PLASTIC ANALYSIS 1.0 INTRODUCTION

Transcription

1 35 PLASTIC ANALYSIS 1.0 INTRODUCTION The elastic design method, also termed as allowable stress method (or Working stress method), is a conventional method of design based on the elastic roerties of steel. This method of design limits the structural usefulness of the material uto a certain allowable stress, which is well below the elastic limit. The stresses due to working loads do not exceed the secified allowable stresses, which are obtained b aling an adequate factor of safet to the ield stress of steel. The elastic design does not take into account the strength of the material beond the elastic stress. Therefore the structure designed according to this method will be heavier than that designed b lastic methods, but in man cases, elastic design will also require less stabilit bracing. In the method of lastic design of a structure, the ultimate load rather than the ield stress is regarded as the design criterion. The term lastic has occurred due to the fact that the ultimate load is found from the strength of steel in the lastic range. This method is also known as method of load factor design or ultimate load design. The strength of steel beond the ield stress is full utilised in this method. This method is raid and rovides a rational aroach for the analsis of the structure. This method also rovides striking econom as regards the weight of steel since the sections designed b this method are smaller in size than those designed b the method of elastic design. Plastic design method has its main alication in the analsis and design of staticall indeterminate framed structures..0 BASIS OF PLASTIC THEORY.1 Ductilit of Steel Structural steel is characterised b its caacit to withstand considerable deformation beond first ield, without fracture. During the rocess of 'ielding' the steel deforms under a constant and uniform stress known as 'ield stress'. This roert of steel, known as ductilit, is utilised in lastic design methods. Fig. 1 shows the idealised stress-strain relationshi for structural mild steel when it is subjected to direct tension. Elastic straining of the material is reresented b line OA. AB reresents ielding of the material when the stress remains constant, and is equal to the ield stress, f. The strain occurring in the material during ielding remains after the load has been removed and is called the lastic strain and this strain is at least ten times as large as the elastic strain, ε at ield oint. Coright reserved

2 When subjected to comression, the stress-strain characteristics of various grades of structural steel are largel similar to Fig. 1 and disla the same roert of ield. The major difference is in the strain hardening range where there is no dro in stress after a eak value. This characteristic is known as ductilit of steel. Strain hardening f Stress f A B Strain hardening commences O. Theoretical Basis As an incremental load is alied to a beam, the cross-section with greatest bending moment will eventuall reach the ield moment. Elsewhere the structure is elastic and the 'eak' moment values are less than ield. As load is incremented, a zone of ielding develos at the first critical section, but due to ductilit of steel, the moment at that section remains about constant. The structure, therefore, calls uon its less heavil stressed ortions to carr the increase in load. Eventuall the zones of ielding are formed at other sections until the moment caacit has been exhausted at all necessar critical sections. After reaching the maximum load value, the structure would siml deform at constant load. Thus it is a design based uon the ultimate load-carring caacit (maximum strength) of the structure. This ultimate load is comuted from a knowledge of the strength of steel in the lastic range and hence the name 'lastic'..3 Perfectl Plastic aterials Strain ε Fig. 1 Idealised stress strain curve for steel in tension The stress-strain curve for a erfectl lastic material uto strain hardening is shown in Fig.. Perfectl lastic materials follow Hook's law uto the limit of roortionalit. The sloes of stress-strain diagrams in comression and tension i.e. the values of Young's modulus of elasticit of the material, are equal. Also the values of ield stresses in tension and comression are equal. The strains uto the strain hardening in tension and comression are also equal. The stress strain curves show horizontal lateau both in tension and comression. Such materials are known as erfectl lastic materials.

3 STRESS f TENSION STRAIN COPRESSIONf Fig. Stress - Strain Curve for erfectl lastic materials.4 Full Plastic oment of a Section The full lastic moment, of a section is defined as the maximum moment of resistance of a full lasticized or ielded cross-section. The assumtions used for finding the lastic moment of a section are: (i) (ii) (iii) (iv) (v) (vi) (vii) The material obes Hooke's law until the stress reaches the uer ield value; on further straining, the stress dros to the lower ield value and thereafter remains constant. The ield stresses and the modulus of elasticit have the same value in comression as in tension. The material is homogeneous and isotroic in both the elastic and lastic states. The lane transverse sections (the sections erendicular to the longitudinal axis of the beam) remain lane and normal to the longitudinal axis after bending, the effect of shear being neglected. There is no resultant axial force on the beam. The cross section of the beam is smmetrical about an axis through its centroid arallel to lane of bending. Ever laer of the material is free to exand and contract longitudinall and laterall under the stress as if searated from the other laers.

