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Chapter 12

Chapter 12. Static Equilibrium and Elasticity. Introduction. Equilibrium- a condition where an object is at rest OR its center of mass moves with a constant velocity. Static Equilibrium (former def.) is a common practice in engineering disciplines, critical for civil, arch, and mech eng.

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Chapter 12

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  1. Chapter 12 Static Equilibrium and Elasticity

  2. Introduction • Equilibrium- a condition where an object is at rest OR its center of mass moves with a constant velocity. • Static Equilibrium (former def.) is a common practice in engineering disciplines, critical for civil, arch, and mech eng. • Elasticity- we will look at how objects deform under load conditions

  3. 12.1 • The conditions for Equilibrium • Translation Eq. (from Ch 5) • Only works (by itself) for objects modeled as particles (point masses) • Rotational Eq- now that we can deal with extended objects… (about ANY axis) • Implies that the object is either not rotating or rotating with a constant speed.

  4. 12.1 • We will be looking at Static Equilibrium only, which implies both • Quick Quizzes p 364

  5. 12.1 • The vector expressions result in six scalar expressions (three for each axis for both Force and Torque) • We will keep motion limited to a single 2D plane for practical purposes.

  6. 12.1 • If the object is in translational equilibrium and the net torque is zero about one axis, then the net torque is zero about any axis. • In other words, when problem solving, any location can be chosen for the axis of rotation.

  7. 12.2 More on Center of Gravity • The location of a force’s application is critical in evaluating equilibrium conditions. • The force of gravity on a given object (assuming a constant gravitational field) acts at the center of mass. • One single gravitational force at the center of mass is equivalent to the sum of all the individual gravitational forces on each particle.

  8. 12.2

  9. 12.2 • The center of gravity can be located via a number of methods both experimental and calculated. • Be careful not to confuse an object’s center of gravity and a system’s center of gravity. • A system will balance so long as the support is underneath the center of gravity of the system. • Quick Quiz p 366

  10. 12.3 Examples of Static Equilibrium • Remember • Examples 12.1-12.5

  11. 12.4 Elastic Properties of Solids • Up to this point we have assumed solid objects remain rigid under external forces. • In reality solid objects deform under external forces. • Two Key Ideas • Stress- the amount of force acting on an object per unit area • Strain- the result of stress, a measure of deformation.

  12. 12.4 • Materials can be rated with an Elastic Modulus, a constant of proportionality between stress and strain. • Depends on the material, and type of deformation • Generally determined by • Relates what is done to an object, to how the object responds.

  13. 12.4 • Different Types of Deformation result in unique elastic moduli. • Young’s Modulus- resistance of a solid to changes in length. • Shear Modulus- resistance of a solid to a shift in parallel planes. • Bulk Modulus- resistance of a solids or fluids to changes in volume (opposite of compressibility)/

  14. 12.4 • Young’s Modulus- (Tensile Modulus) • The bar is stretch from an initial length Li by a change in length ΔL. • The Stress on the bar is the ratio of the tension force and the cross sectional area of the bar.

  15. 12.4 • The strain on the bar is the ratio of the change in length and the initial length. • Youngs Modulus also applies to compression forces.

  16. 12.4 • Objects can be stressed to their elastic limit, at which point it will be permanently deformed, and beyond to their breaking point.

  17. 12.4 • Shear Modulus • When a force acts on the face of an object parallel to a another face held fixed by an opposite force. • The stress is the ratio of force and parallel surface area.

  18. 12.4 • The strain the is ratio of displacement of the sheared face, and the height of the object.

  19. 12.4 • Bulk Modulus • When a force of uniform magnitude is applied perpendicularly to all surfaces. • The object will undergo a change in volume but not shape. • The volume stress is the ratio of the Force to the surface area of the object. (Also known as pressure).

  20. 12.4 • The volume strain is the ratio of the change in volume and the initial volume. • The negative indicates that an increase in pressure, will result in a decrease volume. • The inverse of Bulk Modulus is compressibility, and is more commonly used.

  21. 12.4 • Prestressed Concrete • Quick Quizzes p 375 • Examples 12.6-12.7

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