Initial Stress In A Composite Member Under Tensile Loading Given Allowable Stresses

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Initial Stress In A Composite Member Under Tensile Loading Given Allowable Stresses

A composite member consists of a steel rod shaft in an aluminum tube. The members are fastened together by adjustable nuts. Dimensions are:

Sectional area of steel rod = 1.5 in^{2}; modulus of elasticity = 30,000,000; allowable stress = 15,000 lb/in^{2}.

Sectional area of aluminum tube = 2 in^{2}; modulus of elasticity = 10,000,000; allowable stress = 10,000 lb/in^{2}

Determine initial stresses (prestressed) in the members so that under tensile loading both members will attain their allowable stresses simultaneously.

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

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

Assumptions: *consistent deformation* is applicable, that is, load caried by each member of the composite member is proportional to its *stiffness* if members are uniform.

Where *stiffness* = AE/l (A is area, E is modulus of elasticity, l is length).
*Allowable stress* same as *working stress* usually less than damaging stress.

Steel rod and aluminum tube are of equal length.

Equations of Interest:

Given n members of a composite member with respective area, A_{1}, A_{2}...A_{n}; moduli of elasticity, E_{1}, E_{2} ...E_{n} and lengths, l_{1}, l_{2} ...l_{n}.

Load carried by member k is P_{k} = P[_{total}(A_{k}E_{k})/l_{k}]/[^{n}Σ_{n=1} (A_{n}E_{n})/l_{n}]-------(1).

Where k = 1,2...n.

When composite member is prestressed (i.e.has initial stress), then P_{k} represents the *increment* of force in each member due to the applied load.

Force in memeber = (Allowable Stress)Area -----------(2)

So, force in steel rod, P_{1} = 1.5(15000) = 22,500 lb.

So, force in aluminum = 2(10000) = 20,000 lb.

So, total load on composite member = 22,500 + 20,000 = 42,5000 lb.

In the prestress case P_{k} of equation (1) is an *increment*.

Let P_{i} denote initial tension or compression in the members. Then:

For Steel rod:

P_{isteel} + 42,500[(1.5)(30)/((2)(10) + (1.5)(30))] = 22,500

So, P_{isteel} = -6920 lb (compression if we consider compression to be -ve).

For aluminum tube:

P_{ialuminum} + 42,500[(2)(10)/((2)(10) + (1.5)(30))] = 20,000

So, P_{ialuminum} = +6920 lb (tension if we consider tension to be +ve).

Math

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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