Strain Energy Stored In An Elastic Material

**Strings (S _{i}P_{j}A_{jk}) = S_{7}P_{3}A_{31} Base Sequence = 12735 String Sequence = 12735 - 3 - 31 **

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Strain Energy Stored In An Elastic Material

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A 15 in member is to be designed using a safety factor of 1.50, to withstand a tensile load of 6000 lb. The three choices of material available are:

(a) Aluminum Alloy, for which modulus of Elasticity, E = 10 x 10^{6} psi and tensile stress σ_{y} = 52, 0000 psi

(b) Magnesium Alloy, for which modulus of Elasticity, E = 6.5 x 10^{6} psi and tensile stress σ_{y} = 28,500 psi

(c) Molded Nylon, for which modulus of Elasticity, E = 410,000 psi and tensile stress σ_{y} = 8000 psi.

Calculate the total amount of strain energy stored by each member at the 6000 lb load.

**The strings**:
S_{7}P_{3}A_{31} (Force - Pull).
**The math**:

Pj Problem of Interest is of type *force* (pull).

Formulas of interest:

Stress, σ = Load/Area; Strain, ε = δ/Length; where δ is total elongation

Stress = (modulus of elasticity)(strain)

Poisson's ratio, μ = lateral strain/longitudinal strain

Offset yield strength σ_{y} = (working stress σ_{w})(safety factor)

Total strain energy, U = ∫_{0}^{δ} P(x) dx.

Where x is elongation and P(x) = kx is force as a function of elongation.

Strain energy per unit volume = σ^{2}/(2E)

(a) yield strength, σ_{y} = [Load(safety factor)]/(Cross-section Area)

So, 52,000 = 6,000(1.50)/A

So, A = 6,000(1.5)/52,000 = 0.173 in^{2}.

Total strain energy, U at Load can be calculated by integrating P(x) = kx from 0 to δ

Where k, (load/δ) is the slope of the graph relating load and elongation when material is elastic.

So, U = ∫_{0}^{δ} kx dx = (Load)(δ/2) = (Load)[(ε)(length]/2.

So, U = 6,000[(σ_{y})/(1.5E)](length/2)

So, total strain energy, U = 6,000[(52,000/(1.5 x 10 x 10^{6})](15/2)

= 156 lb-in.

(b) A = 6,000(1.5)/28,500 = 0.316 in^{2}.

So, total strain energy, U = 6,000[(28,500/(1.5 x 6.5 x 10^{6})](15/2)

= 132 lb-in.

(c) A = 6,000(1.5)/8000 = 1.125 in^{2}

So, total strain energy, U = 6,000[(8,000/(1.5 x 410,000)](15/2)

= 586 lb-in.

Total strain energy can also be calculated by first calculatin strain energy per unit volume and then multiplying by total volume.

So, total strain energy = [σ^{2}/(2E)]V.

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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