This document summarizes important laws and equations in structural analysis. It covers stresses and strains in bars, deflection of beams using integration, Mohr's circle for stress analysis, thin-walled vessel equations, and references several university lecture notes on the topic. Key equations presented include Hooke's law relating stress and strain, beam deflection as an integral of bending moment, and thin vessel wall stress as a function of internal pressure and radius.

1. Important Laws in StructureVIP Revision

2. ∈=𝜎𝐸𝛾=−∈𝑦∈𝑥∈𝑥=𝜎𝑥𝐸−𝛾𝜎𝑦𝐸+𝜎𝑧𝐸+𝛼𝑇∈𝑦=𝜎𝑦𝐸−𝛾𝜎𝑥𝐸+𝜎𝑧𝐸+𝛼𝑇∈𝑧=𝜎𝑧𝐸−𝛾𝜎𝑥𝐸+𝜎𝑦𝐸+𝛼𝑇

3. Bars𝜎=𝑃𝐴 Average Stress∈=𝑃𝐸𝐴𝑑𝑢𝑑𝑥=∈∆𝑙=∈𝑙

4. shafts𝜃=𝑇𝑙𝐺𝐽𝐺=𝐸2(1+𝛾)𝜏𝑚𝑎𝑥=𝑇𝑅𝐽𝐽=𝜋2𝑅4 For Circular Sections.=𝑏𝑡33 For Thin Structure.

5. Deflection of Beams𝜎=−𝑑2𝑉𝑑𝑥2𝐸 𝑦=−𝑀𝑦𝐼 Special Case when 𝐸=𝑐𝑜𝑛𝑠𝑡.𝑆=𝑑𝑀𝑑𝑥𝑓=𝑑𝑆𝑑𝑥𝐸𝐼𝑑2𝑉𝑑𝑥2=𝑀𝑑2𝑑𝑥2𝐸𝐼𝑑2𝑉𝑑𝑥2=𝑀

6. Mohr circle𝜏𝑚𝑎𝑥 C = 𝜎𝑥+𝜎𝑦2R = (𝜏𝑥𝑦)2+(𝜎𝑦−𝜎𝑥2)2 𝜏𝑥𝑦 c𝜎𝑚𝑎𝑥 𝜎𝑥 𝜎𝑚𝑖𝑛 𝜎𝑦

7. Thin walled Vessel𝜎h=2𝜎𝑎=𝑃𝑟𝑡

8. ReferencesLecture’s notes of Dr. Mohammad Tawfik:Cairo universityAssistant Professor · February 2010 to present · Giza, Al Jizah, EgyptAerospace Engineering DepartmentEmirates Aviation CollegeAssistant Professor · October 2008 to January 2010 · Dubai, United Arab EmiratesCairo universityAdjunct Professor · November 2005 to January 2010 · Cairo, EgyptThe British University in EgyptAssistant Professor · September 2007 to August 2008German University in CairoAssistant Professor · August 2003 to September 2007

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