Short Notes: Plastic Behaviour | Short Notes for Civil Engineering - Civil Engineering (CE) PDF Download

 Page 1


 
 
 
 
Plastic Behaviour of Structural Steel 
 
 
CONTENTS 
 
 
Introduction 
Plastic theory 
Plastic hinge concept 
Plastic collapse load 
Conditions of Plastic analysis 
Theorems of Plastic collapse 
Methods of Plastic analysis 
Plastic analysis of continuous beams 
 
 
 
 
 
 
 
 
 
 
 
Page 2


 
 
 
 
Plastic Behaviour of Structural Steel 
 
 
CONTENTS 
 
 
Introduction 
Plastic theory 
Plastic hinge concept 
Plastic collapse load 
Conditions of Plastic analysis 
Theorems of Plastic collapse 
Methods of Plastic analysis 
Plastic analysis of continuous beams 
 
 
 
 
 
 
 
 
 
 
 
 
Introduction 
      The traditional analysis of structures is based on the linear elastic behaviour of 
materials, implying that the material follows Hooke’s law. (Stress is proportional to 
strain) It is also assumed that the deformations are small, implying that the original 
dimensions of the structure  can  be  used  in  the  analysis.  This  is  also  known  as  first  
order elastic analysis. (Cl. 4.4.2 pp - 24) 
 
      IS  800 - 2007  permits  plastic   analysis   as   per   the Cl. 4.5 (pp 25 and 26). 
However, the requirements specified in Cl. 4.5.2 shall be satisfied unless otherwise 
specified. 
• The yield stress of the grade of structural steel used shall not exceed 450 MPa. 
• The stress - strain characteristics of steel shall comply with IS : 2062 to ensure 
complete  plastic moment redistribution.  
• The stress - strain diagram shall have a plateau at the yield stress level extending for 
at least six times the yield strain. 
• The ratio of ultimate tensile stress to the yield stress for the specified grade of steel 
shall not be less than 1.2 
• The percentage elongation shall not be less than 15 and the steel shall exhibit strain - 
hardening capabilities. (Steel confirming to IS : 2062 shall be deemed to satisfy the above 
requirements) 
• The members shall be hot - rolled or fabricated using hot - rolled plates and sections. 
• The cross section of the members shall be plastic (class 1 section) at plastic hinges 
and elsewhere at least compact sections. (class 2 section) Table 2 shall be followed in this 
regard. 
• The cross section shall be symmetrical about the axis perpendicular to the axis of the 
plastic hinge rotation indicating that the beams shall be symmetrical about y-y axis and 
columns shall be symmetrical about both y-y and z-z axes. 
• The members shall not be subjected to impact and fluctuating loading requiring 
fracture and fatigue assessment.  
 
Stress - strain curves of structural steel  
 A typical stress - strain curve of steel confirming to IS : 2062 is shown in the figure.  
where, 
 
f
y  
= yield stress in MPa 
e
y
 = yield strain  
f
u
 = Ultimate stress in MPa 
e
sh 
= strain hardening strain 
e
max
 = ultimate strain 
e
sh 
= 6 
*
 e
y 
, e
max
 = 180 
* 
e
y 
and 
 
f
u 
= 1.2 f
y 
 (Typical) 
 
 From the stress - strain curve, steel yields considerably at a constant stress due to 
large flow of the material. This property known as ductility enables steel to undergo large 
deformations beyond the elastic limit without danger of fracture. This unique property of 
steel is utilized in plastic analysis of structures. 
Page 3


 
 
 
 
Plastic Behaviour of Structural Steel 
 
 
CONTENTS 
 
 
Introduction 
Plastic theory 
Plastic hinge concept 
Plastic collapse load 
Conditions of Plastic analysis 
Theorems of Plastic collapse 
Methods of Plastic analysis 
Plastic analysis of continuous beams 
 
 
 
 
 
 
 
 
 
 
 
 
Introduction 
      The traditional analysis of structures is based on the linear elastic behaviour of 
materials, implying that the material follows Hooke’s law. (Stress is proportional to 
strain) It is also assumed that the deformations are small, implying that the original 
dimensions of the structure  can  be  used  in  the  analysis.  This  is  also  known  as  first  
order elastic analysis. (Cl. 4.4.2 pp - 24) 
 
