Total Longitudinal Strain Of A Rectangular Steel Block Under Combined Stress

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Total Longitudinal Strain Of A Rectangular Steel Block Under Combined Stress

The dimensions of a rectangular steel block are:

Length = 12 in, height = 4 in, thickness = 2 in.

Member is subjected to the following stresses:

Longitudinal tensile stress σ_{x} = 12,000 lb/in^{2}

Vertical compressive stress σ_{y} = 15,000 lb/in^{2}

Lateral compressive stress σ_{x} = 9,000 lb/in^{2}

Poisson ratio μ, = 0.30.

Modulus of Elasticity of Steel E, = 30, 000,000.

Determine total longitudinal strain (i.e, total change in length).

**The strings**:
S_{7}P_{5}A_{51} (Physical Change).
**The math**:

Pj Problem of Interest is of type *change* (Physical Change).

Assumptions: Principles of *consistent deformation* and *superposition* are applicable. Elasticity holds.
*Combined stress* are generally of two types:

(1) Combination of tensile and commpressive stresses (usually referred to as biaxial or triaxial stress).

(2) Combination of tensie, compressive and shear stresses (usually referred to as combined stress).

Equations of Interest:

Poisson ratio μ = (lateral unit strain)/(longitudinal unit strain)--------(1)

Strain = stress/Modulus of Elasticity = σ/E---------(2)

Total longitudinal strain = (length of member)x(sum of unit strains due to individual stresses)

(a) So, Total longitudinal strain = 12[12000/(30x10^{6}) + 0.30(1500)/(30x10^{6}) + 0.30(9000)/(30x10^{6})]

= 12(+ 0.000400 + 0.000150 + 0.000090) = 12(0.00064) = 0.00768 in.

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The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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