Structural Analysis and Design I : Deflection of Trusses, Beams and Frames by the Virtual Work Method

 

Lecture - 06, 07 & 08

Deflection of Trusses, Beams and Frames by the Virtual Work Method 

Work:

Work is done when a force causes a displacement in the direction of the force.

Positive Work:

• When the force and displacement are in the same direction.

Negative Work:

• When the force and displacement are in opposite directions.

 

Deformation under Force

• A structure is loaded with a system of forces, one of which is the force P.
• As the load acts, the structure deforms → the point of application of force P moves from its original position A (undeformed) to A′ (deformed).

 

Work Done by a Force (P)

• General Case:
• Small work done during an infinitesimal displacement:
•                                                                                                                                                                                                dW = P d🛆

• Total work:
•                       
  

  
  

• Graphically, this equals the area under the force–displacement curve.

• Case 1: Variable Force (linear increase with displacement)
• Force increases from 0 to P.
• Force–displacement graph = triangle.
• Work = area of triangle:
• 

• Case 2: Constant Force
• Force remains constant at P.
• the force PPP itself is not responsible for creating the displacement — something else moves the point where P is applied.
• Force–displacement graph = rectangle.
• Work = area of rectangle:
•                                         W = P 🛆

Work Done by a Couple (Moment M)

• General Case:
• Small work done during an infinitesimal rotation:
•                                                                              dW = M dθ         

• Total work:
•                  
  

• Case 1: Variable Couple (linear increase with rotation)
• Moment increases from 0 to M.
• Moment–rotation graph = triangle.
• Work =
•                    W = 1/2 * Mθ

• Case 2: Constant Couple
• Moment remains constant at M.
• Moment–rotation graph = rectangle.
• Work =
•                      W = Mθ

Principle of Virtual Work

§  Origin: Introduced by John Bernoulli in 1717.

·         Purpose: It’s a very powerful analytical tool used in structural mechanics to solve equilibrium and deformation problems.

Two Formulations of the Principle

1.     Principle of Virtual Displacements (Rigid Bodies)

o    Applies to rigid bodies (bodies assumed not to deform).

2.     Principle of Virtual Forces (Deformable Bodies)

o    Applies to deformable bodies (bodies that can deform under load).

 

Principle of Virtual Displacements for Rigid Bodies

 “If a rigid body is in equilibrium under a system of forces and if it is subjected to any small virtual rigid-body displacement, the virtual work done by the external forces is zero.’’

Virtual means “assumed imaginary displacement,” not an actual movement.

 

§  The term virtual means imaginary, not real.

§  Consider the beam shown in Fig. (a).

§  The free-body diagram of the beam is shown in Fig. (b).

§  The external load P is resolved into components:

• Px​ → in the x-direction
• Py​ → in the y-direction.

§  Suppose the beam is given an arbitrary small virtual rigid-body displacement:

·         From position ABC (initial equilibrium)

·         To position A′B′C′ (virtually displaced).

§  This virtual displacement can be decomposed into:

• Translation in the x-direction → Δvx​
• Translation in the y-direction → Δvy​
• Rotation about point A → θv

ü The subscript v indicates that these are virtual (imaginary) displacements.

ü As the beam undergoes this virtual displacement, the external forces (Px, Py​) perform work.

ü This work is called virtual work.

Total Virtual Work

• The total external virtual work is the sum of work from:

1.     Translation in x-direction

2.     Translation in y-direction

3.     Rotation about point A

Wve = Wvx + Wvy + Wvr ​

Virtual work due to x and y-translation

Virtual work due to rotation, θv

The total virtual work done as

Equilibrium Condition

• For equilibrium of the beam:
• 
  
  Hence,

Wve​=0

This proves the principle of virtual displacements: A rigid body in equilibrium does zero total virtual work under any virtual displacement.

 

Principle of Virtual Forces for Deformable Bodies

 ‘‘If a deformable structure is in equilibrium under a virtual system of forces (and couples) and if it is subjected to any small real deformation consistent with the support and continuity conditions of the structure, then the virtual external work done by the virtual external forces (and couples) acting through the real external displacements (and rotations) is equal to the virtual internal work done by the virtual internal forces (and couples) acting through the real internal displacements (and rotations).’’

 

§  Consider a two-member truss (as in Fig. (a)).

§  The truss is in equilibrium under a virtual external force Pv​.

