Determine the deflection at free end, Mechanical Engineering

Assignment Help:

Determine the deflection at free end:

For the beam illustrated in Figure, determine the deflection at free end and the maximum deflection.

67_Determine the deflection at free end.png

Figure

Solution

RA + RB  = 48 × 6 + 60 = 348 kN         ----------- (1)

Σ M about A = 0.

48 × 6 × (2 + 3) + 60 × 16 = RB  × 12

 ∴         RB  = 200 kN (↑)                        ---------- (2)

From Eqs. (1) and (2),

RA  = 148 kN (↑)                         ------ (3)

Apply the Udl in downward and upward directions in the portion EC

2085_Determine the deflection at free end1.png

Figure

 

M = 148 x - (48/2) [ x - 2] [ x - 2] + (48/2) [ x - 8] [ x - 8] + 200 [ x - 12]

= 148 x - 24 [ x - 2]2  + 24 [ x - 8]2  + 200 [ x - 12] ----------. (4)

EI d 2 y/ dx2      = M = 148 x - 24 [ x - 2]2  + 24 [ x - 8]2  + 200 [ x - 12]          -------- (5)

EI (dy/ dx )= 74 x2 - 8 [ x - 2]3 + 8 [ x - 8]2 + 100 [ x - 12] 2+ C1          ------- (6)

EIy (74/3) x3  - 2 [ x - 2]4  + 2 [ x - 8]4  + 100 [ x - 12]3 + C1 x + C2   ---------- (7)

The boundary conditions are :

At A,       x = 0,           y = 0                 ------- (8)

At B,          x = 12 m,        y = 0                 ------- (9)

From Eqs. (7) and (8),

C2 = 0   ------------- (10)

From Eqs. (7) and (9),

0 = (74 /3)× 123 - 2 [12 - 2]4  + 2 [12 - 8]4  + C1 × 12

∴          C1 = - 1928                              ------------ (11)

∴          EIy = 74 x3  - 2 [ x - 2]4  + 2 [ x - 8]4  + 100 [ x - 12]3  - 1928 x

Deflection at free end, x = 16 m

EIy C   = (74 /3)× 163  - 2 [16 - 2]4  + 2 [16 - 8]4  + 100 [16 - 12]3  - 1928 × 16 = 3680

∴          yC  =+  3680 / EI

Maximum Deflection : It occurs between D and E.

dy / dx = 0

0 = 74 x2  - 8 [ x - 2]3  - 1928

= 74 x2  - 8 x3  + 64 + 48 x2  - 96 x - 1928

= 8 x3  - 122 x2  + 96 x + 1928

By trial and error, x = 5.7 m

EIy max     = 74 × 5.73  - 2 [5.7 - 2]4  - 1928 × 5.7 = - 6796.3

∴          ymax = - 6796.3 / EI


Related Discussions:- Determine the deflection at free end

Define pinion cutter generating process, Define Pinion Cutter Generating Pr...

Define Pinion Cutter Generating Process? It is shown in diagram and it is same as rack cutter process, and instead of using a rack cutter it uses a pinion to generating the too

Under damping, Under Damping If    the restoring force is larger than ...

Under Damping If    the restoring force is larger than friction, and the oscillation is classified as under-damped. In this case,  ω  is real and behaves exactly like the solu

MATERIAL SCIENCE, IF ONE SOLID PHASE SPLITS INTO TWO SOLID PHASES ON HEATIN...

IF ONE SOLID PHASE SPLITS INTO TWO SOLID PHASES ON HEATING THE REACTION IS CALLED

Advantages of two stroke engine over a four stroke engine, Advantages of tw...

Advantages of two stroke engine over a four stroke engine: The advantages of two stroke engine over a four stroke engine are stated below: 1. A two stroke engine has twice

Screw conveyors, Screw Conveyors: Screw conveyors are used principally f...

Screw Conveyors: Screw conveyors are used principally for transmitting materials in the form of powder or paste with the application of rotating screw. For example feedin

Show the tensile properties of material, Q. Show the tensile properties of ...

Q. Show the tensile properties of material? Tensile properties indicate how the material will react to forces being applied in tension. A tensile test is a fundamental mechanic

Rotary vacuum, Explain with the aid of a sketch how a rotary vacuum (ROVAC)...

Explain with the aid of a sketch how a rotary vacuum (ROVAC) filter works. Derive describing equations to predict the ROVAC's filtration performance in terms of the effect of drum

Calculate the magnitudes of the force, The end gates of a lock are 5m high ...

The end gates of a lock are 5m high and when closed include an angle of 120 o . The width of the lock is 6.25 m. Each gate is carried on two hinged placed at the top and the bottom

Design for dynamic and wear loads, A gear drive is needed to transmit a max...

A gear drive is needed to transmit a maximum power of 22.5KW. The velocity ratio is 1:2 and r.p.m. of the pinion is 200. The centre distance between the shafts can be taken as 600m

Program evaluation and review techniques, Program Evaluation and Review Tec...

Program Evaluation and Review Techniques: The Rola and Cola Corporation is a professional II nursing home aspiring to become I nursing home. It desires to provide the ultimate

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