2 marks questions and answers - sri venkateswara college ... of... · 1. define the term strain...

1. Define the term strain energy.

A: Strain Energy of the elastic body is defined as the internal work done by the external load in deforming or straining the body.

2. Define the terms: Resilience and Modulus of resilience.

A: Resilience: Strain energy per unit volume of the material is

known as strain energy density or resilience.

• Modulus of resilience: When the stress 𝑓 is equal to proof stress, 𝑓𝑝 at the elastic limit, the corresponding resilience is known as

proof resilience, up = 𝑓𝑝2

2𝐸

The proof resilience is known as modulus of resilience. It is the

property of the material. It’s unit is N-m/m3= N/m2

Dr. P.Venkateswara rao, Associate Professor, Dept. of Civil Engg., SVCE

1

2 marks Questions and Answers

3. Write down the expression for strain energy stored in a bar of cross sectional area ‘A’ and length ‘l’ subjected to a tensile load ‘W’.

A: Strain energy due to axial tensile load ‘W’ is 𝑈 =𝑊2𝐿

2𝐴𝐸

4. Write the expression for strain energy due to bending.

A: Strain energy due to bending is,


2


dxEI

MU

L

0

2

2

5. State Castigliano’s first and second theorem for strain energy.

A: Castigliano’s first theorem:

For linearly elastic structure, the Castigliano’s first theorem may be defined as the first partial derivative of the strain energy of the structure with respect to any particular force gives the displacement of the point of application of that force in the direction of its line of action.


3


∴ 𝛿𝑖 =𝜕𝑈

𝜕𝑊𝑖

Castigliano’s second theorem:


4


2

2

1

1

WU

WU

i

i

The energy Ui can be express in terms of spring stiffnesses

k11, k12 (or k21), & k22 and deflections δ1 and δ2; then it can

be shown that

This is Castigliano’s second theorem.

6. What is the strain energy due to axial force of a tapering bar of circular section?

A:


5


P P

d

D

U = 2𝑃2𝐿

𝜋𝐸𝑑𝐷

7. What is the strain energy due to axial force of a tapering bar of rectangular section?

A:


6


P P b B

t L

𝑥

∴ 𝑈 =𝑃2𝐿

2 𝐵 − 𝑏 𝑡𝐸log𝑒

𝐵

𝑏

8. Calculate the strain energy stored in a cantilever beam subjected to a point load ‘W’ at the free end.


7


PxM

dxEI

MU

L

0

2

2

EI

LPdx

EI

xPL

62

32

0

22

EI

LPU

6

32

9. State Maxwell’s reciprocal theorem.

A: In any beam (or) truss, the deflection at any point C due to load W at any point B is the same as the deflection at B due to the same load W applied at C.


8


𝛿𝐶 = 𝛿𝐵

A B C D

𝛿𝐶

W

A B C D

𝛿𝐵

W =

10. State the principle of virtual work.

A: The partial derivative of internal energy with respect to a load applied at a point where the deflection is zero is called principle

of virtual work. And hence, 𝜕𝑈

𝜕𝑊= 0.


9


11. Define fixed beam? A: A fixed beam is a beam whose end supports are such that the end slopes remain zero (or unaltered) and is also called a built-in or encaster beam. 12. What are the disadvantages of fixed beam? A: (i). It is practically difficult to maintain the two ends of the beam at exactly at the same level. Any subsidence of one of the supports, however small it may be, will set up considerable stresses. (ii). Temperatute variations also produce large stresses in a fixed beam. 13. Define theorem of three moments. A: If AB and BC are any two consecutive spans of a continuous beam subjected to an external loading, the support moments 𝑀𝐴, 𝑀𝐵 and 𝑀𝐶 at the supports A, B and C are given by the relation,

𝑀𝐴(𝑙1) + 2𝑀𝐵 𝑙1 + 𝑙2 +𝑀𝐶 𝑙2 =6𝑎1𝑥1𝑙1

+6𝑎2𝑥2𝑙2

Is known as theorem of three moments. 14. What is meant by prop ? A: Simply supported at the free end of cantilever beam is known as prop.


10


15. Find the fixed end moments of a fixed beam subjected to a point load at the center.

A:

• 𝐴′ = 𝐴


11

𝑀 =𝑊𝑙

8= 𝑀𝐴 = 𝑀𝐵

𝑙/2

W

A B

𝑊𝑙

4

Free BMD

𝑙/2

Fixed BMD

M M

+

-

𝑀 × 𝑙 =1

2× 𝑙 ×

𝑊𝑙

4

+

- -

𝑊𝑙

4

Resultant BMD


16. For the fixed beam subjected to an eccentric point load ‘W’, write end moments.


12


𝑀𝐴 =𝑊𝑎𝑏2

𝑙2

𝑀𝐵 =𝑊𝑏𝑎2

𝑙2

𝑎 W

A B

+

𝑊𝑎𝑏2

𝑙2 𝑊𝑏𝑎2

𝑙2

𝑊𝑎𝑏

𝑙

Resultant BMD

- -

𝑏

𝑙

• 17. A fixed beam of span ‘L’ is subjected to UDL of W/m throughout the span. What are end moments?

• A:


13

∴ 𝑀𝐴 = 𝑀𝐵 =𝑊𝑙2

12

A = A’ ‘𝑙′ 𝑚

A B

W kN/m

𝑤𝑙2/8

+

-

M M

2

3× 𝑙 ×

𝑤𝑙2

8= M × 𝑙

𝑀 =𝑤𝑙2

12


18. What are the advantages and limitations of theorem of three moments?

A: Advantages: It is useful to find support moments of a indeterminate beams like continuous beam, fixed beam and propped cantilever beam.

The effects of sinking of support will be taken into consideration for a continuous beam.

It is also useful if moment of inertia is different for different spans of continuous beams.

Limitations: To use this theorem, minimum two spans are required.


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19. What is the value of prop reaction in a propped cantilever of span ‘L’, when it is subjected to a UDL of w/m over the entire length?

A:


15


𝑤𝑙2/8

W kN/m

‘L’ m 𝑅𝐵

B A

𝑀𝐴=0,

𝑅𝐵 × 𝑙 +𝑤𝑙2

8−𝑤𝑙2

2= 0

∴ 𝑅𝐵 =3

8𝑤𝑙

W kN/m

‘L’ m 𝑅𝐵

B A

𝑤𝑙2/8 𝑅𝐴

By using clayperon’s theorem of three moments for the spans A’A and AB, one can find

𝑀𝐴 =𝑤𝑙2

8

L=0 A’

20. What is the value of prop reaction in a propped cantilever of span ‘L’, when it is subjected to a point load of ‘W’ at the centre ?

A:


16


3

16𝑤𝑙

𝑀𝐴=0,

𝑅𝐵 × 𝑙 +3

16𝑤𝑙 −

𝑤𝑙

2= 0

∴ 𝑅𝐵 =5

16𝑤

‘L’ m 𝑅𝐵

B A

𝑅𝐴

By using clayperon’s theorem of three moments for the spans A’A and AB, one can find

L=0 A’

‘L’ m 𝑅𝐵

B A

W

𝑀𝐴 =3

16𝑤𝑙

3

16𝑤𝑙

W

𝑤𝑙

4

2 marks questions and answers - sri venkateswara college ... of... · 1. define the term strain...

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