4 In order to find out the full lastic moment of a ielded section of a beam as shown in Fig. 3, we emlo the force equilibrium equation, namel the total force in comression and the total force in tension over that section are equal. f G 1 A 1 C f.a 1 1 G A T f. A f Fig. 3 Total comression, C Total tension, T Plastic oment of resistance, f. A 1 f. A A 1 A A A 1 + A A 1 A A/ where Z, the lastic modulus of the section ( + ) The lastic modulus of a comletel ielded section is defined as the combined statical moment of the cross-sectional areas above and below the neutral axis or equal area axis. It is the resisting modulus of a comletel lasticised section. 3.0 BENDING OF BEAS SYETRICAL ABOUT BOTH AXES The bending of a smmetrical beam subjected to a graduall increasing moment is considered first. The fibres of the beam across the cross section are stressed in tension or comression according to their osition relative to the neutral axis and are strained in accordance with Fig. 1. f A A A f f Z ( + ) 1 f A (1)

5 f Neutral Axis f f Fig. 4 Elastic stresses in beams While the beam remains entirel elastic the stress in ever fibre is roortional to its strain and to its distance from the neutral axis. The stress (f) in the extreme fibres cannot exceed f. (see Fig. 4) When the beam is subjected to a moment slightl greater than that, which first roduces ield in the extreme fibres, it does not fail. Instead the outer fibres ield at constant stress (f ) while the fibres nearer to the neutral axis sustain increased elastic stresses. Fig. 5 shows the stress distribution for beams subjected to such moments. Such beams are said to be 'artiall lastic' and those ortions of their cross-sections, which have reached the ield stress, are described as 'lastic zones'. f Plastic Zone (Comression) f Neutral Axis Plastic Zone (Tension) (a) Rectangular section (b) I - section f (c) Stress distribution for (a) or (b) Fig. 5 Stresses in artiall lastic beams

6 The deths of the lastic zones deend uon the magnitude of the alied moment. As the moment is increased, the lastic zones increase in deth, and, it is assumed that lastic ielding can occur at ield stress (f ) resulting in two stress blocks, one zone ielding in tension and one in comression. Fig. 6 reresents the stress distribution in beams stressed to this stage. The lastic zones occu the whole of the cross section, and are described as being 'full lastic'. When the cross section of a member is full lastic under a bending moment, an attemt to increase this moment will cause the member to act as if hinged at the neutral axis. This is referred to as a lastic hinge. The bending moment roducing a lastic hinge is called the full lastic moment and is denoted b ' '. Note that a lastic hinge carries a constant moment, P. b f d Neutral Axis (a) Rectangular section (b) I - section f (c) Stress distribution for (a) or (b) Fig. 6 Stresses in full lastic beams 4.0 GENERAL REQUIREENTS FOR UTILISING PLASTIC DESIGN CONCEPTS Generall codes (such as IS 800, BS 5950) allow the use of lastic design onl where loading is redominantl static and fatigue is not a design criterion. For examle, in order to allow this high level of strain, BS 5950 rescribes the following restrictions on the roerties of the stress-strain curve for steels used in lasticall designed structures (clause 5.3.3). 1. The ield lateau (horizontal ortion of the curve) is greater than 6 times the ield strain.. The ultimate tensile strength must be more than 1. times the ield strength. 3. The elongation on a standard gauge length is not less than 15%. These limitations are intended to ensure that there is a sufficientl long lastic lateau to enable a hinge to form and that the steel will not exerience a remature strain hardening.

