Plastic Analysis

PLASTIC ANALYSIS OF STEEL STRUCTURES

Student - BS GrewalGuide - Prof PN Rao

PHASESPhase I - Introduction Phase II - Brief discussionPhase III - Structure under considerationPhase IV - Analysis of structurePhase V - Conclusion

PHASE IINTRODUCTION

INTRODUCTIONSteel structures have peculiar behaviour compared to concrete structures in the sense that on the steel structures if the load is increased, some of the sections in the structure develop yield stress. Any further increase in the stress causes the structure to undergo elasto plastic deformations and some of the sections may develop a fully plastic condition at which a sufficient number of plastic hinges are formed, transforming the structure into a mechanism. This mechanism collapses without noticeable additional loading. A study of the mechanism of failure and knowledge of the load causing the mechanism is necessary to determine the load factor. A structure is designed so that its collapse load is equal to or higher than the working load multiplied by the load factor specified.

PHASE IIBRIEF DISCUSSION

• The material is homogeneous and isotropic.• Member cross-section is symmetrical about the axis at

right angles to the axis of bending.• Cross-section which were plane before bending

remain plane after bending.• The value of modulus of Elasticity of the material

remains the same in tension as well as in compression.

• Effects of temperature, fatigue, shear and axial force are neglected.

• Idealized bi-linear stress-strain curve applies.

ASSUMPTIONS IN PLASTIC BENDING

PLASTIC DESIGN CHARACTERISTICS• Results in a consistent margin of safety for all structures, independent

of the degrees of static indeterminacy.• Results in a considerable cost savings for bending type indeterminate

structures.• Determinate beams yields the same design as by structural design.• Tension and compression members yields same design.• Is not suitable for situations where fatigue stresses are a problem.• Must have a ductile material to employ plastic design.• Plastic hinge is assumed to be confined at a point along the beam

length.

• Relatively simpler procedures are involved.

• Ultimate loads for structures and their components may be determined.

• Sequence and final mode of failure may be known and the capacity at relevant stages may be determined.

ADVANTAGES OF PLASTIC ANALYSIS

• Mechanism condition: The ultimate or collapse load is reached when a mechanism is formed. The number of plastic hinges developed should be just sufficient to form a mechanism.

• Equilibrium condition : ∑Fx = 0, ∑Fy = 0, ∑Mxy = 0

• Plastic moment condition: The bending moment at any section of the structure should not be more than the fully plastic moment of the section.

CONDITIONS FOR PLASTIC ANALYSIS

STRESS STRAIN CURVE

ε

σ

σy

εy σy Yield stress

εy Yield strain

AB - Elastic rangeBC – Yield pointsCD – Stain hardening

A

B

C

D

E

DE - Failure

Actual curve

Idealised curve

MOMENT CURVATURE RELATIONSHIP

Ø

M

M MomentØ Curvature

BC – Yield points

Max attainable moment

Idealised inelastic behaviour

Bilinear idealisationM P

M Y

1.5M P

0.4M P

ElasticInelastic Perfectly

plasticStrain

hardened

STRESS DISTRIBUTION

M Moment corresponding to working loadMy Moment at which section develops yield stress

MP Moment at which entire section is under yield stress

Beam Section

Stress under M

Stress under

My

Stress under

My<M<Mp

Stress under

Mp

• In structural engineering beam theory the term, plastic hinge, is used to describe the deformation of a section of a beam where plastic bending occurs.

• It is that cross-section of a member where bending stresses are equal to yield stresses.

PLASTIC HINGE

• The number of Plastic Hinges required to convert a structure or a member into a mechanism is one more than the degree of indeterminacy in terms of redundant moments usually.

• Thus a determinate structure requires only one more plastic hinge to become a mechanism, a stage where it deflects and rotates continuously at constant load and acquires final collapse.

Therefore N = n+ 1N = Total number of plastic hinges required to convert a structure into a mechanism.n = degree of indeterminacy

Number of Plastic Hinges

• The ratio of the load causing collapse to the working load is called load factor.

• The load factor is dependent upon the shape of the section as the working load is dependent on I and Z values and the collapse load is dependent upon the shape of the section.

