BDA 3033 - Solid Project

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

BDA 3033 - Solid Project

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.

BDA 3033 - Solid Project

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

BDA 3033 - Solid Project

2.0 Problem Statement

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

BDA 3033 - Solid Project

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.

BDA 3033 - Solid Project

5.3 Example: A Cantilever beam is subjected to a bending moment M at the force

end.

2

2

dx

ydEI = Ma…………….(1)

By Integrating of equation 1 (first integration)

dx

dy=

2

2

dx

ydEI

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

At X = 0; dx

dy= 0

Which is C1 = 0

By Integrating of equation 2 (second integration)

y = dx

dyEI = C1 +Max

EI

1

EI y = 2

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

Since the value of C1= 0

At X = 0; y = 0

So the maximum deflection equation will be:

y = EI

Max

2

2

……………….(3) (maximum elastic curve equation)

BDA 3033 - Solid Project

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 quarter circuits, the bridge

is completed with precision resistors.

BDA 3033 - Solid Project

1. Transformation equation:

1 = x cos2

1 + x sin

2

1 +

xysin

1.cos

1

2 = x cos2

2 + x sin

2

2 + xy sin

2.cos

2

3 = x cos2

3 + x sin2

3 + xy sin1.cos 3

2. Principal strain equation

2,1 = 2

+ yx))

2()

2

-(( 22y xyx

3. Max Shear Strain

2

max = ))2

()2

-(( 22y xyx

4. Principal planes

pTan2yx

xy

BDA 3033 - Solid Project

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

Table: specification of Frontier III - Cantilever Diving Board in three various lengths

(Website: http://divingboard.net/info/selection_chart.asp)

Raw data which is use in calculation method

BDA 3033 - Solid Project

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 of elastic is

200GPa.

BDA 3033 - Solid Project

Max. Deflection

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

5.7.1 Analysis of case:

Case 1: deflection of beam

Figure: before the swimmer stand on the diving plate

Figure: After the swimmer stand on the diving plat

Max. Load =

113 kg/1108.53 N

BDA 3033 - Solid Project

1108.53 N

Ray

M

Rax

0.8 m

0.44 m

L = 1.83m

x = 0 x = L

1108.53 N

X

V M

x = 0

Solution for deflection of beam

When x = L

I = 12

3bd(moment inertia) 0

dx

dy

y = 0

I = 12

)44.0)(8.0( 3

= 5.679 x 10-3 m4

Find out support reaction

Rax = 0

yF = yF

Ray = 1108.53 N / 1.109 KN

Find out slope of beam

0Ma

Ma = (1.109 KN) (1.83m) - M

= 2029.47 Nm + M

M = - 2029.47Nm

• Sectioning method

2

2

dx

ydEI -1108.53 N(X)

2

2

dx

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

BDA 3033 - Solid Project

The Slope equation , Ө

dx

dy = -

EI2

)N(X 1108.53 2

+ C1

When X = L, 0dx

dy..……… (Applying boundary condition)

C1 = EI2

)N(L 1108.53 2

The maximum deflection, y

dx

dy = -

EI2

)N(X 1108.53 2

+ C1…………….. (From slope equation)

y = 1

2

2

)(53.1108C

EI

X

y = - 21

3

6

)(53.1108CXC

EI

X

When X = L, y = 0.

C2 = EI

XL

EI

L

2

)()(53.1108

6

)(53.1108 23

= EI

L

EI

L

2

)(53.1108

6

)(53.1108 33

= EI

L

3

53.1108 3

The specific deflection equation:

y = -EI

X

6

)(53.1108 3

+EI2

)(X)N(L 1108.53 2

EI

L

3

53.1108 3

BDA 3033 - Solid Project

When X = 0, y = Maximum.

y = -EI

X

6

)(53.1108 3

+EI2

)(X)N(L 1108.53 2

EI

L

3

53.1108 3

y = EI

L

3

53.1108 3

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

The Slope

dx

dy =

EI2

)N(X 1108.53 2

When X = 1.83 m

dx

dy =

)10679.5)(200(2

)N(1.83 1108.533

2

xG

dx

dymm00163.0

The maximum deflection, y

y = EI

L

3

53.1108 3

= )10679.5)(200(3

)83.1(53.11083

3

xG

= mm00199.0

BDA 3033 - Solid Project

Case 2: Strain rosette

Solution for Strain rosette

400 = x cos20 + y sin

20 + xy sin0 .cos0 ……….. (1)

200 = x cos245 + y sin

245 + xy sin45 .cos45 ……… (2)

350 = x cos290 + y sin

290 + xy sin90 .cos90 ……… (3)

From equation (1):

400x mm …………..(4)

From equation (2):

xyy )5.0()5.0()5.0)(400(200 ……………… (5)

From equation (3):

350y mm……………(6)

1 = 400 x 10-6 mm

2 = 200 x 10-6 mm

3 = 350 x 10-6 mm

BDA 3033 - Solid Project

From equation 4 & 5, substitute 350ymm & 400x mm to equation 4

xy)5.0()5.0)(350()5.0)(400(200

xy)5.0()1075.1()102(200 44

5.0

375200xy

350xy mm

Principal strain equation

2,1 = 2

350+400))

2

350()

2

350400(( 22

2,1 37522 )175()25(

Ans:

78.5511 mm

22.1982 mm

Max. Shear Strain

2

max = ))2

350()

2

350-400(( 22

2

max 22 )175()25(

2

max78.176

mm

max 55.353mm

BDA 3033 - Solid Project

Principal planes

pTan2

350400

55.353

pTan2 7.071

95.812 p

1

40.97°

2

130.98°

6.0 Results

Methods

Type of calculation Results

Macaulay’s method

Reaction of force, Ray 1.109 KN

Slope of beam 0.00163 mm

Max. deflection of beam 0.00199 mm

Strain rosette

Principal strain

ξ1 = 551.78µ mm

ξ 2 = 198.22 µ mm

Max Shear Strain γmax = 353.55 µ mm

Principal planes 1 40.97°

2 130.98°

BDA 3033 - Solid Project

1.7 Conclusion

The analysis gives out the maximum defection by using Macaulay’s method and 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.

BDA 3033 - Solid Project

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