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FE Review
Mechanics of Materials
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1
First Concept – Stress
Normal Stress (normal to surface)
Shear Stress (along surface)
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Second Concept – Strain
Normal Strain – length change
Mechanical
Thermal
Shear Strain – angle change
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2
Material Properties
Hooke’s Law
Normal (1D)
Normal (3D)
Shear
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Material Properties
Poisson’s ratio
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3
Axial Loading F
F σx
F
Stress σ x = P A
Deformation δ = ∑ PL
AE
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Torsional Loading T
T
Stress τ =T ρ
J
τ max =Tc J
ρ
τmax τ
Deformation θ = ∑ TL
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JG
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4
Bending Stress M
M
Stress σ x =− M r y
I
σ max = M rc I
σx
Find centroid of cross-section Calculate I about the Neutral Axis
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Transverse Shear Equation τ ave =V
Average over entire cross-section
A
τ ave =VQ Ib
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Average over line V = internal shear force b = thickness I = 2nd moment of area Q = 1st moment of area of partial section Mechanics of Materials
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5
Partial 1st Moment of Area (Q)
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Max. Shear Stresses on Specific Cross-Sectional Shapes Rectangular Cross-Section
τ max = 3V
2A
τ
Circular Cross-Section
τ max = 4V
3A
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τ Mechanics of Materials
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Max. Shear Stresses on Specific Cross-Sectional Shapes Wide-Flange Beam
τ max ≈ V
Aweb
τ
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8
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V & M Diagrams V
w = dV dx
M
V = dM dx
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Six Rules for Drawing V & M Diagrams 1.
2.
3. 4. 5. 6.
w = dV/dx The value of the distributed load at any point in the beam is equal to the slope of the shear force curve. V = dM/dx The value of the shear force at any point in the beam is equal to the slope of the bending moment curve. The shear force curve is continuous unless there is a point force on the beam. The curve then “jumps” by the magnitude of the point force (+ for upward force). The bending moment curve is continuous unless there is a point moment on the beam. The curve then “jumps” by the magnitude of the point moment (+ for CW moment). The shear force will be zero at each end of the beam unless a point force is applied at the end. The bending moment will be zero at each end of the beam unless a point moment is applied at the end. FE Review
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11
Deflection Equation d2y = M dx2 EI
y = deflection of midplane M = internal bending moment E = elastic modulus I = 2nd moment of area with respect to neutral axis
To solve bending deflection problems (find y): 1. Write the moment equation(s) M(x) 2. Integrate it twice 3. Apply boundary conditions 4. Apply matching conditions (if applicable) FE Review
Combined Loading We have derived stress equations for four different loading types:
σx =
P A
τ max = k
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τ=
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Tc J
σx = −
Mc I
σx = +
Mc I
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Method for Solving Combined Loading Problems 1. Find internal forces and moments at cross-section of concern. 2. Find stress caused by each individual force and moment at the point in question. 3. Add them up.
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Thin-Walled Pressure Vessels
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Column Buckling
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Maximum Shear Stress Theory σp2
σY
−σY
σY −σY
σp1 Failure occurs when:
σ p1 > σ Y σ p1 − σ p 2 > σ Y
if σp1 and σp2 have the same sign if σp1 and σp2 have different signs
where σp1 is the largest principal stress. FE Review
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Maximum Distortion Energy Theory This theory assumes that failure occurs when the distortion energy of the material is greater than that which causes yielding in a tension test.
