PLASTIC ANALYSIS AND DESIGN (FUNDAMENTALS) General Requirement of Plastic Design: The following are the assumptions are made in plastic design to simplify computations: 1) The material obeys Hooke, Law till the stress reaches fy. 2) The yield stress and modulus of elasticity have the same value in compression and tension. 3) The material is homogeneous and isotropic in both the elastic and plastic states. 4) The material is assumed to be sufficiently ductile to permit large rotation of the section to take place. 5) Plastic hinge rotation is large compare with the elastic deformations so that all the rotations are concentrated at the plastic hinges. The segments between the plastic hinges are rigid. 6) The magnitude of bending moment caused by the external loads will at the most be equal to the plastic moment reached the capacity of the section. 7) The influence of normal and shear forces on plastic moments is not considered. 8) Plane sections remain plane even after bending and the effect of shear is neglected. 9) The equilibrium of forces at the time of collapse is considered for the undeformed state of the structure. 10) No instability occurs in any member of the structure upto collapse. IS:800 stipulates that the following conditions should be satisfied in order to use the plastic method of analysis: 1) The yield stress of steel used should not be greater than 450MPa. 2) The stress-strain characteristic of the steel used should obey the following conditions, in order to ensure plastic moment redistribution. a) The yield plateau (horizontal portion of the stress-strain curve) should be greater than 6 times the yield strain. b) The ratio of the ultimate tensile stress to the yield stress should be more than 1.2. c) The elongation on the standard gauge length should be more than 15%. d) The steel should exhibit strain-hardening capacity. 3) The members shall be hot-rolled or fabricated using hot-rolled plates.
4) The cross section of the members not containing plastic hinges should be ‘compact’ and those of member containing plastic hinges should be ‘plastic’. 5) The cross-section should be symmetrical about its axis perpendicular to the axis of the plastic hinge rotation. These limitations are intended to ensure that there is a sufficiently long plastic plateau to enabling a hinge to form and that the steel will not experience premature strain hardening. Advantages of Plastic Design Plastic design methods offer the following advantages: 1) Realization of uniform and realistic F.O.S for all parts of the structures(in contrast to elastic methods, where the safely factor varies) 2) Simplified analytical procedure and readily of obtaining design moments, since there is no need to satisfy elastic strain compatibility conditions. 3) Saving of material over elastic methods resulting in lighter structures. 4) No effect due to temperature changes, settlement of supports, imperfection, erection method, etc. (because their only effect is to change the amount of rotation required). This is in contrast to the elastic method, where extra calculation are required. However, calculation for instability and elastic deflection required careful considerations in plastic method. The plastic design method is very popular for design of some structure, e.g, beams and portal frames. 5) Gives some idea of collapse mode and strength of structure. 6) In the elastic method of design, the design process is repeated several times to obtain an optimum solution, where the plastic method of design produces a balanced section in a single attempt. Dis-advantages of plastic design: The disadvantages of plastic design method are the following: 1) 2) 3) 4) 5) 6)
Obtaining collapse load is difficult if the structure is reasonably complicated. There is little saving in column design. Difficult to design for fatigue. Lateral bracing requirements are more than stringent than elastic design. Calculations for elastic deformations require careful considerations. When more than one loading condition occurs, it is necessary to perform separate calculations, one for each loading condition; the section requiring the largest plastic moment is selected. Unlike the elastic method of design, wherein the moment produced by different loading condition can be added together, the plastic moment obtained by different loading conditions cannot be combined(i.e,
