7
Structural Deflections: Energy Methods

The teaching objectives for this chapter are as follows:

– understanding the notion of strain energy;
– formulating the internal energy of a structure;
– using the energy principle to calculate deflections;
– applying the principle of virtual work, Castigliano’s theorem and the Maxwell–Betti law.

This chapter is devoted to the presentation of energy methods to calculate elastic structure deflections. First, we present the concepts of the work of external actions, internal energy and the principle of virtual work. The latter is widely used to calculate deflections in trusses, beam and frames. Finally, we describe Castigliano’s theorem and the Maxwell–Betti law.

7.1. Work of external actions

The work of an external action is defined by the scalar product of the forces vector (moments) and the displacements vector (slopes), which are generated. The expression of the elemental work of a force is given by

[7.1]

The vectors images and images form an angle α. The work expression is

[7.2]

The work of a force is considered to be positive if images , negative if images and zero if images .

The expression of work depends on the applied external action.

Axial force

A bar of length L is stressed by an axial force F (Figure 7.1), which causes a displacement Δ at its point of application.

The elementary mechanical work of the axial force is written knowing that the force and displacement are in the same direction.

[7.3]

In linear elastic behavior, displacement is proportional to the applied force. It is therefore possible to write

[7.4]

When we substitute the relationship [7.4] with [7.3], the elementary mechanical work becomes

[7.5]

Thus the total mechanical work of force F that caused displacement Δ is

[7.6]

The total mechanical work of force F is the area between curve F-Δ and the axis of abscissas (Figure 7.2).

Bending moment

In the same way, we define the mechanical work of the bending moment by the scalar product of the bending moment M and the elementary slope dω (Figure 7.3).

[7.7]

The applied slope and the bending moment rotate in the same direction. The relationship [7.7] is written as

[7.8]

The bending moment and the slope generated are proportional.

[7.9]

Substituting the relationship [7.9] into equation [7.8], we obtain

[7.10]

The total work of a bending moment creates a slope ω which becomes

[7.11]

The total bending moment work of force M is the area between curve M-ω and the axis of abscissas (Figure 7.4).

**Figure 7.4.** *Mechanical work of moment M*

Shear force

The fundamental relationship of the work of external forces [7.1] is always applicable to structures subjected to shear stresses (Figure 7.5).

To generate a transverse slipping, it is necessary to provide an elementary work equal to

[7.12]

Knowing that

[7.13]

hence

[7.14]

The total mechanical work is

[7.15]

The total work of the shear force T is shown schematically in Figure 7.6.

**Figure 7.6.** *Work of transversal force*

Torque moment

A circular transversal section of a bar is stressed by a torque moment M_t (Figure 7.7).

The torque moment M_t turns on the plane (YZ). The fiber (AB) is horizontal in the initial state. It rotates with the angle dφ when the torque moment M_t is applied (Figure 7.7).

**Figure 7.7.** *Angular deformation due to torque moment*

The elementary work needed to rotate the section and the plane (YZ) of angle dφ is

[7.16]

Knowing that

[7.17]

Integrating the relationships [7.16] and [7.17], we can write

[7.18]

The total work is

[7.19]

In the same way, the total work of torque moment M_t is the area between curve M_T-φ and the axis of abscissas (Figure 7.8).

In the case of a structure which is subjected simultaneously to a normal force, a bending moment, a shear stress and a torque moment, the total mechanical work can be deduced using the superposition principle.

[7.20]

In this area, there are two different cases:

Case 1: When an applied action varies in some way with the corresponding deflection, the mechanical work of this action is written as

[7.21a]

So the total work of action F is the area between curve F-Δ and the axis of abscissas (Figure 7.9).

This action can be a normal force, a bending moment, a shear stress or a torque moment.

Case 2: If applied action F is independent of the deflection which is generated, we deduce the mechanical work by the following expression:

[7.21b]

Figure 7.10 shows the graphical aspect of the mechanical work of the applied action.

