9
Exercises

9.1. Rosette analysis

A rosette is comprised of three extensometric gages, each of which can measure linear strain relative to its own direction. These instruments, in combination with a measurement chain containing a Wheatstone bridge, are used to determine local strain state through experimentation.

When loaded by a structure with the characteristics E = 70 GPa and ν = 0.3, the gages indicate the following relative elongations:

Remember that strains have no unit, and since they are small, we often multiply them by 10⁶; thereby obtaining the microstrains denoted by με.

The material is assumed to be elastic, linear, homogeneous and isotropic.

QUESTION 1.–

Determine the strain tensor in 2D.

QUESTION 2.–

Writing that the (x, y) plane is a free surface, determine the stress tensor in 3D.

QUESTION 3.–

Deduct the strain tensor in 3D.

QUESTION 4.–

Determine the principal stresses and directions.

QUESTION 5.–

Determine the maximal principal stress together with the associated facet.

QUESTION 6.–

Considering this material as elastic brittle with a tension limit of 100 MPa, will it rupture and in what direction will a crack propagate?

QUESTION 7.–

Determine the maximal shear stress together with the direction of the associated facet.

QUESTION 8.–

Determine the equivalent Von Mises stress.

QUESTION 9.–

Considering this material as ductile with an elastic limit of 140 MPa, will there be plasticity when using a Tresca criterion? And when using a Von Mises criterion?

Further application:

Under the effect of loading on a steel structure (E = 210 GPa, ν = 0.3), a rosette with its three gages at 120° indicates the following relative elongations:

Same questions.

9.2. Pure shear

Show that this shear state is a state of pure shear:

[9.1]

Next write the stress tensor in the principal stress coordinate system and in the maximum shear coordinate system.

9.3. Compression of an elastic solid

A material with the characteristics E and ν can be compressed in a cylindrical envelope by means of a piston with cross-section S. Both the envelope and the piston are assumed to be undeformable. The material is assumed to be linear, homogeneous and isotropic.

QUESTION 1.–

Determine the strain tensor and the stress tensor at any point P of the compressed material.

QUESTION 2.–

Calculate the piston displacement.

QUESTION 3.–

On what condition is this displacement zero?

9.4. Gravity dam

For a gravity dam made of concrete:

It is delimited by three lines: OA, AA’ and OA’. The dam is subjected to its own weight and to forces due to water pressure δ.x in the y-direction. The forces and stresses in relation to z are assumed to be negligible.

QUESTION 1.–

Determine δ.

QUESTION 2.–

Assuming the stresses to be linear along x and y, determine the stress tensor (in 2D) at all points of the structure.

QUESTION 3.–

At the edges of the dam (for present purposes, and in practice, we can show that it is at the edges that stress is maximum), what are the points at which we find maximum and minimum normal stress and for which directions. Deduce how the dam will break. In practice, how can we reinforce the dam at this point?

9.5. Shear modulus

Here, we will consider a square subjected to a state of stress:

[9.2]

Let us assume that the material is linear elastic, homogeneous and isotropic; where E is the Young modulus and ν the Poisson ratio.

QUESTION 1.–

Determine the strain tensor when there is only σ_x.

QUESTION 2.–

Determine the strain tensor when there is only σ_y.

QUESTION 3.–

Using the superposition principle, deduce the strain tensor when there is both σ_x and σ_y.

QUESTION 4.–

Now, we will consider a situation where σ_y = –σ_x.

Determine the strain tensor.

QUESTION 5.–

Determine the stress and strain tensors in the (u, v) basis with

Deduce the shear modulus G, defined by τ_xy = G.γ_xy, in relation to E and ν.

9.6. Modulus of a composite

We will now consider a composite made of carbon fibers (E_f = 200 GPa) in the x-direction and of epoxy resin (E_r = 10 GPa). Here V_f is the volume fraction of fiber and V_r the volume fraction of resin (V_f + V_r = 1).

We apply tensile force to this composite in the x-direction.

QUESTION 1.–

Assuming that both materials undergo the same strain, determine the modulus of the composite.

QUESTION 2.–

Assuming that both materials undergo the same stress, determine the modulus of the composite.

QUESTION 3.–

Which of these two hypotheses seems more reasonable? What if we apply tensile force in the y-direction?

QUESTION 4.–

Plot the modulus obtained in relation to the volume fraction of fiber for both of these hypotheses.