Load factor = Mp / M

LOAD FACTOR

This is defined as :

Factor of Safety = First yield load/ Working Load

The FoS is an elastic analysis measure of the safety of a design.

FACTOR OF SAFETY

TYPES OF PLASTIC COLLAPSEComplete Collapse • In the cases considered so far, collapse

occurred when a hinge occurred for each of the number of redundants, r, (making it a determinate structure) with an extra hinge for collapse.

• Thus the number of hinges formed, h =r + 1 (the degree of indeterminacy plus one).

TYPES OF PLASTIC COLLAPSEPartial Collapse • This occurs when h < r+1 , but a collapse

mechanism, of a localised section of the structure can form. A common example is a single span of a continuous beam.

TYPES OF PLASTIC COLLAPSEOver-Complete Collapse • For some frames, two (or more) possible

collapse mechanisms are found (h = r+1 ) with the actual collapse load factor.

• Therefore they can be combined to form another collapse mechanism with the same collapse load factor, but with an increased number of hinges,

h > r+1 .

COLLAPSE MECHANISM• When a system of loads is applied to an elastic

body, it will deform and will show a resistance against deformation. Such a body is known as a structure.

• On the other hand if no resistance is set up against deformation in the body, then it is known as a mechanism.

• Various types of independent mechanisms are identified to enable prediction of possible failure modes of a structure.

COLLAPSE MECHANISMStatically determinate beam with point load

Collapse Mechanism PC = 4 MP /L

PC represents the theoretical max load that the beam can support

L

P

PC

Real hinge Real hinge

Plastic hinge

COLLAPSE MECHANISMStatically determinate beam with UDL



L

W/m


Plastic hinge

W/m

COLLAPSE MECHANISMStatically indeterminate beam with point load



L

P

PC

Plastic hinge

Plastic hinge

Plastic hinge

Statically indeterminate beam with UDL

Collapse Mechanism PC = 16 MP /L2


COLLAPSE MECHANISM

L

W/m

Plastic hinge

Plastic hinge

W/mPlastic hinge


Collapse Mechanism

PC = 6 MP /L


Plastic hingePC

Real hinge

Plastic hinge

L

P

COLLAPSE MECHANISMCantilever with point load

Collapse Mechanism

PC = MP /L


PC

Plastic hinge

L

P

PLASTIC ANALYSIS OF STRUCTURE• Methods used to perform plastic analysis for

a given structure.– Statical method of analysis.– Mechanism method of analysis.

STATICAL METHOD OF ANALYSIS• This method is based on the lower bound theorem.• According to the theorem, a load computed on the

basis of the assumed equilibrium moment diagram in which the moments are not greater than Mp is less than or equal to the true ultimate load.

• The objective of this method as to find an equilibrium moment diagram in which IMI < Mp and a failure mechanism is formed.

STATICAL METHOD OF ANALYSIS• This is called the unsafe theorem because for an

arbitrarily assumed mechanism the load factor is either exactly right (when the yield criterion is met) or is wrong and is too large, leading a designer to think that the frame can carry more load than is actually possible.

STATICAL METHOD OF ANALYSISThe following procedure is adopted in this method:• Release redundants which can be either moments or

forces and make the structure a determinate one.• Obtain moment diagram for the determinate

structure.• Draw the moment diagram for the structure due to

redundant moments or forces.• Sketch the combined moment diagram so that a

mechanism is formed.

STATICAL METHOD OF ANALYSIS• Sketch the combined moment diagram so that a

mechanism is formed.• Compute the magnitude of redundants by solving

equilibrium equations.• Check whether sufficient number of hinges are

formed for the mechanism of failure.

MECHANISM METHOD OF ANALYSIS• This method is based on the upper bound theorem.• The theorem states that a load computed on the

basis of an assumed mechanism will always be greater than or equal to the true failure load.

• The correct mechanism is the one which results in the lowest possible load for which the moment IMI does not exceed the plastic moment Mp at any section.

MECHANISM METHOD OF ANALYSIS• This is a safe theorem because the load factor will be

less than (or at best equal to) the collapse load factor once equilibrium and yield criteria are met leading the designer to think that the structure can carry less than or equal to its actual capacity.

