When we discussed the design of transmission shafts in Sec. 3.7, we considered only the stresses due to the torques exerted on the shafts. However, if the power is transferred to and from the shaft by means of gears or sprocket wheels (Fig. 8.11a), the forces exerted on the gear teeth or sprockets are equivalent to forcecouple systems applied at the centers of the corresponding cross sections (Fig. 8.11b). This means that the shaft is subjected to a transverse loading, as well as to a torsional loading.
The shearing stresses produced in the shaft by the transverse loads are usually much smaller than those produced by the torques and will be neglected in this analysis.^{†} The normal stresses due to the transverse loads, however, may be quite large and, as you will see presently, their contribution to the maximum shearing stress _{max} should be taken into account.
Consider the cross section of the shaft at some point C. We represent the torque T and the bending couples M_{y} and M_{z} acting, respectively, in a horizontal and a vertical plane by the couple vectors shown (Fig. 8.12a). Since any diameter of the section is a principal axis of inertia for the section, we can replace M_{y} and M_{z} by their resultant M (Fig. 8.12b) in order to compute the normal stresses σ_{x} exerted on the section. We thus find that σ_{x} is maximum at the end of the diameter perpendicular to the vector representing M (Fig. 8.13). Recalling that the values of the normal stresses at that point are, respectively, σ_{m} = Mc/I and zero, while the shearing stress is _{m} = Tc/J, we plot the corresponding points X and Y on a Mohrcircle diagram (Fig. 8.14) and determine the value of the maximum shearing stress:
Recalling that, for a circular or annular cross section, 2I = J, we write
It follows that the minimum allowable value of the ratio J/c for the cross section of the shaft is
where the numerator in the righthand member of the expression obtained represents the maximum value of in the shaft, and _{all} the allowable shearing stress. Expressing the bending moment M in terms of its components in the two coordinate planes, we can also write
Equations (8.6) and (8.7) can be used to design both solid and hollow circular shafts and should be compared with Eq. (3.22) of Sec. 3.7, which was obtained under the assumption of a torsional loading only.
The determination of the maximum value of will be facilitated if the bendingmoment diagrams corresponding to M_{y} and M_{z} are drawn, as well as a third diagram representing the values of T along the shaft (see Sample Prob. 8.3).
SAMPLE
A 160kN force is applied as shown at the end of a W200 × 52 rolledsteel beam. Neglecting the effect of fillets and of stress concentrations, determine whether the normal stresses in the beam satisfy a design specification that they be equal to or less than 150 MPa at section AA′.
SAMPLE
SAMPLE
At section AA′, we have
Referring to the table of Properties of RolledSteel Shapes in Appendix C, we obtain the data shown and then determine the stresses σ_{a} and σ_{b}.
At point a:
At point b:
We note that all normal stresses on the transverse plane are less than 150 MPa.
At point a:
At point b:
The state of stress at point b consists of the normal stress σ_{b} = 103.0 MPa and the shearing stress _{b} = 96.5 MPa. We draw Mohr's circle and find
For this beam and loading, the principal stress at point b is 36% larger than the normal stress at point a. For L ≥ 881 mm, the maximum normal stress would occur at point a.
SAMPLE
SAMPLE
The overhanging beam AB supports a uniformly distributed load of 3.2 kips/ft and a concentrated load of 20 kips at C. Knowing that for the grade of steel to be used σ_{all} = 24 ksi and _{all} = 14.5 ksi, select the wideflange shape that should be used.
SAMPLE
SAMPLE
We draw the freebody diagram of the beam. From the equilibrium equations ΣM_{D} = 0 and ΣM_{A} = 0 we find the values of R_{A} and R_{D} shown in the diagram.
Using the methods of Secs. 5.2 and 5.3, we draw the diagrams and observe that
For M_{max} = 2873 kip · in. and σ_{all} = 24 ksi, the minimum acceptable section modulus of the rolledsteel shape is
From the table of Properties of RolledSteel Shapes in Appendix C, we compile a list of the lightest shapes of a given depth that have a section modulus larger than S_{min}.
Shape  S (in^{3}) 

W24 × 68  154 
W21 × 62  127 
W18 × 76  146 
W16 × 77  134 
W14 × 82  123 
W12 × 96  131 
We now select the lightest shape available, namely
Since we are designing the beam, we will conservatively assume that the maximum shear is uniformly distributed over the web area of a W21 × 62. We write
We check that the maximum principal stress at point b in the critical section where M is maximum does not exceed σ_{all} = 24 ksi. We write
Conservatively,
We draw Mohr's circle and find
SAMPLE
SAMPLE
The solid shaft AB rotates at 480 rpm and transmits 30 kW from the motor M to machine tools connected to gears G and H; 20 kW is taken off at gear G and 10 kW at gear H. Knowing that _{all} = 50 MPa, determine the smallest permissible diameter for shaft AB.
SAMPLE
SAMPLE
Observing that f = 480 rpm = 8 Hz, we determine the torque exerted on gear E:
The corresponding tangential force acting on the gear is
A similar analysis of gears C and D yields
We now replace the forces on the gears by equivalent forcecouple systems.
By computing at all potentially critical sections, we find that its maximum value occurs just to the right of D:
For _{all} = 50 MPa, Eq. (7.32) yields
For a solid circular shaft of radius c, we have
SAMPLE
8.1  A W10 × 39 rolledsteel beam supports a load P as shown. Knowing that P = 45 kips, a = 10 in., and σ_{all} = 18 ksi, determine (a) the maximum value of the normal stress σ_{m} in the beam, (b) the maximum value of the principal stress σ_{max} at the junction of the flange and web, (c) whether the specified shape is acceptable as far as these two stresses are concerned.

