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Chapter 12

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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

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.

- Translation Eq. (from Ch 5)

12.1

- We will be looking at Static Equilibrium only, which implies both
- Quick Quizzes p 364

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.

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.

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.

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

12.3 Examples of Static Equilibrium

- Remember
- Examples 12.1-12.5

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.

12.4

- 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.

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)/

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.

- The bar is stretch from an

12.4 Youngs Modulus also applies to compression forces.

- The strain on the bar is the ratio of the change in length and the initial length.

12.4

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

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.

- When a force acts on the

12.4

- The strain the is ratio of displacement of the sheared face, and the height of the object.

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).

- When a force of uniform

12.4 The inverse of Bulk Modulus is compressibility, and is more commonly used.

- 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.

12.4

- Prestressed Concrete
- Quick Quizzes p 375
- Examples 12.6-12.7