=From (2),
Dept. of CE, GCE Kannur Dr.RajeshKN
l
44. Problem 1: For the beam, determine the design plastic moment
icapacity.
50 kN 75 kN
1.5 m 1.5 m
7.5 m
D f d 3 2 1• Degree of Indeterminacy, N = 3 – 2 = 1
• No. of hinges, n = 3
• No. of independent mechanisms ,r = n - N = 2
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44
46. 50 75
1.5 m 1.5 m
θ
θ1
4.5 m
θ1
Mechanism 2
θ + θ1
16 1.5θ θ=
1
1.5
6
θ θ⇒ =
( )1 1 1 1 1
1.5 1.5 1.5
50 1.5 75 1.5
6 6 6
pMθ θ θ θ θ
⎛ ⎞ ⎛ ⎞
× + = + +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠6 6 6⎝ ⎠ ⎝ ⎠
87.5pM kNm∴ =
87.5 kNm=Design plastic moment (Highest of the above)
Dept. of CE, GCE Kannur Dr.RajeshKN
g p ( g )
47. Problem 2: A beam of span 6 m is to be designed for an ultimate UDL
f 25 kN/ Th b i i l t d t th d D iof 25 kN/m. The beam is simply supported at the ends. Design a
suitable I section using plastic theory, assuming σy= 250 MPa.
25 kN/m25 kN/m
66 m
• Degree of Indeterminacy, N = 2 – 2 = 0
• No. of hinges, n = 1
• No. of independent mechanisms, r = n-N = 1
Mechanism25 kN/m
θ θ
3 m
2θ
Dept. of CE, GCE Kannur Dr.RajeshKN
47
3 m 3 m
48. Internal work done 0 2 0 2IW M Mθ θ= + × + =te a o k do e 0 2 0 2I p pW M Mθ θ+ +
External work done 0 3
2 25 2253EW
θ
θ
+⎛ ⎞= × × =×⎜ ⎟
⎝ ⎠2
E ⎜ ⎟
⎝ ⎠
2 225I EW W M θ θ= ⇒ = 112.5pM kNm∴ =2 225I E pW W M θ θ⇒ p
Plastic modulus P
P
M
Z =
6
112.5 10×
= 5 3
4.5 10 mm= ×Plastic modulus P
y
Z
σ 250
Z
5
4.5 10× 5 3
3 913 10PZ
Z
S
=
.5 0
1.15
= 5 3
3.913 10 mm= ×Section modulus
Assuming shape factor S = 1.15
Adopt ISLB 275@330 N/m (from Steel Tables – SP 6)
Dept. of CE, GCE Kannur Dr.RajeshKN
Adopt ISLB 275@330 N/m (from Steel Tables SP 6)
49. Problem 3: Find the collapse load for the frame shownProblem 3: Find the collapse load for the frame shown.
W
F
/ 2 / 2B C
/ 2
W/2
Mp
E
2Mp
/ 2
W/2
2Mp
/ 2
D
AA
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49
50. • Degree of Indeterminacy, N = 5 – 3 = 2
• No. of hinges, n = 5 (at A, B, C, E & F)
• No. of independent mechanisms ,r = n - N = 3
•Beam Mechanisms for members AB & BC•Beam Mechanisms for members AB & BC
•Panel MechanismPanel Mechanism
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50
51. Beam Mechanism for AB
2 2 (2 ) 7W M M M Mθ θ θ θ+ +B M
Beam Mechanism for AB
2 2 (2 ) 7I p p p pW M M M Mθ θ θ θ= + + =
/ 2
B
θ
Mp
2 2
E
W
W θ=
W/2
E
/ 2 θ
2θ
2Mp
28 p
E I c
M
W W W= ⇒ =
W/2
/ 2
p
θ
2Mp
Dept. of CE, GCE Kannur Dr.RajeshKN
51
52. Beam Mechanism for BC
W
F/ 2 / 2B
C
θ θ
Mp
θ θ
2θ
2
θMp
Mp
2θ
Mp
(2 ) 4I p p p pW M M M Mθ θ θ θ= + + =
2
EW W θ=
8 p
E I c
M
W W W= ⇒ =
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52
53. Panel Mechanism
2 4I p p p pW M M M Mθ θ θ θ= + + =
Panel Mechanism
W
/ 2Mp Mp/ 2 2 2
E
W
W θ=
F
/ 2
θ
θ
16 p
E I c
M
W W W= ⇒ =
W/2 2
θ
E
/ 2
E
2Mp
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53
54. W
Combined Mechanism
2 ( ) (2 )
( )
I p pW M M
M
θ θ
θ θ
= +
+ +
W
/ 2 Mp/ 2
θ θ ( )
6
p
p
M
M
θ θ
θ
+ +
=
/ 2
Mp
θ
θ
2
θ
2θ
θ θ
3
2 2 2 4
E
W
W W Wθ θ θ= + =
W/2 2
θ
p
E
2 2 2 4
8M
/ 2
8 p
E I c
M
W W W= ⇒ =2Mp
( )
8
TrueCollapseLoad, , p
c
M
WLowest of the above =
Dept. of CE, GCE Kannur Dr.RajeshKN
54
55. Problem 4: A portal frame is loaded upto collapse. Find
the plastic moment capacity required if the frame is of
uniform section throughout.
10 kN/m
25 kN
B C
25 kN
Mp
8m
Mp
4 m Mp
DA
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55
56. • Degree of Indeterminacy, N = 4 – 3 = 1
• No. of possible plastic hinges, n = 3
(at B, C and between B&C)
• No. of independent mechanisms ,r = n - N = 2
•Beam Mechanism for BC
•Panel Mechanism
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56
57. Beam Mechanism for BC B
C
10 kN/m
C
θ θ
2θ
4θMp
Mp
0 4θ+⎛ ⎞ 2θ
Mp
0 4
2 10 1604
2
EW
θ
θ
+⎛ ⎞= × × =×⎜ ⎟
⎝ ⎠
( )2 4I p pW M Mθ θ θ θ= + + =
40pM kNm∴ =
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57
59. Combined Mechanism 10kN/m
4θ 8x θ
25kN 4θ 8 x−
Mθ
Mp
x
xθ
θ+ θ1
θ
θ1
Mp
4m
θ
It is required to locate the
plastic hinge between B & C
Assume plastic hinge is
formed at x from B
( ) 18x xθ θ= −( ) 1
( )8θ θ⎛ ⎞⎛ ⎞
( ) ( ) 18
25 4 10 10 8
2 2
E
x x
W x x
θ θ
θ
−⎛ ⎞⎛ ⎞= × + × + × − ×⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
Dept. of CE, GCE Kannur Dr.RajeshKN
59
60. ( ) 2
x
W M Mθ θ θ θ θ θ
⎡ ⎤
= + + + = +⎡ ⎤⎣ ⎦ ⎢ ⎥
( )( )5 5 2 8x x+
( )1 1
8
2I p p
x
W M Mθ θ θ θ θ θ= + + + = +⎡ ⎤⎣ ⎦ ⎢⎣ − ⎥⎦
( )( )5 5 2 8
4
E I p
x x
W W M
+ −
= ⇒ =
For maximum, 0PdM
dx
=
2.75x m⇒ =
( )( )5 5 2 8
68.91
4
p
x x
M kNm
+ −
∴ = =
( )Design plastic moment of resistance, ,largest of the ab 68.o 91ve pM kNm=
4
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60
61. Problem 5: Determine the Collapse load of the continuous beam.Problem 5: Determine the Collapse load of the continuous beam.
P
/ 2 / 2
P
A B C/ 2 / 2A B C
D E
4 2 2SI = − =A collapse can happen in two ways:
1 D t hi d l i t A B d D1. Due to hinges developing at A, B and D
2. Due to hinges developing at B and E
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61
62. Equilibrium:
Hinges at A, B and D
Equilibrium:
pM>
M
pM
pM
uP
pM
P
E
8MP
4
u
4
uP
8
4
pu
p p u
MP
M M P= + ⇒ =
Moment at E is greater than Mp. Hence this mechanism is
not possible.