**Figure 7.10.** *Work of an action independent of deflection*

7.2. Internal or strain energy

The expression of a structure’s internal energy depends on the internal action, as well as the geometric and mechanical characteristics of the material constituting the structure. Depending on the action inside the structure, these are as follows.

Normal force

Consider an element of a truss bar of length L that is stressed by an axial internal force (Figure 7.11).

We define the internal elementary work of axial force N(x) by

[7.22]

The stress–strain relationship is described by Hooke’s law.

[7.23]

Knowing that

[7.24]

[7.25]

With E being the modulus of longitudinal elasticity of the material used and Ω the cross-section area.

Substituting relationships [7.24] and [7.25] in the formula [7.23], we obtain the expression of normal force.

[7.26]

By substituting the relationship [7.26] into the relationship [7.22], the internal elementary work becomes

[7.27]

The internal work required to deflect the bar by a quantity ΔL is written as

[7.28]

Internal work can also be written as

[7.29]

Using the relationship [7.26], the lengthening of the bar is

[7.30]

and the work of internal actions is written as

[7.31]

In the integral form, the expression [7.31] is written as

[7.32]

The mechanical work of internal actions is equal to the internal energy of deflections. Hence

[7.33]

Bending moment

Consider a structural element of length L stressed by a bending moment M(x) (Figure 7.12).

The elementary work due to the effect of the bending moment M (x) is

[7.34]

with

[7.35]

where k is the rigidity factor of the bar.

Substituting the expression [7.35] into the relationship [7.34], the internal work of the bending moment is

[7.36]

Using the relationship [6.14], the differential slope is

[7.37]

In substituting equation [7.37] with equation [7.34], we obtain

[7.38]

The internal work to deflect a bar with angle ω is

The internal strain energy is

Shear force

The same procedure is applied to calculate the internal strain energy of a bar subjected to a shear force.

where G is Young’s modulus of transverse elasticity of the material of the bar.

Torque moment

The internal energy of a bar subjected to a torque moment is deduced by

where I₀ is the moment of polar inertia of the section.

Note that when a section is stressed by a set of actions, the internal energy is equal to the sum of the internal energies of each action, using the principle of superposition.

7.3. Principle of energy conservation

Sections 7.1 and 7.2 above have illustrated the expressions of the work of external forces and the internal energy of a bar stressed by an internal action. In this section, we use the principle of energy conservation to calculate the deflections of elastic structures.

EXAMPLE 7.1.– Bar stressed by a normal force (Figure 7.13)

Using the relationship [7.6], the applied force F generates external work equal to

Δ is the displacement that corresponds to the equilibrium limit (Figure 7.14).

The relationship [7.33] allows us to evaluate the energy strain on the bar.

Applying the principle of energy conservation and taking into account relationships [7.6] and [7.33], we can write

This equality makes it possible to calculate the displacement of the application point of the external force.

This result can be deduced using Hooke’s fundamental relationship.

EXAMPLE 7.2.– Bending bar (Figure 7.15)

The force P generates a mechanical work equal to

Δ is the vertical displacement of the application point of force P (Figure 7.16).

The total work of the external force is

The internal strain energy, taking into consideration just the bending effect, is

Knowing that the bending moment M(x)= Px, the internal energy becomes

Using the principle of energy conservation (U_in = W_ext) allows us to write

We deduce the vertical displacement of the application point of force P by

EXAMPLE 7.3.– Cantilever beam under pure bending (Figure 7.17)

The mechanical work of moment M₀ is equal to

where ω₀ is the slope of the free section of the cantilever beam (Figure 7.18).

**Figure 7.18.** *Cantilever deflection*

The internal strain energy of the bar is

According to the applied moment M₀, internal energy is written as

Using the principle of energy conservation, the slope of the section at the free end is

7.4. Principle of virtual work

John Bernoulli introduced the principle of virtual work in 1717. It is also called the unit action method. This principle is a very powerful tool for calculating deflections at specified points on trusses, beams and plane frames.

In this chapter, we focus on deformable structures. This part presents the development of the principle of virtual forces of deformable structures. The principle of virtual work is one of the general methods of calculating structural deflections.