7 4.1 Shae Factor As described reviousl there will be two stress blocks, one in tension, the other in comression, both of which will be at ield stress. For equilibrium of the cross section, the areas in comression and tension must be equal. For a rectangular cross section, the elastic moment is given b, The lastic moment is obtained from, bd 6 f (.a) d d. b. f 4 bd 4 f (.b) Here the lastic moment is about 1.5 times greater than the elastic moment caacit. In develoing this moment, there is a large straining in the extreme fibres together with large rotations and deflection. This behaviour ma be lotted as a moment-rotation curve. Curves for various cross sections are shown in Fig S a (S 1.00) b (S 1.15) 0.67 (S 1.50) (S 1.80) bd 6 f o Rotation Fig.7 oment rotation curves The ratio of the lastic modulus (Z ) to the elastic modulus (Z) is known as the shae factor (S) and will govern the oint in the moment-rotation curve when non-linearit starts. For the theoreticall ideal section in bending i.e. two flange lates connected b a web of insignificant thickness, this will have a value of 1. When the material at the centre of the section is increased, the value of S increases. For a universal beam the value is about 1.15 increasing to 1.5 for a rectangle.

8 5.0 PLASTIC HINGES In deciding the manner in which a beam ma fail it is desirable to understand the concet of how lastic hinges form where the beam is full lastic. At the lastic hinge an infinitel large rotation can occur under a constant moment equal to the lastic moment of the section. Plastic hinge is defined as a ielded zone due to bending in a structural member at which an infinite rotation can take lace at a constant lastic moment of the section. The number of hinges necessar for failure does not var for a articular structure subject to a given loading condition, although a art of a structure ma fail indeendentl b the formation of a smaller number of hinges. The member or structure behaves in the manner of a hinged mechanism and in doing so adjacent hinges rotate in oosite directions. Theoreticall, the lastic hinges are assumed to form at oints at which lastic rotations occur. Thus the length of a lastic hinge is considered as zero. The values of moment, at the adjacent section of the ield zone are more than the ield moment uto a certain length L, of the structural member. This length L, is known as the hinged length. The hinged length deends uon the te of loading and the geometr of the cross-section of the structural member. The region of hinged length is known as region of ield or lasticit. 5.1 Hinged Length of a Siml Suorted Beam with Central Concentrated Load W h L/ L/ x b Y Y Fig. 8 In a siml suorted beam with central concentrated load, the maximum bending moment occurs at the centre of the beam. As the load is increased graduall, this moment reaches the full lastic moment of the section and a lastic hinge is formed at the centre.

9 Let x ( L) be the length of lasticit zone. From the bending moment diagram shown in Fig. 8 (3) Therefore the hinged length of the lasticit zone is equal to one-third of the san in this case. 6.0 FUNDAENTAL CONDITIONS FOR PLASTIC ANALYSIS (i) echanism condition: The ultimate or collase load is reached when a mechanism is formed. The number of lastic hinges develoed should be just sufficient to form a mechanism. (ii) Equilibrium condition : F x 0, F 0, x bh f 6 bh f 4 bh Z 4 bh f 4 W l 4 Q ( ) ( ) L 3 1 x 3 L x L L x L L x L x L L x L 1

10 (iii) Plastic moment condition: The bending moment at an section of the structure should not be more than the full lastic moment of the section. 6.1 echanism When a sstem of loads is alied to an elastic bod, it will deform and will show a resistance against deformation. Such a bod is known as a structure. On the other hand if no resistance is set u against deformation in the bod, then it is known as a mechanism. Various tes of indeendent mechanisms are Beam echanism Fig. 9 sketches three simle structures and the corresonding mechanisms. (a) A siml suorted beam has to form one lastic hinge at the oint of maximum bending moment. Redundanc, r 0 (b) A roed cantilever requires two hinges to form a mechanism. Redundanc, r 1 No. of lastic hinges formed, r + 1 (c) A fixed beam requires three hinges to form a mechanism. Redundanc, r No. of lastic hinges Fig. 9 From the above examles, it is seen that the number of hinges needed to form a mechanism equals the statical redundanc of the structure lus one.

