Steel Design Test 1

As per IS 800: 1984, Clause 3.8.2

For structural member exposed to weather directly and non-accessible for painting, the minimum thickness of structural member its 8 mm.

Using Unwin’s equation:

\(\phi \ge 6.04{\rm{\;}}\sqrt {{{\rm{t}}_{{\rm{mm}}}}} \) 

\(\therefore {\phi _{{\rm{min}}}} \ge 6.04 \times \sqrt 8 \ge 17.08 \approx 18{\rm{\;mm}}\)

Concept:

The throat thickness of the welded connection can be written as:

t = KS

Where,

K is the factor that depends on the angle between fusion faces.

S is the size of the weld

For t to be minimum, the size of the weld is to be minimum and the minimum size of the weld, as per IS 800:2007 codal provisions, depends  on the  thickness of the thicker plate being connected as shown in below tabulated form:

For ‘t’ to be maximum, the size of the weld is to be maximum and as per IS 800: 2007, the maximum Size of the weld for square edge is given as:

S_max = Thickness of plate - 1.5

Calculation:

As per IS 800:2007, if the angle between fusion faces lies between 91-100^o then K = 0.65.

Thickness of thicker plate = 12 mm ⇒ S­_min = 5 mm

Minimum throat thickness, t_min = 0.65 × 5 = 3.25 mm

Maximum size of weld, S_max = max (12, 8) – 1.5 = 10.5 mm

Maximum throat thickness, t_max = 0.65 × 10.5 = 6.825 mm

For the calculation of Design wind load on structure IS 875-1987 part-III the relates the intensity of

wind pressure to the basic maximum basic wind speed Vb in m/sec. This wind speed is modified to

include risk level, terrain roughness, height and size of structure and local topography,

The design wind velocity Vz at any height for the structure is given below

Vz = k1. k2. k3.Vb

k1 = Risk coefficient or probability coefficient

k2 = Terrain, height and structure size factor.

Using standard empirical formula for this:

\(\frac{M}{{{M_y}}} = {\rm{\;Shape\;Factor}} - \frac{1}{{2{\rm{\;}}{{(\frac{{{\rm{Strain\;at\;yielding}}}}{{{\rm{Strain\;beyond\;yielding}}}})}^2}}}\)

For rectangular section, Shape factor = 1.5

Strain at yielding = ε_y

Strain beyond yielding = 2ε_y

\(\frac{{\rm{M}}}{{{{\rm{M}}_{\rm{y}}}}} = {\rm{}}1.5 - \frac{1}{{2{{(\frac{{{ \in _{\rm{y}}}}}{{2{ \in _{\rm{y}}}}})}^2}}}\)

\(\frac{{\rm{M}}}{{{{\rm{M}}_{\rm{y}}}}} = {\rm{}}1.5 - \frac{1}{8} = 1.375\;\)

Or

M = 1.375M_y

Concept

IS 800:2007 classified the cross-sections into four classes depending upon the material yield strength and width to thickness ratio of the individual components within the cross-section and loading arrangement. The characteristics of these four classes are given below:

Plastic or Class I section:   

1. It can develop a plastic hinge and collapse mechanism.

2. They are fully effective in pure compression and capable of reaching the full plastic moment in bending and hence, used in plastic design.

Compact or Class 2 Section:

1. It can form a plastic hinge but does not have the capacity to develop a collapse mechanism because of local buckling.

2. They have lower deformation capacity but also fully effective in pure compression and are capable of reaching their full plastic moment in bending.

Semi- Compact or Class 3 Section:

1. It has the capacity to develop yield moment only and local buckling is liable to prevent the development of the plastic moment resistance.

2. They are also fully effective in pure compression but local buckling prevents the attainment of the full plastic moment in bending.  Bending moment resistance in these cross-sections is limited to yield moment only.

Slender or Class 4 Section:

1. This section fails before reaching the yield stress i.e. local buckling will occur even before the attainment of the yield stress in extreme fiber.

ISA 90 × 90 × 10 mm

Diameter of rivet, (d) = 18 mm

Diameter sectional area, A_g = (90 + 90 - 10) × = 1700 mm

For Fe 410 grade steel,

f_u. = 410 Mpa, γ_m1 = 1.25, f_y = 250 Mpa, γ_mo = 1.1

Tensile strength due to gross section yielding,

\({T_{dg}} = \frac{{{A_g}\; \times \;{f_y}}}{{{\gamma _{mo}}}} = \frac{{1700\; \times \;250}}{{1.1}} = 386.36kN\) 

Concept:

The moment capacity of the base plate is given by,

\(M_d = 1.2 \times \frac{{{f_y}}}{{{\gamma _{m0}}}} \times {Z_e}\) 

Calculation:

Given: f_y = 250 mpa, γ_m0 = 1.1

\(M_d = 1.2 \times \frac{{250}}{{1.1}} \times \left( {\frac{{1 \times b }}{6}} \right)\) 

\(\therefore \;M_d = 1.2 \times \frac{{250}}{{1.1}} \times \frac{{1\; \times \;{{30}^2}}}{6} = 40.91kNm\) 

The design of battened column is prescribed under section 7.7 of IS 800 : 2007. As per

1) 7.7.1.3 – The number of battens shall be of a minimum of three bays members.

