Explanation Given information: The effective length of the column is L e = 2.2 m . The magnitude of axial load is P = 41 kN . The allowable stress for compression is σ C = 8.1 MPa . The depth of the cross section is d = 190 mm . The modulus of elasticity of the material is E = 8.3 GPa . The eccentricity of the load in the column is e = 80 mm . Calculation: The parameter c for sawn lumber columns is 0.8. Find the cross sectional area ( A ) of the rectangular cross section using the equation. A = b d Here, the width of the cross section is b and the depth of the cross section is d . Substitute 190 mm for d . A = b × 190 = 190 b Determine the moment of inertia ( I ) of the cross section using the equation. I = b d 3 12 Substitute 190 mm for d . I = b × 190 3 12 = 571 , 583.333 b Find the distance between the neutral axis and the distance between the extreme fibre ( c ) using the relation. c = d 2 Substitute 190 mm for d . c = 190 2 = 95 mm Find the ratio of the effective length to the width ( L b ) ; L b = 2.2 m × 1,000 mm 1 m b = 2 , 200 b Find the stress ( σ C E ) using the equation. σ C E = 0.3 E ( L b ) 2 Substitute 8.3 GPa for E and 2 , 200 b for L b . σ C E = 0.3 × 8.3 GPa × 1,000 MPa 1 GPa ( 2 , 200 b ) 2 = 5.14463 × 10 − 4 b 2 Find the column stability factor C p using the Equation. C p = 1 + ( σ C E / σ C ) 2 c − [ 1 + ( σ C E / σ C ) 2 c ] 2 − σ C E / σ C c Here, the allowable stress for compression grain is σ C . Substitute 5.14463 × 10 − 4 b 2 for σ C E , 8.1 MPa for σ C , and 0.8 for c . C p = 1 + ( 5.14463 × 10 − 4 b 2 8.3 ) 2 × 0.8 − [ 1 + ( 5.14463 × 10 − 4 b 2 8.3 ) 2 × 0.8 ] 2 − ( 5.14463 × 10 − 4 b 2 8.3 ) 0.8 Calculate the allowable stress ( σ a l l ) using the relation. σ all = σ C C p Substitute 8

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Author: BEER

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Axial Load

When a bar of the uniform cross-section is acted by a load, such that the load acts along the longitudinal axis of the bar, such kinds of loadings are known as axial loadings. In general terms, a load is an external force acting on a component. An axial …

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Chapter 10.4, Problem 107P

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Find the width of the lightest rectangular cross section.

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Chapter 10.4, Problem 106P

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Find the width of the lightest rectangular cross section.

Solution Summary: The author calculates the cross sectional area ( A ) of the rectangular cross-section using the equation.

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