The principle of virtual forces can be established as follows:

When a set of external actions, F_ex, is applied to a deformable structure, internal actions, F_in, develop at any point of the structure. Equilibrium equations ensure the relationship between external and internal actions.

External actions cause external displacement Δ at any point on the structure’s surface. In the same context, internal actions generate internal displacements δ verifying the compatibility relationships.

In general, the principle of virtual work applies when the work of external actions is equal to that of internal actions.

[7.39]

Considering any solid body, we calculate, for example, the displacement at point A on the structure’s surface (Figure 7.19).

Under the effect of applied forces, a solid body deforms and obtains a deformed equilibrium configuration, which differs from its initial position (Figure 7.20).

Now, a virtual unit force is applied at point A in the direction (AA') (Figure 7.21). The unit virtual force generates virtual internal forces N_in in any representative element of the structure. The equilibrium relationship is ensured between the unit virtual force and the internal virtual actions.

Once the real actions (M, F₁, F₂) are applied, point A displaces by Δ, generating a displacement dL of the representative point. The reciprocity of the virtual work of internal actions and of external forces through the displacements Δ and dL allows us to write

[7.40]

Where F_ex=1, N_in is the internal virtual stress acting on the considered element, Δ and dL are, respectively, the displacements caused by the real forces and the internal action N_in of the representative element.

The relationship [7.40] allows us to calculate the displacement at point A.

[7.41]

The relationship [7.41] can also be used to calculate the slope at point A by applying a unit virtual moment. This moment causes internal virtual forces N_in and the slope at point A can be in this case

[7.42]

Hence, the slope at point A is calculated as

[7.43]

Where M_ex=1, N_in is the internal virtual stress acting on the element, ω is the slope caused by real actions and dL is the internal displacement of the representative element.

Relationships [7.41] and [7.43] represent the fundamental equation of virtual work. They can be applied to calculate displacements and slopes.

7.4.1. Method of virtual work: trusses

In trusses, the virtual work method is used to calculate the displacement of a joint. It is necessary to calculate the vertical displacement of joint B and the horizontal displacement of joint C of the truss (Figure 7.22). It is assumed that the membrane rigidity E Ω is constant for all bars.

The forces on the structure’s bars can be determined by one of the truss analysis methods (Chapter 4).

N_k represents the internal force on the bar k; the corresponding axial deflection is calculated by

[7.44]

where L_k, E Ω are, respectively, the length and the axial rigidity of bar k.

To calculate the vertical displacement of joint B, a virtual force is applied to joint B along the positive directions of the axes (Figure 7.23).

The work of the unit virtual force through displacement at the desired point in the real system is

[7.45]

**Figure 7.23.** *Virtual deflection of the structure*

In the same way, the work of the internal virtual forces through the displacements of the bars of the real system is written as

[7.46]

where n_k is the virtual force in bar k due to the unit force.

By applying the fundamental principle of virtual work, we can write

[7.47]

Substituting the relationship [7.44] into relationship [7.47], the displacement at point B is given as

[7.48]

7.4.1.1. Analysis procedure

The steps for calculating the displacement of statically determinate trusses using the virtual work method are as follows:

– the internal forces N_k in the bars of the statically determinate truss are calculated by one of the truss analysis methods;
– in the virtual system, where we want to calculate the displacement in a positive direction, we apply a unit virtual force to the joint. Calculating internal forces due to the unit virtual action is feasible by using another method of truss analysis;
– the displacement can be calculated by applying the relationship [7.48]. The expression of the virtual work due to the real and virtual forces evaluated in 1 and 2 allows us to calculate the displacement. If the displacement obtained is positive, it means that the displacement and the unit force follow the same direction.

EXAMPLE 7.4.–

Calculate the vertical displacement in joint B of the truss (Figure 7.24). E Ω is assumed to be constant.

Study of the real structure

Calculating internal forces (Figure 7.24) is carried out using the method of joint equilibrium. The results obtained are grouped in Table 7.1.

images — **Table 7.1.** *Forces on the bars*

Study of the virtual structure

A unit virtual force is applied to joint B (Figure 7.25) and then the internal forces in the bars are calculated by the joint equilibrium method.

