Dr S R Satish Kumar, IIT Madras 1
PLATE GIRDERS
Built-up sections with deep thin webssusceptible to buckling in shear
Dr S R Satish Kumar, IIT Madras 2
Types of Plate Girders• Unstiffened Plate Girder
• Transversely Stiffened Plate Girder
• Transversely and Longitudinally Stiffened Plate Girder
web plate flange plates
ITS BS
LS
Dr S R Satish Kumar, IIT Madras 3
SHEAR RESISTANCE OF STIFFENED GIRDER
Shear 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
Dr S R Satish Kumar, IIT Madras 4
Shear resistance of a web - 1
Aq
q
45o
B
DC
q
E
q
Unbuckled Shear panel
Dr S R Satish Kumar, IIT Madras 5
Shear buckling of a plate
BUCKLING OF WEB PLATES IN SHEAR
cr
Dr S R Satish Kumar, IIT Madras 6
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 is“critical shear stress”
– Critical shear stress in such a case is given by
– Boundary conditions assumed to be simply supported
2
dt
2112
E2skcrq
Dr S R Satish Kumar, IIT Madras 7
Shear resistance of a web - 3
• shear buckling coefficient (ks) given by
panelswideforeidcwhere
cdks ..,1435.5
2
stiffenerstransversespaced
closelywithwebsforeidcwhere
cdks ..,1435.5
2
c
d
Dr S R Satish Kumar, IIT Madras 8
• 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
Dr S R Satish Kumar, IIT Madras 9
Post buckled behaviour - 1
Anchoring of Tension Field
Dr S R Satish Kumar, IIT Madras 10
Tension field actionTension field action
Dr S R Satish Kumar, IIT Madras 11
• 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
Dr S R Satish Kumar, IIT Madras 12
Collapse behaviour - 1
Collapse of the panel
Tensile membrane stress at yield
Dr S R Satish Kumar, IIT Madras 13
Three phases of tension field actionThree phases of tension field action
Pre-buckling post-buckling collapse
Dr S R Satish Kumar, IIT Madras 1414
ULTIMATE BEHAVIOUR OF TRANSVERSE WEB STIFFENERSULTIMATE BEHAVIOUR OF TRANSVERSE WEB STIFFENERS
Transverse 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
Dr S R Satish Kumar, IIT Madras 1515
ULTIMATE BEHAVIOUR OF TRANSVERSE WEB STIFFENERS - 1ULTIMATE BEHAVIOUR OF TRANSVERSE WEB STIFFENERS - 1
Force imposed on transverse stiffeners by tension field
Dr S R Satish Kumar, IIT Madras 1616
GENERAL BEHAVIOUR OF LONGITUDINALLY STIFFENED GIRDERSGENERAL BEHAVIOUR OF LONGITUDINALLY STIFFENED GIRDERS
Generally located in compression zones of girder Main 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 failure
Once one of sub panels has buckled, post buckling tension field develops over whole depth of web panel and influence of stiffeners may be neglected
Dr S R Satish Kumar, IIT Madras 17
GENERAL BEHAVIOUR OF LONGITUDINALLY STIFFENED GIRDERS – 1
Longitudinal and Transverse stiffeners
Dr S R Satish Kumar, IIT Madras 18
8.4 Shear The factored design shear force, V, in a beam due to external actions shall satisfy
V Vd
Vd = design strength calculated as , Vd = Vn / γm0
8.4.1 The nominal plastic shear resistance under pure shear is given by: Vn = Vp
Av = shear area
Cont…
3ywv
p
fAV
Dr S R Satish Kumar, IIT Madras 19
8.4.2 Resistance to Shear Bucklingfor an unstiffened web
for a stiffened web
a) Simple Post-Critical Method The nominal shear strength is
Vn = Vcr Vcr = d twb
b = shear stress corresponding to buckling, b) Tension Field Method The nominal shear strength is V n = V tf
67wt
dyf/250
vk
Dr S R Satish Kumar, IIT Madras 20
8.4.2.2 Shear Buckling Design Methods
a) Simple Post-Critical Method -The nominal shear strength is Vn = Vcr Vcr = d twb
b = shear stress corresponding to buckling, determined as follows:
a) When w < 0.8
b) When 0.8 < w < 1.25
c) When w 1.25 b =0.9 fyw/(3w
2)
Cont…
3/ywb f
3/8.0625.01 ywwb f
0.8 1.25 w
b
Dr S R Satish Kumar, IIT Madras 21
λw = non -dimensional web slenderness ratio for shear buckling stress, given by
The elastic critical shear stress of the web, cr is given by:
kv = 5.35 when transverse stiffeners are provided only at supports = 4.0 +5.35 /(c/d)2 for c/d < 1.0 = 5.35+4.0 /(c/d)2 for c/d 1.0
Cont…
)3( ,ecryww f
22
2
/112 w
vcr td
Ek
Dr S R Satish Kumar, IIT Madras 22
b) Tension Field Method - the nominal shear resistance, Vn, should be Vn=Vtf
Vnp
fv = yield strength of the tension field obtained from
=1.5 b sin 2
= inclination of the tension field
The width of the tension field, wtf, is given by: wtf = d cos – (c-sc-st) sin
5.0222 3 bywv ff
cd1tan
ctf
Ms
wy
fr
5.0
sin2
202 //125.0 myffffyffffr ftbNftbM
sin9.0 vwtfbwtf ftwtdV
sc
stc
wtf
Dr S R Satish Kumar, IIT Madras 23
8.6 Design of Beams and Plate Girders with Solid Webs
8.6.1 Minimum Web Thickness
8.6.1.1 Serviceability Requirement
a) when transverse stiffeners are not provided
(web connection by flanges along both longitudinal edges)
(web connection by flanges along one longitudinal edge only)
b) when transverse stiffeners only are provided;
i) when c d
ii) when 0.74 d < c < d
iii) when c < 0.74 d
Cont…
180wtd
90wtd
wwtd 200
wwtc 200
wwtd 270
Dr S R Satish Kumar, IIT Madras 24
c) when transverse and longitudinal stiffeners are provided at one level only (0.1 d from compression flange)
i) when c > d
ii) when 0.74 d < c < d
iii) when c < 0.74 d
d) when a second longitudinal stiffener (located at neutral axis is provided )
Cont…
wwtd 250
wwtc 250
wwtd 340
wwtd 400
Dr S R Satish Kumar, IIT Madras 25
Design ProcedureInitial Sizing1) Taking L/d as 15, calculate min. d and provide suitably
2) Afreqrd. = BM/ (fy/mo)d ; using bf = 0.3d select flange plateAlso calculate Nf = axial force in the flange
3) Check that flange criteria gives a plastic sectionb = (bf – tw)/2 and b/ tf < 7.9
4) Web thickness for serviceability 67 < d/ tw < 200choose such that tw > d/200
5) Check for flange buckling into webAssuming c >1.5d , d/ tw < 3452
Dr S R Satish Kumar, IIT Madras 26
Design Procedure
6) Check for shear capacity of webV < Vd = Vn/ mo; Vn = A (fyw /3) or Vcr
7) Check for calculating resistance to shear buckling d/ tw > 67 (kv/5.35) use kv for c/d > 1
8) Simple post-critical methodVcr = d tw b where b = (w) and w = (cr )
9) If V < Vcr/ mo then safe else tension field calculation reqrd.
10) Vn = Vtf = (fv and ); also calculate Mfv = (Nf )If V < Vn/ mo safe ! else revise design
Dr S R Satish Kumar, IIT Madras 27
Design Procedure
• 8.7 Stiffener design– a) Intermediate Transverse Web Stiffener To improve
the buckling strength of slender web due to shear.
– b) Load Carrying Stiffener To prevent local buckling of the web due to concentrated loading.
– c) Bearing Stiffener To prevent local crushing of the web due to concentrated loading .
– d) Torsion Stiffener To provide torsional restraint to beams and girders at supports.
– e) Diagonal Stiffener To provide local reinforcement to a web under shear and bearing.
– f) Tension Stiffener To transmit tensile forces applied to a web through a flange.
Dr S R Satish Kumar, IIT Madras 28
Design Procedure11) End panel design – check as a beam between flanges
Rtf = Hq/2
Av = c t and Vtf = Av (fy /3) > Rtf
12) Mtf = Hqd/10
MR = tc3/12*fyd / (c/2) > Mtf
13) Intermediate Transverse Stiffener Design i) decide to provide stiffener on one side or both sides
ii) choose tq > tw ; outstand bs < 14tq also < b
14) check for minimum stiffness Cl.8.7.2.4 p91for c = 1.5d, c > 2 d giving
I prov. = (bs-tw/2)3 tq/12 > 0.75dtw3
)/1(.25.1 dpcrdpq VVVH Rtf
c
bs
tq
Dr S R Satish Kumar, IIT Madras 29
Design Procedure15) Check for Buckling Cl.8.7.2.5
p91Stiffener force, Fq = V - Vcr/mo Fqd Buckling Resist. Pq with 20tw on either side Cl.8.7.1.5
p90Calculate Ixx and A, rxx = (Ixx/A)Leff = 0.7d, = Leff/rxx, Find fcPq = fc A > Fq
16) Connection to web Cl.8.7.2.6 p92
shear = tw2 / 8bs kN/mm choose appropriate weld size
19) Check for Intermediate Stiffener under Load Cl.8.7.2.5 p91
1
ys
s
xd
x
qd
xq
MM
FF
FFF
bs
tq