the plastic moment calculated for a given set of loads is valid only for that loading condition). This is because the ‘principle of super position’ becomes invalid when certain parts of the structure have yielded. Plastic Analysis of Steel Structures: An elastic analysis is useful to study the performance of the structure, especially with regards to serviceability under working load. However, in steel structure when the load is increased, some of the sections in the structure may develop yield stress. Any further increase in load causes the structure to undergo elasto-plastic deformations and some of the section may develop a fully plastic condition at which a sufficient number of plastic hinges are formed transforming the structure into a mechanism. The mechanism may collapse 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 design so that its collapse load is equal to or higher than the working multiplied by the load factor specified. Further, a structure may reach its limit of usefulness through instability, fatigue or excessive deflection. Alternatively, if none of these failure modes occurs, then the structure will continue to carry load beyond the elastic limit until it reaches its ultimate load through plastic deformation, and then collapse. Plastic analysis is based on this mode of failure. The concept of ductility of structural steel forms the basis for the plastic theory of bending. The rigorous analysis of a structure according to the theory of elasticity demands that the stress satisfy two sets of conditions: (1) the equilibrium conditions and (2) compatibility conditions. The first set of conditions must be invariably satisfied in any material. However, the second condition ceases to be valid as soon as plastic yielding occurs. The elastic method of design assumes that a frame will become useless as soon as yield stress is reached. The working stress is, therefore, kept much below the yield stress. The design so produced gives a structure of unknown ultimate strength. The elastic methods of analysis are also very cumbersome, specially for redundant frames. In plastic method of design, the limit load of a system is a statically determinate quantity. The plastic method of design gives an economical design. The margin of safety provided in this method is not less than that provided in elastic method of design. Furthermore, steel has unique physical property, ductility, because of which it is able to absorb large deformations beyond the elastic limit without fracture. Due to this property, steel posses a reserve of strength beyond its yield, as it evident from the stress-strain curve of mild steel. Plastic design is an aspect of the limit design that extends the structural usefulness upto the plastic strength or ultimate load carrying capacity. The other terms referred to the plastic method of design are Limit Design, Collapse Method of Design and Ultimate Design. In the plastic method, the design criterion is the ultimate strength and hence the behaviour of members beyond the
yield stress in the inelastic or plastic range is considered. The working loads are multiplied with specified factors known as load factors to obtain ultimate loads under which a structure collapses. The plastic method of structural analysis, therefore, find their principal application in the design of redundant rigid framed mild steel structure, when deformation is relatively small compare to significant bending stress. Furthermore, using the elastic design method (also termed as the working stress or allowable stress method) to satisfy strength & stiffness requirements, the stresses and deformations are computed at working loads using the elastic properties of steel. This method of design limits the structural usefulness of steel upto an allowable stress level, which is well below the elastic limit. Hence the structures designed using this method may be heavier than those designed by the plastic method, which uses the ultimate load rather than the yield stress as the design criterion. The plastic design method, which utilizes the strength of steel beyond the yield stress, is also called the ultimate load design or load factor design method. The plastic method is often used in the analysis of statically indeterminate structures and provides a rapid and rational approached for the analysis of such structures. Ductility: Ductility may be described as the ability of a material to change its shape without fracture. In other words, the ductility of a structure or its members is the capacity to undergo large inelastic deformation without significant loss of strength or stiffness. Further, the plastic theory is based on the ductility of steel. Through ductility, structural steel has capacity of absorbing large deformation beyond elastic limit without the danger of failure. Plastic Design: Plastic design is a special case of limit state design, where the limit state is attained when the members reach plastic moment strength Mp and the structure is transformed into a mechanism. Plastic moment strength is the moment strength when all the fibres of the cross section of a member are at yield stress Fy (one side of the neutral axis is in tension and the other side in compression. The safety measure in the design is introduced by an appropriate choice of the load factor, defined as the ratio of the ultimate load (design load) to the working load. The ultimate strength design makes it possible for different types of loads to be assigned different load factors under combined loading conditions, thereby overcoming the related shortcoming of Working Stress Method. It is to be noted that the satisfactory ‘strength’ performance at ultimate loads does not guarantee satisfactory ‘serviceability’ performance at normal service loads. Hence IS codes suggest that the deflections under working loads should be within