11 6.1. Panel or Swa echanism Fig. 10 (A) shows a anel or swa mechanism for a ortal frame fixed at both ends. (A) Panel echanism (B) Gable echanism (C) Joint echanism Gable echanism Fig. 10 Fig. 10 (B) shows the gable mechanism for a gable structure fixed at both the suorts Joint echanism Fig. 10 (C) shows a joint mechanism. It occurs at a joint where more than two structural members meet Combined echanism Various combinations of indeendent mechanisms can be made deending uon whether the frame is made of strong beam and weak column combination or strong column and weak beam combination. The one shown in Fig.11 is a combination of a beam and swa mechanism. Failure is triggered b formation of hinges at the bases of the columns and the weak beam develoing two hinges. This is illustrated b the right hinge being shown on the beam, in a osition slightl awa from the joint. Two hinges develoed on the beam Fig. 11 Combined echanism 6. LOAD FACTOR AND THEORES OF PLASTIC COLLAPSE Plastic analsis of structures is governed b three theorems, which are detailed in this section.

12 The load factor at rigid lastic collase (λ ) is defined as the lowest multile of the design loads which will cause the whole structure, or an art of it to become a mechanism. In a limit state aroach, the designer is seeking to ensure that at the aroriate factored loads the structure will not fail. Thus the rigid lastic load factor λ must not be less than unit. The number of indeendent mechanisms (n) is related to the number of ossible lastic hinge locations (h) and the number of degree of redundanc (r) of the frame b the equation. n h r (4) The three theorems of lastic collase are given below for reference Lower Bound or Static Theorem A load factor (λ s ) comuted on the basis of an arbitraril assumed bending moment diagram which is in equilibrium with the alied loads and where the full lastic moment of resistance is nowhere exceeded will alwas be less than or at best equal to the load factor at rigid lastic collase, (λ ). λ is the highest value of λ s which can be found. 6.. Uer Bound or Kinematic Theorem A load factor (λ k ) comuted on the basis of an arbitraril assumed mechanism will alwas be greater than, or at best equal to the load factor at rigid lastic collase (λ ) λ is the lowest value of λ k which can be found Uniqueness Theorem If both the above criteria are satisfied, then the resulting load factor corresonds to its value at rigid lastic collase (λ ). 7.0 RIGID PLASTIC ANALYSIS As the lastic deformations at collase are considerabl larger than elastic ones, it is assumed that the frame remains rigid between suorts and hinge ositions i.e. all lastic rotation occurs at the lastic hinges. Considering a siml suorted beam subjected to a oint load at midsan, the maximum strain will take lace at the centre of the san where a lastic hinge will be formed at ield of full section. The remainder of the beam will remain straight, thus the entire energ will be absorbed b the rotation of the lastic hinge. (See Fig. 1)

13 W Plastic zone Yield zone Stiff length L B.. D. oment rotation curve Fig. 1 Siml suorted beam at lastic hinge stage Considering a centrall loaded siml suorted beam at the instant of lastic collase (see Fig. 1) Workdone at the lastic hinge (5a) Workdone b the dislacement of the load W L. (5b) At collase, these two must be equal. WL 4 W L. The moment at collase of an encastre beam with a uniform load is similarl worked out from Fig. 13. It should be noted that three hinges are required to be formed at A, B and C just before collase. (6) W / unit length P0 P0 A C B L B Loading P P Collase P P Fig. 13 Encastre Beam

14 Workdone at the three lastic hinges ( + + ) 4 Workdone b the dislacement of the load W/L. L/. L/. (7.a) (7.b) WL 4 WL 4 16 WL 16 In other words the load causing lastic collase of a section of known value of is given b eqn. (8). All the three hinges at A, B and C will have a lastic moment of as given in eqn. (9). 7.1 Continuous Beams Consider next the three san continuous beam of uniform section throughout (constant ) as shown in Fig. 14(a). Here a conventional aroach is more laborious but the collase load ma be readil determined b consideration of the collase atterns. Each attern reresents the conversion of each of the three sans into mechanism. (8) (9) 10W 10W 6W L L 3L L L 3L Collase attern 1: Fig. 14 (a) 10W 10W 6W Fig. 14(b) Work done in hinges ( + ) 3