8.2  Solve Prob. 8.1, assuming that P = 22.5 kips and a = 20 in. 
8.3  An overhanging W920 × 449 rolledsteel beam supports a load P as shown. Knowing that P = 700 kN, a = 2.5 m, and σ_{all} = 100 MPa, determine (a) the maximum value of the normal stress σ_{m} in the beam, (b) the maximum value of the principal stress σ_{max} at the junction of the flange and web, (c) whether the specified shape is acceptable as far as these two stresses are concerned.

8.4  Solve Prob. 8.3, assuming that P = 850 kN and a = 2.0 m. 
8.5  and 8.6(a) Knowing that σ_{all} = 24 ksi and _{all} = 14.5 ksi, select the most economical wideflange shape that should be used to support the loading shown. (b) Determine the values to be expected for σ_{m}, _{m}, and the principal stress σ_{max} at the junction of a flange and the web of the selected beam.

8.7  and 8.8(a) Knowing that σ_{all} = 160 MPa and _{all} = 100 MPa, select the most economical metric wideflange shape that should be used to support the loading shown. (b) Determine the values to be expected for σ_{m}, _{m}, and the principal stress σ_{max} at the junction of a flange and the web of the selected beam.

through 8.14Each of the following problems refers to a rolledsteel shape selected in a problem of Chap. 5 to support a given loading at a minimal cost while satisfying the requirement σ_{m} ≤ σ_{all}. For the selected design, determine (a) the actual value of σ_{m} in the beam, (b) the maximum value of the principal stress σ_{max} at the junction of a flange and the web.


8.15  The vertical force P_{1} and the horizontal force P_{2} are applied as shown to disks welded to the solid shaft AD. Knowing that the diameter of the shaft is 1.75 in. and that _{all} = 8 ksi, determine the largest permissible magnitude of the force P_{2}.

8.16  The two 500lb forces are vertical and the force P is parallel to the z axis. Knowing that _{all} = 8 ksi, determine the smallest permissible diameter of the solid shaft AE.

8.17  For the gearandshaft system and loading of Prob. 8.16, determine the smallest permissible diameter of shaft AE, knowing that the shaft is hollow and has an inner diameter that is the outer diameter. 
8.18  The 4kN force is parallel to the x axis, and the force Q is parallel to the z axis. The shaft AD is hollow. Knowing that the inner diameter is half the outer diameter and that _{all} = 60 MPa, determine the smallest permissible outer diameter of the shaft.

Neglecting the effect of fillets and of stress concentrations, determine the smallest permissible diameters of the solid rods BC and CD. Use _{all} = 60 MPa.


8.20  Knowing that rods BC and CD are of diameter 24 mm and 36 mm, respectively, determine the maximum shearing stress in each rod. Neglect the effect of fillets and of stress concentrations. 
8.21  It was stated in Sec. 8.3 that the shearing stresses produced in a shaft by the transverse loads are usually much smaller than those produced by the torques. In the preceding problems their effect was ignored and it was assumed that the maximum shearing stress in a given section occurred at point H (Fig. P8.21a) and was equal to the expression obtained in Eq. (8.5), namely,

8.22  Assuming that the magnitudes of the forces applied to disks A and C of Prob. 8.15 are, respectively, P_{1} = 1080 lb and P_{2} = 810 lb, and using the expressions given in Prob. 8.21, determine the values of _{H} and _{K} in a section (a) just to the left of B, (b) just to the left of C. 
8.23  The solid shafts ABC and DEF and the gears shown are used to transmit 20 hp from the motor M to a machine tool connected to shaft DEF. Knowing that the motor rotates at 240 rpm and that _{all} = 7.5 ksi, determine the smallest permissible diameter of (a) shaft ABC, (b) shaft DEF.

8.24  Solve Prob. 8.23, assuming that the motor rotates at 360 rpm. 
The solid shaft AB rotates at 360 rpm and transmits 20 kW from the motor M to machine tools connected to gears E and F. Knowing that _{all} = 45 MPa and assuming that 10 kW is taken off at each gear, determine the smallest permissible diameter of shaft AB.


8.26  Solve Prob. 8.25, assuming that the entire 20 kW is taken off at gear E. 
8.27  The solid shaft ABC and the gears shown are used to transmit 10 kW from the motor M to a machine tool connected to gear D. Knowing that the motor rotates at 240 rpm and that _{all} = 60 MPa, determine the smallest permissible diameter of shaft ABC.

8.28  Assuming that shaft ABC of Prob. 8.27 is hollow and has an outer diameter of 50 mm, determine the largest permissible inner diameter of the shaft. 
The solid shaft AE rotates at 600 rpm and transmits 60 hp from the motor M to machine tools connected to gears G and H. Knowing that _{all} = 8 ksi and that 40 hp is taken off at gear G and 20 hp is taken off at gear H, determine the smallest permissible diameter of shaft AE.


8.30  Solve Prob. 8.29, assuming that 30 hp is taken off at gear G and 30 hp is taken off at gear H. 