Present a complete force analysis for a composite bar made of two materials, A a

Question

Present a complete force analysis for a composite bar made of two materials, A and B as shown in the following diagram. If the two constituents are perfectly bonded, what is the force in each constituent when the composite bar is loaded with an axial force P. The Young’s moduli of the constituents are EA and EB while the cross-sectional areas are AA and AB for the constituents A and B, respectively.

Also find the torque and bending moment in each constituent when the composite is loaded with a torque T and a bending moment M, respectively. The shear moduli for the constituents are GA and GB. Use J to denote the polar second moment of area (polar moment of inertia) and I for second moment of area (moment of inertia). You do not have to print your derivation if your handwriting is readable (to me).

Explanation / Answer

(1) For axial load, the deflections of both parts are the same

= FAL/(EAAA) = FBL/(EBAB)     (1)

Also, FA+FB = P                         (2)

From (1) and (2), we hAVE

FA=P EAAA/(EAAA + EBAB)          FB=P EBAB/(EAAA + EBAB)

(2) FOr bending, the bending angle at each end is same, i.e.,

MAL/(EAIA) = MBL/(EBIB)            (1)

Also, MA+MB = M                        (2)

From (1) and (2), we hAVE

MA=M EAIA/(EAIA + EBIB)          MB=M EBIB/(EAIA + EBIB)

(2) FOr torsion, the twist angle at each end is same for both materials, i.e.,

TAL/(GAJA) = TBL/(GBJB)            (1)

Also, TA+TB = T                        (2)

From (1) and (2), we hAVE

TA=T GAJA/(GAJA +GBJB)          TB=T GBJB/(GAJA +GBJB)

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