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Determine for each case the resultant stress at P on a plane through P whose normal is coincident with the X axis.

8.10 (C). At a point in a material the stresses are:

Calculate the shear stress on a plane whose normal makes an angle of 48° with the X axis and 71° with the Y axis.

8.11 (C). At a point in a stressed material the cartesian stress components are:

<Txx = -40 MN/m2 (jyy = 80 MN/m2 aZ7 = 120 MN/m2

Calculate the normal, shear and resultant stresses on a plane whose normal makes an angle of 48° with the X axis and 61° with the Y axis. [135, 86.6, 161 MN/m2.]

8.12 (C). Commencing from the equations defining the state of stress at a point, derive the general stress relationship for the normal stress on an inclined plane:

(J„ = oxx 12 + azz n 2 + <jyy m2 + 2 axylm + 2 ayzmn + 2a7xln

Show that this relationship reduces for the plane stress system (cr^ = a„ = azy = 0) to the well-known equation an = j (ajx + cryy) + j {p^ — <jyy) cos 29 + oxy sin 29

8.13 (C). At a point in a material a resultant stress of value 14 MN/m2 is acting in a direction making angles of 43°, 75° and 50°53' with the coordinate axes X, Y and Z.

(a) Find the normal and shear stresses on an oblique plane whose normal makes angles of 67°13', 30° and 71°34', respectively, with the same coordinate axes.

(b) If axy = 1.5 MN/m2, oyz = —0.2 MN/m2 and axl = 3.7 MN/m2 determine axx, oyy and azz.

8.14 (C). Three principal stresses of 250, 100 and -150 MN/m2 act in a direction X, Y and Z respectively. Determine the normal, shear and resultant stresses which act on a plane whose normal is inclined at 30° to the Z axis, the projection of the normal on the XY plane being inclined at 55° to the XZ plane.

8.15 (C). The following cartesian stress components exist at a point in a body subjected to a three-dimensional complex stress system:

axx = 97 MN/m2 oyy = 143 MN/m2 a77 = 173 MN/m2 axy =0 oyz = 0 o7X = 102 MN/m2

Determine the values of the principal stresses present at the point. [233.8, 143.2, 35.8 MN/m2.]

8.16 (C). A certain stress system has principal stresses of 300 MN/m2, 124 MN/m2 and 56 MN/m2.

(a) What will be the value of the maximum shear stress?

(b) Determine the values of the shear and normal stresses on the octahedral planes.

(c) If the yield stress of the material in simple tension is 240 MN/m2, will the above stress system produce failure according to the distortion energy and maximum shear stress criteria?

8.17 (C). A pressure vessel is being tested at an internal pressure of 150 atmospheres (1 atmosphere = 1.013 bar). Strains are measured at a point on the inside surface adjacent to a branch connection by means of an equiangular strain rosette. The readings obtained are:

Draw Mohr's circle to determine the magnitude and direction of the principal strains. E = 208 GN/m2 and u = 0.3. Determine also the octahedral normal and shear strains at the point.

[0.235%, 0.083%, -0.142%, 9°28'; e<k, = 0.0589%, yoct = 0.310%.]

8.18 (C). At a point in a stressed body the principal stresses in the X, ¥ and Z directions are:

Calculate the resultant stress on a plane whose normal has direction cosines I = 0.73, m = 0.46, n = 0.506. Draw Mohr's stress plane for the problem to check your answer. [38 MN/m2.]

8.19 (C). For the data of Problem 8.18 determine graphically, and by calculation, the values of the normal and shear stresses on the given plane.

Determine also the values of the octahedral direct and shear stresses. [30.3, 23 MN/m2; 23.4, 22.7 MN/m2.]

8.20 (C). During tests on a welded pipe-tee, internal pressure and torque are applied and the resulting distortion at a point near the branch gives rise to shear components in the r, 8 and z directions.

A rectangular strain gauge rosette mounted at the point in question yields the following strain values for an internal pressure of 16.7 MN/m2:

Use the Mohr diagrams for stress and strain to determine the state of stress on the octahedral plane. E = 208 GN/m2 and v = 0.29.