the prescribed limit of the codes. Plastic design does not permit using other limit states such as instability, fatigue, or brittle fracture. Plastic Hinge: A plastic hinge is a zone of yielding due to flexure in a structural member. Although hinges do not actually form, it cal be seen that large changes of slope occurs over small length of the member at position of maximum moments. A strain hardening action usually occurs at these hinges so that large deflections are accompanied by a slight increase in load. A structure can support the computed ultimate load due to the formation of plastic hinges at certain critical sections. The member remain elastic until the moment reaches a value Mp, the maximum moment of resistance of a fully yielded cross section or fully plastic moment of a section(Mp = fy Zp). Any additional moment will cause the beam to rotate with little increase in stress. The rotation occurs at a constant moment (Mp). The zone acts as if it was hinges except with a constant restraining moment (Mp). The plastic hinge, therefore, can be defined as a yielded zone due to flexure in a structure in which infinite rotation can take place at a constant restraining moment (Mp) of the section. It is represented normally by a black dot. The value of the moment at the adjacent sections of the yield zone for a certain length is more than the yield moment. This length is known as hinge length, depends upon the loading and geometry of the section. To simplify the analysis, this small length is neglected and the plastic hinge is assumed to be formed at discrete points of zero length. But, it cannot be neglected for the calculation of deflections and the design of bracings as the length over which yielding extends is quite important. The plastic hinges are formed first at the sections subjected to the greatest deformation( curvature). The possible places for plastic hinges in a structure with prismatic members are points of concentrated loads, at the ends of member meeting at a connection involving a change in geometry and at the point of zero shear in a span under distributed load. Hinge Length: Consider a simply supported rectangular beam subjected to a gradually increasing concentrated load P, at the centre. A plastic hinge will be formed at the centre. Mp = PL/4; My = fy x Ze = fy x bd^2/6 = fy (1/6) x {4x(1/4)} bd^2 = (2/3) x fy xbd^2/4 = (2/3) x fy x Zp = (2/3)Mp, i.e, Mp is 1.5 times more than My. From the BM diagram, Mp / (L/2) = My / (L/2-x/2) => x=L/3. Therefore, the hinge length of the plasticity zone is equal to 1/3rd of the span.
Similarly, the hinge length of the plasticity zone for a simple beam subjected to uniformly distributed load is L/sqrt(3). For a uniformly loaded fixed end beam, Bowels(1980) showed that the hinge length at the ends (which is half the mid-span hinge length) is given by xends = (L/2.83) x sqrt[1-(1/v)], Where v is the shape factor is equal to 1.12 for a Isection. And xcentre = 2L / 8.645, i.e, 23% of the span length. Plastic modulus of a section: May be defined as the combined statical moment of the cross-sectional area above and below the equal-area axis. It is also referred to as the resisting modulus of the completely plasticized section. For a rectangular section: Zpz = (A/2) x (y1-bar + y2-bar) = (bd/2) x (d/4+d/4) = bd^2/4 Zpy = (A/2) x (y1-bar + y2-bar) = (bd/2) x (b/4+b/4) = db^2/4 For a I-section with equal flanges: Zpz = bf tf (h-tf) + tw(h/2-tf)^2 Zpy = {(bf)^2tf }/2 + (tw)^2/4(h-tf) The above Zpz and Zpy for I-section have been developed assuming the flanges and webs of rectangular plates, i.e, by ignoring the fillet. Note: The symbol has usual meaning Shape Factor (v): The ratio Mp / My is a property of a cross sectional shape and is independent of the material properties. This ration is known as the shape factor v and is given by v = Mp / My = = fy Zp / fy Ze.= Zp / Ze For wide-flange I-section in flexure about the strong axis, the shape factor ranges from 1.09 to about 1.18 with the average value being 1.14. One may conservatively take the plastic moment strength Mp of I-section bent about their strong axis to be at least 15% greater than the strength My. On the other hand, the shape factor for I-section bent about their minor axis is about the same as for a rectangular section, i.e, about 1.5. Shape factor of different cross sections: Cross section
Shape factor Max.
