4.14^4.14. GENERAL CASE OF ECCENTRIC AXIAL LOADING^284^293^,,^6141^6367%
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4.14
GENERAL CASE OF ECCENTRIC AXIAL LOADING

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 My and Mz of moments My = Pa and Mz = Pb represented in Fig. 4.64b. Similarly, the eccentric force P′ is equivalent to the centric force P′ and the couples My and Mz.

Fig. 4.64 Eccentric axial loading.

By virtue of Saint-Venant'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 My and Mz, 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,

(4.58)

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 σx from Eq. (4.58), care should be taken to correctly determine the sign of each of the three terms in the right-hand 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 σx 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 σx = 0 in Eq. (4.58), we obtain the equation of a straight line, which represents the neutral axis of the section:

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EXAMPLE 4.09

A vertical 4.80-kN 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.

Fig. 4.65
(a)
Stresses.

The given eccentric load is replaced by an equivalent system consisting of a centric load P and two couples Mx and Mz represented by vectors directed along the principal centroidal axes of the section (Fig. 4.66). We have

Fig. 4.66

We also compute the area and the centroidal moments of inertia of the cross section:

The stress σ0 due to the centric load P is negative and uniform across the section. We have

The stresses due to the bending couples Mx and Mz 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 Mx are positive at C and D, and negative at A and B, and that the stresses due to Mz are positive at B and C, and negative at A and D, we obtain

 
Fig. 4.67
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(b)
Neutral Axis.

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).

Fig. 4.68

The distribution of the stresses across the section is shown in Fig. 4.69.

Fig. 4.69
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SAMPLE

SAMPLE PROBLEM 4.9

A horizontal load P is applied as shown to a short section of an S10 × 25.4 rolled-steel member. Knowing that the compressive stress in the member is not to exceed 12 ksi, determine the largest permissible load P.

SAMPLE

SAMPLE

SOLUTION
Properties of Cross Section.

The following data are taken from Appendix C.

Force and Couple at C.

We replace P by an equivalent force-couple system at the centroid C of the cross section.

Note that the couple vectors Mx and My are directed along the principal axes of the cross section.

Normal Stresses.

The absolute values of the stresses at points A, B, D, and E due, respectively, to the centric load P and to the couples Mx and My are

Superposition.

The total stress at each point is found by superposing the stresses due to P, Mx, and My. We determine the sign of each stress by carefully examining the sketch of the force-couple system.

Largest Permissible Load.

The maximum compressive stress occurs at point E. Recalling that σall = −12 ksi, we write

SAMPLE

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*SAMPLE PROBLEM 4.10

A couple of magnitude M0 = 1.5 kN · m acting in a vertical plane is applied to a beam having the Z-shaped 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

SOLUTION
Principal Axes.

We draw Mohr's circle and determine the orientation of the principal axes and the corresponding principal moments of inertia.

Loading.

The applied couple M0 is resolved into components parallel to the principal axes.

a. Stress at A.

The perpendicular distances from each principal axis to point A are

Considering separately the bending about each principal axis, we note that Mu produces a tensile stress at point A while Mv produces a compressive stress at the same point.

b. Neutral Axis.

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

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PROBLEMS
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.
Fig. P4.127
Fig. P4.128
Fig. P4.129
Fig. P4.130
Fig. P4.131
Fig. P4.132
Fig. P4.133
Fig. P4.134
 
4.135
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.
Fig. P4.135
Fig. P4.136
Fig. P4.137
Fig. P4.138
Fig. P4.139
Fig. P4.140
 
*4.141
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.
Fig. P4.141
Fig. P4.142
Fig. P4.143
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.
Fig. P4.144
4.145 Solve Prob. 4.144, assuming that the 28-kN force at point E is removed.
4.146 A rigid circular plate of 125-mm radius is attached to a solid 150 × 200-mm rectangular post, with the center of the plate directly above the center of the post. If a 4-kN 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.
Fig. P4.146
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.
 
4.148
Knowing that P = 90 kips, determine the largest distance a for which the maximum compressive stress does not exceed 18 ksi.
Fig. P4.148 and P4.149
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 M0 acting in a vertical plane. Determine the largest permissible value of the moment M0 of the couple if the maximum stress is not to exceed 80 MPa. Given: Imax = 2.28 × 10−6 m4, Imin = 0.23 × 10−6 m4, principal axes 25.7° and 64.3° .
Fig. P4.150
4.151 Solve Prob. 4.150, assuming that the couple M0 acts in a horizontal plane.
4.152 A beam having the cross section shown is subjected to a couple M0 that acts in a vertical plane. Determine the largest permissible value of the moment M0 of the couple if the maximum stress in the beam is not to exceed 12 ksi. Given: Iy = Iz = 11.3 in4, A = 4.75 in2, kmin = 0.983 in. (Hint: By reason of symmetry, the principal axes form an angle of 45° with the coordinate axes. Use the relations Imin = Ak2min and Imin + Imax = Iy + Iz.)
Fig. P4.152
4.153 Solve Prob. 4.152, assuming that the couple M0 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 M0 of the couple if the maximum stress is not to exceed 12 ksi. Given: Imax = 0.957 in4, Imin = 0.427 in4, principal axes 29.4° and 60.6°
Fig. P4.154
4.155 A couple M0 acting in a vertical plane is applied to a W12 × 16 rolled-steel beam, whose web forms an angle θ with the vertical. Denoting by σ0 the maximum stress in the beam when θ = 0, determine the angle of inclination θ of the beam for which the maximum stress is 2σ0.
Fig. P4.155
 
4.156
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.
Fig. P4.156
4.157 A beam of unsymmetric cross section is subjected to a couple M0 acting in the horizontal plane xz. Show that the stress at point A, of coordinates y and z, is
where Iy, Iz, and Iyz denote the moments and product of inertia of the cross section with respect to the coordinate axes, and My the moment of the couple.
Fig. P4.157 and P4.158
4.158 A beam of unsymmetric cross section is subjected to a couple M0 acting in the vertical plane xy. Show that the stress at point A, of coordinates y and z, is
where Iy, Iz, and Iyz denote the moments and product of inertia of the cross section with respect to the coordinate axes, and Mz the moment of the couple.
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
where rz and rx denote the radius of gyration of the cross section with respect to the z axis and the x axis, respectively. (b) Further show that, if a vertical force Q is applied at any point located on line BD, the stress at point A will be zero.
Fig. P4.159
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
(b) Further show that, if no tensile stress is to occur in the member, the force P must be applied at a point located within the area bounded by the line found in part a and three similar lines corresponding to the condition of zero stress at B, C, and D, respectively. This area, shown in Fig. P4.160b, is known as the kern of the cross section.
Fig. P4.160
†See Ferdinand F. Beer and E. Russell Johnston, Jr., Mechanics for Engineers, 5th ed., McGraw-Hill, New York, 2008, or Vector Mechanics for Engineers–9th ed., McGraw-Hill, New York, 2010, Secs. 9.8–9.10.