Gage length over which φ is developed distance

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gage length over which φ is developed (distance between collars of troptometer) L'- fiber length after deformation τ - shearing stress at extreme fibers G - modulus of rigidity (modulus of elasticity in shear) c - outside radius of cylindrical test piece r - inside radius of tubular test piece Figure 6.2 Tinius Olsen 10k Torsional Machine
53 The following formulae hold true only for cylindrical specimens: γ = cφ/L τ = cT/J (Torsion Formula) J = πc 4 /2 (for cylindrical bar) J = π(c 4 - r 4 )/2 (for tubular sections, general) J = 2πc 3 t (for thin wall, t = thickness of tube wall) G = TL/Jφ = τ / γ Percent of ductility = [(L' - L)/L]100 L' 2 = (c φ ) 2 + L 2 There is a theoretical relationship between the modulus of rigidity (G) and the tensile modulus of elasticity (E) for homogeneous, isotropic materials, G = 0.4 E. Experimental testing of steel shows this relationship to hold true. For a ductile steel the shearing proportional limit is approximately 60% of the tensile proportional limit. Ductility in torsion is determined by comparing the final fiber length (L') at rupture to the original fiber length or gage length (L). The ductility is expressed as percentage of elongation. The modulus of rupture in torsion is understood to be the nominal extreme fiber stress at rupture as determined by the torsion formula. This value is actually larger than the true maximum stress. An approximate correction to the value determined for the modulus of rupture in torsion for ductile materials is 0.75. For a ductile material the ratio of shear strength to tensile strength is approximately 0.8. The torsion test is not applicable to determining the shearing strength of brittle materials, such as cast iron, since the specimen would fail in diagonal tension before the shearing strength was reached. For further information see Mechanics of Materials by R. C. Hibbeler or another strength of materials book. Procedure 1. With a micrometer caliper determine the mean diameter of the steel specimen near its mid-length. Assume the shearing proportional limit is 0.6 of the tensile proportional limit and the shearing modulus of rupture as equal to the tensile strength. Check with instructor to ascertain tensile strength of specimen. 2. Compute loading increments that will give at least 10 observations below the shearing proportional limit, several close together near the proportional limit, and at least 10 beyond the proportional limit. 3. Note the gage length and least reading of the troptometer.
54 4. Securely clamp the troptometer to the specimen. Note: Make certain that the axes of the troptometer and the test piece coincide and that the troptometer is in proper position for ease of reading. Check to ensure that troptometer has a gage length of approximately 8", record L. 5. Insert the specimen into the two heads, and tighten the chuck grips.

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