Methods of Structural Analysis MCQ Quiz - Objective Question with Answer for Methods of Structural Analysis - Download Free PDF

Explanation:

Stiffness (S): It is defined as the force per unit deflection or moment required per unit rotation.

Case 1:

For the far end being Hinged Supported or freely supported:

If a unit rotation is to be caused at an end A for the far end being hinged support. Moment of 3EI/L is to be applied at end and hence stiffness for the member is said to be 3EI/L

Case 2:

For the far end being Fixed Supported:

If a unit rotation is to be caused at an end A for the far end being fixed supported. Moment of 4EI/L is to be applied at end and hence stiffness for the member is said to be 4EI/L

Case 3: when the far end is free

Beam offers no resistance to the moment.

S = 0

 Concept: 

The principle of virtual work for deformable bodies says that the external virtual work applied to a structure must equal the internal virtual work that is caused within the structure,  Wv,e = Wv,i

External Virtual Work = Internal Virtual Work

Virtual External Forces×  Real external deflections = Virtual Internal forces × Real Internal Deformations

Explanation:

• In the Moment Distribution Method, the Distribution Factor (DF) for a member is a measure of how the joint's moment is distributed among the various connected members.
• It helps in determining the proportion of the joint's moment that will be distributed to each member.

The formula to calculate the Distribution Factor (DF) for a member is:

 \(DF = \frac{K_i}{\sum K} \)

Where:

• Ki K_i" id="MathJax-Element-28-Frame" role="presentation" style="position: relative;" tabindex="0">KiKi K_i" id="MathJax-Element-1-Frame" role="presentation" style="position: relative;" tabindex="0">Ki$K_i$ $K_i$ $K_i$  is the stiffness of the considered member at the joint.

• ∑K \sum K" id="MathJax-Element-29-Frame" role="presentation" style="position: relative;" tabindex="0">∑K∑K \sum K" id="MathJax-Element-2-Frame" role="presentation" style="position: relative;" tabindex="0">∑K$\sum K$ $\sum K$ $\sum K$  is the total stiffness of all the members meeting at the joint.

 Additional Information

1. The Moment Distribution Method is an approximate iterative technique used to analyze indeterminate structures, especially continuous beams and frames, that cannot be solved easily using simple static equilibrium equations.
2. It was developed by Dr. Hardy Cross in the early 20th century.
3. The method works by redistributing the bending moments in a structure in such a way that equilibrium is gradually achieved through an iterative process.

Key Concepts in the Moment Distribution Method:

1. Fixed-End Moment (FEM):
   The moment at the ends of a beam or frame when both ends are assumed to be fixed, considering external loads applied to the structure. It is an initial moment, calculated without considering the flexibility of the joints.

2. Distribution Factor (DF):
   The factor used to distribute the moment at a joint to the connected members, based on the stiffness of the members. It represents the proportion of the joint's moment each member will take.

3. Carry-Over Factor:
   After distributing the moment, the unbalanced moment at the joint is "carried over" to the connected members in the next iteration. The carry-over factor is typically 0.5 0.5" id="MathJax-Element-30-Frame" role="presentation" style="position: relative;" tabindex="0">0.50.5 0.5" id="MathJax-Element-3-Frame" role="presentation" style="position: relative;" tabindex="0">0.5$0.5$ $0.5$ $0.5$ for beams with simple supports.

4. Convergence:
   The iterative process continues until the moment values at the joints do not change significantly between iterations. This indicates that the structure is in equilibrium, and the final moment distribution is achieved.

Explanation:

• The Slope-Deflection Method is more efficient than the Force Method because it reduces the number of equations and unknowns.
• It directly works with the displacement (rotation and deflection) at the supports, which simplifies the solution process when compared to the Force Method, where you need to deal with unknown forces and compatibility equations.

 Additional InformationSummary of Slope-Deflection Method 

• Used for analyzing indeterminate beams and frames.