      IS  800 - 2007  permits  plastic   analysis   as   per   the Cl. 4.5 (pp 25 and 26). 
However, the requirements specified in Cl. 4.5.2 shall be satisfied unless otherwise 
specified. 
• The yield stress of the grade of structural steel used shall not exceed 450 MPa. 
• The stress - strain characteristics of steel shall comply with IS : 2062 to ensure 
complete  plastic moment redistribution.  
• The stress - strain diagram shall have a plateau at the yield stress level extending for 
at least six times the yield strain. 
• The ratio of ultimate tensile stress to the yield stress for the specified grade of steel 
shall not be less than 1.2 
• The percentage elongation shall not be less than 15 and the steel shall exhibit strain - 
hardening capabilities. (Steel confirming to IS : 2062 shall be deemed to satisfy the above 
requirements) 
• The members shall be hot - rolled or fabricated using hot - rolled plates and sections. 
• The cross section of the members shall be plastic (class 1 section) at plastic hinges 
and elsewhere at least compact sections. (class 2 section) Table 2 shall be followed in this 
regard. 
• The cross section shall be symmetrical about the axis perpendicular to the axis of the 
plastic hinge rotation indicating that the beams shall be symmetrical about y-y axis and 
columns shall be symmetrical about both y-y and z-z axes. 
• The members shall not be subjected to impact and fluctuating loading requiring 
fracture and fatigue assessment.  
 
Stress - strain curves of structural steel  
 A typical stress - strain curve of steel confirming to IS : 2062 is shown in the figure.  
where, 
 
f
y  
= yield stress in MPa 
e
y
 = yield strain  
f
u
 = Ultimate stress in MPa 
e
sh 
= strain hardening strain 
e
max
 = ultimate strain 
e
sh 
= 6 
*
 e
y 
, e
max
 = 180 
* 
e
y 
and 
 
f
u 
= 1.2 f
y 
 (Typical) 
 
 From the stress - strain curve, steel yields considerably at a constant stress due to 
large flow of the material. This property known as ductility enables steel to undergo large 
deformations beyond the elastic limit without danger of fracture. This unique property of 
steel is utilized in plastic analysis of structures. 
 
Stress - Strain Curve (Typical) 
 
 
 
Perfectly Elasto - Plastic Material (Typical) 
 
 
 
Page 4


 
 
 
 
Plastic Behaviour of Structural Steel 
 
 
CONTENTS 
 
 
Introduction 
Plastic theory 
Plastic hinge concept 
Plastic collapse load 
Conditions of Plastic analysis 
Theorems of Plastic collapse 
Methods of Plastic analysis 
Plastic analysis of continuous beams 
 
 
 
 
 
 
 
 
 
 
 
 
Introduction 
      The traditional analysis of structures is based on the linear elastic behaviour of 
materials, implying that the material follows Hooke’s law. (Stress is proportional to 
strain) It is also assumed that the deformations are small, implying that the original 
dimensions of the structure  can  be  used  in  the  analysis.  This  is  also  known  as  first  
order elastic analysis. (Cl. 4.4.2 pp - 24) 
 
      IS  800 - 2007  permits  plastic   analysis   as   per   the Cl. 4.5 (pp 25 and 26). 
However, the requirements specified in Cl. 4.5.2 shall be satisfied unless otherwise 
specified. 
• The yield stress of the grade of structural steel used shall not exceed 450 MPa. 
• The stress - strain characteristics of steel shall comply with IS : 2062 to ensure 
complete  plastic moment redistribution.  
• The stress - strain diagram shall have a plateau at the yield stress level extending for 
at least six times the yield strain. 
• The ratio of ultimate tensile stress to the yield stress for the specified grade of steel 
shall not be less than 1.2 
• The percentage elongation shall not be less than 15 and the steel shall exhibit strain - 
hardening capabilities. (Steel confirming to IS : 2062 shall be deemed to satisfy the above 
requirements) 
• The members shall be hot - rolled or fabricated using hot - rolled plates and sections. 
• The cross section of the members shall be plastic (class 1 section) at plastic hinges 
and elsewhere at least compact sections. (class 2 section) Table 2 shall be followed in this 
regard. 
• The cross section shall be symmetrical about the axis perpendicular to the axis of the 
plastic hinge rotation indicating that the beams shall be symmetrical about y-y axis and 
columns shall be symmetrical about both y-y and z-z axes. 
• The members shall not be subjected to impact and fluctuating loading requiring 
fracture and fatigue assessment.  
 
