Chapter 12

Static Equilibriumand 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.

• 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.

0F

0

12.1

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

• Quick Quizzes p 364

0cmv 0

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.

0 xF 0 yF 0 z

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

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

0 xF 0 yF 0 z

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.

StrainStressModulus Elastic

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 aninitial length Li by a change in length ΔL.– The Stress on the bar is theratio of the tension force andthe cross sectional area of the bar.

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.

iLLAFY

Strain TensileStress Tensile

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 theface of an object parallel to a another face held fixed byan opposite force. – The stress is the ratio of force and parallel surface area.

12.4

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

hxAFG

StrainShear StressShear

12.4

• Bulk Modulus– When a force of uniformmagnitude 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).

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.

ii VVP

VVAFB

Strain VolumeStress Volume

BK 1

12.4

• Prestressed Concrete

• Quick Quizzes p 375• Examples 12.6-12.7