Explanation:

Solid circular section is shown below.

Moment of area A₁ and A₂ about neutral axis gives Z_p.

Also, A₁ = A₂ = $\frac{\mathrm{A}}{2}=\frac{\pi {\mathrm{D}}^2}{8}$

Hence, ${\mathrm{Z}}_{\mathrm{p}}=2\times \left(\frac{\pi {\mathrm{D}}^2}{8}\right)\times \frac{4}{3\pi }\times \frac{\mathrm{D}}{2}$

= $\frac{{\mathrm{D}}^3}{6}$

Now, Z about neutral axis is given by

$\mathrm{Z}=\frac{\pi {\mathrm{D}}^3}{32}$

So, shape factor

= $\frac{{\mathrm{Z}}_{\mathrm{p}}}{\mathrm{Z}}=\frac{16}{3\pi }$

Mechanism for given beam will be

W × $\frac{L}{2}\theta$ = M_p × θ + M_p × 2θ + M_p × θ

W = $\frac{8\mathrm{M}\mathrm{p}}{L}$

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

Explanation:

Load factor in plastic design:

• Load factor is ratio of ultimate collapse load to working load that can be applied on structure.
• Load Factor = Factor of safety x Shape factor.
• As factor of safety at member level largely depends upon nature of loading, support conditions and mode of failure, so load factor also depends upon those factors.
• The shape factor of a member solely depends upon geometry of cross sectional area, so load factor also depends on geometrical shape of member.
• Normally a load factor of 1.7 to 2 is used in plastic design method.

Explanation:

According to IS 1893:2002, Clause 6.3.1, following load combinations are allowed in earthquake resistant design

1) Load factor for plastic design of steel structures

• 1.7 (DL + IL)
• 1.7 (DL ±  EL)
• 1.3 (DL + IL ± EL)

2) Partial safety factor for limit state design of reinforced concrete and prestressed structure

• 1.5 (DL+ IL)
• 1.2 (DL + IL ± EL)
• 1.5 (DL ± EL)
• 0.9 DL ± 1.5 EL

Concept-

When a structure is subjected to a system of loads, it is stable and hence functional until a sufficient number of plastic hinges have been formed to render structure unstable. As soon as structure reaches an unstable condition, it is considered to have been failed. The segments of beam between plastic hinges are able to move without an increase of load. This condition in a member is called mechanism.

If an indeterminate structure has redundancy r, insertion of r plastic hinges makes it statically determinate. Any furr hinge converts this statically determinate structure into mechanism. Hence for collapse number of plastic hinges required are r + 1.

Number of independent mechanism-

The number of independent mechanism can be determined as follows,

Let N = Number of possible plastic hinges, r = Number of redundancies

Then number of possible independent mechanism will be N - r.

Given data and Calculation-

Possible location of plastic hinges → 4. (A, B, C, D).

At E, No plastic hinge will form as it is already a hinge joint.

Redundancy of structure = 2.

No. of independent mechanism = 4 – 2 = 2.

(Beam mechanism & Sway mechanism)

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:

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:

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,

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