Stress - strain curves of structural steel  
 A typical stress - strain curve of steel confirming to IS : 2062 is shown in the figure.  
where, 
 
f
y  
= yield stress in MPa 
e
y
 = yield strain  
f
u
 = Ultimate stress in MPa 
e
sh 
= strain hardening strain 
e
max
 = ultimate strain 
e
sh 
= 6 
*
 e
y 
, e
max
 = 180 
* 
e
y 
and 
 
f
u 
= 1.2 f
y 
 (Typical) 
 
 From the stress - strain curve, steel yields considerably at a constant stress due to 
large flow of the material. This property known as ductility enables steel to undergo large 
deformations beyond the elastic limit without danger of fracture. This unique property of 
steel is utilized in plastic analysis of structures. 
 
Stress - Strain Curve (Typical) 
 
 
 
Perfectly Elasto - Plastic Material (Typical) 
 
 
 
 
 
Calculation of failure loads in simple systems 
      Consider a three bar system shown below of length and area of C/S of each bar as 
indicated. E is  the modulus of elasticity of the material. 
 
Elastic analysis (Strength of Materials approach): 
P
1
 is the force in outer bars 
P
2 
 is the force in the middle bar 
Using SV = 0, (Vertical equilibrium equation) 
2P
1 
+ P
2
 = P      (1) 
By compatibility, elongation of each bar is same - 
P
1
L/AE = P
2
L/2AE 
from which 
P
1
 = P
2
/2  or  P
2 
= 2P
1  
      (2) 
Substituting (2) in (1), 
2P
1 
+ 2P
1
 = P  or  P
1
 = P/4 and P
2 
= P/2 
In elastic analysis, as P
2
 > P
1  
the middle bar reaches the yield stress first and the system is 
assumed to fail. 
P
2
 = f
y
A  and P
1
 = f
y
A/2 
Yield load = 
 
2P
1 
+ P
2
 = 2f
y
A  -----  Maximum load by elastic analysis 
 
Plastic analysis: 
      In plastic analysis, it will be assumed that even though the middle bar reaches the 
yield stress, they start yielding until the outer bars also reaches the yield stress. (Ductility 
of steel and redistribution of forces) With this, all the bars would have reached yield 
stress and the failure load (or ultimate load or collapse load) is given by 
Collapse load, P
u
 = 2 f
y 
A + f
y 
A = 3f
y
A ------  Maximum load by plastic analysis   
 
 
 
Page 5


 
 
 
 
Plastic Behaviour of Structural Steel 
 
 
CONTENTS 
 
 
Introduction 
Plastic theory 
Plastic hinge concept 
Plastic collapse load 
Conditions of Plastic analysis 
Theorems of Plastic collapse 
Methods of Plastic analysis 
Plastic analysis of continuous beams 
 
 
 
 
 
 
 
 
 
 
 
 
Introduction 
      The traditional analysis of structures is based on the linear elastic behaviour of 
materials, implying that the material follows Hooke’s law. (Stress is proportional to 
strain) It is also assumed that the deformations are small, implying that the original 
dimensions of the structure  can  be  used  in  the  analysis.  This  is  also  known  as  first  
order elastic analysis. (Cl. 4.4.2 pp - 24) 
 
      IS  800 - 2007  permits  plastic   analysis   as   per   the Cl. 4.5 (pp 25 and 26). 
However, the requirements specified in Cl. 4.5.2 shall be satisfied unless otherwise 
specified. 
• The yield stress of the grade of structural steel used shall not exceed 450 MPa. 
• The stress - strain characteristics of steel shall comply with IS : 2062 to ensure 
complete  plastic moment redistribution.  
• The stress - strain diagram shall have a plateau at the yield stress level extending for 
at least six times the yield strain. 
• The ratio of ultimate tensile stress to the yield stress for the specified grade of steel 
shall not be less than 1.2 
• The percentage elongation shall not be less than 15 and the steel shall exhibit strain - 
hardening capabilities. (Steel confirming to IS : 2062 shall be deemed to satisfy the above 
requirements) 
• The members shall be hot - rolled or fabricated using hot - rolled plates and sections. 
• The cross section of the members shall be plastic (class 1 section) at plastic hinges 
and elsewhere at least compact sections. (class 2 section) Table 2 shall be followed in this 
regard. 
• The cross section shall be symmetrical about the axis perpendicular to the axis of the 
plastic hinge rotation indicating that the beams shall be symmetrical about y-y axis and 
columns shall be symmetrical about both y-y and z-z axes. 
• The members shall not be subjected to impact and fluctuating loading requiring 
fracture and fatigue assessment.  
 