MECHANISM METHOD OF ANALYSIS• The objective is to find a mechanism in which the

plastic moment condition is not violated.• This method is adopted for a structure with large

number of redundants. The possible number of failure mechanisms increase and construction of correct equilibrium moment diagrams becomes difficult.

MECHANISM METHOD OF ANALYSISThe following procedure is adopted in this method:• Determine the required number of plastic hinges

necessary for the mechanism. The number of hinges is n+1 where n is the degree of indeterminacy.

• Select possible mechanisms : elementary or independent mechanisms and combinations thereof.

• For each possible mechanism calculate the collapse load. • Lowest collapse load is the correct ultimate load.• Check to see that nowhere IMI > Mp.

PLASTIC ANALYSIS FOR MULTIPLE LOADING • When more than one condition of loading can be applied to a

beam or structure, it may not always be obvious which is critical.

• It is necessary then to perform separate calculations, one for each loading condition, the section being determined by the solution requiring the largest plastic moment.

• Unlike the elastic method of design in which moments produced by different loading systems can be added together, plastic moments obtained by different loading systems cannot be combined, i.e. the plastic moment calculated for a given set of loads is only valid for that loading condition. This is because the 'Principle of Superposition' becomes invalid when parts of the structure have yielded.

PHASE IIISTRUCTUREs UNDER

CONSIDERATION

STRUCTURE UNDER CONSIDERATION

Mp

2Mp

Mp

A

B

L

L

3P

D

CP

STRUCTURE UNDER CONSIDERATION4P 5P

3P

Mp

2Mp 3Mp

2Mp 2Mp

A

B C D

E F

3L2L

2L

Two bay portal frame.

PHASE IVANALYSIS

METHOD• Draw the structure and visualize different alternative

mechanisms by putting plastic hinges at various crucial points of the structure.

• Calculate the value of critical load for each mechanism by equating virtual internal and external work done during the displacement of the mechanism.

• The mechanism giving the lowest value of the critical load is the critical mechanism.

• Calculate value of moments at the joints where the plastic hinge has not been formed using statics.

STRUCTURE UNDER CONSIDERATION

Mp

2Mp

Mp

A

B

L

L

3P

D

CP

MECHANISMS FORMED1. Beam mechanism in section BC.2. Sway mechanism in complete structure.3. Beam and sway mechanism combined.

BEAM MECHANISM

Mp

2Mp

Mp

A

B

L

L

3P

D

C

BEAM MECHANISM

Mp

2Mp

Mp

A

B

L

L

3P

θ

2θ

θ

D

C

External virtual work done = Wext = 3P(L/2)Ѳ

Internal virtual work done = Wint = Mp Ѳ + Mp Ѳ + 2Mp(2Ѳ)

Wext = Wint

=> (3/2)PLѲ = 6Mp Ѳ

=> PC1 = 4Mp /L

BEAM MECHANISM - I

SWAY MECHANISM

P

Mp

2Mp

Mp

A

B

L

L

D

C

SWAY MECHANISM

P

Mp

2Mp

Mp

A

B

L

Lθ

θ

θ

θ

D

C

External virtual work done = Wext = PLѲ

Internal virtual work done = Wint = Mp(Ѳ+Ѳ)+Mp(Ѳ+Ѳ)

Wext = Wint

=> PLѲ = 4Mp Ѳ

=> PC2 = 4Mp /L

SWAY MECHANISM

SWAY AND BEAM MECHANISM

P

Mp

2Mp

Mp

A

B

L

3P

D

C

L


L

2θ

θ θ

2θ

P

Mp

2Mp

Mp

A

3P

D

C

L

External virtual work done = Wext = 3P(L/2)Ѳ + PLѲ

Internal virtual work done = Wint = Mp(Ѳ)+Mp(Ѳ+2Ѳ)+2Mp(2Ѳ)

Wext = Wint

=> (5/2)P.LѲ = 8Mp Ѳ

=> PC3 = 3.2Mp /L


CRITICAL LOAD VALUES1. PC1 = 4 Mp /L

2. PC2 = 4 Mp /L

3. PC3 = 3.2 Mp /L

The combined sway and beam mechanism gives the least value for critical load and hence PC3 is considered the critical load for this structure in plastic analysis.