• Based on the relationship between end moments of members and the rotations and displacements at their ends.

• Primary unknowns are joint rotations and displacements, not internal forces.

• Uses slope-deflection equations to express member end moments in terms of rotations, deflections, and fixed-end moments.

• Moment equilibrium is applied at joints to solve for unknown rotations.

• Involves fewer equations than the Force Method, making it more efficient for analyzing complex frames.

• Best suited for structures with multiple spans, frames with or without lateral sway, and beams under various loading conditions.

• Can become tedious for very large or highly complex structures, where matrix methods are preferred.

• Commonly introduced in structural analysis as a foundational method before moving to advanced techniques.

Explanation:

• Clapeyron's Theorem of Three Moments is a method used in structural analysis to calculate the bending moments at the supports of a continuous beam.
• It is a fundamental tool in analyzing statically indeterminate structures, specifically for continuous beams with multiple spans.

 Additional InformationClapeyron's Theorem of Three Moments

• Purpose:

  • It is a method used to calculate the bending moments at the supports of a continuous beam.

  • Helps in analyzing statically indeterminate structures.

• Conditions:

  • Applicable to continuous beams (i.e., beams with more than two supports).

  • Requires knowledge of the beam's span lengths, external loads, and reactions.

• Theorem Statement:

  • The theorem relates the bending moments at three consecutive supports of a continuous beam and provides an equation for each of the internal moments.

Mohr’s Theorem I:

The angle between the two tangents drawn on the elastic line is equal to the area of the Bending Moment Diagram between those two points divided by flexural rigidity.

\(\theta = \frac{{\left[ {Area\;of\;bending\;moment\;diagram} \right]}}{{EI}}\)

Mohr’s Theorem II:

The deviation of a point away from the tangent drawn from the other point is given by the moment of area of bending moment diagram about the first point divided by flexural rigidity.

\(\delta = BB' = \frac{{\left[ {Area\;of\;bending\;moment\;diagram} \right] \times \bar x}}{{EI}}\)

I^st condition:

For a cantilever beam subjected to load W at distance of L/3 from free end, the deflection is given by:

\({\rm{y}}{{\rm{c}}_1} = \frac{{\rm{W}}}{{3{\rm{EI}}}} \times {\left( {\frac{{2{\rm{L}}}}{3}} \right)^3} + \frac{{\rm{W}}}{{2{\rm{EI}}}}{\left( {\frac{{2{\rm{L}}}}{3}} \right)^2} \times \frac{{\rm{L}}}{3} = {\frac{{28{\rm{WL}}}}{{162{\rm{EI}}}}^3}\)

II^nd condition:

For a cantilever beam subjected to load W at distance of 2L/3 from free end:

\({\rm{y}}{{\rm{c}}_2} = \frac{{\rm{W}}}{{3{\rm{EI}}}} \times {\left( {\frac{{\rm{L}}}{3}} \right)^3} + \frac{{\rm{W}}}{{2{\rm{EI}}}}{\left( {\frac{{\rm{L}}}{3}} \right)^2} \times \frac{{2{\rm{L}}}}{3} = \frac{{8{\rm{W}}{{\rm{L}}^3}}}{{162{\rm{EI}}}}\)

\({\rm{Ratio\;}}\left( {\rm{r}} \right) = \frac{{{\rm{y}}{{\rm{c}}_1}}}{{{\rm{y}}{{\rm{c}}_2}}} = \frac{{{{\frac{{28{\rm{WL}}}}{{162{\rm{EI}}}}}^3}}}{{\frac{{8{\rm{W}}{{\rm{L}}^3}}}{{162{\rm{EI}}}}}} = \frac{{28}}{8} = \frac{7}{2}\)

\(\therefore \frac{{{\rm{y}}{{\rm{c}}_2}}}{{{\rm{y}}{{\rm{c}}_1}}} = \frac{2}{7}\)

Explanation:

• The column analogous method is the force method used to analyze the indeterminate structure.
• In the method of column analogous, the actual structure is considered under the action of applied loads and the redundant acting simultaneously.The load on the top of the analogous column is usually the B.M.D. due to applied loads on simple spans and therefore the reaction to this applied load is the B.M.D. due to redundant on simple spans.
• It is based on the analogy between moments at ends and pressure at the edges of the short column. If both ends of the beam are fixed then the end moments are called fixed end moments.

