1. Design of Beams for Flexure

2. Design of Beams for Flexure
Introduction
Moment Curvature Response
Sectional Properties
Serviceability Requirements (Deflections)
Compact, Non-compact and Slender Sections
Lateral Torsional Buckling
Design of Beams

3. Beams under Flexure
Members subjected principally to transverse gravity loading
Girders (important floor beams, wide spacing)
Joists (less important beams, closely spaced)
Purlins (roof beams, spanning between trusses)
Stringers (longitudinal bridge beams)
Lintels (short beams above window/door openings)

4. Design for Flexure Limit states considered Yielding
Lateral-Torsional Buckling
Local Buckling
Compact
Non-compact
Slender

5. Design for Flexure Commonly Used Sections:
I – shaped members (singly- and doubly-symmetric)
Square and Rectangular or round HSS
Tees and Double Angles
Rounds and Rectangular Bars
Single Angles
Unsymmetrical Shapes

6. Section Force-Deformation Response & Plastic Moment (MP)
A beam is a structural member that is subjected primarily to transverse loads and negligible axial loads.
The transverse loads cause internal SF and BM in the beams as shown in Fig. 1
SF & BM in a SS Beam

7. Section Force-Deformation Response & Plastic Moment (MP)
These internal SF & BM cause longitudinal axial stresses and shear stresses in the cross-section as shown in the Fig. 2
Curvature =  = 2/d (Planes remain plane)
Longitudinal axial stresses caused by internal BM

8. Section Force-Deformation Response & Plastic Moment (MP)
Steel material follows a typical stress-strain behavior as shown in Fig 3 below. E = 200 GPa
Typical steel stress-strain behavior.

9. Section Force-Deformation Response & Plastic Moment (MP)
If the steel stress-strain curve is approximated as a bilinear elasto-plastic curve with yield stress equal to y, then the section Moment - Curvature (M-) response for monotonically increasing moment.
In Fig. 4, My is the moment corresponding to first yield and Mp is the plastic moment capacity of the cross-section.
The ratio of Mp to My - the shape factor f for the section.
For a rectangular section, f = 1.5. For a wide-flange section, f ≈ 1.1.

10. Moment-Curvature
Beam curvature  is related to its strain and thus to the applied moment e y 
(1)
(2)
(3)
(4)

11. Moment-Curvature When the section is within elastic range
Where S is the elastic section modulus
When the moment exceeds the yield moment My
Then
Where Z is the plastic section modulus S

12. Ex. 4.1 – Sectional Properties
Determine the elastic section modulus, S, plastic section modulus, Z, yield moment, My, and the plastic moment MP, of the cross-section shown below. What is the design moment for the beam cross-section.
300 mm
15 mm
400 mm
10 mm
25 mm
400 mm

13. Ex. 4.1 – Sectional Properties
Ag = 300 x 15 + ( ) x x 25 = mm2
Af1 = 300 x 15 = 4500 mm2
Af2 = 400 x 25 = mm2
Aw = 10 x ( ) = 3600 mm2
distance of elastic centroid from bottom =
Ix = 400x253/ ( )2 + 10x3603/ ( ) x153/ ( )2 = 503.7x106 mm4
Sx = 503.7x106 / ( ) = x103 mm3
My-x = Fy Sx = kN-m.
Sx - elastic section modulus

14. Ex. 4.1 – Sectional Properties
distance of plastic centroid from bottom =
y1 = centroid of top half-area about plastic centroid
= mm
y2 = centroid of bottom half-area about plas. cent.
= mm
Zx = A/2 x (y1 + y2) = 9050 x ( ) = mm3
Zx - plastic section modulus

15. Ex. 4.1 – Sectional Properties
Mp-x = Zx Fy = x 344/106 = kN.m
Design strength according to AISC Spec. F1.1= bMp= 0.9 x = kN.m
Check = Mp  1.5 My
Therefore, kN.m < 1.5 x = kN.m - OK!

16. Flexural Deflection of Beams - Serviceability
The AISC Specification gives little guidance other than a statement, “Serviceability Design Considerations,” that deflections should be checked. Appropriate limits for deflection can be found from the governing building code for the region.
The following values of deflection are typical max. allowable deflections.
LL DL+LL
Plastered floor construction L/360 L/240
Unplastered floor construction L/240 L/180
Unplastered roof construction L/180 L/120
DL deflection – normally not considered for steel beams

17. Local Buckling of Beam Section – Compact and Non-compact
Mp, the plastic moment capacity for the steel shape, is calculated by assuming a plastic stress distribution (+ or - y) over the cross-section.
The development of a plastic stress distribution over the cross-section can be hindered by two different length effects:
Local buckling of the individual plates (flanges and webs) of the cross-section before they develop the compressive yield stress y.
Lateral-torsional buckling of the unsupported length of the beam / member before the cross-section develops the plastic moment Mp.
The analytical equations for local buckling of steel plates with various edge conditions and the results from experimental investigations have been used to develop limiting slenderness ratios for the individual plate elements of the cross-sections.

18. Local Buckling of Beam Section – Compact and Non-compact
Local buckling of flange due to compressive stress (s)

19. Local Buckling of Beam Section – Compact and Non-compact
Steel sections are classified as compact, non-compact, or slender depending upon the slenderness (l) ratio of the individual plates of the cross-section.
Compact section if all elements of cross-section have   p
Non-compact sections if any one element of the cross-section has p    r
Slender section if any element of the cross-section has r  
It is important to note that:
If   p, then the individual plate element can develop and sustain y for large values of e before local buckling occurs.
If p    r, then the individual plate element can develop y at some locations but not in the entire cross section before local buckling occurs.
If r  , then elastic local buckling of the individual plate element occurs.

20. Classification of Sections
Classifications of bending elements are based on limits of local buckling
The dimensional ratio l represents
Two limits exist p and r
p represents the upper limit for compact sections
r represents the upper limit for non-compact sections

21. Local Buckling of Beam Section – Compact and Non-compact
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.
Stress-strain response of plates subjected to axial compression and local buckling.