Analysis diving board by Macaulay’s methods and Strain rosette

Analysis diving board by Macaulay’s methods and Strain rosette

Project Study For BDA 3033 Solid Mechanics II

By MAGENTHRAN KUPPUSAMY

Department of Engineering Mechanics Faculty of Mechanical and Manufacturing Engineering University Tun Hussein Onn Malaysia Johor

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Analysis diving board by Macaulay’s methods and Strain rosette 1.1

Introduction of Diving On a roof slab of a vast burial vault south of Naples is a painting of a young man diving from

a narrow platform. The discovery of the "Tomba Del Tuffatore" (The Tomb of the Diver) shows us that the excitement and grace of diving from high places into water has lured people from at least 480 BC - the date established for the construction of the tomb. As with most sports dating back to ancient times, little information on competitive diving has survived. The origins of modern diving can be traced to two European venues - Halle in Germany and Sweden. It was a traditional specialty of the guild of salt boilers, called Halloren to practise certa in swimming and diving skills. The Halloren used to perform a series of diving feats from a bridge into the River Saale. In 1840 in contact with the German gymnastics movement the world's first diving association was formed. Most of its members were gymnasts starting their tumbling routines as a kind of water gymnastic. Thus diving became very popular in Germany. In Sweden wooden scaffolding was erected around many lakes, inviting courageous fellows to perform diving feats. Somersaulting from great heights and swallow-like flights of a whole team are common. The beginning of competitive diving corresponded to the rise of swimming clubs and associations. In Germany, the oldest club called "Neptun" started international diving contests from a lower board and from a tower in 1882. In 1891 the first diving rules were adopted and the following year the first tables were published in Germany. At the turn of the century, another branch of diving found numerous followers in the USA the bridge and artistic leaping. However, its development was stopped due to the high number of serious accidents. In 1940 in Saint-Louis, with the support of the Germans, diving was added to the Olympic programme. German divers dominated the springboard scene during the first two decades. When high diving from a platform was introduced in 1908, the Swedish athletes dominated these contests.

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1.2

Introduction of Frontier III - Cantilever Diving Board

Figure 1: Frontier III - Cantilever Diving Board

The Frontier board is timber reinforced and encased in fiberglass for durability and appearance. A non-slip top ensures maximum safety. There are no unusual climate restrictions to consider, the boards are designed to be exposed to the elements and live for years.

Product features The diving board includes a streamlined and cantilevered stand with spring. The units are powder-coated Radiant White as Standard color Made of strong steel, powder coated for increased corrosion resistance. Stainless Steel Hardware - resists corrosion (the type of material) Matching, slip-resistant sand tread - for maximum safety Weight limit: 113 kg (maximum load) Various Length of diving broad: 1.83m, 2.44m, 3.05m (maximum length) All diving board and diving stand equipment is supplied with a comprehensive instruction manual Installation of all board and stand apparatus can be carried out without special skills or materials by any home handyman

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2.0

Proble m State ment

A springboard or diving board is used for diving and is a board that is itself a spring, i.e. a linear flex-spring, of the cantilever type. Springboards are commonly fixed by a hinge at one end (so they can be flipped up when not in use), and the other end usually hangs over a swimming pool, with a point midway between the hinge and the end resting on an adjustable fulcrum. Diving board is used in Olympic Games or other diving game. This study analyses which diving board is have more deflection when 113 kg/1108.53 N loads applied. This study also analyses the principle strain in the plane of rosette and the maximum in plane shearing strain.

3.0

Objective The main objective of this project study is to analyze the Frontier III - Cantilever Diving

Board using solid mechanics principles. The solid mechanic method use is stress & strain rosette to find out the principle strain in the plane of rosette and the maximum in plane shearing strain.

By using Macaulay’s methods the maximum deflection in various length of diving board also can calculate.

4.0

Scope

The analysis on air plane wing is carried out using the following basic concepts of solid mechanics only (i)

Deflection of Beam

(ii)

Principle strain in the plane of rosette

(iii)

Maximum in plane shearing strain

The following assumptions are made in this study with respect to Frontier III - Cantilever Diving Board •

The board is assumed to be horizontal

•

The self weight of board is neglected

•

The cross section is assumed as rectangular instead of air foil geometry

•

Material is assumed to be Stainless steel with high strength

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5.0

Analysis of method are use  Deflection of Beams (first method)

The deflection of a spring beam depends on its length, its cross-sectional shape, the material, where the deflecting force is applied, and how the beam is supported. The equations given here are for homogenous, linearly elastic materials, and where the rotations of a beam are small. In the following examples, only loads applying at a single point or single points are considered - the application point of force F in the diagrams is intended to denote a model locomotive horn block (or vehicle axle box) able to move vertically in a horn guide, and acting against the force of the spring beam fixed to or carried by the locomotive or vehicle mainframes. The proportion of the total weight acting on each axle of a loco or vehicle will depend on the position of its centre of gravity in relation to the axle (or the chassis fixing points of equalizing beams where these are used).

