1. CTC 422 Design of Steel Structures
Beams - Flexure

2. Objectives of Structural Design
Structure is adequate to support loads which will be applied during its life
Strength provided ≥ strength required
Structure will meet serviceability requirements
Deflection
Vibration
Structure will meet functional requirements
Structure will meet economic requirements

3. Beam Design Student Objectives
Analyze a beam to calculate load, shear, moment and deflection and to determine if a given beam is adequate
Design (select) a beam to safely to support a load considering moment, shear and deflection

4. Beam Design
Beam
A structural member which carries loads applied perpendicular to its longitudinal axis
These loads cause shear and bending (moment)
Different terms used for beams depending on application or location
Girder, stringer, joist, lintel, spandrel, purlin, girt
Behavior of all is the same.
All are beams

5. Load and Resistance Factor Design - LRFD
Design strength ≥ Required strength
ΦRn ≥ Ru
For bending
Φb Mn ≥ Mu
Where:
Mn = Nominal moment strength
Φb = Strength reduction factor for bending = 0.9
Mu = Required moment strength based on factored loads

6. Load and Resistance Factor Design - LRFD
Nominal moment capacity, Mn, depends on the failure mechanism of the beam
Beam can fail by:
Full yielding of the cross-section
Lateral torsional buckling (LTB)
Can be inelastic or elastic buckling
Flange local buckling (FLB)
Web local buckling (WLB)
Failure mechanism is related to:
Lateral bracing of the beam
Whether or not the beam cross-section is compact

7. Failure Mechanism and Nominal Moment Capacity, Mn
If beam remains stable up to its full plastic moment capacity
Failure is by yielding of the full section
Mn = Mp
Instability could be overall beam instability
Lateral torsional buckling (elastic or inelastic)
Prevented by adequate lateral bracing of the beam’s compression flange
Instability could also be local instability
Flange local buckling or web local buckling
Dependent on width / thickness ratios of compression elements
Compactness, non-compactness or slenderness of section

8. Compactness
Structural shapes are classified as compact, non-compact, or slender
Compact
Section reaches its full strength (yield) before local buckling occurs
Strength of section is governed by material strength
Non-compact
Only a portion of the cross-section reaches its full strength (yield) before local buckling occurs
Slender
Cross-section does not yield before local buckling occurs
Strength is governed by buckling
Compactness, non-compactness, or slenderness is a property of the cross-section itself
A function of the width / thickness ratios of its flanges and its web
Flange width / thickness = bf / 2tf
Web width / thickness = h / tw

9. Compactness Classification is given in Table B4.1
Notation:
λ = width / thickness ratio
λp = upper limit for compact category
λr = upper limit for non-compact category
If λ ≤ λp and the flange is continuously attached to the web, the shape is compact
If λp ≤ λ ≤ λr, the shape is non-compact
If λ > λr, the shape is slender
Category is based on the worst width / thickness ratio
Example: If web is compact and flange is non-compact, section is classified as non-compact
Most standard W, M, S, and C sections are compact
A few are non-compact because of their flanges, but none are slender

10. Bending Strength of Compact Shapes
Moment strength of a compact shape is a function of, Lb, the unbraced length of its compression flange
Lb – distance between points braced against lateral displacement of compression flange
Lp – limiting laterally unbraced length for limit state of yielding
Lr – limiting laterally unbraced length for limit state of inelastic lateral torsional buckling
Compression flange may be braced by:
Perpendicular framing
Steel roof deck or floor deck
Concrete slab
Cross-bracing

11. Bending Strength of Compact Shapes
If the compression flange is continuously braced (Lb ≤ Lp)
Failure will be by yielding at full plastic moment
Nominal moment capacity, Mn = Mp = Fy Zx (AISC Eq. F2-1)
Design strength Φb Mn = Φb Mp
For unbraced length Lb > Lp
Failure will be by inelastic lateral torsional buckling
Nominal moment capacity, Mn < Mp
At Lb = Lp, Mn = 0.7 Fy Sx
For Lp < Lb < Lr , linear interpolation from Mn = Mp to Mn = 0.7 Fy Sx (AISC Eq. F2-2)
For unbraced length Lb > Lr
Failure will be by elastic lateral torsional buckling
Rapid reduction in Mn (AISC Eq. F2-3)

12. Bending Strength of Non-compact Shapes
Most standard W, M, S, and C sections are compact
A few are non-compact because of their flanges, but none are slender
Shapes with noncompact flanges are listed in User note on page
Sections with compact webs and noncompact (or slender) flanges
Nominal moment capacity, Mn < Mp
Calculate Mn using provisions of Code Section F3
Sections with noncompact webs
Calculate Mn using provisions of Code Section F4

13. Design Aids – Braced Beams
Table 3-2, W-Shapes – Selection by Zx
Applies to wide flange shapes with Fy = 50 ksi
Applies mainly to sections which are adequately braced (Lb ≤ Lp)
Can be used for unbraced length up to Lb = Lr
Best to use this table only if fully braced
Table lists Zx, Lp, Lr, and Moment Capacity, Φb Mp
Also lists Ix, and Shear Capacity Φv Vnx
Non-compact sections indicated by the footnote “f”
Moment capacity in table has been adjusted for non-compactness
Sections in table are grouped by weight
Lightest section in group is in bold
Choose this section if there is no depth restriction

14. Design Aids – Unbraced Beams
Table 3-10, Available Moment vs. Unbraced Length
Applies to wide flange shapes with Fy = 50 ksi
Also applies to channel shapes with Fy = 36 ksi
Table is a plot of available flexural strength, ΦbMnx, versus unbraced length Lb
Bending Coefficient in Table conservatively taken as Cb = 1
See Table 3-1 for values of Cb
Choose beam that has available moment strength ΦbMnx ≥ Mu at an unbraced length Lb ≥ Design Lb
Choose a beam above and to right of (Lb, Mu)
Solid line – Beam chosen is lightest section available for the given combination of Mu and Lb
Dashed line – A lighter section is available

15. Design Aids – Channels Braced Channels Unbraced Channels
Table 3-8, Maximum Total Uniform Load – C Shapes
Applies to channel shapes with Fy = 50 ksi
Applies only to sections which are adequately braced (Lb ≤ Lp)
Best to use this table only if fully braced
Table lists Zx, Lp, Lr, and Moment Capacity, Φb Mp
Also lists Shear Capacity Φv Vnx
Unbraced Channels
Table 3-10, Available Moment vs. Unbraced Length
Applies to channel shapes with Fy = 36 ksi