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Chapter 12 - PowerPoint PPT Presentation

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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Presentation Transcript

Chapter 12

Static Equilibrium

and Elasticity

• 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

• 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…

• Implies that the object is either not rotating or rotating with a constant speed.

• We will be looking at Static Equilibrium only, which implies both

• Quick Quizzes p 364

• 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.

• 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.

• 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.

• 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

• Remember

• Examples 12.1-12.5

• 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.

• 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.

• 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)/

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

• Youngs Modulus also applies to compression forces.

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

• 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.

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

• 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).

• 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.

• Prestressed Concrete

• Quick Quizzes p 375

• Examples 12.6-12.7