Mechanics of Materials Formula Sheet

𝜎𝑦 𝜎 𝜀𝑥 = 𝐸𝑥 − 𝑣 𝐸 + 𝛼∆𝑇 𝜎𝑦 𝜎 𝜀𝑦 = 𝐸 − 𝑣 𝐸𝑥 +𝛼∆𝑇 𝜏𝑥𝑦 𝛾𝑥𝑦 = 𝐺 𝐸 𝐸𝛼∆𝑇 𝜎𝑥 = 1−𝑣2 (𝜀𝑥 + 𝑣𝜀𝑦 ) − 1−𝑣 𝐸 𝐸𝛼∆𝑇 𝜎𝑦 = 1−𝑣2 (𝜀𝑦 + 𝑣𝜀𝑥 ) − 1−𝑣 𝐸 𝐾= 3(1 − 2𝑣) 2 𝑣𝐸 λ= (1 + 𝑣)(1 − 2𝑣) 2 𝜀𝑎 = 𝜀𝑥 𝑐𝑜𝑠 𝜃𝑎 + 𝜀𝑥 𝑠𝑖𝑛 𝜃𝑎 + 𝛾𝑥𝑦 𝑠𝑖𝑛𝜃𝑎 𝑐𝑜𝑠𝜃𝑎 2 𝜎𝑏 = 𝑀𝑟 𝐼 𝑇𝑟 𝜏𝑡 = − 𝐽 𝜀𝑐 = 𝜀𝑥 𝑐𝑜𝑠 2 𝜃𝑐 + 𝜀𝑦 𝑠𝑖𝑛2 𝜃𝑐 + 𝛾𝑥𝑦 𝑠𝑖𝑛𝜃𝑐 𝑐𝑜𝑠𝜃𝑐 𝜕𝑢 𝜕𝑥 𝜀𝑦 = 𝜕𝑣 𝜕𝑦 𝛾𝑥𝑦 = ∝𝑥 −∝𝑦 = 𝜀𝑦 ′ = 𝜀𝑥 + 𝜀𝑦 𝜀𝑥 − 𝜀𝑦 𝛾𝑥𝑦 − cos 2𝜃 − sin 2𝜃 2 2 2 𝜕𝑢 𝜕𝑣 + 𝜕𝑦 𝜕𝑥 𝜀1,2 = 𝜀𝑥 −𝜀𝑦 2 𝜀𝑥 − 𝜀𝑦 2 𝛾𝑥𝑦 2 + 2 2 𝜀𝑜 = 𝜎𝑥 + 𝜎𝑦 𝜎𝑥 − 𝜎𝑦 + cos 2𝜃 + 𝜏𝑥𝑦 sin 2𝜃 2 2 𝜎12 − 𝜎1 𝜎2 + 𝜎22 = 𝜎𝑦𝑝 𝜎𝑥 + 𝜎𝑦 𝜎𝑥 − 𝜎𝑦 𝜎𝑦 ′ = − cos 2𝜃 − 𝜏𝑥𝑦 sin 2𝜃 2 2 𝜏𝑥 ′ 𝑦 ′ = − 𝜎𝑥 −𝜎𝑦 2 𝜌= 𝜎1 = 𝜎𝑢 or 𝜎2 = 𝜎𝑢 𝜀𝑥 𝜏𝑥𝑦 𝜎𝑦 𝜏𝑧𝑦 𝜏𝑥𝑧 𝜏𝑦𝑧 𝜎𝑧 𝜀𝑖𝑗 = 𝜎𝑥 − 𝜎𝑦 2 2 + 𝜏𝑥𝑦 2 𝑚 𝑉 2 𝜏𝑥𝑦 𝜎𝑥 − 𝜎𝑦 𝑃 𝐴 𝜎𝑥 = 𝐸 𝜀𝑥 𝑃 𝛿 =𝐸 𝐴 𝐿 𝛿= 𝑝𝑟 𝜎𝑡 = 𝑡 𝑝𝑟 𝜎𝑡 = 𝑡𝑐𝑜𝑠𝛼 𝑊 = 𝑚𝑔 4 𝑉 = 𝜋(𝑅 3 − 𝑟 3 ) 3 𝑒= 𝜀1,2 = ∆𝑉 𝑉0 𝜀𝑥 = 𝛿 𝐿 𝑃𝐿 𝐴𝐸 𝑝𝑟 𝜎𝑙 = 2𝑡 𝑝𝑟 𝜎𝑙 = 2𝑡𝑐𝑜𝑠𝛼 1 𝜀 + 𝜀𝑐 ± (𝜀𝑎 − 𝜀𝑐 )2 +(2𝜀𝑏 − 𝜀𝑎 − 𝜀𝑐 )2 2 𝑎 𝐸 𝜀𝑎 + 𝜀𝑐 1 ± (𝜀𝑎 − 𝜀𝑐 )2 +(2𝜀𝑏 − 𝜀𝑎 − 𝜀𝑐 )2 2 1−𝑣 1+𝑣 2𝜀𝑏 − 𝜀𝑎 − 𝜀𝑐 1 𝑡𝑎𝑛2𝜃𝑝 = 𝜀1,2 = 𝜀𝑎 + 𝜀𝑏 + 𝜀𝑐 ± 2 (𝜀𝑎 − 𝜀𝑏 )2 +(𝜀𝑏 − 𝜀𝑐 )2 +(𝜀𝑐 − 𝜀𝑎 )2 𝜀𝑎 − 𝜀𝑐 3 𝜎1,2 = sin 2𝜃 + 𝜏𝑥𝑦 cos 2𝜃 𝜎𝑥 𝜎𝑖𝑗 = 𝜏𝑦𝑥 𝜏𝑧𝑥 𝜏𝑚𝑎𝑥 = ± 𝐴𝑙𝑙𝑜𝑤𝑎𝑏𝑙𝑒 𝑆𝑡𝑟𝑒𝑠𝑠 −1 𝐹𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑆𝑎𝑓𝑒𝑡𝑦 (𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑆𝑡𝑟𝑒𝑠𝑠) 𝜎𝑦𝑝 = 𝑀𝐴𝑋( 𝜎1 − 𝜎2 , 𝜎2 − 𝜎3 , 𝜎3 − 𝜎1 ) 𝜎𝑥 ′ = 𝜎𝑥 + 𝜎𝑦 ± 2 𝜎𝑥 = 𝐿 − 𝐿𝑜 ∆𝐿 𝛿 = = 𝐿𝑜 𝐿𝑜 𝐿𝑜 𝜎𝑥 − 𝜎𝑦 2 2 + 𝜏𝑥𝑦 2 𝜎𝑚𝑎𝑥,𝑚𝑖𝑛 = 𝑡𝑎𝑛2𝜃𝑝 = 𝜀𝑥 − 𝜀𝑦 2 𝛾𝑥𝑦 2 + 2 2 𝛾𝑥𝑦 𝑡𝑎𝑛2𝜃𝑝 = 𝜀𝑥 − 𝜀𝑦 𝑀𝑎𝑟𝑔𝑖𝑛 𝑜𝑓 𝑆𝑎𝑓𝑒𝑡𝑦 = 𝛾𝑥𝑦 sin 2𝜃 + 2 cos 2𝜃 𝜀𝑥 + 𝜀𝑦 ± 2 𝛾𝑚𝑎𝑥 = ±2 2 𝜀𝑏 = 𝜀𝑥 𝑐𝑜𝑠 𝜃𝑏 + 𝜀𝑦 𝑠𝑖𝑛 𝜃𝑏 + 𝛾𝑥𝑦 𝑠𝑖𝑛𝜃𝑏 𝑐𝑜𝑠𝜃𝑏 𝜀𝑥 = 𝜀𝑥 + 𝜀𝑦 𝜀𝑥 − 𝜀𝑦 𝛾𝑥𝑦 + cos 2𝜃 + sin 2𝜃 2 2 2 𝛾𝑥 ′ 𝑦 ′ = − 𝜏𝑥𝑦 = 𝐺𝛾𝑥𝑦 𝐸 𝐺= 2(1 + 𝑣) 𝜀𝑥 ′ = 1 𝛾 2 𝑦𝑥 1 𝛾 2 𝑧𝑥 1 𝛾 2 𝑥𝑦 𝜀𝑦 1 𝛾 2 𝑧𝑦 1 𝛾 2 𝑥𝑧 1 𝛾 2 𝑦𝑧 𝜀𝑧 𝜎1,2 = 𝐸 𝜀𝑎 + 𝜀𝑏 + 𝜀𝑐 2 ± (𝜀𝑎 − 𝜀𝑏 )2 +(𝜀𝑏 − 𝜀𝑐 )2 +(𝜀𝑐 − 𝜀𝑎 )2 3 1−𝑣 1+𝑣 𝑡𝑎𝑛2𝜃𝑝 = 3(𝜀𝑏 −𝜀𝑐 ) 2𝜀𝑏 − 𝜀𝑎 − 𝜀𝑐

Mechanics of Materials Formula Sheet

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Mechanics of Materials Formula Sheet

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