Structural Analysis-IICE2351

ByP.Sangeetha

Assistant ProfessorDepartment of Civil Engineering

SSN College of Engineering

Unit-IFLEXIBILITY METHOD

Equilibrium and compatibility Determinate vsIndeterminate structures Indeterminacy -

Primary structure Compatibility conditions Analysis of indeterminate pin-jointed plane

frames, continuous beams, rigid jointed planeframes (with redundancy restricted to two).

Unit-II

STIFFNESS MATRIX METHOD Element and global stiffness matrices Analysis

of continuous beams Co-ordinatetransformations Rotation matrix Transformations of stiffness matrices, loadvectors and displacements vectors Analysisof pin-jointed plane frames and rigid frames(with redundancy vertical to two)

Unit-III

FINITE ELEMENT METHOD

Introduction Discretisation of a structureDisplacement functions Truss element Beam element Plane stress and plane strain- Triangular elements

Unit-IV

PLASTIC ANALYSIS OF STRUCTURES

Statically indeterminate axial problems Beamsin pure bending Plastic moment ofresistance Plastic modulus Shape factor Load factor Plastic hinge and mechanism Plastic analysis of indeterminate beams andframes Upper and lower bound theorems

Unit-V

SPACE AND CABLE STRUCTURES

Analysis of Space trusses using method oftension coefficients Beams curved in plan

Suspension cables suspension bridges withtwo and three hinged stiffening girders

TEXT BOOKS

1. Vaidyanathan, R. and Perumal, P.,Comprehensive structural Analysis Vol. I &II, Laxmi Publications, New Delhi, 2003

2. L.S. Negi & R.S. Jangid, Structural Analysis,Tata McGraw-Hill Publications, New Delhi,2003.

3. BhaviKatti, S.S, Structural Analysis Vol. 1Vol. 2, Vikas Publishing House Pvt. Ltd., NewDelhi, 2008

IntroductionDeterminate and indeterminate structuresDeterminate:Forces and Moments are determined by statical

equations of equilibrium

Indeterminate structures: Less equations are availablethan the number of unknown forces that constrain thebody in space. Extra conditions of deformationcompatibility have to be introduced to solve theproblem. These conditions will give the extra numberof equations required to solve the problem, which willindicate the degree of indeterminacy

Plastic Analysis of StructuresINTRODUCTIONThe elastic design method, also termed as allowable stress

method (or Working stress method), is a conventional methodof design based on the elastic properties of steel. This methodof design limits the structural usefulness of the material uptoa certain allowable stress, which is well below the elastic limit.The stresses due to working loads do not exceed the specifiedallowable stresses, which are obtained by applying anadequate factor of safety to the yield stress of steel.

The elastic design does not take into account the strength of thematerial beyond the elastic stress. Therefore the structuredesigned according to this method will be heavier than thatdesigned by plastic methods, but in many cases, elastic designwill also require less stability bracing.

Cont...In the method of plastic design of a structure, the ultimate load

rather than the yield stress is regarded as the design criterion.The term plastic has occurred due to the fact that the ultimate

load is found from the strength of steel in the plastic range.This method is also known as method of load factor design or

ultimate load design. The strength of steel beyond the yield stress is fully utilised in this method. This method is rapid and provides a rational approach for the analysis of the structure.

This method also provides striking economy as regards the weight of steel since the sections designed by this method are smaller in size than those designed by the method of elastic design.

Plastic design method has its main application in the analysis and design of statically indeterminate framed structures.

BASIS OF PLASTIC THEORYDuctility of SteelStructural steel is characterised by its capacity to withstand considerable

deformation beyond first yield, without fracture. During the process of'yielding' the steel deforms under a constant and uniform stressknown as 'yield stress'. This property of steel, known as ductility, isutilised in plastic design methods.

Cont..Fig. 1 shows the idealised stress-strain relationship for structural

mild steel when it is subjected to direct tension.Elastic straining of the material is represented by line OA. AB

represents yielding of the material when the stress remainsconstant, and is equal to the yield stress, fy. The strainoccurring in the material during yielding remains after theload has been removed and is called the plastic strain and thisstrain is at least ten times as large as the elastic strain, atyield point.

When subjected to compression, the stress-strain characteristicsof various grades of structural steel are largely similar to Fig. 1and display the same property of yield. The major differenceis in the strain hardening range where there is no drop instress after a peak value. This characteristic is known asductility of steel.

Perfectly Plastic Materials

The stress-strain curve for a perfectly plastic material upto strain hardening isshown in Fig. 2. Perfectly plastic materials follow Hook's law upto the limitof proportionality. The slopes of stress-strain diagrams in compression andtension i.e. the values of Young's modulus of elasticity of the material, areequal. Also the values of yield stresses in tension and compression areequal. The strains upto the strain hardening in tension and compressionare also equal. The stress strain curves show horizontal plateau both intension and compression. Such materials are known as perfectly plasticmaterials.