The forces on the bars are grouped in Table 7.2.

Table 7.3 groups the forces on the bars of the virtual and real structure (Figure 4.13). We can calculate the displacement using the relationship [7.48].

From Table 7.3, vertical displacement of point B is given as

The negative sign shows that joint B is displaced in the opposite direction to that of the unit force.

EXAMPLE 7.5.–

Repeat Example 7.4 to calculate the horizontal displacement in support C.

A unit virtual force is applied to joint C in a positive direction (Figure 7.26). Calculations of internal forces in the bars of the virtual structure are grouped in Table 7.4.

Table 7.4. Forces due to unit action

Bar	AB	BC	AC
n	0	0	1
Nature of n	Neutral	Neutral	Traction

The internal forces on the bars are grouped in Table 7.5.

Hence, the horizontal displacement at point C is

The deformed structure takes the following configuration (Figure 7.27).

**Figure 7.27.** *Deformation of the structure*

7.4.2. Method of virtual work: beams

The expression of the virtual work method is used to calculate the deflections of the determine beams. In this context, we consider a beam subjected to any load q(x) (Figure 7.28) where we want to calculate the vertical displacement at point B of the beam.

To develop the expression giving the vertical displacement Δ_B using the virtual work method, we apply a unit virtual force in a positive direction (Figure 7.29).

The work of the unit virtual force through the deflection generated in point B is

[7.49]

Similarly, the elementary work of virtual internal actions is written as

[7.50]

where m(x) is the virtual moment along the beam.

The variation of slope dω(x) along element dx [7.16] is written as

[7.51]

M(x) is the bending moment due to real loads.

Substituting the relationship [7.51] with [7.50], we obtain

[7.52]

The total virtual work of the whole beam can be deduced by integrating relationship [7.52].

[7.53]

Applying the principle of virtual work leads to formulating the expression of vertical displacement in point B.

[7.54]

[7.55]

The relationship [7.55] is used to calculate the slope of the section passing through point B.

[7.56]

where m(x) is the bending moment due to applying a unit virtual moment at the desired point.

7.4.2.1. Analysis procedure

To calculate the deflection at a point on a statically determinate beam using the virtual work method, it is necessary to follow the following steps:

– formulate the expression of the bending moment along the beam subjected to the force of real loads;
– formulate the expression of the unit virtual moment. If we want to find a displacement, a unit force must be applied. If we want to find a slope, a unit virtual moment must be applied to the point chosen in a positive direction;
– the deflection at a point is calculated using one of equations [7.55] or [7.56].

EXAMPLE 7.6.– Beam

Calculate the deflection and slope at point A of the cantilevered beam (Figure 7.30). We assume that the flexural rigidity EI is constant.

The variation of the bending moment on reference (Axy) is

A unit virtual force is applied at point A (Figure 7.31).

The variation of the unit virtual moment is

The vertical displacement of point A can be deduced by

The slope of the section passing through point A can be obtained by applying a unit virtual moment at the same point (Figure 7.32).

The variation of the virtual moment along the cantilever is

In the same way, the slope of the section passing through point A is

The deformed cantilever is shown in Figure 7.33.

EXAMPLE 7.7.– A beam with many loads

Calculate the deflection and slope at point A of the cantilever (Figure 7.34). We assume that the flexural rigidity EI is constant.

We associate reference (Bxy) with the beam (Figure 7.35).

The variation of the bending moment of external actions is

A unit virtual force is exerted at point A (Figure 7.36).

The expression of the unit virtual moment is

The vertical displacement of point A can be deduced by

To calculate the slope at point A, we apply the unit virtual moment at the same point (Figure 7.37).

The variation of the virtual moment along the cantilever is calculated as

The slope of the section passing through point C is obtained as

[7.57]

The deformed cantilever is shown in Figure 7.38.

