Question

A steel bar and an aluminum bar are bonded together to form the composite beam shown. The modulus of elasticity for aluminum is 70 GPa and for steel is 200 GPa. Knowing that the beam is bent about a horizontal axis by a couple of moment M = 1500 N·m, determine the maximum stress in (a) the aluminum, (b) the steel.

Fig. P11.31

Answer 1

Calculate the ratio of the composite members by using the equation.

Here, is the modulus of elasticity of steel, and is the modulus of elasticity of aluminium.

Substitute 200 GPa for and 70 GPa for .

Calculate the resultant width of the steel section.

Draw the transformed section of the composite beam.

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Calculate the value of for both the components and tabulate them.

Components	Area (mm2)	Centroidal Distance (mm)
Top Flange
Web
Total	2910

Calculate the centroidal distance by using the equation.

Substitute for and .

Consider the section with the centroidal distance.

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Calculate the moment of inertia of steel bar about horizontal axis.

Substitute 85.5 mm for , 20 mm for , for, for, and 37.62 mm for.

Calculate the moment of aluminium of steel bar about horizontal axis.

Substitute 30 mm for , 40 mm for , for, for, and 37.62 mm for.

Calculate of moment of inertia by using the equation.

Here, is moment of inertia of steel, and is moment of inertia of aluminium.

Substitute for and for .

(a)

Calculate the maximum stress in aluminium section by using the equation.

Here, M is the moment applied, is the distance of centroid for aluminium section, and is the moment of inertia.

Substitute for M and 851639.4 for.

Therefore, the maximum stress in the aluminium is.

(b)

Calculate the maximum stress in steel section by using the equation.

Substitute for M and 851639.4 for.

Stress in the original steel section will be larger than transformed section due to increase of width to n times.

Calculate the stress in the actual steel member by using the equation.

Substitute -39.41for and 2.85 for n.

Therefore, the stress in the actual steel member is.

Answer 2