Fully Plastic Moment of a SectionThe fully plastic moment Mp, of a section is defined as the maximum moment

of resistance of a fully plasticized or yielded cross-section. Theassumptions used for finding the plastic moment of a section are:

(i) The material obeys Hooke's law until the stress reaches the upper yieldvalue; on further straining, the stress drops to the lower yield value andthereafter remains constant.

(ii) The yield stresses and the modulus of elasticity have the same value incompression as in tension.

(iii) The material is homogeneous and isotropic in both the elastic and plasticstates.

(iv) The plane transverse sections (the sections perpendicular to thelongitudinal axis of the beam) remain plane and normal to the longitudinalaxis after bending, the effect of shear being neglected.

(v) There is no resultant axial force on the beam.(vi) The cross section of the beam is symmetrical about an axis through its

centroid parallel to plane of bending.(vii) Every layer of the material is free to expand and contract longitudinally

and laterally under the stress as if separated from the other layers.

In order to find out the fully plastic moment of a yielded sectionof a beam as shown in Fig. 3, we employ the force equilibriumequation, namely the total force in compression and the totalforce in tension over that section are equal.

The plastic modulus of a completely yielded section isdefined as the combined statical moment of the cross-sectional areas above and below the neutral axis or equalarea axis. It is the resisting modulus of a completelyplasticised section.

BENDING OF BEAMS SYMMETRICAL ABOUT BOTH AXES

The bending of a symmetrical beam subjected to a graduallyincreasing moment is considered first. The fibres of the beamacross the cross section are stressed in tension orcompression according to their position relative to the neutralaxis and are strained in accordance with Fig. 1. While thebeam remains entirely elastic the stress in every fibre isproportional to its strain and to its distance from the neutralaxis. The stress (f) in the extreme fibres cannot exceed fy. (Fig.4)

Elastic stresses in beams

When the beam is subjected to a moment slightly greaterthan that, which first produces yield in the extreme fibres, itdoes not fail. Instead the outer fibres yield at constant stress(fy) while the fibres nearer to the neutral axis sustainincreased elastic stresses. Fig. 5 shows the stress distributionfor beams subjected to such moments. Such beams are saidto be 'partially plastic' and those portions of their cross-sections, which have reached the yield stress, are described as'plastic zones'.

Stresses in partially plastic beams

The depths of the plastic zones depend upon the magnitude of the appliedmoment. As the moment is increased, the plastic zones increase in depth, and, itis assumed that plastic yielding can occur at yield stress (fy) resulting in twostress blocks, one zone yielding in tension and one in compression. Fig. 6represents the stress distribution in beams stressed to this stage. The plasticzones occupy the whole of the cross section, and are described as being 'fullyplastic'. When the cross section of a member is fully plastic under a bendingmoment, any attempt to increase this moment will cause the member to act asif hinged at the neutral axis. This is referred to as a plastic hinge. The bendingmoment producing a plastic hinge is called the full plastic moment and isdenoted by 'Mp'. Note that a plastic hinge carries a constant moment, MP.

Stresses in fully plastic beams

Shape FactorIn plastic analysis there will be two stress blocks, one in

tension, the other in compression, both of which willbe at yield stress. For equilibrium of the cross section,the areas in compression and tension must be equal.For a rectangular cross section, the elastic moment isgiven by,

Here the plastic moment Mp is about 1.5 times greater than the elastic moment capacity. In developing this moment, there is a large straining in the extreme fibres together with large rotations and deflection.

Plastic Hinge Plastic hinge is defined as a yielded zone due to

bending in a structural member at which an infiniterotation can take place at a constant plastic momentMp of the section. The number of hinges necessary forfailure does not vary for a particular structure subjectto a given loading condition, although a part of astructure may fail independently by the formation of asmaller number of hinges. The member or structurebehaves in the manner of a hinged mechanism and indoing so adjacent hinges rotate in opposite directions.

Theoretically, the plastic hinges are assumed to form atpoints at which plastic rotations occur. Thus the lengthof a plastic hinge is considered as zero.

Principles of plastic analysisFundamental conditions for plastic analysis(i) Mechanism condition: The ultimate or collapse

load is reached when a mechanism is formed. Thenumber of plastic hinges developed should be justsufficient to form a mechanism.

(ii) Equilibrium condition : Fx = 0, Fy = 0, Mxy = 0(iii) Plastic moment condition: The bending

moment at any section of the structure shouldnot be more than the fully plastic moment of thesection.

MechanismWhen 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 ifno resistance is set up against deformation in the body,then it is known as a mechanism.

Various types of independent mechanisms are

Beam MechanismSway /Panel MechanismGable MechanismJoint MechanismCombined Mechanism

(a) A simply supported beam has to form one plastic hinge at the point of maximum bending moment. Redundancy, r = 0

(b) A propped cantilever requires two hinges to form a mechanism. Redundancy, r = 1No. of plastic hinges formed, = r + 1 = 2

(c) A fixed beam requires three hinges to form a mechanism. Redundancy, r = 2No. of plastic hinges = 2 + 1 = 3

Beam Mechanism

From the above examples, it is seen that the number of hinges needed to form amechanism equals the statical redundancy of the structure plus one.

