PLATE GIRDERS -I

INTRODUCTION

Beam with cover platesSteel trussPlate girdersTwo or more regular beams

This presentation is about the behaviour and design of plate girdersFor Heavy Loads And Long Spans, choice is

INTRODUCTION - I

INTRODUCTION - 2

INTRODUCTION - 3

COMPONENTS OF A PLATE GIRDERFlange plate - flexure

Web plate - shear

Stiffener - prevents buckling

INTRODUCTION 4

SHEAR RESISTANCE OF STIFFENED GIRDERShear resistance of a web

Pre-buckling behaviour (Stage 1)

Requirements of equilibrium in an element inside a square web plate subject to a shear stress result in generation of complementary shear stresses

This results in element being subjected to principal compression along one diagonal and tension along the other

Shear resistance of a web - 1

Shear resistance of a web - 2

As the applied loading is incrementally enhanced, plate will buckle along direction of compressive diagonal - corresponding shear stress in plate iscritical shear stress

Critical shear stress in such a case is given by

Boundary conditions assumed to be simply supported

Shear resistance of a web - 3

shear buckling coefficient (ks) given by

Post buckled behaviour (Stage 2)

Compression diagonal is unable to resist any more loading beyond elastic critical stress

Any further increase in shear load is supported by a tensile membrane field, anchored to top and bottom flanges and adjacent stiffener members on either side of web

Total state of stress in web plate may be obtained by superimposing post-buckled membrane tensile stresses upon critical shear stress

Post buckled behaviour - 1

Post buckled behaviour - 2

Collapse behaviour (Stage 3)

When load is further increased, tensile membrane stress continues to exert an increasing pull on flanges

Eventually resultant stress obtained by combining the buckling stress and membrane stress reaches yield value for web - can be determined by Von-Mises yield criterion

Collapse behaviour - 1

Collapse behaviour - 2

Collapse behaviour - 3

Collapse behaviour - 4

Collapse behaviour - 5

Once web has yielded, final collapse of girder will occur when four plastic hinges are formed in flanges By using virtual work method, failure load can be computed from

This is applicable only if flanges are PLASTICCollapse behaviour - 5

Weak flanges

When a plate girder has weak flanges, tension field is NOT supported by flanges

Field anchors entirely on transverse stiffeners

Failure load can be computed from

Weak flanges - 1

Very Strong flanges

When Very Strong flanges are employed, hinges will form at four corners, constituting a picture frame type mechanism

Tension field angle () is 45

Ultimate shear in this case is given by:

Very strong flanges - 1

Very Thick websIn case web is thick, it will yield before buckling failure will form by a picture frame mechanism

Very Slender webs

Very slender webs are rarely used In very slender webs qcr/qyw is extremely small and there will be significant post-buckled tension field

WEB SUBJECTED TO COMBINED BENDING AND SHEARWhen a girder is subjected to bending moments and low shear, its ultimate capacity is conditioned by interaction

Any point on interaction diagram shows co-existent values of shear and bending moment that girder can sustain

When applied moment is high, failure will be triggered by collapse of flanges by one of following:

By yielding of flange material By inward buckling of the compression flange By lateral buckling of the flange

WEB SUBJECTED TO COMBINED BENDING AND SHEAR - 1

In zone ABC, presence of additional bending moment requires following three factors to be considered:

Reduction in web buckling stress due to presence of bending stresses

Influence of bending stresses on value of membrane stress required causing yield in web

Reduction of plastic moment capacity of flanges due to presence of axial flange stresses caused by bending momentWEB SUBJECTED TO COMBINED BENDING AND SHEAR - 2

Modified web buckling stressModified web buckling stress due to coincident bending stress may be computed from following interaction Equation:

qcrm =modified shear buckling stressqcr =elastic critical shear stress in web fmb =compressive bending stress in extreme fibre at mid panel due to the bending momentfcrb =buckling stress for plate due to a pure bending moment

Modified membrane stress for web yieldingModified expression for membrane stress in presence of applied bending moment is given by

A = 3 qcrm sin 2 + pb sin2 - 2pb cos2

pb = Value of bending stress, which varies over both depth and width of the web panel.

Reduction of plastic moment capacity of flangesWhen high axial forces are developed in flanges, their effects in reducing plastic moment capacity of flange plates must be taken into accountFrom plasticity theory, the reduced capacity is given by

pf is the average axial stress for the portion of the flange between hinges

Webs subjected to pure bendingIn a thin walled girder, web subjected to compressive bending stress will buckle, losing its capacity to carry further compressive stresses

Compression flange will carry practically all compressive stresses - girder is unable to develop full plastic moment of resistance

If no lateral buckling occurs girder will fail by inward collapse of compression flange

Webs subjected to pure bending - 1Ultimate moment capacity to be determined by a simple formula:

My = Bending moment required to produce yield in extreme fibre of flange assuming fully effective web (i.e. neglecting web buckling)

ULTIMATE BEHAVIOUR OF TRANSVERSE WEB STIFFENERSTransverse stiffeners play important role by increasing web buckling stressby supporting tension field after web bucklingby preventing tendency of flanges to get pulled towards each otherStiffeners should possess sufficient rigidity to ensure that they remain straight, while restricting buckling to individual web panels

ULTIMATE BEHAVIOUR OF TRANSVERSE WEB STIFFENERS - 1

GENERAL BEHAVIOUR OF LONGITUDINALLY STIFFENED GIRDERSGenerally located in compression zones of girderMain function - to increase buckling resistance of web When it is subject predominantly to shear would develop a collapse mechanism, provided stiffeners remained rigid up to failureOnce one of sub panels has buckled, post buckling tension field develops over whole depth of web panel and influence of stiffeners may be neglected

GENERAL BEHAVIOUR OF LONGITUDINALLY STIFFENED GIRDERS - 1

CONCLUSIONSConsidered ultimate behaviour of plate girdersFundamental theoretical relationship based on buckling and post-buckling theories have been establishedIn some case, semi-empirical procedures have been suggested to ease tedium of lengthy calculationsTransverse stiffeners have been considered in some depthUse of longitudinal stiffeners has also been described