Mechanics of Materials: Exam 1 Review Summary

Summary

This video review provides an overview of key concepts for the upcoming exam in a mechanics of materials course. The instructor covers important topics like stress types (normal, shear, bearing), strain, material properties (including stress-strain diagrams), and axial elongation. Students are encouraged to focus on formulas such as the play equation for axial elongation and the modulus of resilience. The instructor also highlights essential concepts like shear modulus, Poisson’s ratio, and stress concentrations. Additionally, practice problems are included to help students apply these concepts and prepare for the exam.

Takeaways

• 😀 The review covers key concepts for exam one, including stress, strain, material properties, and axial elongation.
• 😀 Chapter 1 focuses on stress, covering sigma stress (normal stress), tau stress (shear stress), and bearing stress, with examples provided for each.
• 😀 You can bring an equation sheet to the exam, which includes formulas like sigma stress (n/a) and tau stress (v/a).
• 😀 Strain is covered in Chapter 2, explaining how to calculate deformation using delta L (change in length) over L (original length).
• 😀 The law of cosines is crucial for solving strain problems involving angles and lengths.
• 😀 Shear strain involves deformations causing angular changes in a material, with the formula for shear strain (gamma) being pi/2 - theta.
• 😀 Chapter 3 is all about material properties, emphasizing the importance of understanding stress-strain diagrams for ductile and brittle materials.
• 😀 You must study energy equations in Chapter 3, including the modulus of resilience and modulus of toughness, to understand energy absorption before failure.
• 😀 Key equations in Chapter 3 include the shear modulus of rigidity (E/2(1+nu)) and Young’s modulus (stress/strain), both valid in the elastic region.
• 😀 Chapter 4 introduces axial elongation, with the pivotal formula delta = pl/ae (change in length = axial load x length / cross-sectional area).
• 😀 Don’t forget about stress concentrations and the elongation caused by changes in temperature, using the formula delta = alpha * deltaT * L (thermal expansion).

Q & A

What is the formula for sigma (σ) stress, and what does it represent?

The formula for sigma (σ) stress is σ = P / A, where P is the applied force and A is the cross-sectional area. It represents average normal stress, which is the pushing or pulling force applied per unit area.

What is tau (τ) stress, and how is it calculated?

Tau (τ) stress is average shear stress, which represents stress causing tearing or sliding along an area. It is calculated as τ = V / A, where V is the shear force and A is the cross-sectional area.

What is bearing stress and how is it calculated?

Bearing stress is a type of compressive stress that occurs when one object pushes against another. It is calculated as σ_b = P / A, where P is the force and A is the area of contact, like when a pin presses against a bar.

How can stress concepts be combined in a single problem?

Stress concepts like sigma stress, tau stress, and bearing stress can be combined in one problem to test your ability to apply different stress calculations to a single system, such as a pin passing through a bar or other mechanical setup.

What is strain, and how is it calculated?

Strain is the measure of deformation or change in shape of an object when subjected to a load. It is calculated using the formula ε = (L_new - L_original) / L_original, where L_new is the new length and L_original is the original length.

What is shear strain, and how is it calculated?

Shear strain (γ) is the deformation that occurs when a force causes an object to distort or shear. It is calculated as γ = π/2 - θ, where θ is the angle of shear and the result is expressed in radians.

What mathematical principle is commonly used in strain problems involving angles?

The law of cosines is commonly used in strain problems where two lengths and the angle between them are given. This helps to calculate the total deformation or change in length of a system.

Why is Chapter 3 on material properties important for the exam?

Chapter 3 is critical because it covers the stress-strain diagrams for both ductile and brittle materials, the modulus of resilience and toughness, and key material properties that form the foundation of mechanical behavior in materials. There are many questions that can be derived from this content.

What is the 0.2% offset rule, and how is it used in material property analysis?

The 0.2% offset rule is used to estimate the yield stress of brittle materials. It involves drawing a line parallel to the linear elastic portion of the stress-strain curve, offset by 0.2% strain, to determine the yield point.

What is the equation for axial elongation, and what does it represent?

The equation for axial elongation is ΔL = PL / AE, where P is the axial force, L is the original length, A is the cross-sectional area, and E is the Young’s modulus. This equation calculates the elongation of a material when subjected to an axial load.

What are stress risers, and why are they important in axial elongation problems?

Stress risers, or stress concentrations, occur in materials at points where the geometry changes, like fillets or holes. These are critical to consider in axial elongation problems because they can cause localized stress increases and potential failure under load.