5.1

Choosing a deflection value For reasonable 4mm scale fine scale track, a recommended value for horn block

deflection, δ, under the final load of a locomotive, is 0.5mm.The above recommendation is known to be an over simplistic and possibly incorrect assumption on what the design value for the deflection should be, and has given rise to considerable debate. Any experience on applying this recommendation to real chassis modeling practice is welcomed - the purpose of this article is a starter for discussion rather than a conclusion of it.

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5.3

Example: A Cantilever beam is subjected to a bending mome nt M at the force end.

EI

d2y = Ma…………….(1) dx 2

By Integrating of equation 1 (first integration) dy = dx

EI

d2y dx 2

EI dy/dx = Ma x + C1…………….(2) (slope equation)

dy =0 dx Which is C1 = 0

At X = 0;

By Integrating of equation 2 (second integration) y=

EI y =

EI

dy 1 = Max + C1 EI dx

Max 2 + C1x + C2 …………….(2) (max. deflection equation) 2

Since the value of C1 = 0 At X = 0; y = 0

So the maximum deflection equation will be:

Max 2 y= ……………….(3) (maximum elastic curve equation) 2 EI

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 Strain gauge and rosette (second method)

The strain gauge has been in use for many years and is the fundamental sensing element for many types of sensors, including pressure sensors, load cells, torque sensors, position sensors, etc. The majority of strain gauges are foil types, available in a wide choice of shapes and sizes to suit a variety of applications. They consist of a pattern of resistive foil which is mounted on a backing material. They operate on the principle that as the foil is subjected to stress, the resistance of the foil changes in a defined way.

The strain gauge is connected into a Wheatstone Bridge circuit with a combination of four active gauges (full bridge), two gauges (half bridge), or, less commonly, a single gauge (quarter bridge). In the half and q uarter circuits, the bridge is completed with precision resistors.

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1. Transformation equation:

1

= x cos 2 2

3

=

x

+

1

cos 2

= x cos 2

x

+

2

+

3

sin2 x

x

1

sin2

sin2

3

+

sin 1 .cos

xy

+

2

+

xy

xy

sin 2 .cos

sin 1 .cos

2. Principal strain equation

1, 2

x

=

+

y

x

((

2

-

y

2

)2

(

3. Max Shear Strain

max

2

-

x

= ((

y

2

4. Principal planes

Tan2

xy

p x

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y

)2

(

xy

2

)2 )

xy

2

1

)2 )

3

2

5.3

Data of Frontier III - Cantilever Diving Board

Figure: Data of Frontier III - Cantilever Diving Board from website: (http://www.interfab.com/userfiles/2009_U-Stand.pdf)

5.4 specification of Frontier III - Cantilever Diving Board

Raw data which is use in calculation method

Table: specification of Frontier III - Cantilever Diving Board in three various lengths (Website: http://divingboard.net/info/selection_chart.asp)

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5.5

Material

Stainless steels resistance to corrosion and staining, low maintenance, relatively low cost, and familiar luster make it an ideal base material for a host of commercial applicat ions. There are over 150 grades of stainless steel, of which fifteen are most common. The alloy is milled into coils, sheets, plates, bars, wire, and tubing to be used in cookware, cutlery, hardware, surgical instruments, major appliances, industrial equipment, and as an automotive and aerospace structural alloy and construction material in large buildings. Storage tanks and tankers used to transport orange juice and other food are often made of stainless steel, due to its corrosion resistance and antibacterial properties. This also influences its use in commercial kitchens and food processing plants, as it can be steam-cleaned, sterilized, and does not need painting or application of other surface finishes. The material is uses for Frontier III Cantilever Diving Board are the stainless steel High strength which is Modulus o f elastic is 200GPa.

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5.6

Loading

Frontier III - Cantilever Diving Board is used to dive when having swimming activities. The maximum load can applied is 1108.53 N/ 113 KG. So by using three different lengths, we can determine the maximum deflection. To determine the maximum deflection, we are using Macaulay’s method which is just sectioning the last section of beam (Frontier III - Cantilever Diving Board).

5.7

Case 1: Maximum deflection

Max. Load =

113 kg/1108.53 N

5.7.1

Analysis of case:

Case 1: deflection of beam

Figure: before the swimmer stand on the diving plate

Max. Deflection

Figure: After the swimmer stand on the diving plat

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 Solution for deflection of beam 1108.53 N

M Rax

0.44 m x=0 x=L

0.8 m

L = 1.83m

Ray When x = L dy 0 dx y= 0

3

I=

bd (moment inertia) 12

I=

(0.8)(0.44) 3 = 5.679 x 10-3 m4 12

 Find out support reaction Rax = 0

Fy =

Fy

Ray = 1108.53 N / 1.109 KN  Find out slope of beam

Ma

0

Ma = (1.109 KN) (1.83m) - M = 2029.47 Nm + M M = - 2029.47Nm 1108.53 N •

Sectioning method M V

d2y EI 2 dx

-1108.53 N(X) X

2

EI

d y dx 2

-1108.53 N(X) ----------- (first Integrating)

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x=0

The Slope equation , Ө

1108.53 N(X 2 ) dy =+ C1 dx 2 EI

When X = L,

C1 =

dy dx

0 ..……… (Applying boundary condition)

1108.53 N(L2 ) 2 EI

The maximum deflection, y

1108.53 N(X 2 ) dy =+ C1…………….. (From slope equation) dx 2 EI 1108.53( X 2 ) C1 2EI

y = y =-

1108.53( X 3 ) C 1X 6EI

C2

When X = L, y = 0.