NOTE.– The model obtained when assuming homogeneous stresses is called the Voigt limit, while that obtained when assuming homogeneous strains is called the Reuss limit. We can show that the Voigt model is an upper limit and the Reuss model is a lower limit. In practice, we obtain values between them: close to the Voigt limit for traction in the direction of the fibers and close to the Reuss limit for a composite based on particulate reinforcements.

9.7. Torsional cylinder

Here a straight cylinder with circular base of radius R and height h is fixed to the ground. The material is assumed to be linear elastic, homogeneous and isotropic.

We subject it to torque C on the top side, observing movement, such as:

Each straight cross-section turns on its plane around z by a slight angle dα = k.z (k is homogeneous inversely to length with k << 1/h)

Cross-section z = 0 remains still.

QUESTION 1.–

Find the displacement field of a point M(r, θ, z).

QUESTION 2.–

Deduce the strain tensor.

QUESTION 3.–

Determine the principal strains and the principal directions.

QUESTION 4.–

Determine the volume expansion. Was the result predicable?

QUESTION 5.–

Determine the stress tensor, and deduce the relationship between k and C.

QUESTION 6.–

Show that the fields of displacement, stress and strain that have been determined are indeed the solution to this problem.

QUESTION 7.–

Assuming that the material is elastic brittle, at what point will it break, for what value of torque C, and in which direction will the crack appear? Test this with a chalk piece.

QUESTION 8.–

Using the same material and retaining the same value of rupture torque, we need to reduce the cylinder’s mass. How can we alter its cross-section?

9.8. Plastic compression

We will now consider an aluminum cube of dimension a subjected to compression stress σ in the x-direction. Let us assume that its behavior is perfectly elastic plastic (flow stress is assumed to be constant), with modulus E = 70 GPa and elastic limit σ₀ = 400 MPa.

QUESTION 1.–

In practice, we may expect stress to tend to minus infinity when cube thickness tends to 0. Show that this result if coherent with the elastic–plastic behavior proposed, while not confusing true stress/strain with engineering stress/strain.

Deduce the effort/displacement curve obtained in this test.

QUESTION 2.–

In real terms, when performing this test we obtain a barrel effect:

**Figure 9.10.** *Compressive barrel effect*

Why is this? How do you propose to resolve this problem? And what if we used a longer sample?

QUESTION 3.–

We will now consider a bi-compression test:

At what stress will we obtain plasticity?

QUESTION 4.–

One means of performing this test consists of using beams to press on the cube. In the example shown here, we press on a 2 cm face cube by means of 256 beams with a cross-section of 1 mm² and length of 12 mm.

What is the advantage of using these beams?

QUESTION 5.–

We will now consider a tri-compression test.

At what stress will we obtain plasticity?

9.9. Bi-material beam tension

A beam made of two homogeneous isotropic materials with the respective characteristics E₁, ν₁ and E₂, ν₂ is subjected to tensile effort F.

Part 1: pure tension

Assuming the stress field is in the form:

QUESTION 1.–

Comment on this choice.

QUESTION 2.–

Write out boundary conditions and deduce a relationship between a₁, a₂ and F.

QUESTION 3.–

Determine the strain field.

QUESTION 4.–

Write out the continuity condition at the boundary between the materials 1 and 2 assuming that they are perfectly bonded. Deduce the stress field and plot it.

APPLICATION.– material 1 is steel (E₁ = 210 GPa, ν₁ = 0.3) and material 2 aluminum (E₂ = 70 GPa, ν₂ = 0.3). To simplify these calculations, take: E₁ = 3.E₂ and ν₁ = ν₂.

QUESTION 5.–

Determine the displacement field.

QUESTION 6.–

Determine the mean apparent modulus of the bi-material.

QUESTION 7.–

Determine the application point of force F. How can we perform the test in reality?

Part 2: tension/bending

Now we need to apply F exactly at the point where y = 0:

QUESTION 8.–

Show that if we assume longitudinal strain in the form:

[9.3]

and stress states such that only σ_x is other than 0 in the two materials, we can then solve the problem. Plot the stress and strain fields in the beam.

9.10. Beam thermal expansion

Part 1: mono-material beam

We will now consider a steel beam, assumed to be linear elastic, homogeneous and isotropic, noted as material 1 below, with characteristics: E₁ = 210 GPa, ν₁ = 0.3, α₁ = 12.10^-6 K^-1.

This beam of height h = 5 mm is at equilibrium at a temperature of T₀ = 20 °C and is heated to a temperature of T = 120 °C.

