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7/29/2019 Related Theory for Gurney http://slidepdf.com/reader/full/related-theory-for-gurney 1/8 RELATED THEORY Stress Stress is a measuring value of force acting on a unit area of one material. It is a property of material which will define the deformation of material by the result of applied force. Strain Strain is a measuring value of deformation length or length that is elongated or shorten in relation with the initial length of that material in which shown by the ratio. Modulus of Elasticity (Tension) Modulus of elasticity or Young’s modulus is a measuring value used to determine the deformation tendency of that material due to the applied force. It can be observed from the stress-strain diagram, table given by the manufacturer, and calculation using of formula  Where E = modulus of elasticity = stress for that material = strain for that material  Note:  Where P = the amount of load applied A = the cross sectional area which load is acting on  Where = the change in length of the material = the initial length of the material Tensile Strength Tensile strength or Ultimate tensile strength (Su) is the amount of stress that each material can withstand without necking (material is being pulled until its cross-sectional area decrease). Normally the tensile strength is the value that came from an experiment for each material and can be pointed out from the stress-strain diagram.

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RELATED THEORY

Stress

Stress is a measuring value of force acting on a unit area of one material. It is a property of material

which will define the deformation of material by the result of applied force.

Strain

Strain is a measuring value of deformation length or length that is elongated or shorten in relation

with the initial length of that material in which shown by the ratio.

Modulus of Elasticity (Tension)

Modulus of elasticity or Young’s modulus is a measuring value used to determine the deformation

tendency of that material due to the applied force. It can be observed from the stress-strain diagram,

table given by the manufacturer, and calculation using of formula

 

Where E = modulus of elasticity

= stress for that material

= strain for that material

 Note:  

Where P = the amount of load applied

A = the cross sectional area which load is acting on

 

Where = the change in length of the material

= the initial length of the material

Tensile Strength

Tensile strength or Ultimate tensile strength (Su) is the amount of stress that each material can

withstand without necking (material is being pulled until its cross-sectional area decrease). Normally

the tensile strength is the value that came from an experiment for each material and can be pointed out

from the stress-strain diagram.

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Stress-Stain diagrams

Yield Strength

Yield strength (Sy) is the amount of stress in which applied to the material and made it starts to

deform plastically. Once the stress that is applied to the material is exceed the yield strength, material

then will not return to its initial form. The yield strength position can be pointed out from the figure

1.1 also.

Shear Strength

Shear strength is the amount of the shear stress or the stress that acts on the parallel direction to the

cross-sectional area of the material in which material is starting to ruptured. The shear strength can be

determined from the formula of 

 

 

Where F = applied loads

A = cross-sectional area that parallel to the load

Shear Strain

Shear is the amount of deformation when the material is supplied by tangential loads which can be

shown by the relation between the displacement of one side of the material that moved and the

distance between both sides. The formula using for calculation is

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The deformation of material under shear stress

Modulus of Elasticity (Shear)

Modulus of elasticity in shear is most likely the same as the modulus of elasticity in tension. It is the

relation between shear stress and shear strain which can be shown in the formula of 

 

 Note: Both modulus of elasticity in tension and shear can be classified in the form of Hooke’s law

which is shown the relation between applied load and deformation of the object.

Bending

Bending is usually occurred at the beams or shafts which consider as the structural elements since

they have to act as the carrier for other parts of the machine. Normally there are several topics that

involved with bending and should be determined so that we can estimate and also select the proper 

size of material, design of the system etc.

Shear and moment diagrams are mostly concern at the first place so that we can determine the load

which supports are going to receive according to the applied load on the object.

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(a) (b)

(a)  Sign convention in bending, (b) Example of a simple shear and moment diagram

After that, the stress due to bending can be determined which can give us the amount maximum and

minimum stress on the object that we are going to consider on the next topic about design factor. The

formula which is normally used to calculate for bending stress is

 

Where  = bending stress

M = moment due to bending

c = measured length from the neutral axis of the center line to the outer surface

I = moment of inertia for that material

Design Factor 

Design factor or factor of safety is a criteria in which determine the amount of stress should be

applied to the material in order for that material can withstand the stress or load applied without

failure. Design factor can be determined from the formula of 

 

Where N = Design factor of that material

S = the material strength or UTS

= the design stress for that material

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Also there are design factors given which are suited for specific type of material and specific purposes

and can be found from many sources.