CHECK To make sure that some other mechanism was not

overlooked, it is necessary to check the plastic moment condition anywhere on the frame to see that

IMI < Mp

CHECK Consider column CD ∑MC = 0

=> MP + MP - HD L = 0

=> HD = 2MP /L Mp

D

C

L

HD

Mp

Mp

CHECK HA + HD = P

=> HA = 1.2MP /L

Consider column AB ∑MB = 0

=> MB + MP - HA L = 0

=> MB = 0.2MP

Since the moment MB < MP , the correct collapse load has been obtained.

Mp

A

B

L

HA

Mp

Mp

P

BENDING MOMENT DIAGRAM

A

B

L

D

C

L

0.2 Mp

Mp Mp

Mp

Mp

0.2 Mp

II ANALYSIS

STRUCTURE UNDER CONSIDERATION4P 5P

3P

Mp

2Mp 3Mp

2Mp 2Mp

A

B C D

E F

3L2L

2L

Two bay portal frame.

MECHANISMS FORMED1. Beam mechanism in section BC.2. Beam mechanism in section CD. 3. Sway mechanism in complete structure.4. Combined mechanisms.5. Beam and sway mechanism combined.

BEAM MECHANISM - I

Mp

2Mp 3Mp

2Mp 2Mp

A

B C D

E F

3L2L

2L

4P

BEAM MECHANISM - I4P

Mp

3Mp

2Mp 2Mp

A

B C D

E F

3L2L

2L

θ θ

2θ

2Mp

BEAM MECHANISM - IExternal virtual work done =

Wext = 4P.LѲ

Internal virtual work done = Wint = Mp Ѳ + 2Mp(2Ѳ+Ѳ)

Wext = Wint

=> 4PLѲ = 7Mp Ѳ

=> PC1 = 1.75Mp /L

BEAM MECHANISM - II5P

Mp

2Mp 3Mp

2Mp 2Mp

A

C D

E F

3L2L

2L

BEAM MECHANISM - II5P

Mp

2Mp 3Mp

2Mp 2Mp

A

B C D

E F

3L2L

2L

θ θ

2θ

BEAM MECHANISM - IIExternal virtual work done =

Wext = 5P.(3/2)LѲ

Internal virtual work done = Wint = 3Mp (Ѳ+2Ѳ) + 2Mp Ѳ

Wext = Wint

=> (15/2)PLѲ = 11Mp Ѳ

=> PC2 = 1.47Mp /L

SWAY MECHANISM

3P

Mp

2Mp 3Mp

2Mp 2Mp

A

B C D

E F

3L2L

2L

SWAY MECHANISM

3P

Mp

2Mp 3Mp

2Mp 2Mp

A

B C D

E F

3L2L

2Lθ

θ

θ

θ

θ

θ

SWAY MECHANISMExternal virtual work done =

Wext = 3P.2LѲ

Internal virtual work done = Wint = Mp(Ѳ+Ѳ)+2Mp(Ѳ+Ѳ)

+2Mp(Ѳ+Ѳ)

Wext = Wint

=> 6PLѲ = 10Mp Ѳ

=> PC3 = 1.67Mp /L

COMBINED MECHANISM -I4P

3P

Mp

2Mp 3Mp

2Mp 2Mp

A

B C D

E F

3L2L

2L

COMBINED MECHANISM -I4P

3P

Mp

2Mp

3Mp

2Mp 2Mp

A

B C D

E F

3L2L

2L

2θ

θ θ

θ

θ

θ 2θ

COMBINED MECHANISM -IExternal virtual work done =

Wext = 3P.2LѲ + 4P.LѲ

Internal virtual work done = Wint = Mp(Ѳ+ Ѳ)+2Mp(Ѳ+2Ѳ)+

2Mp(Ѳ+Ѳ)+2Mp(2Ѳ)