2) 7.7.1.4 – The effective slenderness ratio of should be taken 1.1 times the actual slenderness ratio.

3) 7.7.2.1 – Battens should be designed to carry 2.5% of the axial load of its transverse shear capacity.

4) 7.7.2.3 – The thickness of the batten or the tie plates shall not be less than one-fiftieth of the distance between innermost connecting lines of connection.

5) 7.7.3 – The spacing shall be such that the slenderness ratio shall neither be greater than 50 nor greater than 0.7 × slenderness ratio of member as a whole.

Hence ii, iii and iv are incorrect statements.

As per IS 800:2007, CL 8.6.1.2, in order to avoid buckling of the compression flange into the web, web thickness shall satisfy the following:

1) When transverse stiffeners are not provided:

\(\frac{d}{{{t}_{w}}}\le 345\epsilon _{f}^{2}\)

2) When transverse stiffeners are provided: (C = spacing between stiffners)

a. When C ≥ 1.5d

\(\frac{d}{{{t}_{w}}}\le 345\epsilon _{f}^{2}\)

b. When C < 1.5 d

Concept

As per IS 800: 2007, Cl- 3.8, the Slenderness ratio of steel sections is limited to avoid buckling in compression members and its value is 180 for pure compression members.

Now, to predict in which direction buckling will occur, it is desired to determine the actual slenderness ratio about both major and minor axis.

The slenderness ratio is the ratio of effective length and radius of gyration i.e.

\(\lambda = \frac{{{l_{eff}}}}{r}\)

Further, the effective length under various conditions as per IS 800:2007 is given below:

Calculation

Case 1: Buckling about major axis

Column is restrained in position but no in direction at both ends i.e. its effective length

L_e1 = 1.00 L = 1 × 15 = 15 m

Radius of gyration about major axis, r₁ = 50 mm

Slenderness ratio about major axis, λ₁ = 15000/50 =  300

λ_max = 180 for pure compression member

Since the actual value of slenderness exceeds the maximum permitted value of SR ⇒ Buckling may occur about the major axis.

Case 1: Buckling about the minor axis

the column has fixed against rotation and translation at both ends i.e. its effective length

L_e1 = 0.50 L = 0.5 × 15 = 7.5 m

The radius of gyration about the minor axis, r₂ = 20 mm

Slenderness ratio about major axis, λ₂ = 7500/20 =  375

λ_max = 180 for pure compression member

Since the actual value of slenderness exceeds the maximum permitted value of Slenderness Ratio ⇒ Buckling may also occur about the minor axis.

∴ Buckling may occur either about the major axis or minor axis.

Concept:

As per clause number 8.2 of IS 800: 2007, the non-dimensional lateral torsional slenderness ratio is given by,

\({{\rm{\lambda }}_{{\rm{LT}}}} = \sqrt {\frac{{{{\rm{\beta }}_{\rm{b}}}{{\rm{Z}}_{\rm{p}}}{{\rm{f}}_{\rm{y}}}}}{{{{\rm{M}}_{{\rm{cr}}}}}}}\) 

Where, \({{\rm{\lambda }}_{{\rm{LT}}}}{\rm{\;}}\)is the non-dimensional lateral torsional slenderness ratio

\({{\rm{\beta }}_{\rm{b}}} = 1{\rm{}}\) for plastic section

\({{\rm{Z}}_{\rm{p}}}{\rm{}}\) is plastic sectional modulus

\({{\rm{f}}_{\rm{y}}}{\rm{}}\) is yield stress of the material   

\({{\rm{M}}_{{\rm{cr}}}}{\rm{}}\) is elastic critical moment corresponding to lateral torsional buckling of the beam 

Calculation:

Given, \({{\rm{\beta }}_{\rm{b}}} = 1,{\rm{\;}}{{\rm{Z}}_{\rm{p}}} = 3948812{\rm{\;m}}{{\rm{m}}^3},{\rm{\;}}{{\rm{M}}_{{\rm{cr}}}} = 16866{\rm{\;}} \times {\rm{\;}}{10^6}{\rm{\;N}} - {\rm{mm}},{\rm{\;}}{{\rm{f}}_{\rm{y}}} = 250{\rm{\;N}}/{\rm{m}}{{\rm{m}}^2}\)

\(\therefore {\rm{\;}}{{\rm{\lambda }}_{{\rm{LT}}}} = \sqrt {\frac{{1\; \times \;3948812\; \times \;250}}{{16866{\rm{\;}} \times {\rm{\;}}{{10}^6}}}} = 0.242\) 

∴ The non-dimensional lateral torsional slenderness ratio will be nearly 0.242.