For Ductile materials,

  N = 1.25 – 2.0 is suited for structures which undergo static loads with confidence in all data.

   N = 2.0 – 2.5 is suited for machine elements which undergo dynamic loads with confidence in

all data.

   N = 2.5 – 4.0 is suited for structures or machine elements which undergo dynamic loads with

uncertain in the amount of loads, properties of material, environment etc.

   N = 4.0 or higher is suited for structures or machine elements which also uncertain with loads,

 properties, environment. This value is to provide more safety in the design process.

For Brittle materials,

   N = 3.0 – 4.0 is suited for structures which undergo static loads with confidence in all data.

   N = 4.0 – 8.0 is suited for structures or machine elements which undergo dynamic loads withuncertain in the amount of loads, properties of material, environment etc.

For special cases which the material is supplied with loads that are not balance, other options which

can be used to determine the design factor should be selected such as Goodman method. The

Goodman method is used for the fluctuating stress on material cases in order to determine the design

factor for that material. It can be expressed into the formula of 

 

 

Where  = the theoretical stress-concentration factor 

= alternating stress

= mean stress

  = actual endurance strength of the material

= ultimate tensile strength

 Note: =  

Where = maximum stress applied to the material

= minimum stress applied to the material

=  

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Where   = endurance strength or the amount of stress that material can withstand for a certain

 period of times in cycle

= material factor 

  = stress factor 

= reliability factor 

= size factor 

Columns Design

Columns are the part that carried the loads which are applied to the whole machine and their  properties have to be well considered so that the machine can performs any application without

failure. Column design is normally studies about how to design and choose for the materials in which

their properties are matched with the application or can support the applied loads. Bucking of columns

is used to consider the amount of load that material can withstand. First, checking the column whether 

it is long or short column by comparing the slenderness ration and the column constant. If the

slenderness ration is higher than the column constant, we can conclude that the column is a long

column and the critical load can be determined by the formula of 

 

Where = critical load for the material

E = modulus of elasticity

A = cross-sectional area of the material

K = constant for different types of end fixity

L = length of the material

r = radius of gyration

 Note: r = √  

Where I = moment of inertia for that material

A= cross-sectional area of the material

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K is given for each type of end fixity which are

fixed-free end; K = 2.0 for theoretical value and 2.1 for actual value

 pinned-pinned end; K = 1.0 for theoretical value and actual value

fixed-pinned end; K = 0.7 for theoretical value and 0.8 for actual value

fixed-fixed end; K = 0.5 for theoretical value and 0.65 for actual value

SR = slenderness ration =  

CC = column constant = √  ; Sy = yield strength

After that design factor should be involved in order to calculate for the allowable load of the column

and make sure that it is safe perform the application. By using the formula

Pa =  

Where Pa = allowable load

 N = design factor 

If the slenderness ratio that have been calculated is less than the column constant, then the column is

considered to be short and another formula should be used instead. The formula for short column is

known in the name of J.B. Johnson formula which is

Pcr  = σcr A =   [ ( ) ] For column under the eccentric load, we have to use another formula in order to calculate for the

amount of critical load, stress for the material. By assume that the column is a pinned-pined end, since

we have to consider about the eccentricity of the column as the load is applied to the edge.

(a) (b)

(a) A pinned-pinned column with eccentrically loaded,

(b) Example of secant formula diagram (load per unit area vs. slenderness ratio)

The formula used for this case is

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Where

= maximum stress for the material which is located at the center 

e = eccentricity

P = amount of applied load

This formula can be written in other form which focusing on yield strength instead.

These two formulas are used for different conditions of end fixity. However, the first formula is often

used than the second one.

Rather than the amount of critical load, the amount of maximum deflection should be concerned so

that we can choose and design the proper application. The maximum deflection can be calculated

from the formula

Where ymax = maximum deflection that material can withstand