15 Work done b loads 10 W( L ) 0WL Collase load, W c 3 / 0L 0.15 / L Collase attern : 10W 10W 6W 3 5 Fig. 14 (c) Work done in lastic hinges ( ) 10 Work done b loads 10W( 6L ) 60 WL Collase load, W c 10 / 60 L 0.17 / L Collase attern 3: 10W 10W 6W 3 Fig. 14(d) 5 Work done in hinges ( ) 8 Work done b loads 6W (6L ) 36 WL Collase load, W c 8 / 36L 0. / L Thus collase will occur in the mode of Fig. 14 (b) when W c 0.15 / L. 7. echanism ethod In the mechanism or kinematic method of lastic analsis, various lastic failure mechanisms are evaluated. The lastic collase loads corresonding to various failure mechanisms are obtained b equating the internal work at the lastic hinges to the

16 external work b loads during the virtual dislacement. This requires evaluation of dislacements and lastic hinge rotations. For gabled frames and other such frames, the kinematics of collase is somewhat comlex. It is convenient to use the instantaneous centres of rotation of the rigid elements of the frame to evaluate dislacements corresonding to different mechanisms. In this, roerties of rotations of a rigid bod during an infinitesimall small angle are assumed as follows (see Fig.15): (i) (ii) An oint P will move b distance r to oint P' normal to the radius vector OP for length r, due to the rotation of the rigid bod b an angle about O. The work done b a force F due to the rotation of the rigid bod about O b an angle is given b F*x*, where x is the shortest (erendicular distance) between vector F and the centre of rotation O. F P P r o x x Fig.15 Rigid bod rotation Consider, for examle the structure shown in Fig 16. Let us consider the lastic mechanism b formation of lastic hinges A, C, F and G. Let the virtual rotation of the member FG be about lastic hinge G. Point F moves, normal to line FG to F' due to rotation about G. This would cause a art of the structure ABC to rotate about oint A and oint C would move to C'. Since oint F moves to F' the instantaneous centre of rotation of segment CDF would be along the line FG. Similarl since oint C would move normal to line AC, the instantaneous centre of rotation of element CDF should also be along the line AC. Thus we can locate the instantaneous centre which will be the oint of intersection of line AC and GF, obtained b extending them to meet at I.

17 I /4 /4 4L H B H 3 P 3 D P 5 C E C H 5 F F L/ A A A L C" L L L G L Let us find the rotation of element CDF about instantaneous centre of rotation I, Let < FGF' From FGF' we get FF' L* To find the location of I, consider similar triangles ACC" and IAG, {EBED Equation.3} {EBED Equation.3} Similarl from IFF' (x + L/) I FF' L* (7L/+L/) I L Fig. 16 I è 4 Similarl from ICC' and ACC'

18 {EBED Equation.3} The dislacements of loads in the direction of alication of loads are as follows: Dislacement of horizontal loads {E For H ; BED Equa {EBED For H 3 ; 3 Equation.3} For H 5 ; 5 {EBED Equation.3} Dislacement of vertical loads {E For P 3 ; ' BED Equa {E For P 5 ; ' 5 B ED The rotations at the lastic hinges are as follows: {E A BE {EBED C Equation.3} F G {EBED Equation.3} With these information the virtual work equation can be written. 7.3 Rectangular Portal Framework and Interaction Diagrams The same rincile is alicable to frames as indicated in Fig. 17(a) where a ortal frame with constant lastic moment of resistance throughout is subjected to two indeendent loads H and V.

19 H a This frame ma distort in more than one mode. There are basic indeendent modes for the ortal frame, the ure swa of Fig. 17 (b) and a beam collase as indicated in Fig. 17 (c). There is now however the ossibilit of the modes combining as shown in Fig. 17(d). a V a H V H V H V (a) (b) (c) (d) Fig.17 Possible Failure echanisms From Fig. 17(b) Va / Work done in hinges 4 Work done b loads Ha At inciient collase Ha / 4 (10) 6 From Fig. 17 (c) 4 A B Work done in hinges 4 Work done b loads Va At inciient collase Va / 4 (11) C From Fig. 17(d) Work done in hinges Work done b loads Ha + Va Fig. 18 At inciient collase Ha / + Va / 6 (1) D Ha H a / / The resulting equations, which reresent the collase criteria, are lotted on the interaction diagram of Fig. 18. Since an line radiating from the origin reresents roortional loading, the first mechanism line intersected reresents failure. The failure condition is therefore the line ABCD and an load condition within the area OABCD is therefore safe. 7.4 Frames not of Constant Section Throughout Let us suose however that the beam had an enhanced value of full lastic moment of. The ossible modes of collase are unaltered but wherever a hinge forms at the beam/stanchion joint, it will occur in the weaker member - in this case it will be at the stanchion. For clarit it is customar to draw the hinge location just awa from the joint