QUESTION 1.–

Assuming the beam to be constrained at each end, determine the strain and stress fields.

QUESTION 2.–

Assuming the beam to be free, determine the strain and stress fields.

Part 2: bi-material beam

Now let us consider a bi-material beam made of steel, noted as material 1, and aluminum, noted as material 2, with characteristics:

These two materials are assumed to be linear elastic, homogeneous and isotropic.

To simplify the calculations, take: E₁ = 3.E₂ and ν₁ = ν₂.

QUESTION 3.–

Assuming the beam is to be fixed at each end, determine the strain and stress fields in the two materials.

QUESTION 4.–

Assuming the beam is free, determine the strain and stress fields. In this case, we will assume that there is a plate at the end of the beam bonding the two ends on the left and free to the right. Accordingly, the boundary conditions at the end of the two beams are verified along cross-section x = L (rather than at each point).

9.11. Cube under shear stress

We will now consider a steel cube subjected to two forces F in the x-direction and two moments M in the z-direction.

QUESTION 1.–

Determine M in relation to F so that the cube is in equilibrium.

We provide: and .

QUESTION 2.–

Show that this stress tensor is the solution. Deduce K in relation to F.

QUESTION 3.–

Plot these stress on the four sides of the square.

QUESTION 4.–

Assuming the material to be elastic brittle (for which the maximum stress criteria approximately announce rupture), at what point will it break (if we continue to increase F), and in what direction will the crack appear?

QUESTION 5.–

Assuming the material to be ductile (for which the Tresca criterion approximately announces the end of elasticity), at what point will it begin to become plastic and what will happen if we continue to increase F?

9.12. Spherical reservoir under pressure

The Storable Propellant Stage (Etage à Propergols Stockables – EPS) is the upper stage of the Ariane 5, which has the role of adjusting payload orbital insertion in relation to the target orbit and ensuring orientation and separation. Being located inside the launcher, it is not exposed to external conditions. Its design is simple: EPS is a truncated cone-shaped stage pressurized without turbopumps, fitted between the compartment and the payload adapter. Its production is the responsibility of Airbus Defence and Space. The structure weighs 1,200 kg, made up of four aluminum reservoirs containing a total of 9.7 tons of classic MMH (MonoMethyl Hydrazine) propellant and hypergolic liquids (3,200 and 6,500 kg). The structure of EPS is an aluminum honeycomb with the truncated part, a spherical cap supporting the motor, four reservoirs and two helium reservoirs (carbon fiber under 400 bars).

The propellant reservoirs are pressurized by two helium spheres (34 kg) under 400 bars, reduced to 21 bars by a reducer. Without any turbopumps to draw fuel into the reservoirs, it is by means of pressure that the two propellants are injected into the combustion chamber. Outside of the reservoirs is a vacuum.

Here, the objective is to size the helium spheres. Let us assume that these reservoirs are composed of a homogeneous, linear elastic and isotropic material with characteristics E and ν. So, we may define the material’s Lamé parameters as:

[9.4]

We note R = 300 mm as the interior radius of the reservoir, e as its thickness and p = 400 bars its internal pressure. The material is assumed to be linear elastic, homogeneous and isotropic.

In this problem, we will use spherical coordinates: r, θ and ϕ. We will ignore volume forces.

QUESTION 1.–

Let us assume that there is an acceptable kinematic displacement field in the form:

[9.5]

Justify this result.

QUESTION 2.–

Specify all of the boundary conditions for this problem.

QUESTION 3.–

What is the Navier’s equation? Solve it.

[9.6]

QUESTION 4.–

Express the strain tensor at any point M of the reservoir.

QUESTION 5.–

Express the stress tensor at any point M of the reservoir.

QUESTION 6.–

Determine the integration constants according to the problem’s available data, using question 2.

QUESTION 7.–

Does the reservoir volume increase or decrease with deformation?

QUESTION 8.–

We may consider e/R small for the unit. Give an approximation of σ_r, σ_θ and σ_φ by performing a first-order limited development.

QUESTION 9–

We require minimal thickness e for the reservoir, considering that it is made of aluminum with the characteristics:

Deduce reservoir mass.

QUESTION 10.–

We require minimal thickness e for the reservoir, considering that it is made of an epoxy/carbon fabric with the characteristics:

Attention, these characteristics are isotropic in the plane, but not isotropic in the out-of-plane directions (in practice, only the resin works in the out-of-plane directions, and we see a modulus of around 10 GPa, a tensile limit around 100 MPa and compression limit around –200 MPa!!)