Circular
16/3pi()=1.7
Min
Average
-
-
Rectangular
1.5
-
-
Thin walled circular tube
1.47
1.3
1.35
Thin walled rectangular hollow section
1.33
1.19
1.25
Thin walled circular tube
1.31
115
1.20
Triangular
1.34
-
-
Diamond (equal diagonals)
12.0
-
-
Wide flange I-section(major axis)
1.18
1.09
1.14
Wide flange I-section(minor axis)
1.67
-
-
Equal angles
1.84
1.81
1.82
Unequal angles
1.83
1.75
1.82
Channels (major axis)
1.22
1.16
1.18
Channels (minor axis)
1.80
-
-
For the theoretically ideal section in bending, i.e, two flange plate connected by a web of insignificant thickness, the value of v will be 1. A value of shape factor nearly equal to one shows that the section is efficient in resisting bending. NOTE: When the material at the centre of the section is increased, the value of v increases.
Load Factor : Load factor is defined as the ratio of the collapse load to the working load(service load) and is represented by F, i.e, F = Pc / Pw If a collection of beams having different end conditions (free or fixed) and the working load W were first design elastically and then plastically, the ratio Pc / Pw will not be identical. Only the beams that are simply supported will produce a constant ratio of Pc / Pw and for these cases the values of Pc / Pw will be the lowest. From a practical point of view, a minimum acceptable and constant load factor is required, and that found for a simply supported beam may be regarded satisfactory. For a simple beam the variation of the bending moment with the load is linear.
F = Pc / Pw = Mp / Mw = fyZp / fbcZe = (fy / fbc) / (Zp/Ze ) = (F.O.S) v. Therefore, the load factor may also be defined as the product of the factor of safety and load factor. Let u take the example of an I-section beam whose shape factor v=1.2 and F.O.S = (fy / fbc) = (fy / 0.66fy) = 1.515. Hence, Load factor, F = 1.515x1.12 = 1.7 In actual practice a load factor varying from 1.7 to 2.0 is assumed depending upon the designer’s judgment. When the structures are subjected to wind the corresponding load factor for plastic design is reduced by 25%. The prime function of the load factor is to ensure that the structure will be safe under the collapse load. Therefore, it may be regarded as a factor of safety based upon the collapse load. It depends upon the nature of loading, the support conditions, and the geometrical shape of the structural members. Uncertainty of the loads, imperfection in workmanship and error in fabrication are some of the other factors which influence the choice of the load factor.
Mechanism: When a structure is subjected to a system of loads, it is stable and hence functional until a sufficient number of plastic hinges have been formed to render the structure unstable. As soon as the structure reaches an unstable condition, it is considered to have been failed. The segments of the beams between the plastic hinges are able to move without an increase of load. This condition in a member is called mechanism. The concept of mechanism formation in a structure due to loading beyond the elastic limit and of virtual work is used in the plastic analysis and design of steel structures. If an indeterminate structure has the redundancy r, the insertion of r plastic hinges makes it statically determinate. Any further hinge converts this statically determinate structure into mechanism. Hence, for collapse, the numbers of plastic hinges required are (r+1). Types of Mechanism: Various possible mechanism are listed below: a) b) c) d) e)
Beam mechanism Panel / sway mechanism Joint mechanism Gable mechanism Composite (combined) mechanism
Number of Independent Mechanism: Let,
N = number of possible plastic hinges r = number of redundancies n = possible independent mechanism. Then, n = N – r After finding out the number of independent mechanism all the possible combinations are made in such a way so as to make the external works maximum or the internal work a minimum. This is done to obtain the lowest load.
Plastic Collapse The plastic collapse of a structure depends upon its redundancies. When a sufficient number of plastic hinges are formed to convert a structure into a mechanism, the structure collapses. As such a stage the deflection increases very fast at constant load. The collapse of a structure can be partial, complete or over-complete. These terms can be explained on the basis of redundancy r, and the number of plastic hinges developed N. N < (r+1) – is called as partial collapse, part of the structure may fail. N = (r+1) – is called as complete collapse, such a mechanism has only one degree of freedom N > (r+1) – is called as over-complete collapse, there are two or more mechanism for which the corresponding value of the load is the same, this value being the actual collapse load. Conditions of plastic analysis The conditions to be satisfied for the plastic methods of analysis are as follows: 1) Equilibrium condition: All the equilibrium conditions, i.e, summation of all the forces and moment should be equal to zero. 2) Mechanism Condition: There must be sufficient number of plastic and frictionless hinges for the beam / structure to form a mechanism. The ultimate or collapse load is reached when a mechanism is formed. This is also called continuity condition. 3) Plasticity or Yield conditions: The bending moment in any section of a structure must be less than the plastic moment of the section- Mp <= M <= Mp. This is also called plastic moment condition.