• An analogous column will be a short column with cross-section dimensions L and 1/EI.​

so, area of analogous column = L/EI

Concept:

• The moment distribution method is best suited for the rigid 2D frame as for rigid 2D frame because only one kind of moment (Mz) is acting at every joint of the structure the moment developed at the joint is distributed to all connected members at that particular joint.
• The moment distribution method only takes into account the moment effect not the axial force effect.
• As trusses are designed to carry axial force, so it is not desired to do moment distribution for pin-jointed truss or trussed beam.
• For space frame, there are three kinds of the moment (Mx, My & Mz) acting at every joint. So, it is very difficult and inconvenient to do moment distribution for spaced frame.

Explanation:

As we know that 

Fixed end moments correspond to reactions at support such that the slope at supports will be zero 

As conjugate beam method 

The slope at supports is equal to the reaction at supports of the conjugate beam.

So we can predict that the reaction ay support of the conjugate beam is inversely proportional to the moment of inertia (MOI)

Greater MOI  of the beam, lesser will be support reaction of conjugate beam hence lesser will be the slope.

So slope decreased with an increase in MOI the moment required to make slope zero will be lesser means the fixed end moment will increase. 

Concept:

When in addition to equilibrium equations, compatibility equations are used to evaluate the unknown reactions and internal forces in any structure, such analysis is called indeterminate analysis, and such structure is called indeterminate structure.

We have two distinct methods of analysis for such an indeterminate structure:

a. Force method of analysis

b. Displacement method of analysis

Difference between force method and Displacement method:

Concept:-

The given figure shows the fixed end moments under given loading:-

The fixed end moment at end A

\(M_A=-{wL^2\over 30}\)

\(M_B={wL^2\over 20}\)

\(\frac{{{M_A}}}{{{M_B}}}\; = {{wL^2\over 30}\over{wL^2\over 20}} \;\frac{2}{3}\)

Hence, the ratio of the moment will be 2/3.

\({M_\theta }\; = \frac{W}{{16}} \times \left( {{R^2}\left( {3 + \mu } \right) - {r^2}\left( {1 + 3\mu } \right)} \right)\)

Where M_θ = Circumferential moment

R = radius of slab

μ = Poisson’s ratio

r = any section at a distance r from the centre of the slab

W = load on circular slab

For the maximum circumferential moment at centre

r = 0 and μ = 0

Explanation:

fixed end moments

Fixed end

\(M_{FAB}= \frac{{Pab(L + b)}}{{2{L^2}}}\)

So \(M_{FAB} = \frac{-{Pab(L + b)}}{{2{L^2}}}\)

Negative sign as anticlockwise rotation is taken

Concept:

The FEM’s when a beam is subjected to point load and clear loading such as uDL is given below:

Calculation:

Given:

a = 2m

b = 3m

L = a + b = 5m

P = 50 kN

\(M_{AB}^F = - \frac{{P{b^2}a}}{{{L^2}}} = - \frac{{50\; \times\; {{\left( 3 \right)}^2}\; \times \;2}}{{{{\left( 5 \right)}^2}}} = - 36\;kNm\;\left( \uparrow \right)\)

(-ve sign means that \(M_{AB}^F\) is in Anticlockwises

\(M_{AB}^F = + \frac{{P{a^2}b}}{{{L^2}}} = \frac{{50\; \times \;{{\left( 2 \right)}^2}\;\times\; 3}}{{{{\left( 5 \right)}^2}}} = + 24\;kN\;M\left( \downarrow \right)\)