The matrix method of Structural Analysis is based on replacing indeterminate structure by

The Maxwell’s reciprocal theorem applies to

1. beam only
2. truss only
3. both (A) and (B)
4. none of the above

Option 3 : both (A) and (B)

Explanation:

Maxwell's reciprocal theorem

• It is a technical relationship that equates two separate distortions in an elastic structure under load
• Maxwell's rule is one of the basic tools of structural engineering.
• Maxwell's reciprocal theorem can be applied to beams and any linear elastic body (Truss, Suspension Bridges), including surfaces.
• It doesn't just apply to displacements but also to rotations produced by torques.
• Maxwell's reciprocal theorem says that the deflection at i due to a unit load at J is the same as the deflection at J if a unit load was applied at i. In our notation, Y1 = Y2.

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In moment distribution method the sum of distribution factors for all the members meeting at a joint is always

1. Equal to zero
2. Equal to one
3. Greater than one
4. Smaller than one

Concept:

Distribution factor (D.F.):

• The factor by which the applied moment is multiplied to obtain the end moment of any member is known as the distribution factor (D.F.)
• It is numerically equal to the ratio of the stiffness (or relative stiffness) of the member to the total stiffness (or total relative stiffness) of all the members meeting at a joint.

Here, Joint stiffness = Σ Member stiffness meeting at the joint

• The summation of D.F at a joint where different members are meeting is equal to one.

Additional Information

Moment distribution method:

It was invented to analyze the indeterminate structures by Hardy Cross in 1930. In this method, the internal moments at the joints are distributed and balanced until the joints have rotated to their final or nearly final position.

• It is a displacement method of analysis
• The process is repetitive and easy to apply
• In the moment distribution method, the carryover moment is equal to one-half of its corresponding moment with the same sign.

The steps regarding the moment distribution method are;

1. Find the fixed-end moments
2. Determine the distribution factor [For a member the distribution factor is = , k = Stiffness]
3. Moment balancing by Moment distribution table
4. drawing Bending Moment diagram by super-imposition.

​Absolute stiffness of that member:

• Stiffness for a member at a joint is the moment (or force) required to produce unit rotation (displacement) at that joint.
• Stiffness of a member if the farther end is fixed = 4EI/L
• Stiffness of a member if the farther end is hinged = 3EI/L
• Stiffness of a member if the farther end is free = 0
• The stiffness factor when the far end is guided roller is EI/L

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Flexibility matrix method of a analysis is basically

1. Force method
2. Displacement method
3. Equilibrium method
4. None of the above

Explanation:

Method of structural analysis:

• Building frames can be analyzed by various methods such as force method, displacement method, and approximate method.
• The method of analysis to adopt depends upon the type of frame, its configuration (portal bay or multi-bay) in a multi-storied frame, and the degree of indeterminacy.

Confusion Points

• Castiglano's 1st theorem is in Displacement method but Castiglano's 2nd theorem is in Force method analysis 

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Pick up the correct statement that corresponds to the moment distributions method. 

(i) Unbalanced moment is carried over to the other end of the member when the joint is released. 

(ii) Carry over the moment has the same sign as the distribution end moments.

1. Both (i) and (ii) is correct
2. Only (i) is correct
3. Only (ii) is correct 
4. Both (i) and (ii) are incorrect

Option 1 : Both (i) and (ii) is correct

Concept:

Distribution Factors in Moment Distribution Method:

• The moment distribution method is a structural analysis method for statically indeterminate beams and frames in which every joint of the structure to be analyzed is fixed so as to develop the fixed-end moments.
• Then each fixed joint is sequentially released and the fixed-end moments are distributed to adjacent members until equilibrium is achieved.
• Unbalanced moments are carried over to the other end of the member when the joint is released. Added to that, the ratio of the carried-over moment at the other end to the fixed-end moment of the initial end is the carryover factor.
• Lastly, For prismatic members, the carryover moment in each span has the same sign as the distribution end moment but is one-half as large.
• When a joint is being released and begins to rotate under the unbalanced moment, resisting forces develop at each member framed together at the joint.
• Distribution factors can be defined as the proportions of the unbalanced moments carried by each of the members. In mathematical terms, the distribution factor of member k framed at joint j is given as: where n is the number of members framed at the joint. Here, EI is the flexural rigidity and L is length of the concerned member.
• Hence, the sum of distribution factors of all the members connecting at any joint is always 1.

