1. contents Design of beams (week 11,12,13), (10,17,24 Nov.)
Design of members for combined loads(week 14), (1 Dec.)
Steel structures project design (week 15)(8 Dec.)
Final Exam (week 16), (15 Dec.)

2. Design of beams
TYPES OF BEAMS

3. Steel W-sections for beams and columns
Closer to square
Thicker web & flange
Source: University of Michigan, Department of Architecture
Beams:
Deeper sections
Flange thicker than web

4. In the design of beams, the following considerations are necessary:
 REVIEW OF BEAM THEORY
In the design of beams, the following considerations are necessary:
1.       Bending stresses
2.       Shearing stresses
3.       Local buckling
4.       Lateral torsional buckling
5.       Web crippling
6.       Deflection

5. 1. Deflections are small and change of geometry is negligible.
Elastic bending of beams
In elastic theory of bending, the following assumptions are usually made:
1.       Deflections are small and change of geometry is negligible.
2.       Plane sections remain plane after bending.
3.       Shear deformations are small
4.       Interaction of axial forces and bending is negligible.
5.       Buckling and stability of the beam is not a problem.
6.       The material is linearly elastic; that is, the stress is proportional to the strain.

6. My is the moment corresponding to first yield and Mp is the plastic moment capacity of the cross-section.

7. The ratio of Mp to My is called as the shape factor f for the section.
For a rectangular section, f is equal to 1.5. For a wide-flange section, f is equal to 1.1.
Calculation of Mp: Cross-section subjected to either +sy or -sy at the plastic limit. See Figure below.

9. fMp = 0.90 Z Fy
My = Fy S for homogenous cross-sections
= Fyf S for hybrid sections.
where,
Mp = plastic moment
My = moment corresponding to onset of yielding at the extreme fiber from an elastic stress distribution
Z = plastic section modulus from the Properties section of the AISC manual.
S = elastic section modulus, also from the Properties section of the AISC manual.

10. Example 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? Assume 50 ksi steel.

11. Shear Stresses
Shear stresses are usually not a controlling factor in the design of beams, except for the following cases:
1.       The beam is very short.
2.       There are holes in the web of the beam. These holes may be for passing electrical and mechanical ducts or for increasing the bending strength in case of castellated beams.
3.       The beam is subjected to a very heavy concentrated load near one of the supports.
4.       The beam is coped.

17. LOCAL BUCKLING

19. Compact section if all elements of cross-section have λ λ p
Non-compact sections if any one element of the cross-section has λ p λ  λ r
Slender section if any element of the cross-section has λ rλ

20. 2. Using torsionally strong sections (for example, box sections)
 LATERAL TORSIONAL BUCKLING
The compression flange of a beam behaves like an axially loaded column. Thus, in beams covering long spans the compression flange may tend to buckle. This tendency, however, is resisted by the tension flange to a certain extent. The overall effect is a phenomenon known as lateral torsional buckling, in which the beam tends to twist and displace laterally. Lateral torsional buckling may be prevented through the following provisions:
1.       Lateral supports at intermediate points in addition to lateral supports at the vertical supports
2.       Using torsionally strong sections (for example, box sections)
3.       I-sections with relatively wide flanges

22. Lp = 1.76 ry x unbraced length Lb a plastic length Lp
If the laterally unbraced length Lb is less than or equal to a plastic length Lp then lateral torsional buckling is not a problem and the beam will develop its plastic strength Mp.
Lp = 1.76 ry x
unbraced length Lb
a plastic length Lp

23. Mn = Mcr =
where, Mn = moment capacity
Lb = laterally unsupported length.
Mcr = critical lateral-torsional buckling moment.
E = ksi; G = 11,200 ksi
Iy = moment of inertia about minor or y-axis (in4)
J = torsional constant (in4) from the AISC manual.
Cw = warping constant (in6) from the AISC manual.

24. The deflection limit is:
Flexural Deflection of Beams
The deflection limit is:
Use service loads to check deflections.
Plastered floor construction – L/360
Unplastered floor construction – L/240
Unplastered roof construction – L/180

25. Example 2 Design a simply supported beam subjected to uniformly distributed dead load of 450 lbs./ft. and a uniformly distributed live load of 550 lbs./ft. The dead load does not include the self-weight of the beam. The length of the beam is 30 ft.

26. Example 3 Design the beam shown below
Example 3 Design the beam shown below. The unfactored dead and live loads are shown in the Figure.