1. By: Prof Dr. Akhtar Naeem Khan chairciv@nwfpuet.edu.pk
Lecture 10: Beams
By: Prof Dr. Akhtar Naeem Khan

2. Beam
A beam is generally considered to be any member subjected principally to transverse gravity or vertical loading.

3. Beam

4. Beam

5. Types of Beams Girders usually the most important beams.
Stringers Longitudinal bridge beams spanning between floor beams.
Floor Beams In buildings, a major beam usually supporting joists; a transverse beam in bridge floors.

6. Types of Beams

7. Types of Beams

8. Types of Beams
Joists A beam supporting floor construction but not major beams.

9. Types of Beams

10. Types of Beams

11. Types of Beams
Purlins Roof beam spanning between trusses.

12. Types of Beams
Girts Horizontal wall beams serving principally to resist bending due to wind on the side of an industrial building.
Lintels Member supporting a wall over a window or door opening.

13. Sections used for Beams
Among the steel shapes that are used as beam include:
W shapes, which normally prove to be the most economical beam sections and they have largely replaced channels and S sections for beam usage.
Channels are sometimes used for beams subjected to light loads, such as purlins and at places where clearances available require narrow flanges

14. Sections used for Beams

15. Design Approaches
Elastic Design
For many years the elastic theory has been the bases for the design and analysis of steel structures. This theory is based on the yield stress of a steel structural element.
However, nowadays, it has been replaced by a more rational & realistic theory the ultimate stress design that is based on plastic capacity of a steel structure.

16. Design Approaches
Elastic Design
In the elastic theory the maximum load that a structure could support is assumed to equal the load that cause a stress somewhere in the structure equal the yield stress of the Fy of the material.
The members were designed so that computed bending loads for service loads did not exceed the yield stress divided a factor of safety (e.g. 1.5 to 2)

17. Elastic Design Versus Ultimate Design
Design Approaches
Elastic Design Versus Ultimate Design
According to ASD, one FOS is used that accounts for the entire uncertainty in loads & strength.
According to LRFD(probability-based) different partial safety factors for different load and strength types are used.

18. Elastic Design Versus Ultimate Design
Design Approach
Elastic Design Versus Ultimate Design
Engineering structures have been designed for many years by the allowable stress design(ASD), or elastic design with satisfactory results.
However, engineers have long been aware that ductile members(e.g. steel) do not fail until a great deal of yielding occurs after yield stress is first reached.
This means that such members have great margin of safety against collapse than the elastic theory would seem to suggest.

19. Bending Behavior of Beams
Assumptions & Conditions
Strains are proportional to the distance from the neutral axis.
The stress-strain relationship is idealized to consist of two straight lines.
Deformations are sufficiently small so that
ø = tanø

20. Bending Behavior of Beams

21. Bending Behavior of Beams
Rectangular Beam: Elastic Bending

22. Bending Behavior of Beams
Bending Stresses
If the beam is subjected to some bending moment the stress at any point may be computed by usual flexural formula
It is important to remember that this expression is only applicable when the maximum computed stress in the beam is below elastic limit.

23. Bending Behavior of Beams
Bending Stresses
The value of I/c is a constant for a particular section and is known as section modulus.
The flexural formula may then be written as

24. Bending Behavior of Beams
Bending Stresses

25. Bending Behavior of Beams
Internal Couple Method

26. Bending Behavior of Beams
Internal Couple Method

27. Bending Behavior of Beams
Plastic Moment
Stress varies linearly from neutral axis to extreme fibers.
When moment increases there will also be linear increase in moment and stress until yield.
When moment increases beyond yield moment the outer fiber will have the same stress but will yield.
The process will continue with more and more parts of the beam x-section stressed to yield point until finally a fully plastic distribution is approached.

28. Bending Behavior of Beams
Plastic Moment

29. Bending Behavior of Beams
Plastic Moment

30. Bending Behavior of Beams
Plastic Moment

31. Bending Behavior of Beams
Plastic Moment

32. Bending Behavior of Beams
Plastic Moment

33. Bending Behavior of Beams
Plastic Moment

34. Bending Behavior of Beams
Plastic Moment
Progression of Yield Zone Leading to Fully Plastic Hinge and Collapse
Stresses reach Yield Magnitude at extreme fibers
Yield Zones spreads towards Neutral axis
Yield Zones join, are now spread through entire x-section

35. Bending Behavior of Beams
Plastic Hinges
The effect of plastic hinge is assumed to be concentrated at one section for analysis purpose.
However, it should be noted that this effect may extend for some distance along the beam.

