1. By: Prof Dr. Akhtar Naeem Khan chairciv@nwfpuet.edu.pk
Lecture 13: Plate Girder
By: Prof Dr. Akhtar Naeem Khan

2. Plate Girders
A girder is a flexural member which is required to carry heavy loads on relatively long spans

3. Plate Girder

4. Plate Girder
Plate girders are typically used as long-span floor girders in buildings, as bridge girders, and as crane girders in industrial structures.
Commonly term girder refers to a flexural x-section made up of a number of elements.
They are generally considerably deeper than the deepest rolled sections and usually have webs thinner than rolled sections.

5. Plate Girder
Modern plate girders are normally fabricated by welding together two flanges and a web plate.

6. Plate Girder
Plate girders are at their most impressive in modern bridge construction where main spans of well over 200m are feasible, with corresponding cross-section depths, haunched over the supports, in the range of 5-10m.

7. Plate Girder
Because plate girders are fabricated separately, each may be designed individually to resist the applied actions using proportions that ensure low self-weight and high load resistance.

8. Plate Girder Changes in X-Section
There is also considerable scope for variation of cross-section in the longitudinal direction. A designer may choose to reduce the flange thickness (or breadth) in a zone of low applied moment.
Equally, in a zone of high shear, the designer might choose to thicken the web plate.

9. Plate Girder Changes in Material
Alternatively, higher grade steel might be employed for zones of high applied moment and shear, while standard grade would be used elsewhere. So-called "hybrid" girders with different strength material in the flanges and the web offer another possible means of more closely matching resistance to requirements.

10. Plate Girder

11. Plate Girder

12. Plate Girder
Any cross-section of a plate girder is normally subjected to a combination of shear force and bending moment.
The primary function of the top and bottom flange plates of the girder is to resist the axial compressive and tensile forces arising from the applied bending moment.
The primary function of the web plate is to resist the applied shear force.

13. Plate Girder
Plate girders are normally designed to support heavy loads over long spans in situations where it is necessary to produce an efficient design by providing girders of high strength to weight ratio.
To produce the lowest axial flange force for a given bending moment, the web depth (d) must be made as large as possible. To reduce the self weight, the web thickness (tw) must be reduced to a minimum.
As a consequence, in many instances the web plate is of slender proportions and is therefore prone to buckling at relatively low values of applied shear.

14. Plate Girder
For efficient design it is usual to choose a relatively deep girder, thus minimizing the required area of flanges for a given applied moment, Msd.
This obviously entails a deep web whose area will be minimized by reducing its thickness to the minimum required to carry the applied shear, Vsd.
Such a web may be quite slender (i.e. a high d/tw ratio) and may be prone to local buckling and shear buckling.

15. Plate Girder
Web buckling does not determine the ultimate strength of a plate girder.
Plate elements do not collapse when they buckle; they can possess a substantial post-buckling reserve of resistance.
For an efficient design, any calculation relating to the ultimate limit state should take the post-buckling action into account.

16. Design Criteria Criteria for design of plate girder may be based on
Elastic bend-buckling strength
Elastic shear-buckling strength
Post-bend-buckling strength
Post-shear-buckling(Tension field)strength

17. Design Criteria
The designer has the choice of following four combinations
Elastic bend buckling + Elastic shear buckling (conventional flexural behavior)
Elastic bend buckling + Post shear buckling
Post bend buckling + Elastic shear buckling
Post bend buckling + Post shear buckling

18. Elastic Bend Buckling Strength
The extreme fiber bending stress at which a perfectly flat web buckles is given by

19. Elastic Bend Buckling Strength
Using a FOS of 1.25 w.r.t service load bending stress fb gives an eqnuation which is AASHTO slenderness limit for plat girders webs
Using AASHTO allowable stress fb=0.55Fy
“ h/t=165 for A36 steel “

20. Elastic Bend Buckling Strength
The bend buckling resistance of beam webs can be increased considerably by reinforcing the slender webs with Longitudinal stiffeners.
Means webs thinner than those given by the equation can be used.
A typical longitudinally stiffened girder is shown after failure

21. Web Stiffeners
They usually consists of rectangular bars to welded to web.
Transverse stiffeners may be in pairs, one on each side of web, or they may placed on one side of web.
Longitudinal stiffeners are usually placed on one side of web.

