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Step 1) Calculate each part's L (I = bt? /12) Step 2) Find section's neutral axis. y(w=lAy/Zyi=170/34 = 5.00 Step 3) Calculate I for axis x-x. l„ =1,1^+1,Ay2 =1325.34 Step 4) Equate I at the neutral axis. IN4 = Iwt—)4(y2'M)

= 1325.34 —^34) (25) = 475.34 cm2 Fig. 11-33. Method tor Finding Neutral Axis and Moment for Inertia for Complex Sections.

Step 1) Calculate each part's L (I = bt? /12) Step 2) Find section's neutral axis. y(w=lAy/Zyi=170/34 = 5.00 Step 3) Calculate I for axis x-x. l„ =1,1^+1,Ay2 =1325.34 Step 4) Equate I at the neutral axis. IN4 = Iwt—)4(y2'M)

= 1325.34 —^34) (25) = 475.34 cm2 Fig. 11-33. Method tor Finding Neutral Axis and Moment for Inertia for Complex Sections.

cross-sections remain planes after bending so that stresses will increase linearly away from the neutral axis. To predict stress values above the material's proportional limit, we would use inelastic methods [Bruhn, 1973].

When a column under axial compression suddenly deflects laterally, or bows, we say that it buckles. Such an occurrence is usually catastrophic. Theoretically, a linear-elastic column in compression will buckle at a critical, or Euler buckling load, Pcr, given by

Per(11-49)

where L' is an effective length, dependent on the column's end conditions as shown in Fig. 11-34. This equation applies only if the axial stress at buckling (P^ /A) does not exceed the material's proportional limit Otherwise, we would replace E in this equation with Et, the tangent modulus, which is the slope of the stress/strain curve at the operating stress level (the buckling stress, in this case). Premature column buckling can also occur as a result of imperfect geometry and local buckling of flanges or webs in the column. See Sarafin [1995] or Bnihn [1973] for details.

Free

Fixed wwkw r

S7 Pinned

Pinned i

V Pinned

Fixed W\V\W

Fig. 11 -34. Effective Lengths for Columns with Different End Conditions. The square of a column's effective length, /.', Is Inversely proportional to the force that would cause the column to buckle elasticaliy. Conceptually, the effective length is the length over which the buckled shape would approximate that of a buckled column with pinned ends (center figure). For example. If the cantilevered column shown at left were to buckle, Its free end would deflect laterally, while Its fixed end would not translate or rotate. This shape is the same as half the shape of a buckled plnned-end column, so i/for the cantilever Is ZL

The elastic buckling stress, acn for curved skin panels in compression is given as lai2E ftY acr~l2(l-v2)[b) (H-50)

where t is panel thickness, b is panel width, v is Poisson's ratio (Eq. 11-41) and k is a geometric coefficient. Figure 11 -35 graphs values of k for curved panels where r is the radius of curvature and is used to compute the independent variable on the graph.

We can quickly evaluate combined axial, lateral, and bending loads on a thin-wall cylinder using the equivalent axial load, Peq (Fig. 11-36):

where M and R are defined on the figure.

The basis for Peq is that bending stress will be greatest at the two points farthest from the cylinder's neutral axis (one point in tension, the other in compression). Because lateral and bending loads can usually come from any direction (wind or drag), this peak stress can occur at any point Therefore, we must size the cylinder for the load that would create this peak stress along the cylinder's circumference. Peq is an axial load on a cylinder that would result in a uniform stress equal to a peak stress created by a combination of an axial load and bending moment

1,000

f 100

0 0

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