§  The free-body diagram (FBD) of joint C is shown in Fig. (b).

§  At joint C, equilibrium requires that:

§  The sum of forces in the x-direction = 0.

§  The sum of forces in the y-direction = 0.

 

 

In which F_vAC and F_vBC represent the virtual internal forces in members AC and BC respectively, and θ1 and θ2 denote, respectively, the angles of inclination of these members with respect to the horizontal (Fig. (a)).

§  Assume joint C of the truss undergoes a small real displacement ∆ to the right from its equilibrium position (Fig.(a)).

§  The deformation is consistent with the support conditions:

·         Joints A and B are attached to supports and do not move.

·         Virtual forces at joints A and B do no work because there is no displacement at those joints.

·         Therefore, the total virtual work Wv​ for the truss is equal to the algebraic sum of the work of the virtual forces acting at joint C.

As indicated the equilibrium equations above, the right-hand side is zero; therefore, the total virtual work is Wv = 0. Thus, Eq.  can be expressed as

ü The left-hand side of this Eq.  represents the virtual external work Wve​:

·         Done by the virtual external force Pv​

·         Acting through the real external displacement ∆

ü The terms ∆cosθ1 ​ and ∆cosθ2 ​ represent the real internal displacements (elongations) of members AC and BC, respectively.

ü Therefore, the right-hand side of this Eq. represents the virtual internal work Wvi​:

·         Done by the virtual internal forces

·         Acting through the real internal displacements of the members.

Hence, Wve = Wvi

The method of virtual work is based on the principle of virtual forces for deformable bodies as expressed by Eq. above and which can be rewritten as

virtual external work = virtual internal work

or, more specifically

The method of virtual work uses two separate systems:

• A virtual force system
• The real system of loads (or other effects) that produce the actual deformation

Deflections of Trusses by the Virtual Work Method

ü Consider a statically determinate truss (Fig.(a)) and assume we want the vertical deflection ∆ at joint B due to external loads P1​ and P2​.

ü The axial forces in members can be determined using the method of joints/ section.

ü For an arbitrary member j (e.g., CD), the axial deformation δ is:

δ = FL/AE

ü  F = axial force in the member

ü  L = length of the member

ü  A = cross-sectional area

ü  E = modulus of elasticity

ü  To find vertical deflection ∆ at joint B, select a virtual system:

ü  Apply a unit load at joint B in the direction of desired deflection (Fig. (b)).

ü  The forces in members due to this unit load are the virtual forces Fv​.

ü  Apply the virtual unit load to the truss while considering the real deformations from actual loads.

ü  Virtual external work done by the unit load through the real deflection ∆:

Wve = 1(∆)

ü  Virtual internal work for member j (CD):

Wvi =Fv(δ)

·         Total virtual internal work for all members:

Wvi = 𝛴Fv(δ)   

ü  Using principle of virtual work:

Wve = Wvi 

 ⟹ 1(∆) = 𝛴Fv(δ)    

ü   Substituting δ=FL/AE

ü Since ∆ is the only unknown, its value can be solved directly from this equation.

Temperature Changes and Fabrication Errors

• The virtual work method is general:
• Can determine truss deflections due to temperature changes, fabrication errors, or any effect where member axial deformations δ are known or can be calculated.
• Axial deformation of member j of length L due to temperature change ΔT given by:

δ=α(ΔT) L

• α = coefficient of thermal expansion of member j
• Truss deflection due to temperature changes can be computed by substituting δ into the virtual work equation:

1(∆) = ∑Fv α (ΔT) L

• Truss deflections due to fabrication errors:

·         Truss deflections due to fabrication errors can be determined by simply substituting changes in member lengths due to fabrication errors for δ.

Procedure for Determining Truss Deflections Using Virtual Work Method

1.     Real System

o    If deflection is due to external loads:

§  Compute the real axial forces F in all members using the method of joints or method of sections.

o    Sign convention:

§  Tensile forces, temperature increases, or elongations due to fabrication errors → positive

§  Compressive forces, temperature decreases, or shortenings → negative

2.     Virtual System

o    Remove all real loads from the truss.

o    Apply a unit load at the joint where the deflection is desired, in the direction of desired deflection.

o    Compute the virtual axial forces Fv​ in all members using the method of joints or method of sections.

o    Sign convention:

§  Must match the real system (tension positive, compression negative).