Wext = Wint

=> 10P.LѲ = 16Mp Ѳ

=> PC4 = 1.60Mp /L

COMBINED MECHANISM -II5P

3P

Mp

2Mp 3Mp

2Mp 2Mp

A

B C D

E F

3L2L

2L

COMBINED MECHANISM -II5P

3P

Mp

2Mp 3Mp

2Mp 2Mp

A

B C D

E F

3L2L

2L

2θ

θ

θ

θ

2θ

θ

θ

COMBINED MECHANISM -IIExternal virtual work done =

Wext = 3P.2LѲ + 5P.(3/2)LѲ

Internal virtual work done = Wint = Mp(Ѳ+ Ѳ)+2Mp(Ѳ+ Ѳ)+

2Mp(Ѳ+2Ѳ)+3Mp(2Ѳ)

Wext = Wint

=> 13.5P.LѲ = 18Mp Ѳ

=> PC5 = 1.33Mp /L

4P 5P

3P

Mp

2Mp 3Mp

2Mp 2Mp

A

B C D

E F

3L2L

2L


SWAY AND BEAM MECHANISM4P

3P

Mp

2Mp

2Mp

A

B C

E

2L

2θ

3Mp

2Mp

D

2θ

3L

2L

5P

θ θ θ

2θ2θ

F

SWAY AND BEAM MECHANISMExternal virtual work done =

Wext = 4P.LѲ + 5P.(3/2)LѲ + 3P.2LѲ

Internal virtual work done = Wint = Mp(Ѳ)+2Mp(Ѳ+2Ѳ)+2Mp(Ѳ+2Ѳ)+

2Mp(2Ѳ)+3Mp(2Ѳ)

Wext = Wint

=> (35/2)P.LѲ = 23Mp Ѳ

=> PC6 = 1.31Mp /L

CRITICAL LOAD VALUES1. PC1 = 1.75Mp /L

2. PC2 = 1.47Mp /L

3. PC3 = 1.67Mp /L

4. PC4 = 1.60Mp /L

5. PC5 = 1.33Mp /L

6. PC6 = 1.31Mp /L

The combined sway and beam mechanism gives the least value for critical load and hence PC6 is considered the critical load for this structure in plastic analysis.

MOMENTMoment at A = Mp

Moment at B = UnknownMoment at C = 2Mp

Moment at D =2Mp

Moment at E = 2Mp

Moment at F = 2Mp

CONCLUSION• Plastic analysis makes use of the assumption that the

elastic deformation is so small that it can be ignored. Therefore, in using this method of analysis, the material behaves as if the structure does not deform until it collapses plastically. This behavior is depicted in the stress–strain diagram shown earlier. Although classical rigid plastic analysis has many restrictions in its use, its simplicity still has certain merits for the plastic design of simple beams and frames. However, its use is applicable mainly for manual calculations as it requires substantial personal judgment to, for instance, locate the plastic hinges in the structure. This report has described the classical theorems of plasticity and the application of these theorems to plastic analysis demonstrated by the use of manual calculations of simple structures.

CONCLUSION• Emphasis is placed on the use of the mechanism method in which

rigid plastic behavior for steel material is assumed. In plastic analysis of a structure, the ultimate load of the structure as a whole is regarded as the design criterion but the calculation of the critical load has shown that the minimum value of the critical load from all the calculated ones from different mechanisms is to be considered and these calculated values are not to be superimposed.

THANKYOU

COLLAPSE MECHANISMStatically determinate beam with point load



L

P

PC


Plastic hinge

COLLAPSE MECHANISMStatically determinate beam with UDL



L

W/m


Plastic hinge

W/m




L

P

PC

Plastic hinge

Plastic hinge

Plastic hinge

Statically indeterminate beam with UDL



COLLAPSE MECHANISM

L

W/m

Plastic hinge

Plastic hinge

W/mPlastic hinge


Collapse Mechanism

PC = 6 MP /L


Plastic hingePC

Real hinge

Plastic hinge

L

P

COLLAPSE MECHANISMStatically indeterminate beam with UDL

Collapse Mechanism

PC = 8 MP /L2


Plastic hinge Real hinge

Plastic hinge

W/m

L

W/m

COLLAPSE MECHANISMCantilever with point load

Collapse Mechanism

PC = MP /L


PC

Plastic hinge

L

P

Plastic Analysis

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