Concept:

The design bending strength of such section is given by:

M_d = β_bZ_pf_y/γ_mo

Where

β_b = 1.0 (for plastic & compact section) or Z_e/Z_p (for semi-compact section)

Z_e = Elastic section modulus of the cross section

Z_p = Plastic section modulus of the cross section

f_y = yield stress of the material

γ_mo = 1.1, the partial safety factor

The design strength obtained from the above equation should be for less than 1.2 Z_ef_y/γ_mo for simply supported beam and 1.5 Z_ef_y/γ_mo for cantilever beams.

Calculation:

For laterally supported beam:

Bending capacity (M_d­) is given by:

\({{\text{M}}_{\text{d}}}={{\text{ }\!\!\beta\!\!\text{ }}_{\text{b}}}\times {{\text{Z}}_{\text{pz}}}\times \frac{{{\text{f}}_{\text{y}}}}{{{\text{ }\!\!\gamma\!\!\text{ }}_{\text{mo}}}}\)

For plastic section, β_b = 1

\(\therefore {{\text{M}}_{\text{d}}}=1\times 851.11\times {{10}^{3}}\times \frac{250}{1.1}=193.43~\text{kNm}\)

Check: \({{\text{M}}_{\text{d}}}<1.2{{\text{Z}}_{\text{e}}}\frac{{{\text{f}}_{\text{y}}}}{{{\text{ }\!\!\gamma\!\!\text{ }}_{\text{mo}}}}\)

\(1.2{{\text{Z}}_{\text{e}}}\frac{{{\text{f}}_{\text{y}}}}{{{\text{ }\!\!\gamma\!\!\text{ }}_{\text{mo}}}}=1.2\times 751.9\times {{10}^{3}}\times \frac{250}{1.1}=~205.06~\text{kNm}\)

∵ 193.43 < 205.06 ⇒ Safe

Concept:

The gross-sectional area of the beam (A) is given, by the relationship:

\({\rm{A}} = \frac{{2{{\rm{M}}_{\rm{z}}}}}{{{\rm{d}}{{\rm{f}}_{\rm{y}}}}} + \frac{{{{\rm{d}}^2}}}{{\rm{k}}}\)

The optimum value of d may be obtained by differentiation the above equation with respect to d and equating it to zero.

\(\frac{{\partial \left( {\rm{A}} \right)}}{{\partial {\rm{d}}}} = \frac{\partial }{{\partial {\rm{d}}}}\left( {\frac{{2{{\rm{M}}_{\rm{z}}}}}{{{\rm{d}}{{\rm{f}}_{\rm{y}}}}}} \right) + \frac{\partial }{{\partial {\rm{d}}}}\left( {\frac{{{{\rm{d}}^2}}}{{\rm{k}}}} \right)\)

\(0 = \frac{{ - 2{{\rm{M}}_{\rm{z}}}}}{{{{\rm{d}}^2}{{\rm{f}}_{\rm{y}}}}} + \frac{{2{\rm{d}}}}{{\rm{k}}}\)

\(\frac{{2{{\rm{M}}_{\rm{z}}}}}{{{{\rm{d}}^2}{{\rm{f}}_{\rm{y}}}}} = \frac{{2{\rm{d}}}}{{\rm{k}}}\)

\({{\rm{d}}^3} = \frac{{{{\rm{M}}_{\rm{z}}}{\rm{k}}}}{{{{\rm{f}}_{\rm{y}}}}}\)

\({\rm{d}} = {\left( {\frac{{{{\rm{M}}_{\rm{z}}}{\rm{k}}}}{{{{\rm{f}}_{\rm{y}}}}}} \right)^{0.33}}\)

Optimum depth of plate girder \(\left( {\rm{d}} \right) = {\left( {\frac{{{{\rm{M}}_{\rm{z}}}{\rm{k}}}}{{{{\rm{f}}_{\rm{y}}}}}} \right)^{0.33}}\)

\({\rm{d}} = {\left( {\frac{{12000 \times {{10}^6} \times 180}}{{250}}} \right)^{0.33}} = 2051.97 {\rm{\;mm}} = 2.05{\rm{\;meters}}\)