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If three members meet at a joint and the stiffness of members are K1 = EI, K2 = 2 EI, K3 = 1.5 EI, the distribution factor for member 1 is

Explanation:

Distribution factor 

The DF for a member at a joint is the ratio of the stiffness (or relative stiffness) of the member to the total stiffness (or total relative stiffness) of all the members meeting at a joint.

Given :

K1 = EI

K2 = 2 EI

K3 = 1.5 EI

So DF for member 

Calculation: 

DF for member 

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The ratio of the deflections of the free end of a cantilever due to an isolated load at 1/3rd and 2/3rd of the span is

Ist condition:

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

IInd condition:

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

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In column analogy method, the area of an analogous column for a fixed beam of span L and flexural rigidity EI is taken as

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

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Which of the following is not the displacement method?

1. Slope deflection method
2. Column analogy method
3. Moment distribution method
4. Kani’s method

Option 2 : Column analogy method

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:

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The deflection is ‘δ’, strain energy ‘U’ and load ‘W’ on a truss. These are related by

Concept:

The two theorems of Castigliano's for structural analysis are as follows:

Castigliano’s 1st theorem: The partial derivative of total strain energy of the system with respect to any particular deflection at a point is equal to the force applied at that point in the same direction as that of the deflection.

Castigliano’s 2nd theorem: The partial derivative of strain energy of the system with respect to load at any point is equal to deflection at that point.

Also, the partial derivative of strain energy of the system with respect to couple at any point is equal to slope at that point.

Where,

u is the strain energy of the system.

Note:

These theorems are valid in both the beams and truss, but in truss strain energy is only due to axial loads.

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A fixed beam AB is subjected to a triangular load varying from zero at end A to W per unit length at end B. the ratio of fix end moments at A to B will be

Explanation:

The fixed end moment at end A

The fixed end moment at end B

Additional Information

Fixed end moments developed due to various load combinations is given below:

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A fixed beam AB is subjected to a triangular load varying from zero at end A to W per unit length at end B. What is the ratio of fixed end moment at A to B?

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Slope = area of BMD/EI, is the relation given by:

1. Mohr's first theorem
2. Mohr's second therorem
3. Castigliano's therorem
4. Macaulay's theorem

Option 1 : Mohr's first theorem

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.

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.

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The moment distribution method is best suited for

1. in determinate pin jointed truss
2. rigid frames
3. space frames
4. trussed beam

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.

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Flexibility matrix method is also known as:

1. Displacement method
2. Stiffness method
3. Equilibrium method
4. Compatibility method 

Option 4 : Compatibility method 

Concept:

In the force method of analysis:

The primary unknown forces in the members and compatibility equations are written for displacement and rotations (which are calculated by force-displacement equations) in this method.

Solving these equations, redundant forces are calculated. Once the redundant forces are calculated, the remaining reactions are evaluated by equations of equilibrium.

In the displacement method of analysis:

Primary unknowns are the displacements and initially, force-displacement relations are computed and subsequently, equations are written satisfying the equilibrium conditions of the structure in this method. After determining the unknown displacements, the other forces are calculated satisfying the compatibility conditions and force-displacement relations.

Difference between Force & Displacement Methods

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Which one of the following methods does not fall under the category of force method?

1. Method of consistent deformation
2. Column analogy method
3. Equilibrium method
4. Three moment equation

Option 3 : Equilibrium method

Concept:

In the force method of analysis:

Primary unknown are forces in the members, and compatibility equations are written for displacement and rotations (which are calculated by force displacement equations) in this method.

Solving these equations, redundant forces are calculated. Once the redundant forces are calculated, the remaining reactions are evaluated by equations of equilibrium.