3.     Compute Deflection

o    Use the appropriate virtual work formula:

§  Deflection due to external loads

§  Deflection due to temperature changes

§  Deflection due to fabrication errors

o    Arrange real and virtual forces in a tabular form for clarity.

o    Sign of deflection:

§  Positive → deflection in the same direction as the unit load

§  Negative → deflection in the opposite direction to the unit load

 

Example 1

Determine the horizontal deflection at joint C of the truss shown in Fig. (a) by the virtual work method.

Real System

The member axial forces due to the real loads (F) obtained by using the method of joints.

Virtual System:

A unit load (1 k) is applied at the point and in the direction where the deflection is to be determined. The axial forces in all members caused by this virtual load, Fv​, are calculated using the method of joints (or sections).

Horizontal Deflection at C, ∆_C

To facilitate the computation of the desired deflection, the real and virtual member forces are tabulated along with the member lengths (L), as shown in Table below.

Example 2

Determine the horizontal deflection at joint G of the truss shown in Fig.(a) by the virtual work method.

Real System

The member axial forces due to the real loads (F) obtained by using the method of joints.

Virtual System:

A unit load (1 k) is applied at the point and in the direction where the deflection is to be determined. The axial forces in all members caused by this virtual load, Fv​, are calculated using the method of joints (or sections).

Horizontal Deflection at G, ∆_G

To facilitate the computation of the desired deflection, the real and virtual member forces are tabulated along with the member lengths (L), as shown in Table below. 

Example 3

Determine the horizontal and vertical components of the deflection at joint B of the truss shown in Fig. (a) by the virtual work method.

Real System

The member axial forces due to the real loads (F) obtained by using the method of joints.

(b) Real system – F forces

 Virtual System:

A unit load (1 k) is applied at the point and in the direction where the deflection is to be determined. The axial forces in all members caused by this virtual load, Fv​, are calculated using the method of joints (or sections).

Horizontal Deflection at B, ∆_BH, ∆_BV

To facilitate the computation of the desired deflection, the real and virtual member forces are tabulated along with the member lengths (L), as shown in Table below. 

Example 4

Determine the vertical deflection at joint C of the truss shown in Fig.(a) due to a temperature drop of 15⁰ F in members AB and BC and a temperature increase of 60⁰ F in members AF, FG, GH, and EH. Use the virtual work method.

Real System

The real system consists of the temperature changes (∆T) given in the problem.

 

Virtual System:

A unit load (1 k) is applied at the point and in the direction where the deflection is to be determined. The axial forces in all members caused by this virtual load, Fv​, are calculated using the method of joints (or sections).

Vertical Deflection at C, ∆_C.

The temperature changes (∆T) and the virtual member forces (Fv) are tabulated along with the lengths (L) of the members.

 

Example 5

Determine the vertical deflection at joint D of the truss shown in Fig.(a) if member CF is 0.6 in. too long and member EF is 0.4 in. too short. Use the method of virtual work.

Real System

 The real system consists of the changes in the lengths (δ) of members CF and EF of the truss

Virtual System:

A unit load (1 k) is applied at the point and in the direction where the deflection is to be determined. The axial forces in all members caused by this virtual load, Fv​, are calculated using the method of joints (or sections).

Vertical Deflection at D, ∆_D

 The desired deflection is determined by applying the virtual work expression.

 

Deflections of Beams by the Virtual Work Method

To develop an expression for the virtual work method for determining the deflections of beams, consider a beam subjected to an arbitrary loading, as shown in Fig.(a). Let us assume that the vertical deflection, D, at a point B of the beam is desired. To determine this deflection, we select a virtual system consisting of a unit load acting at the point and in the direction of the desired deflection, as shown in Fig.(b).

 

Now, the virtual external work performed by the virtual unit load as it goes through the real deflection ∆ is

Wve = 1(∆)

To obtain the virtual internal work, take a differential element of the beam of length dx, located at a distance x from the left support A. as shown in Fig. (a) and (b).

Because the beam with the virtual load (Fig. (b)) is subjected to the deformation due to the real loading (Fig.(a)), the virtual internal bending moment, Mv, acting on the element dx performs virtual internal work as it undergoes the real rotation dθ, as shown in Fig.(c).