20 as indicated in Fig. 19, but in the ensuing geometric comutations it is assumed that its location is at the joint. The revious calculation is then modified as follows: H a a V a H V H V H V (a) From Fig. 19 (b) (b) Work done in hinges 4 Work done b loads Ha At inciient collase Ha / 4 (13) Fig.19 Va / 8 6 A (c) B (d) From Fig. 19 (c) Work done in hinges 6 Work done b loads Va At inciient collase Va / 6 (14) From Fig.19 (d) 4 C Work done in hinges 8 Work done b loads Ha + Va 0 At inciient collase Ha / + Va / 8 (15) D 4 6 Fig. 0 The interaction diagram then becomes as shown in Fig. 0. H8 a / Ha / 8.0 STABILITY For lasticall designed frames three stabilit criteria have to be considered for ensuring the safet of the frame. These are 1. General Frame Stabilit.

21 . Local Buckling Criterion. 3. Restraints. 8.1General Frame Stabilit Under loading, all structures move. In some cases this movement is small comared to the frame dimensions and the designer does not need to consider these an further. In other cases, the movement of the structure will be sufficient to cause the factor of safet to dro b a significant amount (for more details readers ma wish to refer to BS:5950 Part 1, clauses 5.1.3, and ). In these cases the designer will need to take this dro in the load carring caacit into account in checking the structure. 8. Local Buckling Criterion At the location of a lastic hinge, there is a considerable strain, and at ultimate load this can reach several times the ield strain. Under these conditions it is essential that the section does not buckle locall, or the moment caacit will dro considerabl. In order to ensure that the sections remain stable, limiting values are rovided for flange outstands and web deth ratios. In no circumstances should sections not comling with the lastic section classification limits given in the code be used in locations where there are lastic hinges; otherwise there is a real risk of a remature reduction in the moment caacit of the member at the hinge location. The limits for the sizing of flanges and webs are discussed in another chater on Local Buckling and Section Classification. 8.3 Restraints In order to ensure that the lastic hinge osition does not become a source of remature failure during the rotation, torsional restraint should be rovided at the lastic hinge locations. These are discussed in the next chater, which covers the design requirements in detail. 9.0 EFFECT OF AXIAL LOAD AND SHEAR If a member is subjected to the combined action of bending moment and axial force, the lastic moment caacit will be reduced. The resence of an axial load imlies that the sum of the tension and comression forces in the section is not zero (Fig. 1). This means that the neutral axis moves awa from the equal area axis roviding an additional area in tension or comression deending on the te of axial load. Consider a rectangular member of width b and deth d subjected to an axial comressive force P together with a moment in the vertical lane (Fig. 1).

22 The values of and P are increased at a constant value of /P until the full lastic stage is attained, then the values of and P become: a 0.5 f b (d 4 ) (16) P b f (17) where f ield stress distance from the neutral axis to the stress change for without axial force, f bd /4 (18) If axial force acts alone -P f bd (19) at the full lastic state. From equations (16) to (19) the interaction equation can be obtained: x / 1 P / P (0) The resence of shear forces will also reduce the moment caacit. b f f d d/ C C C T f T f f Total stresses Bending + Axial comression Fig. 1 Effect of axial force on lastic moment caacit 10.0 PLASTIC ANALYSIS FOR ORE THAN ONE CONDITION OF LOADING When more than one condition of loading can be alied to a beam or structure, it ma not alwas be obvious which is critical. It is necessar then to erform searate calculations, one for each loading condition, the section being determined b the solution requiring the largest lastic moment. Unlike the elastic method of design in which moments roduced b different loading sstems can be added together, lastic moments obtained b different loading sstems cannot be combined, i.e. the lastic moment calculated for a given set of loads is onl

