The structural assemblage in Fig. 1 is made up of three uniform elements, or members. Element (1) is a solid rod. Element (2) is a pipe that surrounds element (3), which is a solid rod that is identical to element (1) and collinear with it. The three elements are all attached at B to a rigid plate of negligible thickness. With no external force at B, the three-element assemblage exactly fits between the rigid walls at A and C; its ends are then attached to the two walls. Determine expressions for the axial forces in the three elements when an external force P_B is applied at B.

Step-by-Step

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**Plan the Solution** We can think of this structure as being composed of three uniform elements and one connecting node (joint), and we can write an equilibrium equation for node B. Using Eq. 3.14, we can write a force-deformation equation for each of the three elements. We can relate the element elongations to each other through the displacement at B.

Finally, we can use the Basic Force Method to combine these three sets of fundamental equations to get expressions for the forces in the individual elements.

e = fF, \text{where} f ≡ \frac{L}{AE} (3.14)

If the external force P_B acts to the right (i.e., if it is positive), we should find that the left-hand element is in tension and the two righthand elements are in compression.

Equilibrium: From the free-body diagram of node B in Fig. 2,

\underrightarrow{+} \sum{F_x}=0: -F_1 + F_2 + F_3 + P_B = 0 Equilibrium (1)

Equation (1) relates the three unknown internal element forces to the known external load. Since there are three unknown forces, but only one equilibrium equation, this system is statically indeterminate and there are **two redundant forces.**

Element Force-Deformation Behavior: We have three uniform, axial-deformation elements, and for each one we can write an element force-deformation equation like Eq. 3.14.We have called the element forces

F_1, F_2, and F_3 (tension positive), so we have

e_1 = f_1F_1, where f_1 = (L_1/A_1E_1)

e_2 = f_2F_2, where f_2 = (L_2/A_2E_2) Element Force-Deformation Behavior (3a–c)

e_3 = f_3F_3, where f_3 = (L_3/A_3E_3)

In Eqs. (2) the e’s are the element elongations. A positive Fi (tension) produces a positive e_i (element gets longer), since the f_i’s are, by definition, positive.

Geometry of Deformation: Referring to Fig. 1, we can easily relate the elongation of each of the three elements to the displacement u_B by using the definition of elongation of an element, that is, e = u(L) – u(0). So,

e_1 = -e_2 = -e_3 = u_BHere we have used the fact that the displacements at joints A and C are zero. Note that, since e is positive when an element gets longer, a displacement of joint B to the right by an amount u_B implies a shortening of elements (2) and (3) by that amount; hence the minus sign for e_2 and e_3.We can eliminate u_B and write the above as two **compatibility equations:**

e_2 = -e_1 Geometry of Deformation (3a,b)

e_3 = e_2The fact that there are **two compatibility equations** is consistent with the fact that there are **two redundant forces** in the equilibrium equation. There will always be as many compatibility equations as there are redundant forces!

Solution of the Equations: If we count equations and unknowns, we find that we have six equations and six unknowns. Rather than just combine Eqs. (1) through (3) in some arbitrary order, we will follow the Basic Force Method.

Substitute Eqs. (2) (element force-deformation) into Eqs. (3) (deformation compatibility) to obtain the compatibility equations written in terms of forces.

f_2F_2 = -f_1F_1 Compatibility in Terms of Element Forces (4a,b)

f_3F_3 = f_2F_2

We now have three equations, Eqs. (1) and (4a,b), in three unknowns, the three element forces. We \underline{\text{solve these equations simultaneously}} to get the following expressions for the three unknown element forces:

F_1 = \left( \frac{f_2 f_3} {f_1 f_2 + f_2 f_3 + f_1 f_3}\right)P_BF_2 = \left( \frac{-f_1 f_3} {f_1 f_2 + f_2 f_3 + f_1 f_3}\right)P_B Ans. (5a–c)

F_3 = \left( \frac{-f_1 f_2} {f_1 f_2 + f_2 f_3 + f_1 f_3}\right)P_B**Review the Solution** As one check of our work, we can substitute Eqs.

(5) back into Eqs. (4a,b) to see if deformation compatibility is satisfied.

Are f_2F_2 = -f_1F_1 and f_3F_3 = f_2F_2? Yes.

The fact that the compatibility equations, Eqs. (4a,b), are satisfied by our answers means that we have probably not made errors in our solution. Also, from Eqs. (5) we see that, when P_B is positive, element (1) is in tension and elements (2) and (3) are in compression. This is what we expected to find.

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