22. Web Stiffeners

23. Web Stiffeners

24. Web Stiffeners
The main function of the longitudinal stiffeners is to increase the buckling resistance of the web with respect of both shear and bending loads. An effective stiffener will remain straight, thereby sub-dividing the web panel and limiting the buckling to the smaller sub-panels. The resulting increase in the ultimate resistance of the girder can be significant.

25. Web Stiffeners
Efficiency of stiffener is a function of its location in the compression zone
The optimum location for a longitudinal stiffener has been determined to be at least h/5 from compression edge.
In this case k=129. The corresponding allowable web slenderness is h/t=330 as compare to 165

26. Web Stiffeners
Stiffener acts as a beam supported at the ends where a vertical stiffener holds the web in line.
Stiffener acts as a beam column and hence must be proportioned in terms of x-sectional area and moment of inertia.
AASHTO specifies Is as

27. Web Stiffeners
The stiffeners must also be proportioned to resist local buckling.
For plates supported on one longitudinal edge AASHTO require b/t<1625/fb
Multiple longitudinal stiffeners are used for large depth webs.
As longitudinal stiffener is also acting as a column so it must be satisfied for critical stress (Fcrs>0.6Fcrf)

28. Post buckling bending strength
If bending strain increases after Fcr, the upper edge of panels shortens and bottom edge lengthens.
If web were to remain flat there will be increase in stress.
Because the web has buckled, the increase in stress is non-linear.

29. Post buckling bending strength
As variation in post-buckled state is not known, simplify assumptions are made.
Non-linear compression is replaced with linear distribution acting on effective depth be.

30. Post buckling bending strength
Point A gives point that enables a girder to reach its full yield moment(925 /Fy=154).
If stiffeners at h/5 is provided gives point B.
h/t
154
360
315 A B
M/My
0.18
0.4
0.82
0.94
Considering the post buckling strength, the point where reduction in web effectiveness begins s taken to be 980/Fy=170.

31. Post buckling bending strength
Equation connecting the revised point A with points corresponding to h/t=360 is

32. Post buckling bending strength
LRFD
Where

33. Compression Flange Vertical buckling

34. Compression Flange Vertical buckling
If plate-girder web is too slender, the compression flange may buckle in vertical plane at stress less than yield stress.
The compression flange is a beam-column continuous over vertical stiffener as supports
Its stability depends on stiffener spacing and relative stiffness of the flange and the web. Fcr is

35. Compression Flange Vertical buckling
Slenderness of webs with vertical stiffeners is taken conservatively
AISC ASD/LRFD limits the h/t by the given equation with Aw/Af =0.5

36. Shear buckling of beam webs
Shear buckling is seldom a determining factor in design of rolled section but plate girders have much larger h/t so it must be considered.

37. Shear buckling of beam webs
Transverse stiffeners are used to increase the buckling strength by increasing factor k through a reduction in aspect ratio a/h.

38. Transverse Stiffeners
Transverse stiffeners play an important role in allowing the full ultimate load resistance of a plate girder to be achieved.
In the first place they increase the buckling resistance of the web;
Secondly they must continue to remain effective after the web buckles, to provide anchorage for the tension field;
finally they must prevent any tendency for the flanges to move towards one another.

39. Transverse Stiffeners
The satisfactory performance of a transverse stiffener can best be illustrated by comparing the girders shown, after testing.
Figure 2
Figure 1

40. Transverse Stiffeners
In Figure 1 the stiffeners have remained straight.
In Figure 2 the stiffener has failed and has been unable to limit the buckling to the adjacent sub-panels of the girder; instead, the buckle has run through the stiffener position extending over both panels. Consequently, significant reduction in the failure load of the girder occurred.
In Figure 1 One can also see the effect of aspect ratio,i.e greater a/h less k and small Fcr.

41. Transverse Stiffeners
The stiffener must be of adequate rigidity in the direction perpendicular to the plane of the web to prevent web buckling. This condition is satisfied provided the stiffener has a second moment of area Is that satisfies the following empirical formulae:

42. Transverse Stiffeners
AISC/LRFD Moment of Inertia of stiffener is:
where

43. Transverse Stiffeners
Transverse stiffeners spacing can be determined from the following

44. Tension Field Action
The resulting shear stresses on an element of a web are equivalent to principal stresses, one Tensile and one Compressive, at 45 to the shear stress.