Panel or Sway MechanismFig. (A) shows a panel or sway mechanism for a portal

frame fixed at both ends.

Gable MechanismFig. (B) shows the gable mechanism for a gable structurefixed at both the supports.

Joint MechanismFig. (C) shows a joint mechanism. It occurs at a joint where more than two

structural members meet.

Combined MechanismVarious combinations of independent mechanisms can be made depending uponwhether the frame is made of strong beam and weak column combination orstrong column and weak beam combination.

Lower bound or Static theorem

A load factor ( s ) computed on the basis of anarbitrarily assumed bending moment diagramwhich is in equilibrium with the applied loadsand where the fully plastic moment ofresistance is nowhere exceeded will always beless than or at best equal to the load factor atrigid plastic collapse, (p).

In other words, p is the highest value of swhich can be found.

Upper bound or Kinematic theorem

A load factor ( k) computed on the basis of anarbitrarily assumed mechanism will always begreater than, or at best equal to the loadfactor at rigid plastic collapse (p ).

In other words, p is the lowest value of kwhich can be found.

Continuous BeamsFind the collapse load for the given continues beam having three span continuous beam of uniform section throughout (constant Mp)

Number of Independent Mechanism = N r = 5-2 = 3N = No of Possible Plastic Hinges = 5(under all loads(3) and at two intermediate support)r = Redundancy of the structure = 2 (4+0-2 =2)

(4+0-2 =2) = ( No of vertical reaction + moments 2 )

Collapse load for the given continuous beam is least of all values from the three mechanism

Beam Mechanism 1

Beam Mechanism-2

Beam Mechanism -3

Beam Mechanism -4

No of Independent Mechanism = N-r = 5 3 =2 N No of plastic hinge=5(at A,B,C,D and under the load)r= Degree of redundancy = 3 (no of closed loop)-No of release=3(1)-0=3

Beam mechanism Sway MechanismCombined mechanism

From (i ),(ii),(iii) we conclude, Mp = 116.67KNm and the combined mechanism is the real mechanism.

Find the collapse load for the given frame

No of mechanism = 5-3 = 2

No of plastic hingesDegree of redundancy

Beam mechanism

Question paper problems

Two Marks Questions 1. What is a plastic hinge?

When a section attains full plastic moment Mp, it acts as hinge which iscalled a plastic hinge. It is defined as the yielded zone due to bending atwhich large rotations can occur with a constant value of plastic momentMp.

2. What is a mechanism?When a n-degree indeterminate structure develops n plastic hinges, itbecomes determinate and the formation of an additional hinge will reducethe structure to a mechanism. Once a structure becomes a mechanism, itwill collapse.

3. What is difference between plastic hinge and mechanical hinge?Plastic hinges modify the behavior of structures in the same way asmechanical hinges. The only difference is that plastic hinges permitrotation with a constant resisting moment equal to the plastic momentMp. At mechanical hinges, the resisting moment is equal to zero.

4. Define collapse load.The load that causes the (n + 1) the hinge to form a mechanism is called collapseload where n is the degree of statically indeterminacy. Once the structure becomesa mechanism

5. List out the assumptions made for plastic analysis.The assumptions for plastic analysis are:

Plane transverse sections remain plane and normal to the longitudinal axisbefore and after bending.

Effect of shear is neglected. The material is homogeneous and isotropic both in the elastic and plastic

state. Modulus of elasticity has the same value both in tension and compression. There is no resultant axial force in the beam. The cross-section of the beam is symmetrical about an axis through its

centroid and parallel to the plane of bending. 6.Define shape factor.

Shape factor (S) is defined as the ratio of plastic moment of the section to the yield moment of the section.

Where Mp = Plastic momentM = Yield moment

Zp = Plastic section modulusZ = Elastic section modulus

7. List out the shape factors for the following sections.(a) Rectangular section, S = 1.5 (b) Triangular section, S = 2.346 (c) Circular section, S = 1.697 (d) Diamond section, S = 2

8. Mention the section having maximum shape factor.The section having maximum shape factor is a triangular section, S = 2.345.

9. Define load factor.Load factor is defined as the ratio of collapse load to working load and is given by

10. What are unsymmetrical frames and how are they analyzed?Un-symmetric frames have different support conditions, lengths and loading conditions on its columns and beams. These frames can be analyzed by:(a) Beam mechanism(b) Column mechanism(c) Panel or sway mechanism(d) Combined mechanism11. Define plastic modulus of a section Zp.The plastic modulus of a section is the first moment of the area above and below the equal area axis. It is the resisting modulus of a fully plasticized section.Zp = A/2 (y1 + y2)