Design of Beams Theory | Structural Design

* Plastic neutral axis divides the cross section into two equal areas.

* When plastic hinge is formed at all fibres at the section all fibres are having yield stress, with

opposite nature on either side of N–A

Zp =

 , for rectangular sections.

* The sections are classified as class 1 (plastic), class 2 (compact) and class (3 semi-compact)

classification of various sections into class 1 to class 3 may be found in Table 2 of IS 800–2007.

Design procedure

1. Select a trial section assuming it is going to be plastic section.

2. Find the class to which it belongs.

3. Check for bending strength, shear strength and deflection. Revise the section if necessary.

Bending strength

If ,

 two cases

(a) If V £ 0 .6 Vd

Md = bb

fy

× for simply supported beams

= for cantilever beam

where bb = 1.0 for plastic and compact sections

= for semi-compact sections

(b) If V > 0.6 Vd

(i) Plastic or compact sections

Mdv =

where b =

Md = Plastic design moment of the whole section

Mfd = plastic design strength of the area of the cross section excluding the shear area

considering partial safety factor gmo

(ii) Semi-compact section:

Mdv =

Shear strength of a laterally supposed beams:

Vd =

where Av = shear area fyw = yield strength of web. Shear area is given

 By

(i) I and channel sections:

Major axis bending: Hot rolled, Av = h tw

Welded, Av = d tw

Minor axis bending Av = 2b t

f

(ii) Rectangular hollow sections:

Loaded parallel to depth: Av =

Loaded parallel to width Av =

(iii) Circular: Hollow tubes Av =

(iv) Solid bars and plates Av = A

Deflection limits: Refer Table 6 in IS 800–2007.

Web Buckling Strength Certain portion of beam at support acts as a column to transfer the load from

beam to support and hence there is a chance for buckling of web. In this case dispersion angle of

beam may be taken as 45°. There is no need to check it for rolled section since the web thickness

are sufficient to avoid such buckling failures web buckling strength is given by

fcdw = (b1 + n1

) tw fc

where b1 = width of stiff bearing on the flange

n = which h is the depth of section

fc = allowable compressive strength ratio corresponding to slenderness ratio

= 

Web crippling Near the support web of the beam may cripple due to lack to bearing capacity.

Crippling occurs at the root of the radius.

Fw = 

where b1 = stiff bearing length

nc = length obtained by assuring dispersion at a slope 1 in 2.5

fyw = yield strength of the web

* IS Table No. 15 gives effective length for simple beams with different end conditions.

Design of Purlins

* The effective length may be taken as centre-to-centre distance between the supports.

* Bending moment

M = in case of simply supported beams

= in case of continuous purlins.

* Design procedure:

1. Resolve factored forces parallel and perpendicular to sheeting

2. Find moment and shear forces about z – z and y – y axis.

3. Zpz

required = gmo

where gmo = 1.1

4. Check for shear capacity

Vdz =

Vdy =

where AVz = h tw and Avy = 2bf

t

f

5. Compute the design capacity of the section in both axes.

Mdz =

Mdy =

6. The condition to be satisfied is

7. Check for deflection. Simplified method of design of purlins: It assumes that the load normal to

sheeting is resisted by purlin and the load parallel to sheeting is resisted by sheeting, if

1. Roof slope is less than 30°

2. Width of angle leg perpendicular to sheeting

3. Width to angle leg parallel to sheeting

In such case bending moment about z – z axis should be taken as and there is need to check for

deflection.