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7/29/2019 Related Theory for Gurney
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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