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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

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…

      (about ANY axis)

      • 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.

12 2 more on center of gravity
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.


  • 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
12.3 Examples of Static Equilibrium

  • Remember

  • Examples 12.1-12.5

12 4 elastic properties of solids
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.


  • 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.

  • 12.4

    • 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



    • 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


      • 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.

  • 12.4

    • Prestressed Concrete

    • Quick Quizzes p 375

    • Examples 12.6-12.7