Thus, the virtual internal work done on the element dx is given by-

dWvi =Mv (dθ)

The change of slope dθ over the differential length dx can be expressed as

dθ = (M/EI)dx

in which M = bending moment due to the real loading causing the rotation dθ

By substituting dθ in above equation, we get,

The total virtual internal work done on the entire beam can now be determined by integrating Eq above over the length L of the beam as

By equating the virtual external work, Wve = 1(∆), to the virtual internal work, we obtain the following expression for the method of virtual work for beam deflections:

Slope

If we want the slope θ at a point C of the beam (Fig. (a)), then use a virtual system consisting of a unit couple acting at the point, as shown in Fig.(d).

When the beam with the virtual unit couple is subjected to the deformations due to the real loading, the virtual external work performed by the virtual unit couple, as it undergoes the real rotation θ , is

Wve  = 1(θ )

The expression for the internal virtual work remains the same as given in Eq. of, virtual work for beam deflections, except that Mv here denotes the bending moment due to the virtual unit couple.

For Wve  = Wvi, we obtain the following expression for the method of virtual work for beam slopes:

Procedure for Analysis

1.     Real System

·         Draw the beam with all given (real) loads.

2.     Virtual System

·         Remove real loads.

·         For deflection → apply a unit load at the desired point, in the direction of deflection.

·         For slope → apply a unit couple at the desired point.

3.     Segmentation of Beam

·         Divide the beam into segments where real loads, virtual loads, and EI are continuous.

4.     Real Bending Moment (M)

·         For each segment, write the equation of bending moment due to real loads using coordinate x.

·         The origin for x may be located anywhere on the beam and should be chosen so that the number of terms in the equation for M is minimum.

·         Follow the beam sign convention.

5.     Virtual Bending Moment (Mv)

·         For each segment, write the bending moment equation due to virtual load/couple, using the same x coordinate as in Step 4.

·         Apply the same sign convention as in real system.

6.     Apply Virtual Work Equations

·         Use the appropriate formula for slope and deflection.

·         If beam is divided into segments, evaluate integral for each and then sum all segments algebraically.

 

Graphical Evaluation of Virtual Work Integrals

When to Use

• Structure has constant EI in its segments.
• Loading is relatively simple.
• Useful as an alternative to lengthy mathematical integration.

Steps

• Step 1: Draw bending moment diagram (M) due to real loads.
• Step 2: Draw bending moment diagram (Mv) due to virtual loads/couples.

·         Step 3: Compare the shapes of M and Mv​ diagrams with standard geometric shapes (triangles, rectangles, trapezoids, parabolas, etc.).

·         Step 4: Use a table of integrals to directly obtain the expression for virtual work integral based on these shapes.

·         Step 5: Evaluate the virtual work integral for each segment and sum if needed.

Advantage

• Avoids complex integration.
• Quick estimation using known area formulas and centroid locations of standard shapes.

 

Integrals (0-L) ∫Mv M dx for moment diagrams of simple geometric shapes

Sign convention (Beam Convention)

Axial Force (Q):

• Positive when it produces tension (pulls the section apart).

Shear Force (S):

• Positive when the left portion moves upward relative to the right.
• Rule of thumb: Clockwise moment about the cut = positive shear.
• Positive if external forces cause the left portion of the section to move upward relative to the right portion.
• Equivalently:
• Upward load on left side → positive shear
• Downward load on right side → positive shear

Bending Moment (M):

• Positive when the beam bends concave upward (compression in top fibers, tension in bottom fibers).
• Using left portion → the forces / couple acting on the portion that produce clockwise moment/couple about the section = positive.
• Using right portion → the forces /couple producing counterclockwise moment/couple about the section = positive.

·         Positive if it produces sagging (concave upward, tension at bottom).

·         Negative if it produces hogging (concave downward, tension at top).

 

Example 1

Determine the slope and deflection at point A of the beam shown in Fig. (a) by the virtual work method.

Slope at A

There are no discontinuities of the real and virtual loadings or of EI along the length of the beam. Therefore, there is no need to subdivide the beam into segments. To determine the equation for the bending moment M due to real loading, we select an x coordinate with its origin at end A of the beam, as shown in Fig.(b).

