Scientific & Analytical Methods Assignment 1

2010

HNC CAD/CAM SCIENTIFIC & ANALITYCAL METHODS ASSIGNMENT 1 DAVID ANTUNA

David Antuna HNC CAD/CAM

Scientific & Analytical Methods Ass.1 ANT07078917

2 Table of Contents

Summary: ...................................................................................... 3

Shear Force, Bending Moment & Stress for Beams: ......................... 4

Beam Calculation Software: ........................................................... 6

Shear Stress for Circular Shafts: ...................................................... 8

Friction & Acceleration: .................................................................. 9

Simple Harmonic Motion: ............................................................. 10

Appendix: .................................................................................... 11



3 Summary: The aim of this report is to demonstrate the ability to apply and calculate the correct formulas for different Scientific and Analytical scenarios like: Shear Force & Bending Moment for a Beam. The use of software for the calculation of the above. Shear Stress for solid shafts. Friction & Acceleration. Simple Harmonic Motion.



4 Shear Force, Bending Moment & Stress for Beams: Determine distribution of shear force, bending moment and stress due to bending

in simply supported beams.

1. A cantilever of rectangular cross-section 74 mm wide and 110 mm deep carries a concentrated load of 3 kN at its free end. Neglecting the mass of the cantilever: Find the maximum length if the greatest permissible stress due to bending is 85 MPa. Draw the shear force and bending moment for the beam.

74mm BENDING EQUATION 110mm

𝑴

𝑰=

𝝈

𝒀=

When: σ max=85 Mpa.

𝑰 =𝒃𝒅³

𝟏𝟐=

𝟎.𝟎𝟕𝟒𝒙𝟎.𝟏𝟏³

𝟏𝟐

W= 3Kn

I= 8.21x10¯⁶

𝑰 =𝒃𝒅³

𝟏𝟐

Y= 0.055m

M= 𝑰· 𝝈𝒎𝒂𝒙

𝒀𝒎𝒂𝒙 =

𝟖.𝟐𝟏𝒙𝟏𝟎¯𝟔𝑿 𝟖𝟓𝒙𝟏𝟎⁶

𝟓.𝟓𝒙𝟏𝟎¯²

M= 1.27x10⁴

M=WL there for L=𝑴

𝑾 =

𝟏.𝟐𝟕𝒙𝟏𝟎⁴

𝟑𝒙𝟏𝟎³

L= 4.23 m



5 2. Find the maximum total uniformly distributed load which can be carried by a timber

beam 300 mm deep, 200 mm wide and 3.5 m long which is simply supported at its ends. Neglect the mass of the beam itself and take the maximum stress due to bending to be 8 MPa. Draw the shear force and bending Moment for the beam.

200mm

300mm 3.5m

BENDING EQUATION

𝑴

𝑰=

𝝈

𝒀=

When:

σ= 8 Mpa = 8x10⁶ N/mm² Y= 150x10¯³

M= 𝒘·𝒍

𝟖

𝑰 =𝒃𝒅³

𝟏𝟐 =

𝟎.𝟐𝒙𝟎.𝟑³

𝟏𝟐 = 4.50x10¯⁴

L= 3.5 m

M= I𝝈

𝒀 .·.

𝑾𝑳

𝟖= I

𝝈

𝒀 .·. W=

𝑰𝒙𝝈𝒙𝟖

𝑳𝒙𝒀

.·. W= 𝟒.𝟓𝟎𝒙𝟏𝟎¯⁴𝑿𝟖𝒙𝟏𝟎⁶𝑿𝟖

𝟑.𝟓𝑿𝟏𝟓𝟎𝒙𝟏𝟎¯³

W= 5.49x10⁴ N



6 Beam Calculation Software: Determine the sectional properties of structural sections.

3. The below screen grabs are an example of how is possible to use different type of

software to calculate Load, Shear Force & Bending Moments for any type of beam. In this case the software is SuperBeam 4.

Cantilever beam with a 3 KN concentrate load.

Figure 1 Use of Software to Calculate Beams



7 UDL Timber Beam 3.5m long with a 8 MPa of bending stress.

Figure 2 Use of Software to Calculate Beams



8 Shear Stress for Circular Shafts: Determine the distribution of shear stress and the angle and deflection due to torsion

in circular shafts. 4. In a torsion experiment, a specimen is used whose shaft is 30 mm diameter. A torsion

meter, which is clamped to the shaft shows that a 200 mm length of the shaft twists through an angle of 25° when a twisting moment of 400 Nm is applied. a. Find the value of the modulus of rigidity of the shaft material. b. Find the value of the maximum shear stress and hence draw a diagram showing the shear stress distribution over the cross section of the shaft.

R=0.015 τ= When: TORSION EQUATION

R= 0.015m = 15x10¯³ 𝐓

𝐉=

𝛕

𝐑=

𝐆𝛉

𝐋

L= 200mm = 0.2m = 200x10¯³

θ= 25° .·. θ=𝝅𝒙𝟐𝟓

𝟏𝟖𝟎= 0.436 rad

T= 400Nm

𝑱 =𝝅𝑫⁴

𝟑𝟐= 𝑱 =

𝝅𝒙𝟎.𝟎𝟑⁴

𝟑𝟐=7.95x10¯⁸

a. Modulus of Rigidity b. Maximum Shear Stress

𝑻

𝑱=

𝑮𝜽

𝑳

𝑻

𝑱=

𝝉

𝑹

.·. 𝑮 =𝑻𝑳

𝑱𝜽 .·. 𝝉 =

𝑻𝑹

𝑱

𝝉 =𝟒𝟎𝟎𝑿𝟎.𝟎𝟏𝟓

𝟕.𝟗𝟓𝒙𝟏𝟎¯⁸

𝑮 =𝟒𝟎𝟎𝒙𝟎.𝟐

𝟕.𝟗𝟓𝒙𝟏𝟎¯𝟖𝑿𝟎.𝟒𝟑𝟔 τ= 7.54x10⁷

G= 438742138.4 N/m² G= 4.38x10⁸ N/m²



9 Friction & Acceleration: Determine the behaviour of dynamic mechanical Systems in which uniform

acceleration is present.

5. A cast iron block of mass 30kg is pulled along a surface and accelerates at 0.3m/s2. If the coefficient of friction is 0.15N, and g=9.81m/s2. a) Find the necessary tractive force.

If the force in the last example is removed after 20 seconds, and the crate started moving from rest.

b) Calculate the final velocity, and how far it moved until the force was removed.

Sketch the crate as it accelerates and indicate all the forces acting upon it. W=30 Kg a=0.3 m/s²

FF=0.15 N F=53.145 N

FRICTION EQUATION F= FF+FI

F= µMg+Ma

a) Find the necessary tractive force.

F= FF+FI FF=µFN F= µMg+Ma =µMg FI =Ma F= 0.15x30x9.81+30x0.3

F= 53.145 N

a) Calculate the final velocity, and how far it moved until the force was removed.

V=u+at 𝑆 =𝑇

2(𝑉 + 𝑎) S= ut+½at²

V=at since U=0 𝑆 =20

2(6 + 0.3) S= ½at²

V=0.3x20 S = 63m S= ½x0.3x20²

V=6 m/s = 21.6 Km/Hr S= 60m



10 Simple Harmonic Motion: Determine the behaviour of oscillating mechanical systems in which simple harmonic

motion is present. 6. The crank of a scotch-yoke mechanism is 52 mm long and rotates at a uniform 250

rev/mm. Find the slider's: (a) acceleration and velocity when 15 mm from the mid-point of its travel; (b) maximum acceleration, and state where it occurs; (c) maximum velocity, and state where it occurs.

ω

SIMPLE HARMONIC MOTION EQUATIONS

X= aCosθ

U= 𝝎 𝒂² − 𝒙² F= ω²X

When: a= 52mm = 52x10¯³ m

ω= 250 rev/min = 250𝑥2𝜋

60 rads/sec

X= 15mm = 15x10¯³ m Find the Acceleration and Velocity.

F= ω² X = 250𝑥2𝜋

60

2

X 15x10¯³ .·. F= 10.28 m/s²

U= 𝜔 𝑎² − 𝑥² = 250𝑥2𝜋

60

2 52𝑥10¯³ 2 − (15𝑥10¯3)² .·. U= 1.36 m/s²

Find the maximum acceleration, and state where it occurs./F=ω²X and occurs when x=a

Fmax=ω²a = 250𝑥2𝜋

60

2

X 52x10¯³ .·. Fmax= 35.64 m/s²

Find maximum velocity, and state where it occurs. Occurs when X=0 From U= 𝜔 𝑎² − 𝑥²

Umax= ωa = (250𝑥2𝜋

60) X 52x10¯³ .·. Umax= 1.36 m/s²



11 Appendix: In this section you will find the diagrams for question 1 & 2 of this report.

Scientific & Analytical Methods Assignment 1

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