23 valid for that loading condition. This is because the 'Princile of Suerosition' becomes invalid when arts of the structure have ielded CONCLUDING REARKS Basic concets on Plastic Analsis have been discussed in this chater and the methods of comutation of ultimate load causing lastic collase have been outlined. Theorems of lastic collase and alternative atterns of hinge formation triggering lastic collase have been discussed. Worked examles illustrating lastic methods of analsis have been rovided. 1.0 REFERENCES 1. Clarke, A. B. and Coverman, S. H. Structural Steelwork, Limit state design, Chaman and Hall Ltd, London, Horne,. R. Plastic Theor of Structures, Pergamon Press Ltd, Oxford, Introduction to Steelwork Design to BS 5950: Part 1, The Steel Construction Institute, Owens G.W., Knowles P.R : "Steel Designers anual", The Steel Construction Institute, Ascot, England, 1994 Structural Steel Design Project Job No. Sheet 1 of Rev. Job title: PLASTIC ANALYSIS Worked Examle. 1 ade b RSP Date Aril 000

24 CALCULATION SHEET Checked b RN Date Aril kn/m (factored) I x /3 φ B x C D 3 m 6 m A E m Fig. Portal with fixed base and smmetrical vertical loading Determine the required for a smmetric single ba itched ortal frame with a factored UDL of 5 kn /m b instantaneous centre method. In the case of gable frames, comutation of the geometrical relationshi of the dislacement in the direction of the load as the structure moves through the mechanism ma become somewhat tedious. In such cases the method of instantaneous centres ma be used. This method is discussed below along with its use for solving ractical roblems. In the following roblem, when the structure moves under loading, the oint B will move in a direction erendicular to line AB. Then its centre of rotation should be along line AB extended. The oint C will move verticall downwards and its centre of rotation should lie in a horizontal line. The oint I satisfies both the conditions. Thus I is the centre of rotation of the member BC. The rotation at both the column bases is taken as. Assume that the hinges will form in the rafter at a distance of x from B and ver close to roof aex. Structural Steel Design Project Job No. Sheet of Rev. Job title: PLASTIC ANALYSIS Worked Examle. 1 ade b RSP Date Aril 000

25 CALCULATION SHEET Checked b RN Date Aril 000 Data (as shown in Fig. 0). Frame centres San of ortal Eves height Eves to ridge height Purlin sacing 5.0 m 18.0 m 6.0 m 3.0 m 1.5 m Solution: (x / 3) φ 6 φ 18 / x ( + 18 / x / x ) 18 / x [ 5 x / + ( 9 - x ) 5x] For maximum value of, d ( 18 + x ) ( - 5x + 45 ) - ( -.5 x + 45 x ) x - 90 x (.5 x + 45 x) ( 18 + x) x 7.5 m Substituting in eqn. for, 69.5 knm dx 0 9 Structural Steel Design Project Job No. Sheet 1 of 6 Rev. Job title: PLASTIC ANALYSIS Worked Examle. ade b RSP Date Aril 000

26 CALCULATION SHEET Checked b RN Date Aril 000 Find for the ortal frame with electricall oerated travelling crane as shown in Fig. 1 b Reactant moment diagram method. The roof itch is 30. Neglect the effect of wind acting verticall on the roof. Horizontal wind ressure is 1 kn/m γ f 1. for the combined effects of wind, crane, dead load and live load. LL DL Purlin locations 4@ Wind C 3 1 B eccentricit e D 4 5 Crane loads ( 3T. E. O. T) E F A in G in Frames at 6 m c/c Fig. 3 Summar of tical loads and dimensions for Examle PRELIINARY CALCULATIONS (1) Forces due to dead load and live load on roof Suerimosed load 0.6 kn/m Dead load 0.5 kn/m Total load ( ) kn/m Structural Steel Design Project Job No. Sheet of 6 Rev. Job title: PLASTIC ANALYSIS Worked Examle. ade b RSP Date Aril 000