What is the direct stress component on planes normal to the direction of zero extension?

8.21 (C). During service loading tests on a nuclear pressure vessel the distortions resulting near a stress concentration on the inside surface of the vessel give rise to shear components in the r, \$ and z directions. A rectangular strain gauge rosette mounted at the point in question gives the following strain values for an internal pressure of 5 MN/m2.

£0 = 150 x 10"6, e45 = 220 x 10~6 and £90 = 60 x 10"6

Use the Mohr diagrams for stress and strain to determine the principal stresses and the state of stress on the octahedral plane at the point. For the material of the pressure vessel E = 210 GN/m2 and v = 0.3.

[B.P.] [52.5, 13.8, -5 MN/m2; a^, = 21 MN/m2, roc, = 24 MN/m2.]

8.22 (C). From the construction of the Mohr strain plane show that the ordinate ¿y for the case of a = /8 = y (octahedral shear strain) is

8.23 (C). A stress system has three principal values:

<7, = 154 MN/m2 o2 = 113 MN/m2 <73 = 68 MN/m2

(a) Find the normal and shear stresses on a plane with direction cosines of / = 0.732, m = 0.521 with respect to the a 1 and a2 directions.

(b) Determine the octahedral shear and normal stresses for this system. Check numerically.

8.24 (C). A plane has a normal stress of 63 MN/m2 inclined at an angle of 38° to the greatest principal stress which is 126 MN/m2. The shear stress on the plane is 92 MN/m2 and a second principal stress is 53 MN/m2. Find the value of the third principal stress and the angle of the normal of the plane to the direction of stress.

8.25 (C). The normal stress a„ on a plane has a direction cosine I and the shear stress on the plane is r„. If the two smaller principal stresses are equal show that

If r„ = 75 MN/m2, <r„ = 36 MN/m2 and I = 0.75, determine, graphically a, and a2. [102, -48 MN/m2.]

8.26 (C). If the strains at a point are £ = 0.0063 and y = 0.00481, determine the value of the maximum principal strain £1 if it is known that the strain components make the following angles with the three principal strain directions:

Fory: a = 128°32' = 45" 10' / = positive [0.0075.]

8.27 (C). What is meant by the term deviatoric strain as related to a state of strain in three dimensions? Show that the sum of three deviatoric strains e\, e'2 and e'3 is zero and also that they can be related to the principal strains

e? + e'i+e\$= £[(«■, - e2)2 + (e2 - s, )2 + (ej - e, )2] [C.E.I.]

8.28 (C). The readings from a rectangular strain gauge rosette bonded to the surface of a strained component are as follows:

e0 = 592 x 10~6 £45 = 308 x 10~6 £go = -432 x 10"6

Draw the full three-dimensional Mohr's stress and strain circle representations and hence determine:

(a) the principal strains and their directions;

(b) the principal stresses;

(c) the maximum shear stress.

[640 x 10"6, -480 x 10~6; at 12° and 102° to A, 109, -63.5, 86.25 MN/m2]

8.29 (C). For a rectangular beam, unit width and depth 2d, simple beam theory gives the longitudinal stress axx = CM y/I where y = ordinate in depth direction (+ downwards)

The shear force is Q and the shear stress zxy is to be taken as zero at top and bottom of the beam. <Tyy = 0 at the bottom and ayy = — w/unit length, i.e. a distributed load, at the top.

Using the equations of equilibrium in cartesian coordinates and without recourse to beam theory, find the distribution of avv and <jxv.

8.30 (C). Determine whether the following strain fields are compatible:

(a) £xx = 2x2 + 3v2 + z + 1 (b) £ja = 3>>2 + xy

8.31 (C). The normal stress a„ on a plane has a direction cosine / and the shear stress on the plane is r. If the two smaller principal stresses are equal show that r r 7 T'

8.32 (C). (i) A long thin-walled cylinder of internal radius Rq, external radius R and wall thickness T is subjected to an internal pressure p, the external pressure being zero. Show that if the circumferential stress (a») is independent of the radius r then the radial stress (nrr) at any thickness t is given by

The relevant equation of equilibrium which may be used is:

(ii) Hence determine an expression for ago in terms of T.

