Concept:

Shape Factor: It is defined as ratio of moment carrying capacity of plastic section over that of elastic section when yielding just starts.

$\text{Shape Factor}=\frac{Z_P{\sigma }_y}{Z_e{\sigma }_y}=\frac{M_P}{M_y}=\frac{Z_P}{Z_e}$

Shape factor values for different sections:

Calculation:

Plastic modulus of section ZP = 5 × 10-4 m3

SF = 1.2

Plastic moment Capacity MP = 120 kNm

Plastic moment capacity is given by,

$M_P=Z_P{\sigma }_y$

Yield Stress is given by,

${\sigma }_y=\frac{M_P}{Z_P}=\frac{120\times 1000}{5\times {10}^{-4}}=240\times {10}^{6}N/m^2=240MPa$

Explanation:

Shape factor:

The shape factor is defined as ratio of fully plastic moment to yield moment of section. Shape factor depend upon cross-section.

Some standard shape factors are as follows:

Explanation:

Shape factor

• It is defined as  ratio of plastic moment to yield moment.
• It is a function of cross-section or shape and is represented by ' f '.
• $f=\frac{M_P}{M_y}=\frac{f_y{Z}_{Pz}}{f_y{Z}_{ez}}=\frac{Z_{Pz}}{Z_{ez}}$

Shape factor for different cross-sectional shapes:

Calculation:

Given, Bending moment = 590.63 KN-m

Partial safety factor = 1.10 and Yield strength of steel (f_y) = 250 N/mm2

Plastic moment(M_p), 

M_p = Bending moment × F.O.S

      = 590.63 × 1.1

      = 649.693 KN-m

Mp = 649.693 × 10⁶ N-mm

Relation between Plastic moment and plastic section modulus of a beam,

Mp = fy × Zp 

649.693 × 106 = 250 × Zp 

Z_p = 2.59877 × 106 mm³

Zp = 2598.77 cm³.

Concept:

• Moment of inertia about neutral axis X-X of a triangular cross-section is,
• $I_{NA}=\frac{\mathrm{b}{\mathrm{h}}^3}{36}$
• The section modulus (Z) of cross-sectional shape is significant in designing beams.
• It is a direct measure of strength of beam.
• A beam that has a larger section modulus than anor will be stronger and capable of supporting greater loads.
• To calculate Z, distance (Y) to extreme fibers from centroid (or neutral axis) must be found as that is where maximum stress could cause failure.
• $\begin{array}{l}\frac{M}{I}=\frac{\sigma }{y}=\frac{E}{R}\\ Z=\frac{I}{Y_{max}}=\frac{M}{\sigma }\end{array}$
• Now, section modulus about neutral axis is,
• $Z_{NA}=\frac{{\mathrm{I}}_{\mathrm{N}\mathrm{A}}}{{\mathrm{Y}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}$

where Ymax = Distance of extreme edge from neutral axis

Since distance between top edge and neutral axis is $\frac{2\mathrm{h}}{3}$ while distance between bottom edge and neutral axis of triangle is $\frac{\mathrm{h}}{3}$

Therefore, $Y_{max}=\frac{2\mathrm{h}}{3}$

$Z_{NA}=\frac{\left(\frac{\mathrm{b}{\mathrm{h}}^3}{36}\right)}{\left(\frac{2\mathrm{h}}{3}\right)}$

$Z_{NA}=\frac{\mathbf{b}{\mathbf{h}}^2}{24}$

Concept:

Plastic moment capacity for simply supported beam with UDL;

$M_p=\frac{w{l}^2}{8}$

Calculation:

Given;

w = 25 KN/m

l= 6 m

From above equation:

$M_p=\frac{25\times 6^2}{8}$ = 112.5

M_p = 112.5 KN - m

Concept:

Imperfection Factor: It takes into account, imperfection that may occur while load transferring, fabrication, or installation.

It depends upon shape of column cross-section under consideration, direction in which buckling can occur, and fabrication process (hot rolled, welded or cold-formed).

Classification of different sections under different buckling class i.e. a, b, c or d used for design of axial compression member.

As per IS 800: 2007, Table 7;

∴ The imperfection factor for buckling class b is 0.34

Concept:

Imperfection Factor: It takes into account, imperfection that may occur while load transferring, fabrication, or installation.

It depends upon shape of column cross-section under consideration, direction in which buckling can occur, and fabrication process (hot rolled, welded or cold-formed).

Classification of different sections under different buckling class i.e. a, b, c or d used for design of axial compression member.

As per IS 800: 2007, Table 7;

∴ The imperfection factor for buckling class C is 0.49

Explanation:

Shape factor:

The shape factor is defined as ratio of fully plastic moment to yield moment of section. Shape factor depend upon cross-section.

Some standard shape factors are as follows:

Explanation:

Shape Factor:

$\mathrm{S}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{e}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}=\frac{\mathrm{P}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{u}\mathrm{s}}{\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{u}\mathrm{s}}=\frac{\mathrm{P}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{m}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}}{\mathrm{E}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{m}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}}$

$f=\frac{M_P}{M_y}=\frac{f_y{Z}_{Pz}}{f_y{Z}_{ez}}=\frac{Z_{Pz}}{Z_{ez}}$

The shape factor of standard rolled beam section varies from 1.10 to 1.20.

It is a function of cross-section or shape and is represented by ' f '

Shape factor for different cross-sectional bar are as follows:

Load factor = Shape factor × FOS

Margin of safety = FOS - 1