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. 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.

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

12.4 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 Δ L. – The Stress on the bar is the ratio of the tension force and the 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.

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.

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).

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.

12.4 Prestressed Concrete Quick Quizzes p 375 Examples