Deflection at the centre - simply supported beam, Mechanical Engineering

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

Posted Date: 1/21/2013 5:32:03 AM | Location : United States







Related Discussions:- Deflection at the centre - simply supported beam, Assignment Help, Ask Question on Deflection at the centre - simply supported beam, Get Answer, Expert's Help, Deflection at the centre - simply supported beam Discussions

Write discussion on Deflection at the centre - simply supported beam
Your posts are moderated
Related Questions
Ceramics Ceramics are mostly inorganic crystalline materials characterized by high melting point and low ductility. Such materials are processed via methods other than melting

What us the duff between direct stress and bending stress

The ARM microprocessor has 32 bit instructions and 32 bit registers. Explain why this means that immediate addressing cannot be used to load a register with a 32 bit constant. T

Determine the increase in pressure in vessel: A thin spherical vessel of diameter 750 mm & wall thickness 8 mm is filled through water at atmospheric pressure. Determine the i

Material S48C thickness 4mm pierce hole daimeter 4 mm can do or not? please advise

Find out Moment of Inertia: Find out Moment of Inertia of an I shaped area around its centroidal axis as illustrated in Figure.  Solution  (i)        The area of

Planning A New Airport: Elements of Airport Planning The elements of planning for a new airport are : (a)        Demand Forecast (b)       Site Selection (c)

Transformer Type Power Sources These are the simplest and cheapest type available for arc welding. These are the mainly used for SMAW, AC SAW and gouging applications. These are

Beam: Sol.: A beam can be defined as a structural element which has one dimension (length) considerable larger As compared to the other two direction, which is breadth an

Q. What is Vertical Spoilers? Vertical Spoilers - Use a three-start system of short vertical spoilers arranged in a helical pattern on the top third of the vessel. The exposed