C2

1108.53( L3 ) 1108.53( L) 2 ( X ) = 6EI 2EI 1108.53( L3 ) 1108.53( L3 ) = 6 EI 2 EI =

1108.53L3 3EI

The specific deflection equation:

y= -

1108.53( X 3 ) 1108.53 N(L2 )(X) + 2 EI 6EI

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1108.53L3 3EI

When X = 0, y = Maximum. y= y=

1108.53( X 3 ) 1108.53 N(L2 )(X) + 2 EI 6EI

1108.53L3 3EI

1108.53L3 3EI

By using, I = 5.679 x 10-3 m4 & E = 200 GPa

The Slope

dy 1108.53 N(X 2 ) = dx 2 EI

When X = 1.83 m

1108.53 N(1.832 ) dy = dx 2(200G )(5.679x10 3 ) dy dx

0.00163mm

The maximum deflection, y

y= =

1108.53L3 3EI 1108.53(1.83) 3 3(200G)(5.679x10 3 )

= 0.00199mm

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Case 2: Strain rosette

1 2 3

= 400 x 10-6 mm = 200 x 10-6 mm = 350 x 10-6 mm

 Solution for Strain rosette

400

= x cos 2 0 +

y

sin2 0 +

200

= x cos 2 45 +

y

sin2 45 +

xy

sin 45 .cos 45 ……… (2)

350

= x cos 2 90 +

y

sin2 90 +

xy

sin 90 .cos 90 ……… (3)

xy

sin 0 .cos 0 ……….. (1)

From equation (1):

x

400 mm …………..(4)

From equation (2):

200

(400 )(0.5) (0.5)

y

From equation (3):

y

350 mm……………(6)

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(0.5)

xy

……………… (5)

From equation 4 & 5, substitute

350 mm &

(400 )(0.5) (350 )(0.5) (0.5)

200

(2 10 4 ) (1.75 10 4 ) (0.5)

200

xy

xy

375 0.5

350 mm

xy

Principal strain equation

=

1, 2

1, 2

400 + 350 2

((

(25 ) 2

375

400

350 2

)2

( 175 ) 2

Ans:



400 mm to equation 4

x

200

xy



y

1

551.78

2

198.22

mm mm

Max. Shear Strain

max

2

max

2

max

2 max

= ((

400 - 350 2 ) 2

(25 ) 2

176.78 353.55

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(

( 175 ) 2

mm mm

350 2 ) ) 2

(

350 2 ) ) 2

 Principal planes

Tan2

p

Tan2

p

2

6.0

353.55 400 350

7.071

81.95

p

1

40.97°

2

130.98°

Results

Methods

Macaulay’s method

Type of calculation

Results

Reaction of force, Ray

1.109 KN

Slope of beam

0.00163 mm

Max. deflection of beam

0.00199 mm ξ1 = 551.78µ mm

Principal strain ξ 2 = 198.22 µ mm Strain rosette Max Shear Strain

Principal planes

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γmax = 353.55 µ mm 1

40.97°

2

130.98°

1.7

Conclusion The analysis gives out the maximum defection by using Macaulay’s method a nd the

Principal strain, Max Shear Strain, Principal Planes by using Strain rosette of Frontier III Cantilever Diving Board. The specification of Frontier III - Cantilever Diving Board is found from the trusted website because they are one of the diving board deliver for big game event such as Olympic Games. So the specification follows the original length and width of Frontier III - Cantilever Diving Board. This diving board use Stainless steels material with 200G (this is I assume own).

Along I did this solid project; I was able to calculate the deflection of beam (Frontier III - Cantilever Diving Board) by Macaulay’s method and strain rosette to find the strain in the beam (Frontier III - Cantilever Diving Board). I also learn how to apply the concept I learn in class, in the real world or our daily life such as deflection occur in bridge by loads (cars).

So this project is really worth it if a student applying the concepts are learn in the class such as buckling of strut, strain energy, Euler theory and many more to apply in our real life.

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Referents: 1.

Ferdinand P. Beer,E Russell Johnston, John T. DeWolf. "Third Edition: Mechanics of Materials”

2.

http://en.wikipedia.org/wiki/Strain_gauge

3.

http://divingboard.net/info/selection_chart.asp

4.

http://diving.about.com/od/divingglossary/g/fulcrumDef.ht

5.

http://www.interfab.com/userfiles/2009_U-Stand.pdf

6.

http://www.aquanet.net/pool-diving-boards- fibredive.htm

7.

http://www.poolwarehouse.net/Catalogs/catDivingBoards/fibreDiveDivingBoa rds.asp

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