Strength of Materials Flashcards
The stress where the Force applied is always perpendicular to the area of application
Normal Stress
The stress where the Force applied is always parallel to the area of application
Shear Stress
Formula for Normal Stress(σ)
σ = P / A
Formula for Shear Stress(τ)
τ = P / A
Another term for Normal Stress
Axial Stress
Another term for Shear Stress
Tangential Stress
Unit for Pascal
Newton / square meter (N/m^2)
Unit for MegaPascal
Newton / square milimeter (N/mm^2)
A joint that fixes two bars with a bolt and is subjected to both normal(bar experiences this) and shear(Bolt experiences this) stresses
Lap Joint
The perpendicular stress that the bolt exerts on the rectangular area of the hole on the bar
Bearing stress
Formulas for the Factor of Safety
F.S. = σult / σallow F.S.= Fult / Fallow F.S.= τult / τallow
The Factor of Safety must be __________
Greater than 1
Formula for Strain(ε)
ε = δ / L
δ - change in length
Formula for the angle of deformation(γ)
γ = arctan(ΔX / L) in RAD
Formula for Hooke’s Law
σ = Eε
E - Modulus of Elasticity
Formula for Shear Strain(τ)
τ = Gγ
G - Modulus of Rigidity
γ - angle of deformation
Derivation of the formula for deformation(δ)
σ = Eε
but σ = P/A and ε= δ/L
P/A = Eδ/L
Answer:
δ = PL / AE
Steel’s Modulus of Elasticity
Esteel = 200 GPa
Steel’s Modulus of Rigidity
Gsteel = 83 GPa
Formula for the elongation of a material due to its own weight
δweight = (ρgL^2) / (2E)
δtotal = δexternalforce + δweight
Formula for Total Strain Energy
U = 0.5F . δ
Note: used only when below proportionality limit
Property of a material that depicts the force required to cause deformation
Stiffness
Another term for Stiffness
Spring Constant(k)
Formula for Spring Constant(k)
k = F/δ k = AE/Lo
Lo - initial length
Formula for Poisson’s Ratio(ν)
ν = -εlat / εlong
consider a cylinder:
εlat = δdiameter / do
εlong = δlength / Lo
do and Lo - initial diameter and length of cylinder
The Poisson’s Ration cannot exceed a value of ________
0.5
Formula for Shear Modulus(G)
G = E / (2(1 + ν))
E - modulus of elasticity
ν - poisson’s ratio
Formula for Bulk Modulus(K or β)
K or β= E / (3(1 + 2ν))
E - modulus of elasticity
ν - poisson’s ratio
Formulas regarding Thermal Stress
δ = αLΔT ε = αAT "eat" σ = EαΔT read invertedly; "Taeyo"
Coefficient of thermal expansion of Steel(α)
αsteel = 11.7 x 10^6 /C
Formula of the Stress of a column under its own weight
σ = ρgh
¯_(ツ)_/¯
How to deal with composite bars (in a sense, connected in parallel) that are fixed together, and Linear deformation is involved(no torqure)
δmat1 = δmat2 Ptotal = Pmat1 + Pmat2
How would you consider a cylindrical/spherical container as thin walled?
When the thickness of the container is Greater than or equal to 1/10ths of the radius of cylindrical/spherical container
What cylindrical stress is experienced by a cylindrical container along its lateral surface area?
Tangential/Hoop/Circumferential Stress
Formula for Cylindrical Tangential stress (σT)
σT = PD / (2t)
P - Internal pressure (Pa)
D - Diameter(m)
t - thickness(m)
If external pressure outside container is accounted for: P = Pexternal - Pinternal
What cylindrical stress is experienced by a cylindrical container along its Top and Bottom Base/Lid?
