Frame Virtual Work

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Introduction

The theory of virtual work is fundamental in the study of mechanics. Applicable to all types of structures including trusses, beams and frames, virtual work provides a way to calculate the deflections of determinate and indeterminate structures. A disadvantage however, is that each application only solves for one variable, slope or deflection, at one location.[1]

This section will explain the concepts of virtual work and show how to apply virtual work to determine the deflection of members. This section will only do this for determinate structures. Virtual work can be used to determine the deflections of indeterminate members such methods will not be explained in this section.

Important Terms

Virtual Force
An imaginary force imposed on a member for the purpose of determining how that member will react or to determine the member's physical characteristics.[1]
Virtual Displacement
An arbitrary, instantaneous and infinitesimal change of position of the system due to a virtual loading and is compatible with the restraint conditions.[2]
Axial Deformation
The change in length of a member due to a loading or temperature change.[1]
Deflection
The bending of a structural member as a result of its self weight or an applied load.[1]
Flexural Rigidity
A measure of stiffness of a member, indicated by the product of the modulus of elasticity (E), and moment of inertia (I), divided by the length of the member.[3]

Principles of Work

Basic work principles are used to formulate the method of virtual work for the analysis of deflections for trusses, beams and frames. The most basic principle of work is defined by the equation
Figure 1: Principle of Work
$dW = P(d\triangle),$ [1]

where P is the force and dΔ is an infinitesimal displacement.

The work W is then found by integration the formula across the displacement

$W = \int_{0}^{\triangle} P d\triangle.$

Work done by a couple force is found in a similar fashion. The work done by a couple is defined as the moment of the couple multiplied by the angle through which the couple rotates. This is represented by the equation

$dW = M (d\theta).$[1]

The work is then found by integrating the equation across the angle

$W = \int_{0}^{\theta} M d\theta.$

Principles of Virtual Work

There are two important formulations to the principle of virtual work. They are the principle of virtual displacements for rigid bodies and the principle of virtual work for deformable bodies. The principle of virtual work for deformable bodies is the method of virtual work that is derived from for this course. This principle identifies the concept of virtual work. The concept identifies virtual work as the external work done by virtual external forces and couples acting through real external displacements and rotations.[1] The conditions for this are that the virtual system of forces must act on a deformable structure in equilibrium and must cause a real deformation consistent with the support and continuity conditions.[1] These conditions must be met in order for the analysis to be valid. A body is in equilibrium if, and only if, the virtual work of all forces acting on the body is zero.[4] The virtual work performed on a system then by virtual forces will always be equal to zero. This relationship is represented by

$W_{ve}=W_{vi},$

or

$\sum\left(\frac{Virtual \quad External \quad Force}{Real \quad External \quad Displacement}\right) =\sum\left(\frac{Virtual \quad Internal \quad Force}{Real \quad Internal \quad Displacement}\right),$

in which the terms forces and displacements are used in a general sense and include moments and rotations respectively. The principle of virtual work is directly related to this relationship.[1]

Virtual Work for Frames

Virtual work is used for frames to determine the slope and deflection at a specific point. This is done by applying a virtual unit load or unit couple at the specified point. The virtual external work performed by these loads is equal to either the extent of the deflection or slope of the deflection.[1] Virtual exterior work is equal to the virtual interior work when using the method of virtual work. The virtual interior work is found using the equation

$W_{vi}=\sum F_{v}\left(\frac{FL}{AE}\right)+ \int\frac{M_{v}M}{EI}dx.$[1]

This equation has two parts, the first being the summation of axial deformations of all members and the second being the summation of bending in all members.The axial deformation of each member is found by using the equation

$W_{via}=\sum F_{v}\left(\frac{FL}{AE}\right),$

where F is axial force, and L, A, and E denote, respectively, the length, cross-sectional area, and modulus of elasticity for each member. The bending in each member is found using the equation

$W_{vib}=\int_{0}^{L}\frac{M_{v}M}{EI}dx,$

where Mv now denotes the bending moment due to the virtual unit couple and M, E, and I denote, respectively, the moment due to the real load, the modulus of elasticity and the moment of inertia for each member integrated over the differential length dx.[1]

For most modern day materials analysed by these methods, the extent of axial deformations that occur within frames tends to be rather insignificant in comparison to the extent of bending deformations in the same frames. Therefore, axial deformations are usually neglected in the analysis of frames.

$1(\Theta)= \int \frac{M_v{_{}}M}{EI}$

The external virtual work is the unit couple multiplied by the rotation. The internal virtual work is equal to the integral of the curvature of the real system multiplied by the virtual system. The external virtual work multiplied by the unit couple is equal to the internal virtual work. The rotation can be found by dividing the internal virtual work by the unit moment.

The same method is used for a the system with a unit displacement.

$1(\Delta)= \int \frac{M_v{_{}}M}{EI}$ [1]

Procedure for Analysing Frames

Step 1: Solve for the internal forces in the frame. This can be done following the analysis of plane frames.

Step 2: Compute the curvature for the real system and solve the internal forces of each section.

Step 3: Create your virtual system and apply a unit load or unit couple in the direction of the desired displacement. Analyse the virtual system and solve for the internal forces for each section as well.

Step 4: Integrate the curvature of the real system multiplied by the virtual system for each section.

Step 5: Neglect axial deformations and end the analysis. [5]

Example Problem

Determine the rotation of joint B of the frame shown below using the virtual work method.

The real system must be solved. The x-coordinates for each segment of the frame can be seen below. They are used to solve for the moment of each segment and are placed in the table for M.

After solving the real system, the virtual system must now be solved. The x-coordinates for each segment of the frame can also be seen below. They are used to solve for the moment of each segment and are placed in the table for Mv.

EXAMPLE IS NOT CORRECT FROM THIS POINT FORWARD

Once the equations for M and Mv are determined, they are placed in the table below.

Segment Origin Limit (m) M (kN-m) Mv (kN-m)
$AB$ $A$ $0-24$ $360x - 20\frac{x^{2}}{2}$ $\frac{x}{24}$
$BC$ $B$ $0-8$ $2880$ $0$
$CD$ $D$ $0-8$ $360x$ $0$

After the table is complete, the rotation of joint B of the frame is determined using the virtual work equation below.

$\quad 1(\Theta _{B}) = \sum \int\frac{M_{v}M}{EI}dx$

$\qquad = \frac{1}{EI}\int_{0}^{24}\left ( \frac{x}{24} \right )\left ( 360x - 20\frac{x^{2}}{2} \right )dx$

$\qquad = \frac{1}{EI}\cdot \frac{-5}{48}(x-48)x^{3}$

$\qquad = \frac{34\ 560 \qquad k\!N^{2}\cdot m^{3}}{EI}$

$\therefore \qquad \qquad \qquad \Theta _{B} =\frac{34\ 560 \qquad k\!N^{2}\cdot m^{3}}{(200\times 10^{6})(1000\times 10^{-6})} = 0.1728 \qquad \qquad rad$

References

1. Kassimali, A. (2011). Structural Analysis: SI Edition (4th ed.). Stamford, CT: Cengage Learning.
2. A.sommerfield, Mechanics,Lectures of Theoretical Physics, vol.I, Acedemic Press, New York, 1952.
3. www.dictionaryofconstruction.com/definition/flexural-rigidity.html
4. http://www.colincaprani.com/files/notes/SAIII/Virtual%20Work%200910.pdf
5. //www.ce.memphis.edu/3121/notes/notes_09c.pdf