Engineering Mechanics: Statics Chapter 11: Virtual Work Engineering Mechanics: Statics

Chapter Objectives To introduce the principle of virtual work and show how it applies to determining the equilibrium configuration of a series of pin-connected members. To establish the potential energy function and use the potential energy method to investigate the type of equilibrium or stability of a rigid body or configuration.

Chapter Outline Definition of Work and Virtual Work Principle of Virtual Work for a Particle and a Rigid Body Principle of Virtual Work for a System of Connected Rigid Bodies Conservative Forces Potential Energy

Chapter Outline Potential-Energy Criterion for Equilibrium Stability of Equilibrium

11.1 Definition of Work and Virtual Work Work of a Force In mechanics, a force F does work only when it undergoes a displacement in the direction of the force Example Consider the force F located in the path s specified by the position vector r

11.1 Definition of Work and Virtual Work Work of a Force If the force moves along the path to a new position r’ = r + dr, the displacement is dr and therefore, work dU is a scalar quantity defined by the dot product dU = F·dr Because dr is infinitesimal, magnitude of dr can be represented by ds, the differential arc segment along the path If the angle between the tails of dr and F is θ, dU = F ds cos θ

11.1 Definition of Work and Virtual Work Work of a Force If 0° ≤ θ ≤ 90°, force component and the displacement have the same sense, so that work is positive, whereas if 90° ≤ θ ≤ 180°, these vectors have an opposite sense and work is negative dU = 0 if the force is perpendicular to displacement since cos 90° = 0 or if the force is applied at a fixed point, in which case, the displacement ds = 0

11.1 Definition of Work and Virtual Work Work of a Couple The two forces of a couple do work when the couple rotates about an axis perpendicular to the plane of the couple Consider body subjected to a couple whose moment has a magnitude M = Fr Any general displacement of the body can be considered as a combination of a translation and rotation

11.1 Definition of Work and Virtual Work Work of a Couple When the body translates such that the component of displacement of the body along the line of action of each force is dst Positive work (F dst) cancels negative work of the other (-F dst)

11.1 Definition of Work and Virtual Work Work of a Couple Consider differential rotation dθ of body about an axis perpendicular to the plane of the couple, which intersects the plane at point O Each force undergoes a displacement dsθ = (r/2) dθ in the direction of the force Hence, for work of both forces, dU = F(r/2) dθ + F(r/2) dθ = (Fr) dθ or dU = M dθ

11.1 Definition of Work and Virtual Work Work of a Couple Resultant work is positive when the sense of M is the same as that of dθ, and negative when they have an opposite sense For moment vector, the direction and sense of dθ are defined by the right hand rule where the fingers of the right hand follow the rotation or the curl and the thumb indicates the direction of dθ

11.1 Definition of Work and Virtual Work Work of a Couple Line of action of dθ will be parallel to line of action of M if movement of the body occurs in the same plane If the body rotates in space, the component of dθ in the direction of M is required In general, work done by a couple is defined by the dot product dU = M·dθ

11.1 Definition of Work and Virtual Work Definition of work of a force and a couple have been presented in terms of actual movements expressed by differential displacements having magnitudes of ds and dθ Consider an imaginary or virtual movement which indicates a displacement or rotation that is assumed and does not exist Movements are first order differential quantities

11.1 Definition of Work and Virtual Work For virtual work done by a force undergoing virtual displacement, δU = F cosθ δs When a couple undergoes a virtual rotation in the plane of the couple forces, for virtual work, δU = M δθ

11.2 Principle of Virtual Work for a Particle and a Rigid Body If the particle undergoes an imaginary or virtual displacement, then virtual work done by the force system becomes δU = ∑F.δr = (∑Fxi + ∑Fyj + ∑Fzk).(δxi + δyj + δzk) = ∑Fx δx + ∑Fy δy + ∑Fz δz For equilibrium, ∑Fx = 0, ∑Fy = 0, ∑Fz = 0 Thus, virtual work, δU = 0

11.2 Principle of Virtual Work for a Particle and a Rigid Body Example Consider the FBD of the ball which rests on the floor Imagine the ball to be displacement downwards a virtual amount δy and weight does positive virtual work W δy and normal force does negative virtual work -N δy For equilibrium, δU = Wδy –Nδy = (W-N)δy =0 Since δy ≠ 0, then N = W

11.2 Principle of Virtual Work for a Particle and a Rigid Body A similar set of virtual work equations can be written for a rigid body subjected to a coplanar force system If these equations involve separate virtual translations in the x and y directions and a virtual rotation about an axis perpendicular to the x-y plane and passing through an arbitrary point O, it can be shown that ∑Fx = 0; ∑Fy = 0; ∑MO=0 Not necessary to include work done by internal forces acting within the body

11.2 Principle of Virtual Work for a Particle and a Rigid Body Consider simply supported beam, with a given rotation about point B Only forces that do work are P and Ay Since δy = lδθ and δy’ = (l/2)δθ, virtual work δU = Ay(lδθ) – P(l/2)δθ = (Ay – P/2)l δθ = 0 Since δθ ≠ 0, Ay = P/2 Excluding δθ, terms in parentheses represent moment equilibrium about B