Longitudinal Stress
Formula for Cylindrical Longitudinal stress (σL)
σL = PD / (4t)
P - Internal pressure (Pa)
D - Diameter(m)
t - thickness(m)
If external pressure outside container is accounted for: P = Pexternal - Pinternal
If a Cylindrical container is said to explode due to internal pressure, what portion of the container will rupture? (Top and bottom Lid/Lateral surface area)
Answer: Lateral Surface Area
Remember: stress felt by the container is greater along the lateral surface for a given pressure P
σT > σL
PD / (2t) > PD / (4t)
Formula for Spherical stress (σS)
Same as Cylindrical Longitudinal stress:
σS = PD / (4t)
P - Internal pressure (Pa)
D - Diameter(m)
t - thickness(m)
If external pressure outside container is accounted for: P = Pexternal - Pinternal
If Stress due to internal pressure of a thin walled cylinder is asked for in a problem, what is the default assumed stress to obtain?(Tangential/Longitudinal)
Tangential
Formula for Torsion(τ)
τ = Tρ/J
T - Torque
J - Polar Moment of Inertia
ρ - radius
In the formula for Torsion(τ = Tρ/J), is ρ always the outermost radius of the rod under torsion?
Not necessarily.
Although, the torsion experienced by the rod is maximum at the outermost surface of the rod, that is why ρ is usually assumed as ‘r’ - the outermost radius
Moment and Torque are (Scalar/Vector)
Vector
Angular Velocity(ω) is a (Scalar/Vector)
Vector
Inertia is a (Scalar/Vector)
Scalar
Formula for the Polar Moment of Inertia(J) for a cylindrical rod
J = (π/32) d^4
or
J = (π/32) (douter^4 - dinner^4)
Formula for the angle of deformation/twist in a rod under torsion(θrad)
θrad = TL/JG
“True Love ko si Johnrae Glodo”
T - Torque
L - Length of rod ( ͡° ͜ʖ ͡°)
J - Polar Moment of Inertia
G - Modulus of Rigidity
Power transmitted by a spinning rod under torsion
P = Tω
or
P = 2πf . T
T -Torque
ω - Angular velocity
f - frequency of rotation
How to deal with composite bars connected in series) that are fixed together, and Angular deformation is involved(with torqure)
Both experience the same torque:
τ1 = τ2
The total angular twist/deformation is the summation of the two angular twists:
θT = θ1 + θ2
Formula for the shearing stress(τ) of a LIGHT helical spring
τ = [ 16PR / (πd^3) ] . [ 1 + d / (4R) ]
“(16) (P)ink (R)oses, (Pi)na(d)ala ng (3) beses sa isang(1) (d)alagang apat(4) na beses ni(R)eject”
P = Force on the spring
R = Radius of the spring(not the radius of the wire that forms the spring)
d - diameter of the wire that forms the spring
Formula for the shearing stress(τ) of a HEAVY helical spring
τ = [ 16PR / (πd^3) ] . [ (4m -1)/(4m -4) + (0.615/m) ]
“(16) (P)ink (R)oses, (Pi)na(d)ala ng (3) beses sa Apat(4) na matanda, may namatay(-) na isa(1). Apat(4) na matanda, namatay(-) pala yung apat(4) noong June 15 (0.6 15)……(m)alungkot”
m = Spring index
P = Force on the spring
R = Radius of the spring(not the radius of the wire that forms the spring)
d - diameter of the wire that forms the spring
Formula for the Spring index(m)
m = D/d
D= diameter of spring(not the diameter of the wire that forms the spring) d = diameter of the wire that forms the spring
Formula for spring Deformation(δ)
δ = 64P(R^3)n / (G(d^4))
“(64) na (P)ari, ni(R)espeto ng (3) beses ang (n)un. (G)ood yon, diba diba diba diba(d^4)?”
n - number of turns on the spring
P = Force on the spring
R = Radius of the spring(not the radius of the wire that forms the spring)
d = diameter of the wire that forms the spring
G - Modulus of Rigidity
Formula for Effective Spring Constant(Keff) when springs are connected in series
1/Keff = 1/K1 + 1/K2 + 1/K3 + …
Formula for Effective Spring Constant(Keff) when springs are connected in Parallel
Keff = K1 + K2 + …
The limit on the Stress-Strain diagram that dictates up to what point the linear formula σ = Eε stay linearly proportional
Proportionality limit
The limit on the Stress-Strain diagram that is beyond the Proportionality Limit (non linear resopnse), but still wont suffer from permanent deformation
Elastic Limit / Upper Yield Point
The Region on the Stress-Strain Diagram between the (0,0) point of the Stress-Strain Diagram and the Elastic Limit
Elastic Region
With reference to the elastic limit, where in the Stress-Strain diagram will the material experience permanent deformation?
Beyond the Elastic Limit
Reaction of the material when it experiences a stress-strain ratio on Stress-Strain Diagram between the upper(elastic limit) and lower yield point
Yielding
When Yielding(between Upper and lower yield point), does increasing the strain(deformation) require an increase in stress?
No. The Stress strain diagram dips downwards from upper to lower yield point
Reaction of the material when it experiences a stress-strain ratio on Stress-Strain Diagram between the lower yield point and the Ultimate stress point
Stream Hardening
When Stream hardening(between the lower yield point and the Ultimate stress point ), does increasing the strain(deformation) require an increase in stress?
Yes. The Stress strain diagram shows an increasing trend from the lower yield point to the ultimate stress point
When a material’s stress-strain ratio on the stress strain diagram operates between the Elastic Limit and the Ultimate Stress, the material is considered as a ______ material.
Brittle
Reaction of the material when it experiences a stress-strain ratio on Stress-Strain Diagram between the Ultimate stress point and the Rupture point
Necking
When a material’s stress-strain ratio on the stress strain diagram operates between the Ultimate stress point and the Rupture point, the material is considered as a ______ material.
Ductile
The point on the Stress-Strain diagram where the material experiences breaking/fractire
Rupture Stress point
The whole region beyond the Elastic limit is generally called as
Plastic Region
A material with a uniform cross-sectional area is said to be ________
Prismatic
A material made with a uniform material throughout is said to be ________
Homogenous
A material that has uniform properties in all directions is said to be ________
(ex. wood is not ______, because fibers are observed to be bonded in long chains only in one direction)
Isotropic
The deformation caused by CONTINUOUS load application
Creep
The deformation caused by REPETITIVE Load application
Fatigue
The bending of the material due to excessive compression
Buckling
The term used to describe the permanent deformation after the elastic limit
Permanent Set
The average value of the Stress-Strain ratio BEYOND Elastic limit in a Stress-Strain Diagram
Secant Modulus
The point on the Stress strain diagram where in the material experiences elongation without an increase in load
Yield point
The property of a material that defines how long a material can be elongated before failure
Ductility
The formula used to determine the Ductility of the material
Ductility(?) = εult / εyield
Ductile Materials are said to have a (High/Low) percentage elongation
High
The property of a material that defines the material’s ability to absorb and release strain without permanent deformation
Resilience
The property of a material that defines the material’s ability withstand occasional High Stress without fracturing
Toughness
Parameter that measures Toughness
Modulus of Toughness
If two safety parameters are given in a problem(example, both a maximum allowable stress(σallow), and a maximum allowwable strain(εallow) are given)
How do you determine which of the two will be the used parameter to obtain the minimum safe value of a parameter asked for(ex. radius, or force)?
Evaluate the problem individually, first using (σallow), and then, create a separate evaluation of the problem using (εallow).
Two answers are obtained, one from each assumed situation.
The answer is the parameter that is generally better, and is a case to case situation. So evaluate the answer upon your own discretion
( ex. a longer radius allows more stress to be handled by the material, so choose the longer radius if asked to maximize the allowable stress)
Effective Cross-sectional Area of a tapered cylinder
A = (π/4) . d1 . d2
d1 and d2 are the diameters on either ends of the tapered cylinder
The principle that states that even if a force applied on a surface is focused onto a single point, there is still a pressure distribution along the whole surface
Saint Venant’s Principle