(iii) What difference in approach would you adopt for a similar treatment in the case of a thick-walled cylinder?

8.33 (C). Explain what is meant by the following terms and discuss their significance:

(a) Octahedral planes and stresses.

(b) Hydrostatic and deviatoric stresses.

(c) Plastic limit design.

(d) Compatibility.

(e) Principal and product second moments of area. [B.P.]

8.34 (C). At a point in a stressed material the cartesian stress components are:

<7,v = 72 MN/m2 cr<: = 32 MN/m2 ay: = 46 MN/m2

Calculate the normal, shear and resultant stresses on a plane whose normal makes an angle of 48° with the X axis and 61° with the Y axis. [B.P.] [135.3, 86.6, 161 MN/m2.]

8.35 (C). The Cartesian stress components at a point in a three-dimensional stress system are those given in problem 8.33 above.

(a) What will be the directions of the normal and shear stresses on the plane making angles of 48° and 61° with the X and Y axes respectively?

[I'm'n' =0.1625, 0.7010, 0.69i4; /.vmv«.v = -0.7375,0.5451,0.4053]

(b) What will be the magnitude of the shear stress on the octahedral planes where I = m = n = 1 /-v/3?

8.36 (C). Given that the cartesian stress components at a point in a three-dimensional stress system are:

<t„ = 20 MN/m2, ctvv = 5 MN/m2, a:: = -50 MN/m2 xxy = 0, rv; = 20 MN/m2, r-t = -40 MN/m2

(a) Determine the stresses on planes with direction cosines 0.8165, 0.4082 and 0.4082 relative to the X, Y and Z axes respectively. [-14.2, 46.1, 43.8 MN/m2]

(b) Determine the shear stress on these planes in a direction with direction cosines of 0, —0.707, 0.707.

8.37 (C). In a finite element calculation of the stresses in a steel component, the stresses have been determined as follows, with respect to the reference directions X, Y and Z:

r,v = -41.3 MN/m2 rv; = -8.9 MN/m2 r-t = 38.5 MN/m2

It is proposed to change the material from steel to unidirectional glass-fibre reinforced polyester, and it is important that the direction of the fibres is the same as that of the maximum principal stress, so that the tensile stresses perpendicular to the fibres are kept to a minimum.

Determine the values of the three principal stresses, given that the value of the intermediate principal stress is 3.9 MN/m2. [-53.8; 3.9; 84.9 MN/m2]

Compare them with the safe design tensile stresses for the glass-reinforced polyester of: parallel to the fibres, 90 MN/m2: perpendicular to the fibres, 10 MN/m2.

Then take the direction cosines of the major principal stress as I = 0.569, m = —0.781, n = 0.256 and determine the maximum allowable misalignment of the fibres to avoid the risk of exceeding the safe design tensile stresses. (Hint: compression stresses can be ignored.) [15.9°]

8.38 (C). The stresses at a point in an isotropic material are:

axx = 10 MN/m2 <rvv = 25 MN/m2 o:z = 50 MN/m2 rvv = 15 MN/m2 r,-= 10 MN/m2 r,t = 20 MN/m2

Determine the magnitudes of the maximum principal normal strain and the maximum principal shear strain at this point, if Young's modulus is 207 GN/m2 and Poisson's ratio is 0.3. [280(ae; 419ji£]

8.39 (C). Determine the principal stresses in a three-dimensional stress system in which:

oxx = 40 MN/m2 ctvv = 60 MN/m2 o,,- = 50 MN/m2 ax: = 30 MN/m2 axy = 20 MN/m2 ayz = 10 MN/m2

8.40 (C). If the stress tensor for a three-dimensional stress system is as given below and one of the principal stresses has a value of 40 MN/m2 determine the values of the three eigen vectors.

0 0