If all the three conditions are satisfied, a unique value, the lowest plastic limit load, is obtained. However, if the equilibrium condition and any of the above two, i.e mechanism or yield conditions is satisfied an approximate value above or below the true plastic limit load is obtained.
Principle of virtual work The principle of virtual work may be stated as follows: if a system of forces in equilibrium is subjected to a virtual displacement, then the work done by the external forces equals the work done by the internal forces, i.e., WE = WI . This is accompanied by allowing rotations of the structure only at points of simple support and at the points where plastic moments are expected to occur in producing the mechanism.
Theorem of plastic collapse The plastic analysis of a structure is govern by three theorem, which are as follows: 1) The static or lower bound theorem: states that a load (P
Pc). Hence the kinematic method represents an upper limit to the true ultimate load and has a smaller factor of safety. The kinematic theorem satisfies the equilibrium and continuity conditions. 3) Uniqueness theorem: The lower and upper bound theorems can be combined to produce the uniqueness theorem, which states that the load that satisfies both the theorems at the same time is the correct collapse load. When both the theorems are satisfied in a given problem then the solution is said to be the correct (unique) one.
Methods of analysis Using the principle of virtual work and the upper and lower bound theorems, a structure can be analysed for its ultimate load by any of the following methods: 1) Static method: This consists of selecting the redundant forces, The free and redundant bending moment diagram is drawn for the structure. A combined bending moment diagram is drawn in such a way that a mechanism is formed. The collapse load is found by working out the equilibrium equation. It is checked that the bending moment is not more than the fully plastic moment at any section. 2) Kinematic method: This consists of locating the possible places of plastic hinges. The possible independent and combined mechanism are ascertained. The collapse load is found by applying the principle of virtual work. A bending moment diagram corresponding to the collapse mechanism is drawn and it is checked that the bending moment is not more than the fully plastic moment at any section. For complicated frames, the static method of analysis is more difficult, and finding the correct equilibrium equation becomes illusive. In these cases, the kinematic method is more practical. Plastic Design of Portal Frame: Plastic design is used extensively for the design of single-storey portal frame structures. The actual application of plastic design involves the two steps, namely, adequate strength and avoidance of secondary failure. In addition, the effect of axial load on vertical members should be considered. The design of portals is a little complicated because there can be more than one mechanism of failure, and the mechanism giving the least collapse load has to be chosen. Single-bay portal frames with fixed bases have three redundancies and require four hinges to produce a mechanism (I=R-r = 6-3 =3). Further more as the number of redundancies increases, so does the number of possible modes of collapse. The possible mechanism are classified into two groups: elementary and combined mechanisms. When the mechanism method is applied to structure with sloping members, the determination of displacements in the direction of applied forces is required. This difficulty may be resolve by the application of centre-of-rotation technique( also called as method of instantaneous centres). When the gravity load case governs the design, a good estimate of the required section (assuming uniform frame) may be obtained by using the following formulae(Horne & Morris 1981):
For pinned-base frames:
Mp = γL(wL^2/8) [1 + k + (1+k)^0.5]
For fixed-base frames:
Mp = γL(wL^2/8) [1 / {1 + k + (1+k)^0.5}]