By applying the method of sections, we determine the equation for M

Similarly, the equation for the bending moment Mv1 due to virtual unit moment in terms of the same x coordinate is

To determine the desired slope θ_A, we apply the virtual work expression given by

Deflection at A

The equation for M remains the same as before, and the equation for bending moment Mv2 due to virtual unit load

By applying the virtual work expression, we determine the desired deflection

Example 2

Determine the deflection at point D of the beam shown in Fig.(a) by the virtual work method.

Real and virtual systems

• Shown in Fig.(b) and (c).
• 
  
  
  
• 

  
  
  

Flexural rigidity (EI)

• Changes abruptly at points B and D.

Loadings

• Real loading is discontinuous at point C.
• Virtual loading is discontinuous at point D.

Effect on MvM/EI

• Discontinuous at points B, C, and D.
• Therefore, beam must be divided into four segments:
• AB
• BC
• CD
• DE

Continuity condition

• In each segment, integration can be carried out segment by segment.

Coordinate system

• x-coordinates chosen as shown in Fig.(b) and (c).
• Important rule: For any segment, use the same x-coordinate for both:
• Real bending moment equation M.
• Virtual bending moment equation Mv​.

Tabulated equations

• Using method of sections, equations for M and Mv​ for all four segments are given in Table.

Finally

• Deflection at point D is computed by applying the virtual work expression.

Example 3

Determine the deflection at point C of the beam shown in Fig. (a) by the virtual work method.

Real and virtual systems

• Shown in Fig. (b) and (c).
• 
  
  
  
• 

  
  
  

Loadings

• Real and virtual loadings are discontinuous at point B.

Segmentation of beam

• Beam is divided into two segments:
• AB
• BC

Coordinate system

• x-coordinates for bending moment equations are shown in Fig. (b) and (c).
• Same x-coordinate is used to write both:
• Real bending moment equation MMM.
• Virtual bending moment equation Mv​.

Equations

• M and Mv​ for segments AB and BC are obtained by the method of sections.
• Results are tabulated in Table.
• 

Finally

• Deflection at point C is computed using the virtual work expression

 

 

Example 4

Determine the deflection at point B of the beam shown in Fig.(a) by the virtual work method. Use the graphical procedure to evaluate the virtual work integral.

Real and virtual systems

• Shown in Figs.(b) and (c).
• Their corresponding bending moment diagrams (M and Mv​) are also shown.
• 
  
  
  
• 

  
  
  

Flexural rigidity (EI)

• Constant along the entire length of the beam.
• Therefore, no subdivision of the beam into segments is required.

Virtual work equation

• For deflection at point B, can be directly applied without breaking the beam into parts.

Integral to evaluate

• ∫(0→L) Mv M dx graphically.

Step 1: Compare M diagram

• From Moment diagrams.
• Matches shape in the Table.

Step 2: Compare Mv​ diagram

• From Moment diagrams.
• Matches shape in the Table.

Step 3: Locate intersection in Table

• Intersection of row and column.
• Gives the required expression for evaluating the integral.
• 

Given values

• Mv1=2.25 kN.m
• M1=630 kN.m
• L=12 m
• l1=3 m
• l2=9 m

Step: Substitution

• Substitute the above numerical values into the derived expression
• 

The deflection at point B applying the virtual work equation

 

 

Deflections of Frames by the Virtual Work Method

 

1.     Determination of the slopes and deflections of frames using virtual work method is similar to the beams.

2.     To determine the deflection, ∆, or rotation, θ, at a point of a frame, a virtual unit load or unit couple is applied at that point.

3.     In frames, members can deform both in bending and in axial direction.

4.     The total virtual internal work equals the sum of bending work and axial work.

5.     Bending contribution comes from flexural deformations of members.

6.     Axial contribution comes from elongation or shortening of members.

7.     The total virtual internal work from bending for the whole frame can be found by adding the bending work of each individual segment.

8.     The total virtual internal work from axial deformations is obtained by summing the contributions from all frame members.

9.     Hence, the total internal virtual work for the frame due to both bending and axial deformations:

10. As the virtual system is subjected to the deformations of the frame due to real loads, the virtual external work performed by the unit load or the unit couple is

Wve = 1(∆), For deflection at the unit load point

 Wve =1(θ), For rotation at the couple location

Now,

The virtual external work = the virtual internal work

ü Axial deformations in frame members are usually much smaller than bending deformations.

ü Therefore, they are neglected here in frame analysis.

ü Axial effects should not be ignored in frame analysis when they are specifically mentioned.  

