Chapter Objectives Chapter Outline Principle of virtual work and how to apply in statics problem for pin-connected members Chapter Outline Definition of Work Principle of Virtual Work Principle of Virtual Work for a System of Connected Rigid Bodies

11.1 Definition of Work Work of a Force The work occurs from the force F when there is a displacement in the direction of the force Consider the force F and the displacement dr, and the angle between the angle and displacement is θ, hence the component force in the direction of the displacement is F cos θ Work can be writtten as dU = F dr cos θ Using dot product dU = F·dr Work is scale with the unit Joule (J = N-m)

11.1 Definition of Work Work of a Couple Moment Consider RBD with the couple moment with the force F and -F and hence the couple is M = Fr For the object moved from A by drA and from B by drB drB = drA + dr’ The work from F and -F with drA is cancelled out Hence only work dr’ (=r d θ ) dU = F dr’ = F r d θ = M d θ The work from Moment couple dU = M d θ

11.1 Definition of Work Virtual Work = Consider that there is a virtual displacement, i.e. what would happen if there is a displacement, but does not actually occur Virtual is represented as The virtual work from the force F with the virtual displacement with is the angle between F and The virtual work from couple moment M

11.2 Principle of Virtual Work Principle of Virtual Work: If the object is in equilibrium, the sum of virtual work must be zero Can be used in stead of equilibrium equations Example, FBD of an object on the floor If virtual displacement is acting downwards δy, there will be virtual work downward Wδy and the virtual work upward –Nδy Total Virtual Work Done; δU = Wδy –Nδy = (W-N)δy =0 Since δy ≠ 0, then N = W = Equilibrium

11.2 Principle of Virtual Work Example, consider equilibrium of a beam Apply a virtual work at point B with the virtual angle δθ The virtual work is from P and Ay only Since δy=L δθ and δy’=(L/2) δθ the virtual work is δU = Ay(Lδθ) – P(L/2)δθ = (AyL– PL/2) δθ = 0 dure to δθ ≠ 0, Ay = P/2 L/2 L/2

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Virtual can be used to solve all equilibrium system with pin-connected member Use when degrees of freedom is 1 or the motion in the system or can use only one unknown θ from the example below

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Procedure for Analysis Free Body Diagram Draw FBD of the system and then use q as one unknow displacement from degrees of freedom of the system Sketch how the system can be moved with the virtual displacement δq

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Procedure for Analysis Virtual Displacements Locate the position coordinate s of the force in the system in the function of q from a static point Write a reletionship s and q in Differentiation form and use δs in δq

11.3 Principle of Virtual Work for a System of Connected Rigid Bodies Procedure for Analysis Virtual-Work Equation Write Virtual-Work equation by assuming each point s moved by δs and use the write sign No work from internal force, except deformable object such as spring Convert δs in the form δq Solve the equation using the virtual work principle

Example 11.1 Determine the angle θ for equilibrium of the two-member linkage. Each member has a mass of 10 kg.

Solution (by equilibrium)

Solution (by virtual work) FBD One degree of freedom since location of both links may be specified by a single independent coordinate. (q= θ) θ undergoes a positive (CW) virtual rotation δθ, only the active forces, F and the 2 units of 9.81N weights do work. Virtual Displacement Position coordinates xB and yW

Solution Virtual Work Equation If δ xB and δyw were both positive, forces W and F would do positive work. For virtual work equation for displacement δθ, Relating virtual displacements to common δθ,

Example 11.2 Determine the required force P needed to maintain equilibrium of the scissors linkage when θ=60°. The spring is unstretched when θ=30°. Neglect the mass of the links.

Solution FBD Virtual Displacement Virtual Work Equation

Example 11.3 If the box has a mass of 10 kg, determine the couple moment M needed to maintain equilibrium when θ=60°. Neglect the mass of the members.

Solution FBD Virtual Displacement Virtual Work Equation

Example 11.4 The mechanism supports the 500-N cylinder. Determine the angle θ for equilibrium if the spring has an unstretched length of 2 m when θ=0°. Neglect the mass of the members.

Solution FBD Virtual Displacement Virtual Work Equation

Solution by equilibrium

Solution by virtual work