Example

Derive the load-deflection equation for the axially loaded member shown in Figure 1.7. The member is made from a material with a uniaxial stress-strain relationship described by the equation e = B(alBnE0)n, where B and n are material constants and E0 is the initial slope of the stress-strain curve (i.e., the slope at a = 0).

The deflection (which for this problem is equal to the elongation) of the axially loaded member can be obtained by integrating the strain over the length of the member; that is,

1.4.3 Inelastic Behavior

For structures subject to uniaxial loading, inelastic behavior occurs once the stress in the structure exceeds the yield stress, sy, of the material. The yield stress is defined as the stress beyond which inelastic or permanent strain is induced, as shown in Figure 1.6. While some materials (e.g., structural steel) exhibit a definitive yield point on the uniaxial stress-strain curve, others do not. For such cases, the yield stress is often determined graphically using the 0.2% offset method. In this method, a line parallel to the initial slope of the uniaxial stress-strain curve is drawn from the 0.2% strain point. The 0.2% yield stress is obtained as the stress at which this line intersects the stress-strain curve.

For structures subject to biaxial or triaxial loading, inelastic behavior is assumed to occur when some combined stress state reaches a yield envelope (for a 2-D problem) or a yield surface (for a 3-D problem). Mathematically, the yield condition can be expressed as n dx dx f (ff,j, ki, k2, k3,...) =0

where k1, k2, k3,... are (experimentally determined) material constants.

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