Plastic Analysis MCQ Quiz - Objective Question with Answer for Plastic Analysis - Download Free PDF

Explanation:

Shape factor:

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

Some standard shape factors are as follows:

Triangular	2.34
Rectangular	1.5
Circular	1.7
Diamond	2
Hollow Circular	1.27

Explanation:

Plastic Moment Condition in Steel Structures

In the plastic method of analysis of steel structures, the term "plastic moment condition" is used to describe the situation where the moment at a section reaches the plastic moment capacity of the section. This means that the entire cross-section has yielded and is capable of sustaining the plastic moment. This condition is critical for the formation of plastic hinges, which are essential for the structure to undergo plastic deformations and redistribute moments. The plastic moment condition is specifically related to:

• Yield Condition

Analyzing the Given Options

1. "Mechanism condition." (NOT the correct answer)

   • The mechanism condition refers to the formation of a collapse mechanism in the structure, which happens after the plastic moment condition is reached but is not the same as the plastic moment condition itself.

2. "Yield condition." (Correct answer)

   • The yield condition is precisely what the plastic moment condition is about. It signifies that the section has yielded completely and can sustain the plastic moment.

3. "Continuity condition." (NOT the correct answer)

   • The continuity condition refers to the requirement that the structure remains continuous and unbroken. It is not directly related to the plastic moment condition.

4. "Equilibrium condition." (NOT the correct answer)

   • The equilibrium condition ensures that the internal forces in the structure are balanced with the external loads. While important, it is not specifically what is meant by the plastic moment condition.

Concept:

The plastic section modulus (Zp" id="MathJax-Element-1-Frame" role="presentation" style="position: relative;" tabindex="0">ZpZp $Z_p$ ) and elastic section modulus (Ze" id="MathJax-Element-2-Frame" role="presentation" style="position: relative;" tabindex="0">ZeZe $Z_e$ ) are related by the shape factor (SF" id="MathJax-Element-3-Frame" role="presentation" style="position: relative;" tabindex="0">SFSF $SF$ ) as follows:

\( SF = \frac{Z_p}{Z_e} \)

Given Data:

• Elastic section modulus (\( Z_e \)) = 66,000 mm³
• Shape factor (\( SF \)) = 1.2

Step 1: Formula for Plastic Section Modulus

Rearranging the formula:

\( Z_p = SF \times Z_e \)

Step 2: Substituting the Given Values

\( Z_p = 1.2 \times 66000 \)

\( Z_p = 79200 \:mm^3\)

Explanation:

• 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 the structure unstable. As soon as the structure reaches an unstable condition, it is considered to have been failed. T
• he segments of the beam between the 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, the insertion of r plastic hinges makes it statically determinate. Any further hinge converts this statically determinate structure into mechanism. Hence for collapse the 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 =  N - r.

Given,

Plastic Hinges (N) = 4, Degree of indeterminacy (D_s) = 2

∴ Number of possible independent mechanism =  4 - 2 = 2 

The degree of static indeterminacy (n) indicates the number of additional equations required to solve the structure's equilibrium. For a structure to become a mechanism (collapse), it needs a sufficient number of plastic hinges to reduce its degrees of freedom to zero. The necessary number of plastic hinges to form a collapse mechanism is given by the degree of indeterminacy plus one (n+1). This concept is crucial in plastic analysis for determining the load-carrying capacity of structures.

Concept:

Shape Factor: It is defined as the ratio of moment carrying capacity of the plastic section over that of the elastic section when the 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:

Shape Factor	Section
1.5	Rectangular
1.7	Circular
2.34	Triangular

 

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,

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

Explanation:

Shape Factor:

\(\rm{Shape\;factor = \frac{{Plastic\;section\;modulus}}{{Section\;modulus}} = \frac{{Plastic\;moment}}{{Elastic\;moment}}}\)

\(f = \frac{{{M_P}}}{{{M_y}}} = \frac{{{f_y}{Z_{Pz}}}}{{{f_y}{Z_{ez}}}} = \frac{{{Z_{Pz}}}}{{{Z_{ez}}}}\)

The shape factor of the 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 the different cross-sectional bar are as follows:

Load factor = Shape factor × FOS

Margin of safety = FOS - 1

Concept:

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

It depends upon the shape of the column cross-section under consideration, the direction in which buckling can occur, and the 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 the design of axial compression member.

As per IS 800: 2007, Table 7;

Buckling Class	a	b	c	d
α	0.21	0.34	0.49	0.76

 

∴ The imperfection factor for buckling class b is 0.34

Explanation:

Shape factor:

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

Some standard shape factors are as follows:

Triangular	2.34
Rectangular	1.5
Circular	1.7
Diamond	2
Hollow Circular	1.27

Concept:

Plastic moment capacity for the 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

Mechanism for the given beam will be

W × \(\frac{L}{2}θ \) = M_p × θ + M_p × 2θ + M_p × θ

W = \(\frac{{8\;{\rm{Mp}}}}{L}\)

Explanation:

Shape factor

• It is defined as the ratio of the 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:

Section	Shape factor
Triangular	2.34
Diamond	2
Rectangle / Square	1.5
I - Section	1.12 to 1.14
H - Section	1.5
Circular hollow section	1.27
Solid Circular section	1.67

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 the structure unstable. As soon as the structure reaches an unstable condition, it is considered to have been failed. The segments of the beam between the 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, the insertion of r plastic hinges makes it statically determinate. Any further hinge converts this statically determinate structure into mechanism. Hence for collapse the 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 the structure = 2.

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

(Beam mechanism & Sway mechanism)

Explanation:

Load factor in plastic design:

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

Explanation

In beam mechanism, the hinges will form at B, E & C.

So,

Let Plastic moment will be M_P.

Internal work done = External work done

\(\theta =\dfrac{\Delta }{2}\Rightarrow \Delta = 2\theta\)

M_P × θ + M_P × 2θ + M_P × θ = 40 × Δ

⇒ 4 M_Pθ = 80 × θ

→ M_P = 20 kNm.