7.4.3. Method of virtual work: frames

The virtual work method applies to frames in the same way to that of beams. Therefore, to calculate the displacement Δ or the slope ω at a point of a statically determinate frame, it is necessary to apply a unit virtual force or a unit virtual moment respectively. The corresponding external virtual work is

Force applied:

[7.58]

Moment applied:

[7.59]

Frames are usually subjected to a normal force, a tangential force and a bending moment. In this case, the virtual work of internal actions is equal to the sum of the internal virtual work.

Internal force:

[7.60]

Internal bending moment:

[7.61]

By neglecting the effect of the shear force, the expression of the total internal virtual work can be obtained by integrating relationships [7.60] and [7.61].

[7.62]

Using the principle of virtual work allows us to write:

Displacement:

[7.63]

Slope:

[7.64]

Axial deflections in the bars of a frame are generally very small compared to bending deflections. The relationships [7.63] are [7.64] are written as:

Displacement:

[7.65]

Slope:

[7.66]

where m_i (x) is the expression of the bending moment due to a unit virtual force [7.65] and a unit virtual moment [7.66].

7.4.3.1. Analysis procedure

To calculate the deflection and/or the slope at a point on a statically determinate frame using the virtual work method, it is necessary to follow the following steps:

– formulate the expression of the bending moment on each bar of a frame subjected to the force of real loads;
– formulate the unit virtual moment. If we are considering a displacement, a unit force must be applied at the desired point. In the case of a slope, a unit virtual moment is applied to the selected point in a positive direction;
– the displacement or slope at the requested point is calculated using relationships [7.65] or [7.66] respectively.

EXAMPLE 7.8.–

Calculate the horizontal displacement and the slope at point C of the frame (Figure 7.39). It gives P = 2.25 qL.

Calculation of the slope at point C

Static analysis of the frames allows us to calculate the support reactions.

Bar (AC)

Bar (CD)

To calculate the slope at point C, we apply the unit virtual moment at point C (Figure 7.40).

Under the effect of the unit moment, static analysis of the frame makes it possible to deduce the variation of the bending moment on the bars of the frame.

Bar (AC)

Bar (CD)

The external work of the unit moment is

In another part, the internal virtual work, taking into account the bending effect, is given as

We substitute the expressions of the bending moments of each bar. We obtain

Applying relationship [7.67] allows us to deduce

Horizontal displacement calculation at point C

A unit virtual force is applied at point C (Figure 7.41).

Static analysis of the frame is given as

**Figure 7.41.** *Effect of the virtual force*

Bar (AC)

Bar (CD)

The external work of the unitary moment is given as

The internal virtual work, considering the bending stress, is given as

Substituting the expressions of the bending moments of each bar, we obtain

The horizontal displacement at point C can be deduced from relationship [7.65].

The deformed frame under the applied load system is shown in Figure 7.42.

7.5. Conservation of energy and strain energy

Internal energy is the ability of a structure to provide internal work when it is subject to external actions. Strain energy is the energy stored in a structure due to its deflection. The statement of the principle of energy conservation is “the external work of an elastic structure is equal to the work of internal actions or the strain energy stored in the structure”.

The principle of energy conservation allows us to write

[7.67]

[7.68]

where U is the internal or strain energy.

In another way, we can define the internal or strain energy according to the work of the external actions [7.1], which is equal to that of the internal actions.

7.6. Castigliano’s theorem

Another method of calculating structural deflections based on the concept of energy is described in this section. This method only applies to elastic structures bearing the name of its author (the Castigliano method).

7.6.1. Displacement theorem statement

The partial derivative of internal energy in relation to an applied force is equal to the displacement of the applied force point in its direction.

[7.69]

7.6.2. Slope theorem statement

The partial derivative of internal energy in relation to an applied moment is equal to the slope of the section passing through this point in the direction of the slope of the moment.

[7.70]

To derive Castigliano’s theorem, we consider a structure subject to n forces; F₁, F₂, F₃, …, F_i, …, F_n-1, F_n (Figure 7.43).

The virtual work of external forces is

[7.71]

By applying the principle of energy conservation, internal energy is equal to the work of external forces.

[7.72]

The internal energy depends on the applied forces. If the external force F_i increases by one increment dF_i, the internal energy increases by quantity dU. The expression of internal energy is written in this case as.