36. Bending Behavior of Beams
Plastic Moment

37. Bending Behavior of Beams
Plastic Modulus
The resisting moment at full plasticity can be determined in a similar manner.
The result is the so-called plastic moment Mp.

38. Bending Behavior of Beams
Plastic Modulus b d
d/2 Fy

39. Bending Behavior of Beams
Plastic Modulus
The plastic moment is equal to the yield stress Fy times the Plastic modulus Z.
From the foregoing expression for a rectangular section, the plastic modulus Z can be seen equal to bd2/4.

40. Bending Behavior of Beams
Shape Factor
The shape factor which is equal to
So, for rectangular section the shape factor is equal to 1.5

41. Bending Behavior of Beams
Shape Factor

42. Bending Behavior of Beams
Shape Factor

43. Bending Behavior of Beams
Neutral Axis for Plastic Condition
The neutral axis for plastic condition is different than its counterpart for elastic condition.
Unless the section is symmetrical, the neutral axis for the plastic condition will not be in the same location as for the elastic condition.
The total internal compression must equal the total internal tension.

44. Bending Behavior of Beams
Neutral Axis for Plastic Condition
As all the fibers are considered to have the same stress Fy in the plastic condition, the area above and below the plastic neutral axis must be equal.

45. Bending Behavior of Beams
Plastic Modulus

46. Bending Behavior of Beams
Plastic Modulus: Unsymmetrical Shape
The areas above and below the neutral axis must be equal for Plastic analysis

47. Bending Behavior of Beams
Plastic Modulus: Assignment
Determine the yield moment My, the Plastic Mp and the plastic modulus Z for the simply supported beam having the x-section as given.
Also calculate the shape factor.
Calculate nominal load Pn acting transversely through the mid span of the beam. Assume the Fy=36 ksi

48. Bending Behavior of Beams
Advantages of Plastic Design

49. Bending Behavior of Beams
Advantages of Plastic Design
There is 50% increase in strength above the computed elastic limit (stage !) due to plasticization of the x-section

50. Bending Behavior of Beams
Advantages of Plastic Design: Wide Flange Section
My = Fy S
Mp = Fy Z

51. Bending Behavior of Beams
Advantages of Plastic Design
Shape factor is one source of reserve strength beyond elastic limit.

52. Bending Behavior of Beams
Advantages of Plastic Design: Shape Factors
Mp/My
f / f y

53. Bending Behavior of Beams
Advantages of Plastic Design
Another source of reserve strength in indeterminate structure loaded beyond the elastic limit is called re-distribution of moments.that is the process of moment transfer due to successive formation of plastic hinges which continues until ultimate load is reached.

54. Bending Behavior of Beams
Advantages of Plastic Design 1 2 3

55. Bending Behavior of Beams
Advantages of Plastic Design 1 2 3
Deflection
Load Wu

56. Thanks

57. Design of Steel Beams
The development of a plastic stress distribution over the cross-section can be hindered by two different length effects:
Lateral Torsional buckling of the unsupported length of the beam/member before the cross-section develops the plastic moment Mp.
Local buckling of the individual plates (flanges and webs) of the cross-section before they develop the compressive yield stress Fy.

58. Lateral Torsional Buckling
A simply supported beam can be subjected to gravity transverse loading.
Due to this loading the beam will deflect downward and its upper part will be placed in compression and hence will act as compression member.

59. Lateral Torsional Buckling
Beams are generally proportioned such that moment of inertia about the major principal axis is considerably larger than that of minor axis.
This is done to make Economical Beams.

60. Lateral Torsional Buckling
As result they are weak in resistance to Torsion and Bending about the Minor axis.
If its Y-axis is not braced perpendicularly, it will buckle laterally at much smaller load than would otherwise have been required to produce a vertical failure.

61. Lateral Torsional Buckling

62. Lateral Torsional Buckling

63. Lateral Torsional Buckling

64. Lateral Torsional Buckling

65. Lateral Torsional Buckling
A laterally unsupported compression flange will behave like a column and tend to buckle out of plane between points of lateral support. However because the compression flange is part of a beam x-section with a tension zone that keeps the opposite flange in line, the x-section twists when it moves laterally. This behavior is referred to as lateral torsion buckling. Simply it is a sidewise buckling of beam accompanied by twist.

