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Dept. of Civil & Urban Eng., iOTec-HU. Hawassa University

Bending Members

The most frequently used structural element for bending is the beam. Beams which carry loads from floors or other beams to the columns are generally called main beams. Secondary beams will be provided to transfer load to the main beams, or in some cases just to give lateral stability to columns, while themselves carrying only their self weight. The manner in which loads are distributed from the floors on to the beams needs careful consideration so that each beam is designed for a realistic proportion of the total load.

Beams can be laterally restrained. Concrete floor slabs and wall or roof cladding, are generally able to give this lateral support or restraint. Timber floors and open steel floors are less certain in providing restraint. Alternatively, lateral restraint may be provided by bracing members at specific points along the beams.

Lateral torsional instability will be prevented if

Adequate bracing or floor slab restraint is present

The section has a high torsional stiffness, e.g. a rectangular hollow section.

An understanding of the behavior of struts will be useful in appreciation the behavior of beams where full lateral restraint is not provided. The compression flange of such members will show a tendency to fail by buckling sideways (laterally) in the most flexible plane. Design factors which will influence the lateral stability can be summarized as:

The length of the member between adequate lateral restraints. The shape of the cross-section. The variation of moment along the beam. The form of end restraint provided

The manner in which the load is applied, i.e. to tension or compression flange.

The buckling resistance (Mb) of a beam may be found by use of a number of parameters and factors:

Effective length (Le), which allows for the effects of end restraint, as well as type of beam, and the existence of destabilizing forces.

Minor axis slenderness () , which includes lateral stiffness in the form of iy, and is defined by ( = . Torsional index ((), which is a measure of torsional stiffness of a cross-section. Slenderness factor ((), which allows for torsional stiffness and includes the ration of .

Slenderness correction factor (n), which is dependent on the moment variation along the beam. Buckling parameter (u), which allows for the section type and includes a factor for warping. Equivalent slenderness (LT), which combines the above parameters and from which the bending strength (Pb) may be derived:LT = nu( In addition, an equivalent moment factor (m) is used which allows for the effect of moment

variation along the beam.

Resistance to Pure Bending

In the absence of shear force, the design value of the bending moment Msd at each cross-section shall satisfy:

For bending about one axis in the absence of shear force, the design moment resistance of a cross-section without holes for fasteners may be determined as follows.(a) Class 1 or 2 cross-sections:

(b) Class 3 cross-sections:

(c) Class 4 cross-sections:

Where:

Wpl = Plastic section modulus.

Wel = Elastic section modulus.

Weff = Effective section modulus.

Fastener holes in the tension flange need not be allowed for, provided that for the tension flange:

Where: Af = Area of flange When is less than the limit in above, a reduced flange area may be assumed which satisfies the limit. Fastener holes in the tension zone of the web need not be allowed for, provided that the limit given in above is satisfied for the complete tension zone comprising the tension flange plus the tension zone of the web. Fastener holes in the compression zone of the cross-section need not be allowed for, except for oversize and slotted holes.

Resistance to Shear The design value of the shear force Vsd at each cross-section shall satisfy:

Where: - the design plastic shear resistance.

Av = the shear area.

Cross-sectionLoad directionShear area, Av

Rolled H sectionsParallel to web

Rolled channel sectionsParallel to web

Welded L, H, and box sectionsParallel to web

Welded L, H, channel and box sectionsParallel to flanges

Rolled RHS of uniform thicknessParallel to depth

Parallel to breadth

CHS and tubes of uniform thickness____

Plates and solid bars____A

Where: A = cross-sectional area r = root radius

b = overall depth tf = flange thickness

d = depth of the web tw = web thickness

h = overall depth

Fastener holes need not be allowed for in shear verification provided that:

Where Av,net is less than this limit, an effective shear area of (fu/fy)Av,net may be assumed.

Resistance to Bending and Shear When Vsd exceeds 50% of Vpl,Rd the design resistance moment of the cross-section should be reduced to Mv,Rd the reduced design plastic resistance moment allowing for the shear force, obtained as follows:

(a) For cross-sections with equal flanges, bending about the major axis:

(b) For other cases Mv,Rd should be taken as the design plastic resistance moment of the cross-section, calculated using a reduced strength (1 - ()fy fot the shear area, but not more than Mc,Rd. Appropriate value of Mc,Rd should be used for class 1, 2, 3 and 4 cross-sections.Lateral Torsional BucklingBeams are generally proportioned so that the moment of inertia about the major principal axis is considerably larger than that about the miner principal axis. As a result, they are relatively weak in resistance to torsion and bending about the minor axis, and if not laterally restrained, they may become unstable under load. The instability manifests itself as a sidewise bending accompanied by twist and is called lateral buckling or lateral torsional buckling. The design buckling resistance moment:

where: (w = 1.0 for class 1 and class 2 cross-sections.

(w = for class 3 cross-section.

(w = for class 4 cross-section.

and the value of (LT is the reduction factor for lateral torsional buckling.

(LT = 0.21 (curve a) for rolled sections.

(LT = 0.49 (curve c) for welded sections. (LT can be obtained from table 2.2.2 with and ( = (LT.

where:

Mcr - the elastic critical moment for lateral-torsional buckling.

For beams with doubly symmetric cross-sections and with end-moment loading and for transverse loads applied at the shear center:

where: c1 is given in table 2.3.1

k = refers to end rotation on plan. Varies from 0.5 (full fixity) to 1.0 (no fixity).

kw = refers to end warping (kw = 1.0).

Iw = warping constant.

It = torsion constant.

G = shear modulus.

for doubly symmetrical cross-sections

hs - distance between shear centers of the flanges.

hs = h - tf

Figure 2.3.1 Load distribution excluding self weight

PAGE 6___________________________________________________________________

Structural Engineering IV (CEng 3312) Chapter 5

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