[7.73]

The differential quantity of the force dF_i generates a deflection that corresponds to an increase in displacement dΔ_i . The increment of external work is

[7.74]

This corresponds to an increase in internal energy such that

[7.75]

The second order differential of dU can be neglected.

The set of applied forces F₁, F₂, F₃, …, F_i, …, F_n-1, F_n, respectively generates displacements Δ₁, Δ₂, Δ₃,…, Δ_i,…, Δ_n−1, Δ_n, which leads to the provision of internal energy equal to:

[7.76]

The quantity dU = dF_i Δ_i is the additional internal energy generated by the force increment dF_i.

The strain energy is independent of the application order of external forces. Equation [7.76] allows us to deduce that

[7.77]

The analysis method can be applied to set out Castigliano’s second theorem relating to slopes. It is

[7.78]

7.6.3. Application of Castigliano’s theorem to truss analysis

The internal energy expression of a truss is

[7.79]

The displacement of a joint in the direction of an applied force is equal to

[7.80]

EXAMPLE 7.9.–

Repeat Example 7.4 to determine vertical displacement in joint C.

We exert force F in the direction of the requested displacement (Figure 7.44).

The stresses on the bars of a truss under the effect of force F₀ have already been established in Example 7.4.

Under the effect of force F, the stresses on the bars of the structure are grouped in Table 7.6 using the superposition principle.

The internal strain energy of the truss is

Substituting the expressions of the stresses in the internal energy formula and taking the partial derivative with respect to force F, we obtain

Given that applied force F is not a real force, we can write

Vertical displacement of point B is given as

The result obtained is the same as in Example 7.4.

7.6.4. Using Castigliano’s theorem to analyze beams and frames

The internal energy of a frame composed of several bars can be deduced from the internal energy formulated from a structural bar.

[7.81]

Applying Castigliano’s theorem allows us to calculate the displacement Δ_i at a point on the bar.

[7.82]

At a point on a frame, the displacement is

[7.83]

In the same way, the slope at a point on a bar can be deduced by

[7.84]

If the desired point is on a frame, the corresponding slope is given as

[7.85]

EXAMPLE 7.10.– Cantilever

Repeat Example 7.6 to calculate the displacement and slope at point A of the cantilever (Figure 7.30). At the same point, a virtual force is exerted (Figure 7.45).

The expression of the bending moment due to applied forces is given as

The strain energy of the bar is written as

Applying Castigliano’s theorem, the displacement at point A is written as

It can also be written as

Force F is zero, so the previous relationship is written as

Similarly, to calculate the slope at point A, we apply a virtual moment M_A (Figure 7.46).

**Figure 7.46.** *Given cantilever beam*

The expression of the bending moment is

The slope at point A using Castigliano’s theorem is written as

Knowing that M_A = 0, the previous relationship is written as

The deformed cantilever beam is shown in Figure 7.33.

EXAMPLE 7.11.– Frame

Repeat Example 7.8 to calculate the horizontal displacement and slope at point C of the frame (Figure 7.39).

Calculation of the slope at point C

We apply a virtual moment M_C at point C (Figure 7.47).

The equilibrium equations allow us to calculate the support reactions.

The variation of the bending moment in each bar is

Bar (AC)

Bar (CD)

The internal energy of the frame is written in the following form:

The slope of point C can be deduced using the presented method.

Given that M_C = 0, the slope of joint C is

Calculation of horizontal displacement at point C

This time, the force F is applied to joint C (Figure 7.48).

Static analysis of the frame makes it possible to determine the support reactions and formulate the bending moment expressions.

Bar (AB)

Bar (BC)

Bar (CD)

The internal energy of the frame is written as

The horizontal displacement at point C can be deduced from Castigliano’s method.

Since F = 0, we can deduce the horizontal displacement at point C as

7.7. Maxwell–Betti law

The statement of the Maxwell–Betti law is: “for any elastic structure, the virtual work of force F₁ (moment M₁) through the displacement (the slope) caused by force F₂ (moment M₂) is equal to the virtual work of force F₂ (moment M₂) through the displacement (the slope) generated by force F₁ (moment M₁)”.

