In Sec. 4.12 you analyzed the stresses produced in a member by an eccentric axial load applied in a plane of symmetry of the member. You will now study the more general case when the axial load is not applied in a plane of symmetry.
Consider a straight member AB subjected to equal and opposite eccentric axial forces P and P′ (Fig. 4.64a), and let a and b denote the distances from the line of action of the forces to the principal centroidal axes of the cross section of the member. The eccentric force P is statically equivalent to the system consisting of a centric force P and of the two couples M_{y} and M_{z} of moments M_{y} = Pa and M_{z} = Pb represented in Fig. 4.64b. Similarly, the eccentric force P′ is equivalent to the centric force P′ and the couples M′_{y} and M′_{z}.
By virtue of SaintVenant's principle (Sec. 2.17), we can replace the original loading of Fig. 4.64a by the statically equivalent loading of Fig. 4.64b in order to determine the distribution of stresses in a section S of the member, as long as that section is not too close to either end of the member. Furthermore, the stresses due to the loading of Fig. 4.64b can be obtained by superposing the stresses corresponding to the centric axial load P and to the bending couples M_{y} and M_{z}, as long as the conditions of applicability of the principle of superposition are satisfied (Sec. 2.12). The stresses due to the centric load P are given by Eq. (1.5), and the stresses due to the bending couples by Eq. (4.55), since the corresponding couple vectors are directed along the principal centroidal axes of the section. We write, therefore,
where y and z are measured from the principal centroidal axes of the section. The relation obtained shows that the distribution of stresses across the section is linear.
In computing the combined stress σ_{x} from Eq. (4.58), care should be taken to correctly determine the sign of each of the three terms in the righthand member, since each of these terms can be positive or negative, depending upon the sense of the loads P and P′ and the location of their line of action with respect to the principal centroidal axes of the cross section. Depending upon the geometry of the cross section and the location of the line of action of P and P′, the combined stresses σ_{x} obtained from Eq. (4.58) at various points of the section may all have the same sign, or some may be positive and others negative. In the latter case, there will be a line in the section, along which the stresses are zero. Setting σ_{x} = 0 in Eq. (4.58), we obtain the equation of a straight line, which represents the neutral axis of the section:
A vertical 4.80kN load is applied as shown on a wooden post of rectangular cross section, 80 by 120 mm (Fig. 4.65). (a) Determine the stress at points A, B, C, and D. (b) Locate the neutral axis of the cross section.
The given eccentric load is replaced by an equivalent system consisting of a centric load P and two couples M_{x} and M_{z} represented by vectors directed along the principal centroidal axes of the section (Fig. 4.66). We have
We also compute the area and the centroidal moments of inertia of the cross section:
The stress σ_{0} due to the centric load P is negative and uniform across the section. We have
The stresses due to the bending couples M_{x} and M_{z} are linearly distributed across the section, with maximum values equal, respectively, to
The stresses at the corners of the section are
where the signs must be determined from Fig. 4.66. Noting that the stresses due to M_{x} are positive at C and D, and negative at A and B, and that the stresses due to M_{z} are positive at B and C, and negative at A and D, we obtain
We note that the stress will be zero at a point G between B and C, and at a point H between D and A (Fig. 4.67). Since the stress distribution is linear, we write
The neutral axis can be drawn through points G and H (Fig. 4.68).
The distribution of the stresses across the section is shown in Fig. 4.69.
SAMPLE
A horizontal load P is applied as shown to a short section of an S10 × 25.4 rolledsteel member. Knowing that the compressive stress in the member is not to exceed 12 ksi, determine the largest permissible load P.
SAMPLE
SAMPLE
The following data are taken from Appendix C.
We replace P by an equivalent forcecouple system at the centroid C of the cross section.
Note that the couple vectors M_{x} and M_{y} are directed along the principal axes of the cross section.
The absolute values of the stresses at points A, B, D, and E due, respectively, to the centric load P and to the couples M_{x} and M_{y} are
The total stress at each point is found by superposing the stresses due to P, M_{x}, and M_{y}. We determine the sign of each stress by carefully examining the sketch of the forcecouple system.
The maximum compressive stress occurs at point E. Recalling that σ_{all} = −12 ksi, we write
SAMPLE
A couple of magnitude M_{0} = 1.5 kN · m acting in a vertical plane is applied to a beam having the Zshaped cross section shown. Determine (a) the stress at point A, (b) the angle that the neutral axis forms with the horizontal plane. The moments and product of inertia of the section with respect to the y and z axes have been computed and are as follows:
SAMPLE
We draw Mohr's circle and determine the orientation of the principal axes and the corresponding principal moments of inertia.^{†}
The applied couple M_{0} is resolved into components parallel to the principal axes.
The perpendicular distances from each principal axis to point A are
Considering separately the bending about each principal axis, we note that M_{u} produces a tensile stress at point A while M_{v} produces a compressive stress at the same point.
Using Eq. (4.57), we find the angle ϕ that the neutral axis forms with the v axis.
The angle β formed by the neutral axis and the horizontal is
SAMPLE
4.127  through 4.134The couple M is applied to a beam of the cross section shown in a plane forming an angle β with the vertical. Determine the stress at (a) point A, (b) point B, (c) point D.