In the displacement method of analysis:

Primary unknowns are the displacements and initially force -displacement relations are computed and subsequently equations are written satisfying the equilibrium conditions of the structure in this method. After determining the unknown displacements, the other forces are calculated satisfying the compatibility conditions and force displacement relations.

Difference between Force & Displacement Methods

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In the structural analysis, the unit load method used is:

1. Another name of stiffness method
2. An extension of Maxwell’s reciprocal theorem
3. Applicable only to statically indeterminate structures
4. Derived from the Castigliano’s theorem

Option 4 : Derived from the Castigliano’s theorem

Explanation

1. The unit load method is extensively used in the calculation of deflection of beams, frames, and trusses.
2. Theoretically, this method can be used to calculate deflections in statically determinate and indeterminate structures.
3. However, it is extensively used in the evaluation of deflections of statically determinate structures only as the method requires a priori knowledge of internal stress resultants.
4. The unit load method used in structural analysis is derived from Castigliano’s theorem.

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For the propped cantilever beam loaded as shown in Figure, the Kani’s Rotation moment at the end B is +8.89 kNm. (taking anticlockwise as +). Hence the end moment at fixed end, A is

1. zero
2. + 8.89 kNm
3. + 17.78 kNm
4. – 17.78 kNm

Concept:

Displacement factor in Kani’s Method:

Kani’s method is a similar method as moment distribution method which is often used in analysis of continuous beam or indeterminate frame

In Kani’s method rotation factor for joints is given by,

Where, Iij and Lij is the moment of inertia and length of the member connecting i th and j ih node of the frame.

Calculations:

Using slope deflection equation

 .......(1)

 .......(2)

We know; θA = 0 

∴ Substituting in (2)

MB = 0 

we get 

Now substituting θB in (1) we get,

∴ At Support, A Moment is -8.889 + 

 = - 17.778 kN-m

Alternate MethodBut as Support B do not carry any moment it is balanced. The moment is carried to Support A

And the carryover factor is - 1/2 (As direction is different)

 ∴ At Support, A Moment is MAB + (- MBA/2)

 ∴ At Support, A Moment is -8.889 +

 = - 17.778 kN-m

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A fixed beam AB of span L is subjected to a clockwise moment at a distance 'a' from end A. Fixed end moment at end A will be-

Concept:

The fixed end moments generated due to the application of Moment M on a beam fixed at both ends are shown below:

⇒ MAB = 

⇒ MBA = 

Mistake Points

Here Moment is in Anticlockwise Direction.

MAB = 

MBA = 

Note: MAB and MBA must have the same sign. a and b, will always be positive which means to have the same sign-on MAB and MBA (3a – L) and (3b – L) must have the same sign.

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From the above figure,

find out the distribution factor for joint ‘B’ using the moment distribution method.

1. Member	D.F
   BA	0.4
   BE	0.3
   BC	0.3

2. Member	D.F
   BA	0.4
   BE	0.4
   BC	0.3

3. Member	D.F
   BA	0.3
   BE	0.3
   BC	0.4

4. Member	D.F
   BA	0.3
   BE	0.4
   BC	0.3

Option 1 :

Concept:

Distribution factor (DF)

• The DF for a member at a joint is the ratio of the stiffness (or relative stiffness) of the member to the total stiffness (or total relative stiffness) of all the members meeting at a joint.
• The summation of DF for all the members at a joint is one.
• DF is a property of rigid joint, it is not the property of hinge joint. So, the DF of a hinge joint is always zero.
   

Distribution factor for any member =

Stiffness: Force required for unit displacement

If the far end is fixed then stiffness = 4EI/L

If the far end is hinged then stiffness = 3EI/L

Calculation:

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Which of the following is NOT the correct method for finding the deflections of determinate beams?

1. Stress energy method
2. Moment area method
3. Conjugate beam method
4. Unit load method

Option 1 : Stress energy method

Concept:

The methods for finding the deflections of determinate beams.

• Strain energy method
• Moment area method
• Conjugate beam method
• unit load method  

Additional Information
Strain Energy Method: It is also known as Castigliano's method. 