27 CALCULATION SHEET Checked b RN Date Aril 000 () Crane loading 3 ton caacit crane, 9.3 m san. Horizontal crane loading This ma be shared between each side of the ortal, based on the assumtion that the crane wheels are flanged, and in effect share the load between the two rails. Check that the crane wheels are flanged when the vendor is selected, or lace entire horizontal crane load at oint B for a more onerous case. Vertical crane loading aximum wheel load 6.5 kn ( wheels) inimum wheel load 7.5 kn ( wheels) aximum reaction at column due to loaded crane inimum reaction at column due to loaded crane kn kn oment due to vertical crane loading (unfactored) oment at B knm oment at F knm Load on the crane bracket is 350 mm eccentric from column centre line. Transverse crane loading Transverse load due to crab and load 0.1 ( ) 3.6 kn Shared between oints B and F, i.e. 1.8 kn each. oment due to transverse crane loading oment at B kNm Slit the frame at the aex, then it can be treated as two cantilevers. Total roof load 7.9 kn/m Load at urlin oint 5.61 kn Job No. Sheet 3 of 6 Rev.

28 Job No. Sheet 3 of 6 Rev. Structural Steel Design Project Job Worked title: Examle. PLASTIC ANALYSIS ade b RSP Date Aril 000 Checked b RN Date Aril Load CALCULATION at urlin oint SHEET kn 1000 oment at urlin oint [ ( ) ] knm oment at A oment due to roof load knm oment due to wind load on ABC 6.5 ( ) knm oment due to vertical crane loading knm 1 oment due to transverse crane load knm oment at G Total 84. knm oment due to roof load 99 knm oment due to vertical crane loading oment due to transverse crane load 6.1 knm 10.8 knm Total knm Summar of the reactant moment diagram method for the ortal frame is shown in Fig. (a) and (b). For solution b calculation, let urlins be numbered 1 to n from roof to aex. Put moment for each urlin oint into the reactant moment diagram equations and solve for successive urlin oints. The largest value of found b this method is the design case.

29 Structural Steel Design Project Job No. Sheet 4 of 6 Rev. Job title: PLASTIC ANALYSIS Worked Examle. ade b RSP Date Aril 000 CALCULATION SHEET Checked b RN Date Aril 000 For this design the equations for are: At A : 84. m R 5 S 0 At oint 3 : 4.59 m R.495 S - At E : 99 - m R + 5 S + At G : m R + 5 S 0 Using the method and equations illustrated in Fig. (b) these equations can be solved simultaneousl (or b matrix) to give R 16. kn, S kn, m 48 kn, and 88.4 knm.

30 Structural Steel Design Project Job No. Sheet 5 of 6 Rev. Job title: PLASTIC ANALYSIS Worked Examle. ade b RSP Date Aril 000 CALCULATION SHEET Checked b RN Date Aril 000 w 1 w w 3 C B W 1 P W 1 P a h 3 D b E F h h A l 1 l L Loading and geometr G w 3 h 1 / w 1 l 1 / w l / A C D E G Free moment diagram for UDLs P 1. h 3 W 1.a W.b P 3.h 3 A B F G Free moment diagram due to crane loads Fig. 4(a) Loadings and free moment diagrams for the ortal frame

31 Structural Steel Design Project Job No. Sheet 6 of 6 Rev. Job title: PLASTIC ANALYSIS Worked Examle. ade b RSP Date Aril 000 CALCULATION SHEET Checked b RN Date Aril 000 R For the urose of analsis, frame is notionall slit at aex, and treated as two cantilevers S S Purlin oints C S R 1 3 n x B x A + (h I + h )R + l 1 S D x R Redundant reactants F G E S R + (h 1 +h )R - l S Equations for fixed bases at C ; Free h R - l 1 S + at n ; Free R xs - at E ; Free h R + l S + at G ; Free (h 1 +h )R + l S - Equations for inned bases are as for fixed bases but with moment at the bases set to zero. Put the factored values of free moment into the equations and solve simultaneousl or b matrix. Solve for successive urlin oints, thus finding the largest value of + (h1 + h)r + l1s + hr + l1s + R + xs x + hr - ls + (h1+h)r - ls A B C D E F G Redundant reactant moment diagram Fig. 4(b) Summar of the reactant moment diagram method for a ortal frame