Where, γL= 1.7 is the global load factor; k= h2/h1,h2=sloping ht. & h1=straight ht. The design of portal frame using haunches, either at the eaves or at the apex, which improve the strength and stiffness of the frame, provide the complete design of haunched portal frame. Haunces shift the eaves plastic hinges to the column top immediately below the lower end of the haunches for a fixed-based frame. Using haunches, about 5 to 10 percent can be saved on rafter size, though it involves additional fabrication cost. Deflection: even If we use the plastic design, the member is actually in the elastic range of behaviour. Hence deformations and deflections under load would still be in the elastic range and the normal limitations of deflection must be applied for structure designed using plastic theory. Serviceability Limit States: Under the combinations of actions to be considered for serviceability limit states, Eurocode 3[5] recommends that the deflections of portal frames do not exceed the following limiting values: a) Horizontal deflections at the top of columns - for portal frames without gantry frames: h/150 - for other single storey building: h/300 b) Total vertical deflection in beams: L/200 Usually, for portal frames with a pitched roof, the deflection criteria for beams is not a critical one. Furthermore, in common with elastic structural analysis, plastic theorey assumes that deflections have little effect on equilibrium equations. Thus additional bending moments in a column, for example, due to axial load combined with frame sway, are usually ignored. Plastic designs, no less than elastic design, must be checked to ensure that the designed assumptions are obeyed. Plastic Mehod Analysis: Plastic method of analysis, as it allows the engineer to analyse the frame easily and design it economically. By taking advantages of the ductility of steel, plastic design produces lighter and more slender structural members than similar rigid frames designed by elastic theory.
While attempting a plastic design, in addition to member strength (capacity) check, the following checks must also be carried out: a) b) c) d) e) f)
Reduction in the plastic moment due to the effect of axial force and shear force. Instability due to local buckling, lateral buckling, and column buckling. Frame stability Brittle fracture Deflection at service loads In addition, the connections should be designed properly such that they are capable of developing and maintaining the required moment until the frame fails by forming a mechanism.
DESIGN OF STEEL STRUCTURES Reference books: 1. Engineering Mechanica - Timoshenko 2. Strength of Material – S Ramamrutham 3. Basic Structural Analysis – C S Reddy 4. Structural Analysis – G S Pandit & S P Gupta 5. Strength of Material and theory of Structures – B.C Punmia, A K Jain & A K Jain 6. Design of Steel Structures – N Subramanian 7. Limit State Design of Steel Structures – S K Duggal 8. Steel Designers’ Manual – ELBS 9. Design of Steel Structures – A S Arya & J L Ajmani 10. Handbook of Structural Engineering – W. F Chen 11. Structural Engineering Handbook – Gaylord & Gaylord – Mc Graw Hill 12. Plastic analysis and Design of Steel Structures – M. Bill Wong 13. Ductile Design of Steel Structures – Michel Bruneau, Chia Ming Uang & S.E Sabelli 14. Structural Steel Designers Hand Books – Roger Brockenbrogh & Frederick Merritt. 15. Plastic Method of Structural Analysis – Neal B G 16. Theory of Elasticity and Plasticity – Westergard H M 17. Steel Design Handbook: LRFD Method Vol.1 – Akbar R Tamboli 18. Stability Analysis and Design of Structures – Murari L. Gambhir 19. Design of Steel Bins for Storage of Bulk Solids – Gaylord and Gaylord 20. Silos – Brown & Nielsen 21. Design and Construction of Silos & Bunkers – Sargis S Safarian, Ernest C Harris. 22. Plastic design of single-storey pitched- roof portal frames to Eurocode 3 - by C. M King 23. Advanced analysis of steel frames by Wai-Fah Chen, S. Toma 24. Stability design of steel frames by Wai-Fah Chen, E. M. Lui 25. Structural Engineer’s Pocket Book - Fiona Cobb 26. The steel skeleton Vol, 1: Elastic behavior and design – by J. F Baker 27. The steel skeleton Vol, 2: Plastic behavior and design – by J. F Baker 28. The steel skeleton Vol, 2: Plastic behavior and design – by J. F Baker 29. Insdag Teaching Material – IIT Madras 30. Statically Indeterminate Structures: Their Analysis And Design - Paul Andersen
