## 321 Unsymmetrical Bending

When the plane of loads acting transversely on a beam does not contain any of the beam's axes of symmetry, the loads may tend to produce twisting as well as bending. Figure 3.53

shows a horizontal channel twisting even though the vertical load H acts through the centroid of the section.

The bending axis of a beam is the longitudinal line through which transverse loads should pass to preclude twisting as the beam bends. The shear center for any section of the beam is the point in the section through which the bending axis passes.

For sections having two axes of symmetry, the shear center is also the centroid of the section. If a section has an axis of symmetry, the shear center is located on that axis but may not be at the centroid of the section. FIGURE 3.53 Twisting of a channel. Figure 3.54 shows a channel section in which the horizontal axis is the axis of sym-

metry. Point O represents the shear center. It lies on the horizontal axis but not at the centroid C. A load at the section must pass through the shear center if twisting of the member is not to occur. The location of the shear center relative to the center of the web can be obtained from b/2

where b = width of flange overhang

Af = tfb = area of flange overhang Aw = tw h = web area

(F. Bleich, Buckling Strength of Metal Structures, McGraw-Hill, Inc., New York.) For a member with an unsymmetrical cross section subject to combined axial load and biaxial bending, Eq. (3.86) must be modified to include the effects of unsymmetrical bending. In this case, stress in the elastic range is given by p + My - M(Ix ix + Mx - My(y Iy)

y xy x xy x xy y xy where A = cross-sectional area Mx, My = bending moment about x-x and y-y axes Ix, Iy = moment of inertia about x-x and y-y axes x, y = distance of stress point under consideration from y-y and x-x axes I^ = product of inertia

Moments Mx and My may be caused by transverse loads or eccentricities of axial loads. An example of the latter case is shown in Fig. 3.55. For an axial load P, Mx = Pex and My = Pey, where ex and ey are eccentricities with respect to the x-x and y-y axes, respectively.

To show an application of Eq. (3.88) to an unsymmetrical section, stresses in the lintel angle in Fig. 3.56 will be calculated for Mx = 200 in-kips, My = 0, and P = 0. The centroidal axes x-x and y-y are 2.6 and 1.1 in from the bottom and left side, respectively, as shown in Fig. 3.56. The moments of inertia are Ix = 47.82 in4 and Iy = 11.23 in4. The product of inertia can be calculated by dividing the angle into two rectangular parts and then applying Eq. (3.89):

where A1 and A2 = cross-sectional areas of parts 1 and 2

x1 and x2 = horizontal distance from the angle's centroid to the centroid of parts 1 and 2

y1 and y2 = vertical distance from the angle's centroid to the centroid of parts 1 and 2

Substitution in Eq. (3.88) gives

This equation indicates that the maximum stresses normal to the cross section occur at the corners of the angle. A maximum compressive stress of 25.43 ksi occurs at the upper right corner, where x = -0.1 and y = 4.4. A maximum tensile stress of 22.72 ksi occurs at the lower left corner, where x = -1.1 and y = -2.6.

(I. H. Shames, Mechanics of Deformable Solids, Prentice-Hall, Inc., Englewood Cliffs, N.J.; F.R. Shanley, Strength of Materials, McGraw-Hill, Inc., New York.)

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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