ü The virtual work expressions then reduce to only bending terms:

 

Procedure for Analysis

Real System

• Find internal forces/moments in frame members due to real loading.

Virtual System

• For deflection: apply a unit load at the desired point/direction.
• For rotation: apply a unit couple at the desired point.
• Determine member forces/moments due to virtual load.

Segmentation

• Divide members into segments where loads and EI are continuous.

Moment Equations

• Express real moment M for each segment.
• Express virtual moment Mv using same coordinate/sign convention.

Deflection/Rotation (Bending / Axial Effects)

• Apply the appropriate virtual work expressions.

 

Example 1

Determine the rotation of joint C of the frame shown in Fig.(a) by the virtual work method.

Considering three segments for this frame, AB, BC, and CD, determine bending moment equations for each segment for real and virtual system.

Real System

Virtual System 

The rotation of joint C of the frame can now be determined by applying the virtual work expression

 

Example 2

 Use the virtual work method to determine the vertical deflection at joint C of the frame shown in Fig.

The real and virtual system are shown in figure below-

 

The vertical deflection at joint C of the frame can now be calculated by applying the virtual work expression

Example 3

Determine the horizontal deflection at joint C of the frame shown in Fig.(a) including the effect of axial deformations, by the virtual work method.

The real and virtual systems for determining the bending moment equations for the three members of the frame, AB, BC, and CD, are also shown in the figures:

                                                                                                                                  

The horizontal deflection at joint C of the frame can be determined by applying the virtual work expression

Here it observed that the magnitude of the axial deformation term is negligibly small as compared to that of the bending deformation term.

Example 4

Determine the vertical deflection of joint A of the frame shown in Fig.(a) by the virtual work method. Use the graphical procedure.

The real and virtual systems, for bending moment diagrams

 

For member AB comparing the diagram, obtain the relevant expression from the eighth row and second column of the diagram table-

By substituting Mv1 = 5 kN.m, M1 = 87.5 kN.m, and L = 5 m into the equation

Similarly, the expression for the integral for member BC is obtained from the second row and second column-

By substituting Mv1 = 5 kN.m, M1 = 87.5 kN.m, and L = 10 m into the equation

Now at joint A the deflection is determined using the numerical values of the integrals for the two members

Home Work

Determine the horizontal and vertical deflection at joint B using virtual work method of the truss shown in Figs. 1–4.

Determine the vertical deflection at joint C using virtual work method of the truss shown in Figs. 5–6.

Determine the vertical deflection at joint E of the Fig-7 and at joint H of the Fig-8 using virtual work method of the truss shown in below.

 

Determine the smallest cross-sectional area required for the members of the truss shown, so that the horizontal deflection at joint D does not exceed 10 mm. Use the virtual work method. (Fig- 9, 10)

Determine the smallest cross-sectional area A for the members of the truss shown, so that the vertical deflection at joint B does not exceed 0.4 inches. Use the method of virtual work. (Fig- 11, 12).

Determine the horizontal deflection at joint E of the truss shown in Fig. 13 due to a temperature increase of 50⁰C in members AC and CE. Use the method of virtual work.

Determine the vertical deflection at joint G of the truss shown in Fig. 14 due to a temperature increase of 65^o F in members AB, BC, CD, and DE, and a temperature drop of 20⁰ F in members FG and GH. Use the method of virtual work.

Use the virtual work method to determine the slope and deflection at point B of the beam shown in Fig- 15,16.

Use the virtual work method to determine the deflection at point C of the beam shown in Figures: 17-20

Use the virtual work method to determine the slope and deflection at point D of the beam shown in Fig- 21-22.

Use the virtual work method to determine the vertical deflection at joint C of the frame shown in Fig- 23-24.

Use the virtual work method to determine the horizontal deflection at joint C of the Figure 25 and horizontal deflection at joint E of the Figure 26.

 

Use the virtual work method to determine the vertical deflection at joint B of the frame shown in Fig.27.

Use the virtual work method to determine the horizontal deflection at joint C of the frame shown in Fig- 28, 29.

Determine the smallest moment of inertia I required for the members of the frame shown in Fig 30 & 31, so that the horizontal deflection at joint C does not exceed1 inch. Use the virtual work method.

Using the method of virtual work, determine the vertical deflection at joint E of the frame shown in Fig. P32.