through 4.140The couple M acts in a vertical plane and is applied to a beam oriented as shown. Determine (a) the angle that the neutral axis forms with the horizontal, (b) the maximum tensile stress in the beam.


through *4.143The couple M acts in a vertical plane and is applied to a beam oriented as shown. Determine the stress at point A.


4.144  The tube shown has a uniform wall thickness of 12 mm. For the loading given, determine (a) the stress at points A and B, (b) the point where the neutral axis intersects line ABD.

4.145  Solve Prob. 4.144, assuming that the 28kN force at point E is removed. 
4.146  A rigid circular plate of 125mm radius is attached to a solid 150 × 200mm rectangular post, with the center of the plate directly above the center of the post. If a 4kN force P is applied at E with θ = 30°, determine (a) the stress at point A, (b) the stress at point B, (c) the point where the neutral axis intersects line ABD.

4.147  In Prob. 4.146, determine (a) the value of θ for which the stress at D reaches its largest value, (b) the corresponding values of the stress at A, B, C, and D. 
Knowing that P = 90 kips, determine the largest distance a for which the maximum compressive stress does not exceed 18 ksi.


4.149  Knowing that a = 1.25 in., determine the largest value of P that can be applied without exceeding either of the following allowable stresses:

4.150  The Z section shown is subjected to a couple M_{0} acting in a vertical plane. Determine the largest permissible value of the moment M_{0} of the couple if the maximum stress is not to exceed 80 MPa. Given: I_{max} = 2.28 × 10^{−6} m^{4}, I_{min} = 0.23 × 10^{−6} m^{4}, principal axes 25.7° and 64.3° .

4.151  Solve Prob. 4.150, assuming that the couple M_{0} acts in a horizontal plane. 
4.152  A beam having the cross section shown is subjected to a couple M_{0} that acts in a vertical plane. Determine the largest permissible value of the moment M_{0} of the couple if the maximum stress in the beam is not to exceed 12 ksi. Given: I_{y} = I_{z} = 11.3 in^{4}, A = 4.75 in^{2}, k_{min} = 0.983 in. (Hint: By reason of symmetry, the principal axes form an angle of 45° with the coordinate axes. Use the relations I_{min} = Ak^{2}_{min} and I_{min} + I_{max} = I_{y} + I_{z}.)

4.153  Solve Prob. 4.152, assuming that the couple M_{0} acts in a horizontal plane. 
4.154  An extruded aluminum member having the cross section shown is subjected to a couple acting in a vertical plane. Determine the largest permissible value of the moment M_{0} of the couple if the maximum stress is not to exceed 12 ksi. Given: I_{max} = 0.957 in^{4}, I_{min} = 0.427 in^{4}, principal axes 29.4° and 60.6°

4.155  A couple M_{0} acting in a vertical plane is applied to a W12 × 16 rolledsteel beam, whose web forms an angle θ with the vertical. Denoting by σ_{0} the maximum stress in the beam when θ = 0, determine the angle of inclination θ of the beam for which the maximum stress is 2σ_{0}.

Show that, if a solid rectangular beam is bent by a couple applied in a plane containing one diagonal of a rectangular cross section, the neutral axis will lie along the other diagonal.


4.157  A beam of unsymmetric cross section is subjected to a couple M_{0} acting in the horizontal plane xz. Show that the stress at point A, of coordinates y and z, is

4.158  A beam of unsymmetric cross section is subjected to a couple M_{0} acting in the vertical plane xy. Show that the stress at point A, of coordinates y and z, is

4.159  (a) Show that, if a vertical force P is applied at point A of the section shown, the equation of the neutral axis BD is

4.160  (a) Show that the stress at corner A of the prismatic member shown in Fig. P4.160a will be zero if the vertical force P is applied at a point located on the line
