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MECHANICS OF MATERIALS: &

Stress:

 There are four types of : o Normal (푁) is perpendicular to the surface o force (푉) is parallel to the surface

o Torsional moment (푀푡) is about the axis normal (perpendicular) to the surface

o moment (푀퐵) is about the axis parallel to the surface  Stress measures the intensity of the force per given area:

o Normal stress (휎) results from the normal force 푁 and/or 푀퐵

o (휏) results from shear stress 푉 and/or torsional moment 푀푡  Stress can occur on oblique planes:

such that: 푁 푃푐표푠2휃 푁 = 푃푐표푠휃 휎 = = 퐴휃 퐴 푉 = 푃푠푖푛휃 퐴 푉 푃푠푖푛휃푐표푠휃 퐴휃 = 휏 = = 푐표푠휃 퐴휃 퐴

 Factor of safety is the ratio of failure load case to the necessary/typical load case:

푢푙푡푖푚푎푡푒 푙표푎푑 푢푙푡푖푚푎푡푒 푠푡푟푒푠푠 퐹푆 = = 푎푙푙표푤푎푏푙푒 푙표푎푑 푎푙푙표푤푎푏푙푒 푠푡푟푒푠푠

Strain:

 Strain is a measure of the material’s response to stress and is expressed as a ratio to the change in length to the original length: Where elongation results in a positive strain; 퐿푓 − 퐿표 ∆퐿 휀푎푣푔 = = results in a negative strain 퐿표 퐿표  Shear strain is based on the rotation of the object, measured in radians: 훾 = angle of  Young’s Modulus defines the relationship between normal stress and lateral strain: 휎 퐸 = 휀 This can be graphically represented on a stress-strain diagram (note it only holds for the elastic region) as the rise-run ratio. Young’s Modulus is material-dependent and can be found in tables.

Elastic Region in which an applied stress will cause a strain that is Region non-permanent. When the stress is removed, the material will return to its original state (reducing any elongation to 0). Plastic Region in which applied stresses cause irreversible strains. Region When the stress is removed, the material will return to the elongation of the point but will not reenter the elastic region. Yield Dividing point between elastic behavior and plastic behavior Strength (“point of no return”). Any stress applied that is greater than the stress at the yield point results in bringing the material into the plastic region. Brittle A brittle material will experience miniscule elongation prior to Material failure. Ductile A ductile material will experience significant elongation Material (“”) before failure. Particularly, it may have a long period in the plastic region before the point. Ultimate The greatest stress that is experienced in a material. Ductile Strength materials will experience less stress as necking increases. Brittle materials will have a fracture point very near (if not identical to) the ultimate strength. Fracture The point at which the material separates, or breaks. This point Point will be close to the ultimate stress for low materials and further from the ultimate stress for high ductility materials.

 Modulus of Rigidity is similar to Young’s Modulus but measures the ratio of shear stress to angle of deformation: 휏 퐺 = 훾

 Poisson’s Ratio measures the rate of lateral strain to axial strain; determining how likely a sample is to “neck” (think of taffy as it is pulled):

푙푎푡푒푟푎푙 푠푡푟푎푖푛 휀푦 휀푧 푣휎푥 ∆퐷푖푎 푣 = − = − = − ; 휀푦 = 휀푧 = − ; −휀푑푖푎 = 푎푥푖푎푙 푠푡푟푎푖푛 휀푥 휀푥 퐸 퐷푖푎표

A typical ratio is 0.2 − 0.4. This indicates the “sideways” strain will be 40% of the “linear” strain.

 Relationship between these three (퐸, 퐺, 푣) can be described using the following, such that knowing any two results in knowing the third:

퐸 = 2퐺(1 + 푣)

 Strain observes how much energy is absorbed before entering the plastic deformation zone and in total before failure: 푥 Because 푤표푟푘 = ∫ 푓 퐹 ⋅ 푑푥 , and Hooke’s Law 퐹 = 푘푥 … 푥0 푥1 1 1 푢 = ∫ (푘푥)푑푥 = 푘푥2 = 퐹푥 2 2 Linear approximation 0 푥1 휀1 of green area 푢 푃 푑푥 = ∫ ⋅ = ∫ 휎 푑휀 푣표푙푢푚푒 퐴 퐿 푥 푥 0 0

휀푦 휀푦 퐸 휎2 푢 = ∫ 휎 푑푥 = 퐸 ∫ 휀 푑휀 = 휀2 = 푦 = 푢 푦 푥 푥 푥 2 푦 2퐸 푦 0 0

푢푡 = 푢푦 + ∆푦 ⋅ ∆푥

For more information, visit a tutor. All appointments are available in-person at the Student Success Center, located in the Library, or online. Adapted from Hibbeler, R.C. (2014). Mechanics of Materials (9th Edition). Boston, MA: Prentice Hall.