Fundamental Formulas for Exam 1 in CE 220 – Mechanics of Solids Average Normal Stress, π π π= π΄ π = internal normal force acting on the desired cross-section π΄ = area of the desired cross-section Note that π and π΄ must correspond to the same cross-section Average Shear Stress, π π π= π΄ π = internal shear force acting on the desired cross-section π΄ = area of the desired cross-section Note that π and π΄ must correspond to the same cross-section Average Bearing Stress, ππ π ππ = π΄π π = internal normal force acting on the desired cross-section where bearing occurs π΄π = area of the desired cross-section where bearing occurs Note that π and π΄π must correspond to the same cross-section For a round pin or bolt in a round hole, the diameter of the pin or bolt is used as the width of the bearing area when calculating the average bearing stress Factor of Safety, F.S. πππππ πππππ πΉ. π. = , πΉ. π. = , ππππππ€ ππππππ€ πΉ. π. = πππππ , ππππππ€ πΉ. π. = πππππ ππππππ€ Average Normal Strain, πΊ πΏπ − πΏπ π= πΏπ πΏπ = final length between two points after deformation πΏπ = initial length between two points before deformation Shear Strain, πΈ π πΎ = − π′ 2 π π′ = final angle between two line segments for which the initial angle was 2 Poisson’s Ratio ππππ‘ π=− πππππ ππππ‘ = lateral strain (perpendicular to applied normal stress, π) πππππ = longitudinal strain (parallel to applied normal stress, π) Hooke’s Law for Stress and Strain For uniaxial normal strain π = πΈπ π = normal stress πΈ = Elastic modulus (Young’s Modulus), slope of the initial linear portion of the π − π diagram π = normal strain For shear strain π = πΊπΎ π = shear stress πΊ = Shear modulus (Modulus of Rigidity), the slope of the initial linear portion of the π − πΎ diagram π = shear strain Generalized Hooke’s Law for multiaxial stress 1 1 ππ₯ = [ππ₯ − π(ππ¦ + ππ§ )], ππ¦ = [ππ¦ − π(ππ₯ + ππ§ )], πΈ πΈ ππ₯π¦ ππ¦π§ ππ₯π§ πΎπ₯π¦ = , πΎπ¦π§ = , πΎπ₯π§ = πΊ πΊ πΊ ππ§ = 1 [π − π(ππ₯ + ππ¦ )] πΈ π§ Change in Length of Axially Loaded Members For homogeneous and prismatic members subjected to an constant internal axial force π⋅πΏ πΏ= π΄⋅πΈ π = internal normal force acting member πΏ = the length of the member π΄ = the cross-sectional area of the member πΈ = the elastic modulus of the material from which the member is made For members made of discrete sections that are individually homogeneous, prismatic, and subjected to an constant internal axial force π ππ ⋅ πΏπ π1 ⋅ πΏ1 π2 ⋅ πΏ2 ππ ⋅ πΏπ πΏ = ∑( )= + + β―+ π΄π ⋅ πΈπ π΄1 ⋅ πΈ1 π΄2 ⋅ πΈ2 π΄π ⋅ πΈπ π=1 ππ = internal normal force acting discrete member i πΏ = the length of discrete member i π΄ = the cross-sectional area of discrete member i πΈ = the elastic modulus of the material from which discrete member i is made For homogeneous members that are non-prismatic and/or have a varying internal axial force πΏ π(π₯) πΏ=∫ ππ₯ 0 π΄(π₯) ⋅ πΈ π(π₯) = a function describing the distribution of internal axial force along the member π΄(π₯) = a function describing the cross-sectional area along the length of the member πΈ = the elastic modulus of the material from which the member is made