66. Lateral Torsional Buckling
Embedment of top flange in concrete slab provides lateral support to beam, except when the beam is cantilever.
Lateral bracing will be adequate (both for strength & stiffness) if each lateral brace is designed for 2% of compressive force in the flange of beam it braces.( this thumb rule is based on lab test results).

67. Lateral Torsional Buckling
Consider a doubly symmetric prismatic beam
Both ends simply supported w.r.t x & y axis but
Held against rotation about z-axis.
It is subjected to a uniform bending moment Mx

68. Lateral Torsional BucklingMx
Moment at which Lateral Torsional buckling begins is given by:
Mn = Mcr =

69. Lateral Torsional Buckling
The critical moments for beams with end moments and beams with transverse loads acting through shear center can be given by
Where Cb is a coefficient which depends on variation in moments along the span and K is an effective length coefficient depending on restraint at supports.
Values of Cb and K are given in table 5-1

70. Lateral Torsional Buckling
Values of Cb developed by curve fitting to data from numerical analysis of LTB of simple beams acted only by end-moments is given by:
Cb= (M1/M2) + 0.3(M1/M2)2  2.3
Another equation obtained by working on numerical test data of beam-column behaviour is
Cb= 1/ [0.6 – 0.4(M1/M2) ]  2.3
Where M1 is smaller of two end moments.
M1/M2 is +ve for reverse curvature. M1 M2

71. Lateral Torsional Buckling
Accurate equation for Cb, if moment diagram within the un braced length is not a straight line

72. Inelastic LTB
If stress is proportional to strain, Mx,cr for elastic LTB is valid as given.
But for critical stress, Fcr exceeding Fy, Mx,cr is given by

73. Inelastic LTB
The equation can be solved in a simplified manner by using an equivalent radius of gyration which is obtained by equating the critical bending stress to the tangent modulus critical stress for columns

74. Local Buckling of Beam Elements
Concept of
Compact,
Non-Compact, And
Slender Elements and Sections.

75. Local Buckling of Beam Elements
For establishing width-thickness ratio limits for elements of compression members, the LRFD classification divides members into three distinct classification as follows.
Compact
Non-compact
Slender

76. Local Buckling Compact Elements lp lr Mp
If the slenderness ratio (b/t) of the plate element is less than lp, then the element is compact. It will locally buckle much after reaching Fy lp Mp lr Mr

77. Local Buckling Non-compact Elements lp lr Mp
If the slenderness ratio (b/t) of the plate element is less than lr but greater than lp, then it is non-compact. It will locally buckle immediately after reaching Fy lp Mp lr Mr

78. Local Buckling Slender Elements lp lr Mp
If the slenderness ratio (b/t) of the plate element is greater than lr then it is slender. It will locally buckle in the elastic range before reaching Fy lp Mp lr Mr

79. Local Buckling Compact Sections Non-compact Sections Slender sections
A section that can develop fully plastic moment Mp before local buckling of any of its compression element occurs.
Non-compact Sections
A section that can develop a moment equal to or greater than My, but less than Mp, before loca buckling of any of its element occurs.
Slender sections
If any one plate element is slender, the section is slender.

80. Local Buckling Important Note
Thus, slender sections cannot develop Mp due to elastic local buckling. Non-compact sections can develop My but not Mp before local buckling occurs. Only compact sections can develop the plastic moment Mp.
All rolled wide-flange shapes are compact with the following exceptions, which are non-compact.
W40x174, W14x99, W14x90, W12x65, W10x12, W8x10, W6x15 (made from A992)

81. Local Buckling contd;
If the beam x-section is to develop the yield moment My, the compression flange must be able to reach yield stress and the web/webs, must be able to develop corresponding bending stresses.
Local Buckling of the flange and/ or web can prevent these limits from being attained.
More restrictive limits must be observed if a beam x-section is to attain the fully plastic moment Mp.

82. Local Buckling
For uniformly compressed laterally simply supported on one unloaded edge and free on the other, the critical stress is
Plated used in structural members are long enough to warrant neglecting the second term, so 2

83. Local Buckling
Following limits of late slenderness (b/t) which preclude premature local buckling of compression flange of beams are available.
Projecting Element
Flange of Box
Since these limits are not well defined, they differ somewhat from one specifications to another refer table 5-3

84. Local Buckling
Limiting values of beam flange and web slenderness