Deduce the reservoir mass and compare it to that of aluminum.

QUESTION 11.–

Does the hypothesis in question 8 seem justifiable?

QUESTION 12.–

We need to find σ_θ, and σ_ϕ using a calculation derived from static of solids. To this end, we will divide the sphere into two hemispheres and apply the fundamental principle of static equilibrium to one of them. Is the result similar to that obtained in question 8?

9.13. Plastic bending

Let us now consider an aluminum beam (E = 70 GPa, σ_e = 300 MPa) of length L with a rectangular cross-section of height h, width b clamped at x = 0 and subjected to a force of F at x = L.

QUESTION 1.–

Let us consider the structure in elasticity. Assuming that the longitudinal stress σ_x of a horizontal cross-section x is linear in y:

[9.7]

determine K(x) coefficient and show that:

[9.8]

Plot the stress field σ_x and strain field ε_x in the beam.

QUESTION 2.–

Determine where the plasticity will appear and the value of the elastic effort limit F_e.

QUESTION 3.–

Assuming the material to be perfectly plastic:

Plot the stress field in a cross-section during a test (as F progressively increases). Here, let us assume that the linearity of the strain field ε_x with regard to y remains valid (which would certainly not be the case for that of stress σ_x!).

How will plasticity develop in the beam?

Next, qualitatively plot the development of effort F in relation to the deflection δ, and in particular determine the maximum effort F_max.

QUESTION 4.–

What would happen if the effort was released following the development of plasticity? Plot this development on the effort curve in relation to the deflection.

Qualitatively determine the stress field in the cross-section x = 0.

9.14. Disc under radial tension

Let us consider a thin steel disc of z-axis (O, z) with interior radius a = 50 mm and exterior radius b = 200 mm. Thickness e is much less than the other dimensions. We assume the material to be linear elastic, homogeneous and isotropic.

On its periphery, the disc is subjected to a pressure P = 100 MPa and is clamped on its interior radius.

Part 1: conventional method

QUESTION 1.–

What hypothesis can be associated with this problem? Deduce the form of stress and strain tensors.

QUESTION 2.–

Justify the choice of displacement field:

[9.9]

Based on the local equilibrium equation expressed in relation to displacement, deduce the differential equation verified by this field.

QUESTION 3.–

Specify the useful boundary conditions to the determination of integration constants. Deduce the expression for radial displacement u(r).

Part 2: energy-based method

QUESTION 4.–

Select an admissible kinematic displacement field (that respects the displacement boundary conditions of the problem) among the linear displacement fields of r.

Discuss the hypothesis of planar stress? Propose a choice of displacement field (but do not use it thereafter) in order to respect the hypothesis of plane stress?

QUESTION 5.–

Calculate the elastic strain energy in the solid.

QUESTION 6.–

Calculate the work of external forces acting on the structure.

QUESTION 7.–

Using the Ritz method, deduce the unknown coefficients of the selected displacement field.

QUESTION 8.–

Compare this approximate solution to the exact solution obtained earlier. In particular, compare the σ_r and σ_θ stresses in r = a and r = b. What is the conclusion?

In reality, a significant part of this difference is due to the hypothesis of plane stress. If you redo the first part of this exercise, selecting the hypothesis of plane stain (hypothesis adopted for thick plate), you will find:

[9.10]

And:

[9.11]

Hence the displacement at r = b is 0.047 mm.

9.15. Bending beam: resolution by the Ritz method

We will now consider a simply supported beam in bending.

QUESTION 1.–

Propose a deformation shape v₀(x) of the beam neutral axis, with sinusoidal shape, depending only on parameter δ (the beam deflection in center).

QUESTION 2.–

Assuming that a cross-section remains straight and perpendicular to the beam neutral axis, deduce the u(x, y) and v(x, y) displacement fields throughout the beam.

QUESTION 3.–

Deduce the strain field.

QUESTION 4.–

Deduce the stress field, plot it and comment.

QUESTION 5.–

Apply the Ritz method to find the deflection δ in relation to bending force F.

Compare the value of this deflection to the exact solution:

[9.12]

9.16. Stress concentration in open hole

We will now consider the stress field in a plate drilled by a circular hole of radius R, considered very small relative to the plate dimensions. This plate is submitted to a tensile stress σ₀ in the x-direction “to infinity”.