45. Tension Field Action
Once a web panel has buckled in shear, it loses its resistance to carry additional compressive stresses.
On the other hand tensile principal stress continues to increase in strain in the diagonal direction.
Such a panel has a considerable post buckling strength, since increase in tension is limited only by yield stress.

46. Tension Field Action
In this post-buckling range, a new load-carrying mechanism is developed, whereby any additional shear load is carried by an inclined tensile membrane stress field. This tension field anchors against the top and bottom flanges and against the transverse stiffeners on either side of the web panel. The load-carrying action of the plate girder than becomes similar to that of the N-truss
In the post-buckling range, the resistance offered by the web plates is analogous to that of the diagonal tie bars in the truss.

47. Phases of behavior up to collapse of a typical panel in shear
Tension Field Action
Phases of behavior up to collapse of a typical panel in shear
Prior to Buckling
Post Buckling
Collapse

48. Tension Field Action
The load-carrying action of the plate girder than becomes similar to that of the N-truss
In the post-buckling range, the resistance offered by the web plates is analogous to that of the diagonal tie bars in the truss.

49. Tension Field Action

50. Tension Field Action

51. Tension Field Action ft V Vt=Tsin Vt = ft ht cos sin
Vt = (1/2)ft ht sin2
Vt =(1/2) ft ht =45
Vty=(1/2) Fy ht………….(1)
T=ft ht cos 

52. Tension Field Action Vty =(1/2) Fy ht = Fy Vy Fvy ht 2Fvy
Vty = 3 Vy = 0.87 Vy 2

53. Tension Field Action
The angle  for which Vt is max

54. Tension Field Action
Where

55. Tension Field Action (1) Taking inelastic and strain hardening range
(2)
(3)

56. Tension Field Action
Codal equations are derived from eqn;(1),(2),(3)

57. Tension Field Action
a/h k
AISC/LRFD

58. Combined Bending & Shear of Webs
Interaction diagram is based on Tension-field of webs
If the web is completely yielded in shear,any accompanying moment must be resisted entirely by flanges.

59. Combined Bending & Shear
Bending & shear Interaction Curve
0.75
0.83
1.0
1.07
1.12
M/My
V/(FvyAw) B B C D E
1/3

60. Combined Bending & Shear
0.2
0.4
0.6
0.8
1.0
Vu/Vn
Mu/Mn
LRFD Interaction Curve

61. Web Proportioning
Notations

62. Web Proportioning Depth of girder is influenced by many factors:
Headroom
Clearance for high water in deck bridges
Traffic passing beneath the bridge

63. Web Proportioning
Depth: Overall girder depth, h, will usually be in the range
Lo/12 £ h £ Lo/8,
occasionally lighter loads may be accommodated with Lo/20.
Flange:
The breadth, b, will usually be in the range
h/5 £ b £ h/3,

64. Design Procedure Maximum Moment & Shear for Factored Load Web Design
Assume depth of girder L/12 £ h £ L/8
Depth of Web hw=h-2tf
Web slenderness
For a/h <5 …………….
and for a/h > 5 ……………………
hw/tw= 970/Fy
Select optimum tw

65. Design Procedure Flange Design Find Af Select suitable tf and bf
Flange slenderness
bf/ 2tf < 65/Fy …………….Compact

66. Design Procedure Check trial girder section
Web local buckling limit state
hw/tw< 640/Fy…………………..Compact
640/Fy< hw/tw < 970/Fy……Non-Compact
hw/tw > 970/Fy…………………..Slender
Flange local buckling limit state
bf/ 2tf < 65/Fy …………….Compact
Lateral Torsional Buckling
Calculate Iy
A=Af+Aw/6
ry= Iy/A
Find Lb/ry
p= 300/Fy ………….. < p ______Compact

67. Design Procedure
Bending strength
Calculate Ix
Calculate Sxt .
Mn Mu

68. Procedure for Design Bending strength Calculate Ix Calculate Sxt .
Mn Mu