Stress - strain curves of structural steel  
 A typical stress - strain curve of steel confirming to IS : 2062 is shown in the figure.  
where, 
 
f
y  
= yield stress in MPa 
e
y
 = yield strain  
f
u
 = Ultimate stress in MPa 
e
sh 
= strain hardening strain 
e
max
 = ultimate strain 
e
sh 
= 6 
*
 e
y 
, e
max
 = 180 
* 
e
y 
and 
 
f
u 
= 1.2 f
y 
 (Typical) 
 
 From the stress - strain curve, steel yields considerably at a constant stress due to 
large flow of the material. This property known as ductility enables steel to undergo large 
deformations beyond the elastic limit without danger of fracture. This unique property of 
steel is utilized in plastic analysis of structures. 
 
Stress - Strain Curve (Typical) 
 
 
 
Perfectly Elasto - Plastic Material (Typical) 
 
 
 
 
 
Calculation of failure loads in simple systems 
      Consider a three bar system shown below of length and area of C/S of each bar as 
indicated. E is  the modulus of elasticity of the material. 
 
Elastic analysis (Strength of Materials approach): 
P
1
 is the force in outer bars 
P
2 
 is the force in the middle bar 
Using SV = 0, (Vertical equilibrium equation) 
2P
1 
+ P
2
 = P      (1) 
By compatibility, elongation of each bar is same - 
P
1
L/AE = P
2
L/2AE 
from which 
P
1
 = P
2
/2  or  P
2 
= 2P
1  
      (2) 
Substituting (2) in (1), 
2P
1 
+ 2P
1
 = P  or  P
1
 = P/4 and P
2 
= P/2 
In elastic analysis, as P
2
 > P
1  
the middle bar reaches the yield stress first and the system is 
assumed to fail. 
P
2
 = f
y
A  and P
1
 = f
y
A/2 
Yield load = 
 
2P
1 
+ P
2
 = 2f
y
A  -----  Maximum load by elastic analysis 
 
Plastic analysis: 
      In plastic analysis, it will be assumed that even though the middle bar reaches the 
yield stress, they start yielding until the outer bars also reaches the yield stress. (Ductility 
of steel and redistribution of forces) With this, all the bars would have reached yield 
stress and the failure load (or ultimate load or collapse load) is given by 
Collapse load, P
u
 = 2 f
y 
A + f
y 
A = 3f
y
A ------  Maximum load by plastic analysis   
 
 
 
 
 
 
 
      The collapse load calculated by plastic analysis is 1.5 times that of the elastic analysis. 
(Reserve strength) Plastic analysis can give economical solutions.   
 
Plastic Theory of Beams: 
      The  simple  plastic  theory  makes  use  of  the  ductility of   steel.   (Large   strain   at   
collapse)  The   following  assumptions are made in plastic bending of beams -  
 
• Structural steel is a ductile material capable of deforming plastically without 
fracture.  
• The material is homogeneous and  isotropic  obeying Hooke’s law upto limit of 
proportionality (yield point) and then the stress is constant with increase in strain. 
• The stress - strain curve can be represented by an ideal elasto - plastic material 
with properties of steel in compression and tension same. (yield stress and yield 
strain, modulus of elasticity etc.,) 
• Cross - sections remain plane and normal to the longitudinal axis before and after 
bending. With this, the effect of shear force is ignored and the distribution of the 
strain across the depth of the c/s of the beam is linear. 
• The effect of axial forces and residual stresses are ignored. 
• The c/s of the beam is symmetrical about an axis parallel to the plane of bending. 
(y -y axis) 
• Members are initially straight and instability does not develop before collapse 
occurs due to the formation of sufficient plastic hinges.  
• Each layer of the beam is free to expand or contract independently with respect to 
the layer above or below it.(each layer is separated from one another) 
• Deformations are sufficiently small so that  ? = tan ? can be used in the 
calculations of the collapse load. 
• The connections provide full continuity so that plastic moment can develop and 
transmitted through the connections. 
• Strain energy due to elastic bending is ignored. 
 
Behaviour of beam under an increasing BM: 
      Consider a beam having a symmetrical C/S subjected to an increasing BM.  
• With BM M
1 
< M
y
 ( yield moment) the stress and strain distributions across the 
depth will follow the elastic bending equation (Euler’s - Bernoulli’s equation) and 
is indicated in the figure below. All the fibres are stressed below the yield stress, 
f
y
.