To show the validity of the Maxwell–Betti law, we consider a deformable body stressed by two forces F₁ and F₂, respectively, generating displacements Δ₁ and Δ₂ (Figure 7.49).

The application of the force is carried out in the following process: force F₁ is applied, the body takes any deformed configuration. We then apply force F₂ and the body is deformed once more and takes another deformed position. The total work due to applying forces F₁ and F₂ is

[7.86]

Force F₁ causes additional work through the displacement Δ₁₂ caused by force F₂, independent of the effect of force F₁.

We reverse the order of applying forces; we first apply force F₂ then force F₁. The total external work due to applying forces F₂ and F₁ is

[7.87]

In the same way, force F₂ causes additional work through the displacement Δ₂₁ caused by applying force F₁, independent of the effect of force F₂.

The total work is independent of the order of applying forces F₁ and F₂. This allows us to write

[7.88]

Equation [7.88] shows that the work of force F₁ through displacement Δ₁₂ caused by force F₂ is equal to the external work of F₂ through displacement Δ₂₁ caused by force F₁ (reciprocity of external work).

Relationship [7.88] also applies to equal forces (F₁ = F₂). The reciprocity of deflections becomes

[7.89]

EXAMPLE 7.12.–

Verify the Maxwell–Betti law applied to the beam in Figure 7.50.

The deflection expressions of each section using the differential equation of the elastic line method are

With L = 3a.

**Figure 7.51.** *Application of force in A*

We first apply force P to point A (Figure 7.51). The displacement of point B is given by

**Figure 7.52.** *Application of force in B*

We then apply force F (Figure 7.52). Point A moves by a value of Δ_A.

The external work of force F through displacement Δ_B is equal to

Similarly, the external work of force P through displacement Δ_A is

Hence, W(P) =W(F)

7.8. Conclusion

First, we presented the formulation of the work of the external forces and the formulation of the strain energy of a structure subjected to an axial force, a bending moment, a shear stress and a torque moment. These expressions lead to the calculation of displacements and/or slopes by applying the principle of energy conservation.

Second, we examined the principle of virtual work. This principle is a powerful tool for calculating displacements and/or slopes at distinctive points. It was introduced to analyze trusses, beams and frames.

In the same context, the calculation of the displacement (slope) using Castigliano’s theorem can be carried out by deriving the internal energy expression in relationship to the applied force (at the applied moment) at the desired point.

Finally, the Maxwell–Betti law allows us to quantify the displacement and/or the slope: “the displacement (slope) at point i due to a force (moment) applied at point j is equal to the displacement (at the slope) at point j when a force (moment) is applied at point i”.

7.9. Problems

Exercise 1

– Express the work of force P and the internal energy of the following structures. Consider that EΩ and EI are constant.
– Calculate the vertical displacement at point A in each case.

Exercise 2

Calculate the slope in the direction of the slope of moment M₀. EI is assumed to be constant.

Exercise 3

Using the principle of virtual work, calculate the vertical and horizontal displacements at point A of the following structures: EI and E Ω are constant.

Exercise 4

Using the principle of virtual work, calculate the vertical and horizontal displacement and the slope at point A of the following structures: EI is assumed to be constant.

Exercise 5

Using the principle of virtual work, calculate the vertical and horizontal displacement and the slope at point A of the following structures: EI is assumed to be constant.

Exercise 6

Determine the vertical and horizontal displacement and the slope at point A of the following structures.

Exercise 7

Using Castigliano’s method, calculate the vertical displacement in A and the slope at point B in the following structures: EI is assumed to be constant.

Exercise 8

Using Castigliano’s method, calculate the unknown, represented by for each structure.

Exercise 9

Using Castigliano’s method, remove the static indeterminacy of the following structures and deduce the corresponding unknown.

Exercise 10

Verify Maxwell’s law for the following structures.

Bar	N^(F₀⁾	N^(F)	L	N^(F₀⁾+N^(F)
AB
BC
AC			2L