Dept. of CE, GCE Kannur Dr.RajeshKN
63. Hinges at B and EHinges at B and E
M pM
pM
pM p
4
uP
4
uP
6p pu
M MP
M P= + ⇒ =
4 2
p uM P= + ⇒ =
True Collapse Load,
6 p
u
M
P =
Dept. of CE, GCE Kannur Dr.RajeshKN
63
64. P PVirtual work:
/ 2 / 2
A B C
D E
θ θ
4 2 2SI = − =
θ
2θ
θ θ
2θ
Hinges at A, B and D
( )
8
2
2
p
u p u
M
P M Pθ θ θ θ
⎛ ⎞
= + + ⇒ =⎜ ⎟
⎝ ⎠ Hinges at B and E
( )
6
2
2
p
u p u
M
P M Pθ θ θ
⎛ ⎞
= + ⇒ =⎜ ⎟
⎝ ⎠
Dept. of CE, GCE Kannur Dr.RajeshKN
64
65. Problem 6: For the cantilever, determine the collapse load.
W
A
L/2 L/2
2 Mp Mp
A
B
C
• Degree of Indeterminacy N = 0
p
Degree of Indeterminacy, N 0
• No of possible plastic hinges n = 2 (at A&B)No. of possible plastic hinges, n 2 (at A&B)
• No of independent mechanisms r = n - N = 2• No. of independent mechanisms ,r = n - N = 2
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65
66. /2 L/2
Wu
L/2
θ Mechanism 1
L/2
Mp
Lθ/2
L
W Mθ θ×
2 pM
W∴ =
2
u pW Mθ θ× = uW
L
∴ =
Wu
L
θ
Mechanism 2
2Mp Lθ2Mp
2 pM
2u pW L Mθ θ× =
p
uW
L
∴ =
( )
2
T C ll L d pM
WL t f th b
Dept. of CE, GCE Kannur Dr.RajeshKN
66
( )TrueCollapseLoad, , p
cWLowest of the above
L
=
67. Problem 7: A beam of rectangular section b x d is subjected to a
bending moment of 0.9 Mp. Find out the depth of elastic core.
yσ
Let the elastic core be of depth 2y0
02yExternal bending moment must be
resisted by the internal couple.
yσ
Distance of CG from NA,
y
0 0 0 0 0
1 2
2 2 2 2 3
y
y
d d
b y y y by y
y
σ
σ
⎡ ⎤⎛ ⎞ ⎛ ⎞
− × × + − +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦′
2 2
03 4d y−
0 0
2 2
y
y
y
d
b y by
σ
σ
⎣ ⎦=
⎛ ⎞
− +⎜ ⎟
⎝ ⎠
( )
0
012
y
d y
=
−
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67
68. I t l l ( t f i t )
2 2
03 4
2 yd d y
b y by
σ
σ
⎧ ⎫ −⎛ ⎞
= × + ×⎨ ⎬⎜ ⎟
Internal couple (moment of resistance)
( )0 0
0
2
2 2 12
yb y by
d y
σ= × − + ×⎨ ⎬⎜ ⎟
−⎝ ⎠⎩ ⎭
2 2
3 4d y03 4
12
y
d y
bσ
−
=
2
bd
External bending moment = 0.9 0.9 0.9
4
p p y y
bd
M Z σ σ= × = ×
2 2 2
3 4d bd
Equating the above,
2 2 2
03 4
0.9
12 4
y y
d y bd
bσ σ
−
= ×
0 0.274y d⇒ =
02 0.548y d= =Hence, depth of elastic core
Dept. of CE, GCE Kannur Dr.RajeshKN
68
02 0.548y dHence, depth of elastic core
69. SummarySummary
Plastic Theory
• Introduction-Plastic hinge concept-plastic section modulus-shape
factor-redistribution of moments-collapse mechanism-
• Theorems of plastic analysis - Static/lower bound theorem;
Kinematic/upper bound theorem-Plastic analysis of beams and
portal frames b equilibrium and mechanism methodsportal frames by equilibrium and mechanism methods.
Dept. of CE, GCE Kannur Dr.RajeshKN
69