Castigliano’s first theorem:

The first partial derivative of the total internal energy (strain energy) in a structure with respect to any particular deflection component at a point is equal to the force applied at that point and in the direction corresponding to that deflection component. This first theorem is applicable to linearly or nonlinearly elastic structures in which the temperature is constant and the supports are unyielding.

Castigliano’s second theorem: 

The first partial derivative of the total internal energy in a structure with respect to the force applied at any point is equal to the deflection at the point of application of that force in the direction of its line of action. 

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Bending moment at any section in a conjugate beam gives the actual beams

1. Curvature
2. Bending moment
3. Deflection
4. Slope

Explanation:

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The Castigliano’s second theorem can be used to compute deflections

1. In statistically determine structures only
2. At the point under the load only
3. For beams and frames only
4. For any type of structure

Option 4 : For any type of structure

Castigliano’s first theorem: The first partial derivative of the total internal energy (strain energy) in a structure with respect to any particular deflection component at a point is equal to the force applied at that point and in the direction corresponding to that deflection component.

This first theorem is applicable to linearly or non-linearly elastic structures in which the temperature is constant, and the supports are unyielding.

Castigliano’s second theorem: The first partial derivative of the total internal energy in a structure with respect to the force applied at any point is equal to the deflection at the point of application of that force in the direction of its line of action.

The second theorem of Castigliano is applicable to linearly elastic structures with constant temperature and unyielding supports. Note that in the above statements, force may mean point force or a couple (moment) and displacement may mean translation or angular rotation.

∴ These theorems are applicable to any structure for which the force deformation relations are linear.

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In the moment distribution method, the carry over moment at fixed end is equal to 

1. double of its corresponding distributed moment and has same sign
2. one-half of its corresponding distributed moment and has same sign
3. one-half of its corresponding distribution moment and has opposite sign
4. None of the above

Option 2 : one-half of its corresponding distributed moment and has same sign

Explanation:

Moment distribution method:

It was invented to analyze the indeterminate structures by Hardy Cross in 1930. In this method, the internal moments at the joints are distributed and balanced until the joints have rotated to their final or nearly final position.

• It is a displacement method of analysis
• The process is repetitive and easy to apply
• In the moment distribution method, the carryover moment is equal to one-half of its corresponding moment with the same sign.

The steps regarding the moment distribution method are;

1. Find the fixed end moments
2. Determine the distribution factor [For a member the distribution factor is = , k = Stiffness]
3. Moment balancing by Moment distribution table
4. drawing Bending Moment diagram by super-imposition.

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Members AB and BC shown in the figure below are identical. Due to a moment, M applied at B, what is the value of axial force in the member AB?

1. M/L (Compression) 
2. M/L (Tension)
3. 0.75M/L (Compression)
4. 0.75M/L (Tension)

Option 4 : 0.75M/L (Tension)

Explanation

The applied moment at B =M.

As the members are identical, so they will be shared equally by two members. (Distribution factor = 0.5)

So end moment at B = 0.5 × M = M/2

The moment carried to A = M/4

Similarly, moment carried to  C = M/4

FBD of BC.

Writing shear equation, we have,

HB = - 0.75M/L, (-ve sign indicates the opposite direction) 

So HB will be in down ward direction

FBD of structure

Axial force in BA = 0.75/L (Tensile)  

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Which one of the following methods is classifiable as a Displacement Method?

1. Theorem of Three Moments
2. Method of Consistent Deformation
3. Castigliano’s Theorem
4. Moment Distribution Method

Option 4 : Moment Distribution Method

Concept:

In the force method of analysis:

Primary unknown forces in the members and compatibility equations are written for displacement and rotations (which are calculated by force-displacement equations) in this method.

Solving these equations, redundant forces are calculated. Once the redundant forces are calculated, the remaining reactions are evaluated by equations of equilibrium.

In the displacement method of analysis:

Primary unknowns are the displacements and initially, force-displacement relations are computed and subsequently, equations are written satisfying the equilibrium conditions of the structure in this method. After determining the unknown displacements, the other forces are calculated satisfying the compatibility conditions and force-displacement relations.

Difference between Force & Displacement Methods

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