Deflection at the centre - simply supported beam, Mechanical Engineering

Assignment Help:

Deflection at the centre:

A simply supported beam of span 6 m is subjected to Udl of 24 kN/m for a length of 2 m from left support. Discover the deflection at the centre, maximum deflection & slopes at the ends and at the centre. Take EI = 20 × 106 N-m2.

Solution

∑ Fy  = 0, so that RA  + RB  = 24 × 2 = 48 kN          --------- (1)

 

2109_Deflection at the centre - simply supported beam.png

Taking moments around A,

24 × 2 × 1 = RB  × 6

RB  = 8 kN (↑)                     -------- (2)

RA  = 48 - 8 = 40 kN (↑).         ------------(3)

By apply the Udl over the portion DB downwards and upwards,

 

                                 Figure

M = 40 x - 24 x × (x/2) + 24 ( x - 2) ( (x - 2)/2)

Note down that the third term vanishes if x < 2 m.

= 40 x - 12 x2  + 12 ( x - 2)2               ------- (4)

EI d 2 y/ dx2 = 40 x - 12 x 2  + 12 ( x - 2)2          ------- (5)

EI dy / dx = 40 x2/2- 12 x3 /3+ 12 ( x - 2)3/3 + C1

= 20 x2 - 4 x3 + 4 ( x - 2)3 + C1           -------- (6)

EIy = 20 x 2/3 - x4 + (x - 2)4 + C1 x + C2            -------- (7)

Here again note that the third term vanishes for x < 2 m.

at A,      x = 0,    y = 0  ∴ C2  = 0

at B,  x = 6 m,     y = 0         

0 = 20 × 63 /3 - 64  + (6 - 2)4 + C1 × 6

C1 =- 20 × 12 + 36 × 6 - ((16 × 16 )/6)=- 200/3

∴          EI dy/dx = 20 x2  - 4 x3  + 4 ( x - 2)3  - 200/3         -------- (8)

The third term vanishes.

Slope at A, (x = 0),     27

θA  = -200/3EI =- (200 × 103)/ (3 × 20 ×106)

            = -(1/300) rad = - 3.33 × 10- 3  rad

 

Slope at B, (x = 6 m),

EI θ B = 200 × 62  - 4 × 63  + 4 (6 - 2)3  - (200/3)

 θ  = 136/ 3 EI = (136 × 103 )/(3 × 20 ×106)

= + 2.27 × 10- 3  radian

Slope at C, (x = 3 m), i.e. x > 2 m

EI θ C = 20 × 32  - 4 × 33  + 4 (3 - 2)3  - (200/3)

θC = 20 /3 EI = 0.47 × 10- 3  radians

EIy =( 20 x 3/3)- x4  + ( x - 2)4  - (200/3) x                   -------- (9)

Deflection at centre, (x = 3 m),

EIyC = (20/3) × 33  - 34  + (3 - 2)4  - (200 /3)× 3

yC  = - 100 / EI =  - 100 × 103 × 103 / (20 × 106)

= - 5 mm

For maximum deflection,

dy/ dx  = 0

0 = 20 x2  - 4x3  + 4 ( x - 2)3  - (200/3)

= 20 x2  - 4x3  + 4x3  - 32 - 24 x2  + 48 x - (200 /3)

=- 4x2  + 48 x - (296 /3)

∴          x2  - 12x + (74 /3 )= 0

x = 2.63 m , x > 2m

EIy max = (20/3) × 2.633  - 2.634  + (2.63 - 2)4  - (200/3) × 2.63 = - 101.7

∴ ymax  = - 5.087 mm;  - 5.1 mm


Related Discussions:- Deflection at the centre - simply supported beam

Evaluate power transmitted by shaft, Evaluate power transmitted by shaft: ...

Evaluate power transmitted by shaft: How do you evaluate power transmitted by shaft (P)? Sol.: Consider force 'F' Newton's acting tangentially on shaft of radius ' R '.

Compression ignition engines - thermodynamics, Compressio n Ignition Engin...

Compressio n Ignition Engines (or C.I. Engine): It works on the basis of dieses cycle. In diesel engines, energy addition occurs at constant pressure but energy rejection at

Types of coated electrodes-iron powder electrodes, Iron Powder Electrodes ...

Iron Powder Electrodes Iron powder is a widely used covering ingredient. It is incorporated in certain coverings to the extent of one half of the total weight. Iron Powder impart

Mathematics, solve the following differential equation: dy/dx-(1+1/x)y=y^2

solve the following differential equation: dy/dx-(1+1/x)y=y^2

Steps of inspection of electrical equipment, Inspection of Electrical Equip...

Inspection of Electrical Equipment STEPS All electrical equipment are checked for proper functioning. The starting and lighting of the motorcycle should be prope

Explain thermodynamics system, Explain Thermodynamics System In thermod...

Explain Thermodynamics System In thermodynamics the system is illustrated as the quantity of matter or region in space upon which the concentration is concentrated for the sake

Winch gearbox design, An engine repair workshop in County Durham has identi...

An engine repair workshop in County Durham has identified a need  for a winch which is  to lift complete engine assemblies from floor level up to workbench level  to allow  remedia

Organisation set-up - construction of railway line, Organisation set-up: ...

Organisation set-up: The civil engineering department of the Railways is responsible for the maintenance of railway lines. The maintenance organisation consists of Chief Engi

Find out the moment of inertia of a semicircle, Find out the moment of ine...

Find out the moment of inertia  of a semicircle: Find out the M. I. of a semicircle ABM around its periphery AB and also around centroidal axes x and y as illustrated in Figur

Mechanical design and development, The major assignment for the unit is to ...

The major assignment for the unit is to design the final drive for the F-SAE race car. This is a group project to be completed with groups of up to 5 people. The design should be r

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd