Strength Of Materials
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STRENGTH OF MATERIALS
KRISTUFFER DARCYTH TAN PRING, CE,RMP,ME I Batch 2016 Instructor, 2JP Review and Tutorial Center Instructor/Reviewer, Bicol University
STRESS AND STRAIN STRESS – force per unit area STRAIN – change in length divided by the original/gauge length SIMPLE STRESSES
A. Normal Stress
Where:
σ = Normal stress P = load effects/applied load A = area perpendicular to the plane of loading
P σ A B. Shearing Stress
Where:
V τ A Where:
C. Bearing Stress
Pb σb Ab
τ = Shearing stress V = Shearing force A = area parallel to the plane of loading
σb = Bearing stress Pb = Load produced by contact pressure Ab = area of contact
DEFORMATION OF MEMBER UNDER AXIAL LOADING
PL δ AE
Normal stress
Normal strain
P σ A
δ ε L
Where: δ = elongation/contraction P = internal load effect L = Original length A = cross sectional area E = Elastic modulus AE = axial rigidity HOOKE’S LAW
σ εE P δ E A L PL δ AE
STRAIN
NORMAL/ AXIAL STRAIN
Where: ξ = Normal strain δ δ = change in length ε L L = Original length HOOKE’S LAW – states that the within the elastic limit, stress is proportional to the strain HOOKE’S LAW
σ εE Where: σ = Normal stress ε = Normal strain E = Young’s Modulus of Elasticity = (the slope of the stress-strain diagram)
THIN -WALLED PRESSURE VESSEL A- CYLINDRICAL VESSEL
CYLINDRICAL PRESSURE VESSEL
Where: σ = Longitudinal stress p = applied pressure d = diameter r = radius t = thickness
FBD FOR CIRCUMFERENCIAL/ TANGENTIAL STRESS σC
FBD FOR LONGITUDINAL/ GIRTH STRESS σL
pr pd σc t 2t
pr pd σL 2t 4t
THIN -WALLED PRESSURE VESSEL A- SPHERICAL VESSEL
SPHERICAL PRESSURE VESSEL
pr pd σ 2t 4t
FBD FOR STRESS σ
Where: σ = Longitudinal stress p = applied pressure d = diameter r = radius t = thickness
pr pd σ 2t 4t
Where: σc = circumferencial stress σL = Longitudinal stress p = applied pressure d = diameter r = radius t = thickness
TORSION FORMULAS TORSIONAL SHEARING STRESS
Where: τ = shearing stress T = internal torque ρ = radial distance J = Polar moment of inertia MAXIMUM SHEARING STRESS
Sign convention for torque T and angle of twist θ
POLAR MOMENT OF INERTIA OF CIRCULAR AREAS
HELICAL SPRINGS STRESSES IN SPRINGS
LIGHT SPRINGS
DIRECT SHEARING STRESS
16 PR d 3
P 1 A
d 1 4 R
TORSIONAL SHEARING STRESS
2
Tr J
T PR
where: P =applied load R = mean radius of the spring d = cross sectional diameter
HEAVY SPRINGS considering curvature 16 PR d 3
4m 1 0.615 4m 4 m
where:
SPRING DEFLECTIONS where:
64 PR 3 n Gd 4
SPRING CONSTANT
P =applied load R = mean radius of the spring d = cross sectional diameter m = 2R/d (4m – 1)/(4m – 4) = spring index (Wahl factor) P =applied load R = mean radius of the spring n = number of turns G = shear modulus d = cross sectional diameter
SPRING IN SERIES SPRING IN PARALLEL
Gd 4 k 64 R 3n P
STRESSES IN BEAMS BENDING STRESS Moment-Curvature Relationships
FLEXURAL NORMAL STRESS
Where: ρ = radius of curvature 1/ρ = curvature M = bending moment I = centriodal moment of inertia
1 M ρ EI
NORMAL STRESS/STRAIN DUE TO BENDING b
εt
σt
ct h cb εb CROSS SECTION
STRAIN DIAGRAM
σb STRESS DIAGRAM
x max
M(y) I Mc M I S
Where: σx = bending stress at any surface from NA σmax = maximum bending stress y = distance from NA where bending stress is required c = distance from NA tot the outermost fiber I = centriodal moment of inertia S = elastic section modulus = I/c
STRESSES IN BEAMS SHEAR STRESS
Horizontal Shear Stress
VQ It Q Ay
Shear flow
VQ VQ q τt t It I Rivet capacity
Spacing of rivet Where: τ = horizontal shear stress V = Shear force Q = A.y = Statical moment of area I = centriodal moment of inertia t = thickness
R τ r Ar RI s VQ
q = shear flow R = Shear capacity of rivet τr = allowable shear stress of rivet Ar = cross sectional area of rivet s = spacing of rivet
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KRISTUFFER DARCYTH TAN PRING, CE,RMP,ME I Batch 2016 Instructor, 2JP Review and Tutorial Center Instructor/Reviewer, Bicol University
STRESS AND STRAIN STRESS – force per unit area STRAIN – change in length divided by the original/gauge length SIMPLE STRESSES
A. Normal Stress
Where:
σ = Normal stress P = load effects/applied load A = area perpendicular to the plane of loading
P σ A B. Shearing Stress
Where:
V τ A Where:
C. Bearing Stress
Pb σb Ab
τ = Shearing stress V = Shearing force A = area parallel to the plane of loading
σb = Bearing stress Pb = Load produced by contact pressure Ab = area of contact
DEFORMATION OF MEMBER UNDER AXIAL LOADING
PL δ AE
Normal stress
Normal strain
P σ A
δ ε L
Where: δ = elongation/contraction P = internal load effect L = Original length A = cross sectional area E = Elastic modulus AE = axial rigidity HOOKE’S LAW
σ εE P δ E A L PL δ AE
STRAIN
NORMAL/ AXIAL STRAIN
Where: ξ = Normal strain δ δ = change in length ε L L = Original length HOOKE’S LAW – states that the within the elastic limit, stress is proportional to the strain HOOKE’S LAW
σ εE Where: σ = Normal stress ε = Normal strain E = Young’s Modulus of Elasticity = (the slope of the stress-strain diagram)
THIN -WALLED PRESSURE VESSEL A- CYLINDRICAL VESSEL
CYLINDRICAL PRESSURE VESSEL
Where: σ = Longitudinal stress p = applied pressure d = diameter r = radius t = thickness
FBD FOR CIRCUMFERENCIAL/ TANGENTIAL STRESS σC
FBD FOR LONGITUDINAL/ GIRTH STRESS σL
pr pd σc t 2t
pr pd σL 2t 4t
THIN -WALLED PRESSURE VESSEL A- SPHERICAL VESSEL
SPHERICAL PRESSURE VESSEL
pr pd σ 2t 4t
FBD FOR STRESS σ
Where: σ = Longitudinal stress p = applied pressure d = diameter r = radius t = thickness
pr pd σ 2t 4t
Where: σc = circumferencial stress σL = Longitudinal stress p = applied pressure d = diameter r = radius t = thickness
TORSION FORMULAS TORSIONAL SHEARING STRESS
Where: τ = shearing stress T = internal torque ρ = radial distance J = Polar moment of inertia MAXIMUM SHEARING STRESS
Sign convention for torque T and angle of twist θ
POLAR MOMENT OF INERTIA OF CIRCULAR AREAS
HELICAL SPRINGS STRESSES IN SPRINGS
LIGHT SPRINGS
DIRECT SHEARING STRESS
16 PR d 3
P 1 A
d 1 4 R
TORSIONAL SHEARING STRESS
2
Tr J
T PR
where: P =applied load R = mean radius of the spring d = cross sectional diameter
HEAVY SPRINGS considering curvature 16 PR d 3
4m 1 0.615 4m 4 m
where:
SPRING DEFLECTIONS where:
64 PR 3 n Gd 4
SPRING CONSTANT
P =applied load R = mean radius of the spring d = cross sectional diameter m = 2R/d (4m – 1)/(4m – 4) = spring index (Wahl factor) P =applied load R = mean radius of the spring n = number of turns G = shear modulus d = cross sectional diameter
SPRING IN SERIES SPRING IN PARALLEL
Gd 4 k 64 R 3n P
STRESSES IN BEAMS BENDING STRESS Moment-Curvature Relationships
FLEXURAL NORMAL STRESS
Where: ρ = radius of curvature 1/ρ = curvature M = bending moment I = centriodal moment of inertia
1 M ρ EI
NORMAL STRESS/STRAIN DUE TO BENDING b
εt
σt
ct h cb εb CROSS SECTION
STRAIN DIAGRAM
σb STRESS DIAGRAM
x max
M(y) I Mc M I S
Where: σx = bending stress at any surface from NA σmax = maximum bending stress y = distance from NA where bending stress is required c = distance from NA tot the outermost fiber I = centriodal moment of inertia S = elastic section modulus = I/c
STRESSES IN BEAMS SHEAR STRESS
Horizontal Shear Stress
VQ It Q Ay
Shear flow
VQ VQ q τt t It I Rivet capacity
Spacing of rivet Where: τ = horizontal shear stress V = Shear force Q = A.y = Statical moment of area I = centriodal moment of inertia t = thickness
R τ r Ar RI s VQ
q = shear flow R = Shear capacity of rivet τr = allowable shear stress of rivet Ar = cross sectional area of rivet s = spacing of rivet
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