7 dx

Note: Ab = 0.9 AB

Figure 8.32e. Idealized column cross-sectional areas.

Figure 8.32f. Axial shortening of columns; closed-form solution: (1) axial shortening Az; (2) column area; (3) column axial load; (4) unit load at height z; (5) axial strain; (6) axial displacement.
Figure 8.32g. Example 1, column-shortening calculations: (1) axial load variation; (2) actual and assumed variation in column cross-sectional areas.

Evaluating the integrals we get the following final expression for Ar

Ab j

Example 1.

Height of building: L = 682 ft = 8184 in. (207.8 m)

Modulus of elasticity: E = 29,000 ksi (200 x 103 MPa)

Axial load at base: Pb = 2770 kips (12.32 x 103 kN)

Actual column area at base: AB = 147 in2. (94.84 x 103 mm2)

Reduced column area at base: Ab = 0.9 x 147 = 133.3 in2. (86.0 x 103 mm2)

Required. Axial shortening of column at top.

Solution. Since column shortening is calculated at top, z = L.

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