# Beams with Different Materials

From assumption no. (3) in the previous page: The strains of any two adjacent materials at their junction point are equal.

From assumption no. (3) in the previous page: The strains of any two adjacent materials at their junction point are equal.

Flexure formula do not apply directly to composite beams because it was based on the assumption that the beam was homogeneous. It is therefore necessary to transform the composite material into equivalent homogeneous section. To do this, consider a steel and wood section to be firmly bolted together so that they can act as one unit. Shown below are the composite wood and steel section and the corresponding equivalent in wood and steel sections.

The quantity n is the ratio of the moduli of elasticity of stronger material to the weaker material. In the above case, n = E_{s} / E_{w}.

**Problem**

The diameter of a sphere is 18 in. Find the largest volume of regular pyramid of altitude 15 in. that can be cut from the sphere if the pyramid is (a) square, (b) pentagonal, (c) hexagonal, and (d) octagonal.

**Problem**

A closed cylindrical tank measures 12 ft. long and 5 ft. in diameter. It has to contain water to a depth of 3 ft when lying in the horizontal position. Find the depth of water when it is in vertical position.

**Problem 1**

A circle is tangent to the line 2x - y + 1 = 0 at the point (2, 5) and the center is on the line x + y = 9. Find the equation of the circle.

**Definition of circle**

The locus of point that moves such that its distance from a fixed point called the center is constant. The constant distance is called the radius, r of the circle.

**Problem 1**

A closed cylindrical vessel 3 m. in diameter and 6 m high is filled with water to a height of 4.5 m. The rest is filled with air, the pressure of which is 105 kPa. If the vessel is rotated at 191 rpm about its axis, determine the maximum and minimum inside pressure at the base.

**Problem 31**

One section of the mainline sewer pipe for the city of Annapolis passes through a tunnel 484 ft. long. The cross section of the tunnel is shown in the figure. The sewer pipe is 16 in. in external diameter and is partially embedded, in a mattress of concrete, a cross section of which is shown in the figure. Find the total cost (a) of digging the tunnel at $5.00 per cubic yard, (b) of pouring of concrete at $9.00 per cubic yard.

**Problem 30**

Window glass formerly was made by gathering a lump of molten glass on the end of a hollow rod and blowing it into the form of large hollow cylinder about 6 ft. long and 1-1/2 ft. in diameter. This was cut longitudinally and then placed in an oven and heated until it softened, when it was flattened out into plates and cut into desired sizes. If this plate was 0.12 in. thick, find the amount of glass in the original lump.

**Problem 29**

A certain Chinese coin, 1/2 in. in diameter and 1/16 in. thick, is pierced by a square hole 1/8 in. on a side. Find the amount of metal in a coin.

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