# Restrained beams part 1

##### OBJECTIVE/SCOPE

To derive and discuss the procedures used to design restrained compact beams for bending, shear, and deflection according to the principles of Eurocode 3 [1].

##### RELATED WORKED EXAMPLES

Worked example 7.8: Laterally restrained beams

##### SUMMARY

This lecture is restricted to beams whose design may be based on simple strength of materials considerations. Behaviour in simple bending is discussed, leading to the concept of section modulus as the basis for strength design. Subsidiary considerations of shear strength, resistance to local loads, and adequate stiffness against deflection are also mentioned. Behaviour under complex loading, producing bending about both principal axes, and combined bending and torsion are introduced.

##### ADDITIONAL NOTATION

A area

A_{v} shear area

d depth of section

F applied load

f_{d} limiting stress in material

f_{y} material yield strength

f_{yd} material design strength

I second moment of area

M moment

M_{pl} plastic moment

t thickness

t_{f} thickness of flange

t_{w} thickness of web

V_{s} shear due to applied load

W elastic section modulus

W_{max} maximum value of W

W_{min} minimum value of W

W_{pl} plastic section modulus

α,β coefficients (see Equation (7))

σ normal stress

## Related content

### 1. INTRODUCTION

Probably the most basic structural component is the beam, spanning between two supports and transmitting loads principally by bending action. Steel beams, which may be drawn from a wide variety of structural types and shapes, can often be designed using little more than the simple theory of bending. However, situations will arise in which the beam's response to its loading will be more complex with the result that other forms of behaviour must also be considered. The main purpose of this lecture is to concentrate on the design of that class of steel beam for which strength of materials forms the basis of the design approach. These are termed 'restrained compact beams'. In order to fall within this category, the beam must not be susceptible to either local instability (see the Cross-section classification for Class 4 beams lecture). A further limitation is that the beams are assumed to be statically determinate or, if statically indeterminate, that the distribution of internal bending moments has been obtained on a simple linear elastic basis.

The first requirement will be met if the width of the individual plate elements of the cross-section are limited relative to their thickness. Sections for which the ratios of flange width/flange thickness, web depth/web thickness, etc. have been limited so that their full plastic moment capacity may be achieved correspond to either Class 1 or Class 2 according to Eurocode 3 [1]. The majority of hot rolled sections, e.g. HEs and IPEs, meet these requirements. However, care is necessary when using fabricated welded sections and suitable rules are provided in codes of practice. This lecture assumes that the beam cross-section will at worst correspond to Class 2 or Class 3 and that, if it meets the Class 1 limits, then the global analysis of the structure will be conducted using plastic methods.

Lateral-torsional instability will not occur if any of the following conditions apply:

- the section is bent about its minor axis
- full lateral restraint is provided, e.g. by positive attachment of the top flange of a simply supported beam to a concrete slab
- closely spaced, discrete bracing is provided so that the weak axis slenderness (L/i
_{z}) of the beams is low - adequate torsional restraint of the compression flange is provided, e.g. by profiled sheeting
- the section has high torsional and lateral bending stiffness, e.g. rectangular box sections bent about their major axis are unlikely to fail in this way

For the special case of continuous beams supporting a roof or floor, care must be taken to ensure adequate stability of those regions in which the bottom flange is in compression, e.g.the support region under gravity loading, the mid-span region under wind uplift, etc. For the purpose of this lecture, beams within any of these categories will be classed as 'restrained'.

Read more