Learning Objectives:

• Determine the deformations and/or normal stress in a member due to application of forces.

Elastic Deformation wherein body recovers its original shape after removal of force whereas Plastic Deformation permanent deformation (body does not recover its shape after forces are removed.

Learning Objectives:

• Calculate deflection using the equation of the elastic curve.

• Utilize the direct determination of the elastic curve.

• Derive the deflection and slope curves for a beam through integration of the moment-curvature relationship.

• Apply discontinuity functions and standardized solutions to simplify the calculation of deflection and slope curves for beams.

Differential Equation of deflection curve,

EI = M

Integrating once, we get \[EI \dfrac{dy}{dx} = \int M dx\] Integrating once more, we get \[EI y = \int \int M dx dx\] The constants of integration can be eliminated by applying boundary conditions, such as: At (unyielding) supports, \[y = 0\] At fixed supports \[\dfrac{dy}{dx} = \mathrm{slope\ of\ deflected\ curve} = 0\]

Solved Example: 47-2-01

Slope at a point in a beam is the:

A. Vertical displacement

B. Angular displacement

C. Horizontal displacement

D. None of these.

Correct Answer: B

Solved Example: 47-2-02

Which of the following is an elastic curve equation for shear force? (EI = flexural rigidity)

A. $S = EI \dfrac{dy}{dx}$

B. $S = EI \dfrac{d^2y}{dx^2}$

C. $S = EI \dfrac{d^3y}{dx^3}$

D. $S = EI \dfrac{d^4y}{dx^4}$

Correct Answer: C

Solved Example: 47-2-03

Which of the following statements is/are true for a simply supported beam?

A. Deflection at supports in a simply supported beam is maximum

B. Deflection is maximum at a point where slope is zero

C. Slope is minimum at supports in a simply supported beam

D. All of the above.

Correct Answer: B

Solved Example: 47-2-04

Which of the following is a differential equation for deflection?

A. $\dfrac{dy}{dx} = \dfrac{M}{EI}$

B. $\dfrac{dy}{dx} = \dfrac{MI}{E}$

C. $\dfrac{d^2y}{dx^2} = \dfrac{M}{EI}$

D. $\dfrac{d^2y}{dx^2} = \dfrac{ME}{I}$

Correct Answer: C

Solved Example: 47-2-05

The equation of deformation is derived to be $y = x^3 - xL$ for a beam as shown in the figure. Curvature of the beam at the mid-span (in units, in integer) will be: (GATE Civil 2021)

A. $2$

B. $2L$

C. $\dfrac{L}{2}$

D. $\dfrac{L}{3}$

\[y = x^2 - xL\] Curvature at the midsection is: \[\dfrac{1}{R} = \dfrac{d^2y}{dx^2}\] \[\dfrac{dy}{dx} = 2x - L\] \[\dfrac{d^2y}{dx^2} = 2\]

Correct Answer: A

Learning Objectives:

• Calculate deformation of simple structural idealizations/models.

1. Cantilever Beam – Concentrated load P at the free end

   \[\delta_{max} = \dfrac{PL^3}{3EI}\]

2. Cantilever Beam – Concentrated load P at any point

   \[\delta_{max} = \dfrac{Pa^2}{6EI}(3l-a)\]

3. Cantilever Beam – Uniformly distributed load w(N/m) \[\delta_{max} = \dfrac{wL^4}{8EI}\]

4. Cantilever Beam – Uniformly varying load: Maximum intensity \(w_0\)(N/m) \[\delta_{max} = \dfrac{w_0L^4}{30EI}\]

5. Cantilever Beam – Couple Moment M at the free end \[\delta_{max} = \dfrac{ML^2}{2EI}\]

Solved Example: 47-3-01

The simply supported beam 'A' of length l carries a central point load W. Another beam 'B' is loaded with a uniformly distributed load such that the total load on the beam is W. The ratio of maximum deflections between beams A and B is:

A. 5/8

B. 8/5

C. 5/4

D. 4/5

Correct Answer: B

Solved Example: 47-3-02

A cantilever beam with length 100 mm and flexural rigidity of 200 $N.m^2$ is loaded with a point load of 500 N at the midpoint. The deflection in (mm) at the tip of the beam is :

A. 0.26mm

B. 2.12mm

C. 0.061 mm

D. 0.97 mm

Correct Answer: A

Solved Example: 47-3-03

A cantilever of length 2 m carries a uniformly distributed load of 2.5kN/m for a length of 1.25m from the fixed end and a point load of 1kN at the free end. The deflection at the free end if the section is rectangular 12 cm wide and 24 cm deep and E = $1 \times 10^4 N/mm^2$

A. 2.9224mm

B. 0.9934 mm

C. 1.929 mm

D. 2.4419 mm

L = 2 m, w = 2.5 kN/m, W = 1 kN a = 1.25 m, b = 12 mm d = 24 mm \[I = \dfrac{bd^3}{12} = \dfrac{12 X 24^3}{12} = 1.3824 X 10^4 mm^4\] E = $1 \times 10^8 N/mm^2$ Deflection at free end due to point load of 1 kN \[y_1 = \dfrac{WL^3}{3EI} = \dfrac{1000 \times 2000^3}{3 \times 10^4 \times 1.3824 \times 10^8} = 1.929\ mm\] Deflection of the free end due to uniformly distributed load of 2.5 N/mm on a length of 1.25 m \[y_2 = \dfrac{wa^4}{8EI} + \dfrac{wa^3}{6EI (L-a)} = 0.5519 + 0.4415 = 0.9934\ mm\] Total deflection = 1.929 + 0.9934 = 2.9224 mm

Correct Answer: A

Solved Example: 47-3-04

Determine the load capacity in kN on a 25 mm diameter $\times$ 1200 mm long steel shaft if its maximum elongation shall not exceed 1 mm. Assume E = 200,000 MPa.

A. 88.2 kN

B. 78.3 kN

C. 83.2 kN

D. 81.8 kN

Correct Answer: D

Solved Example: 47-3-05

A solid shaft 2.0 m long is transmitting 27 kN-m torque. If the shear modulus of the shaft material is 85 GPa and the allowable shearing stress is 70 MPa, determine the angle of twist between the two ends of the shaft.

A. 1.52$^\circ$

B. 1.02$^\circ$

C. 1.44$^\circ$

D. 1.32$^\circ$

Correct Answer: A

Solved Example: 47-3-06

The deflection of a cantilever beam under load W is 8. If its width is halved, then the deflection under load W will be:

A. 28

B. 8/2

C. 48

D. 8/4

Correct Answer: C

Solved Example: 47-3-07

A non-yielding support implies that the:

A. Support is frictionless

B. Support can take any amount of reaction

C. Support holds member firmly

D. Slope of the beam at the support is zero

Correct Answer: D

Solved Example: 47-3-08

The ratio of elongation in a prismatic bar due to its own weight (W) as compared to another similar bar carrying an additional weight (W) will be:

A. 1 : 2

B. 1 : 3

C. 1 : 4

D. 1 : 2.5

Correct Answer: B

Solved Example: 47-3-09

In a prismatic member made of two materials so joined that they deform equally under axial stress, the unit stresses in two materials are:

A. Equal

B. Proportional to their respective moduli of elasticity

C. Inversely proportional to their moduli of elasticity

D. Average of the sum of moduli of elasticity

Correct Answer: B

Solved Example: 47-3-10

The deformation of a bar under its own weight compared to the deformation of same body subjected to a direct load equal to weight of the body is:

A. Same cross-section throughout the beam

B. Double

C. Half

D. Four times

Correct Answer: C