The plate is of low thickness and we will assume the plane stress:

[9.13]

We can show that the stresses are written as:

[9.14]

Students interested in the proof are referred, for example, to [CHE 08], [AGA 08], [BAM 08], etc. or to the examination that inspired this exercise: “http://mms2.ensmp.fr/mmc_paris/annales/examen2007.pdf”.

QUESTION 1.–

Show that this stress field is the solution to the problem.

QUESTION 2.–

Determine and plot the stress field in r = R.

QUESTION 3.–

Determine the stress concentration factor at the edge of the hole:

[9.15]

Considering that the material is brittle with a tension limit stress much less than its compression limit (in absolute value), determine the σ₀ value at which rupture will appear, together with the crack location and the crack propagation direction.

In particular, show that the response is different for tension and compression.

Below are several of the results obtained by finite elements calculation (aluminum material, dimensions 400 × 400 mm², hole radius 10 mm, σ₀ = 100 MPa). Of course, only a quarter of the structure is meshed, and symmetries are imposed:

In cases where the hole size is not negligible (dimensions 200 × 40 mm²), stress concentration increases still further. Essentially, the structure is better without holes. Incidentally, you may also observe that this geometry is representative of a plate drilled with holes spaced by twice hole diameter (in aeronautics, this distance, called pitch distance, is actually four times diameter):

**Figure 9.25.** *Results of the FE calculation for open hole tension. For a color version of this figure, see* *www.iste.co.uk/bouvet/aeronautical.zip*

9.17. Bending beam

Part 1: strain tensor

We will now consider a beam of dimension L × h × b bending while clamped at one end and subjected to force F at the other end (face x = L).

Let us assume the stress be on the (x, y) plane, solving the problem in 2D.

QUESTION 1.–

Determine the torsor (that is, the force and moment) of internal forces of the left part on the left part of the beam. Show that it can be decomposed into a force in the y-direction, called shear force F_y, and a moment in the z-direction called bending moment M_z Determine (without calculating them) what stresses are created by F_y and M_z.

QUESTION 2.–

Write out all of the conditions that stress tensor must verify (in 2D) to solve this problem.

QUESTION 3.–

Assuming:

[9.16]

show that we can then solve the problem.

Now solve the problem, determining the stress tensor throughout the beam. In particular, determine K in relation to F and the geometric characteristics of the beam.

Keep this result in relation to K in order to help with writing the following.

QUESTION 4.–

Physically justify the shape of σ_x given by equation [9.16].

QUESTION 5.–

Determine and plot the stress in a horizontal cross-section with an x-coordinate. Deduce the force (force and moment) of these stresses, showing that they are equal to the cohesive forces determined in question 1.

QUESTION 6.–

This beam is made of aluminum with characteristics: E = 70 GPa; ν = 0.3; σ_e = 250 MPa

Select a sizing criterion for this beam and justify it. Determine at what points this criterion is maximum (assuming L > h).

QUESTION 7.–

Next determine the force F_lim for which this criterion is reached. What would happen in reality at such a force, and what would happen if we continued increasing F?

QUESTION 8.–

This beam is made of glass with characteristics: E = 70 GPa; ν = 0.3; σ_r = 60 MPa.

Select a sizing criterion for this beam and justify it. Determine at what points this criterion is maximum (assuming L > h).

QUESTION 9.–

Next determine the force F_lim for which this criterion is reached. What would happen in reality at such a force, and what would happen if we continued increasing F?

Part 2: strain tensor and displacement

QUESTION 10.–

Determine the strain tensor throughout the beam in relation to E and ν.

QUESTION 11.–

Assuming that v = 0 (to simplify the calculations) and taking the boundary condition as:

[9.17]

determine the displacement field throughout the beam in relation to K.

NOTE.– We can show that the clamping boundary condition is not coherent with the hypothesis of the stress field found in equation 9.16, but that it induces other stresses approaching the clamping boundary condition.

QUESTION 12.–

Show that the boundary conditions proposed in the previous question let us block any rigid-body displacement field in the plane.

QUESTION 13.–

Deduce the relationship between the beam deflection δ and force F, showing that in the beam case, i.e. where length L is much greater than the other dimensions, we find:

[9.18]

with I_z being the quadratic bending moment of inertia in the z-direction:

[9.19]

Part 3: finite element comparison

A finite element study performed using the Abaqus program and the following dimensions, gave the results below.

QUESTION 14.–

Comment on each figure, comparing it with the analytical solution (for which you will have made the main numerical applications).

**Figure 9.28.** *Results of the FE calculation for a bending beam*