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For the spring assemblages shown in Figures P2–8 through P2–16, determine the nodal displacements, the forces in each element, and the reactions. Use the direct stiffness method for all problems.

For the five-spring assemblage shown in Figure P2–17,
determine the displacements at nodes 2 and 3 and the reactions at nodes 1 and
4. Assume the rigid vertical bars at nodes 2 and 3 connecting the springs
remain horizontal at all times but are free to slide or displace left or right.
There is an applied force at node 3 of 1000 N to the right.

Use the principle of minimum potential energy developed in Section 2.6 to solve the spring problems shown in Figure P2–18. That is, plot the total potential energy for variations in the displacement of the free end of the spring to determine the minimum potential energy. Observe that the displacement that yields the minimum potential energy also yields the stable equilibrium position.

1. Reverse the direction of the load in Example 2.4 and
recalculate the total potential energy. Then use this value to obtain the
equilibrium value of displacement.
2. The nonlinear spring in Figure P2–20 has the
force/deformation relationship f = kd
2
. Express the total potential
energy of the spring, and use this potential energy to obtain the equilibrium
value of displacement.

Resistor type elements are often used in electrical
circuits. Consider the typical resistor element shown in Figure P2–23 with
nodes 1 and 2. One form of Ohm’s law says that the potential voltage difference
across two points is equal to the current I through the conductor times the
resistance R between the two points. In equation form, V = IR where I denotes
the current in units of amperes (amps) and V is the potential or voltage drop
in units of volts (V) across the conductor of resistance R in units of ohms ().
Use the method in Section 2.2 to derive the ‘‘stiffness’’ matrix relating
potential drop to current at the nodes shown as

a. Compute the total stiffness matrix ½K of the assemblage
shown in Figure P3–1 by superimposing the stiffness matrices of the individual
bars. Note that ½K should be in terms of A
1
; A
2
; A
3
;
E
1
; E
2
; E
3
; L
1
; L
2
, and
L
3
. Here A; E, and L are generic symbols used for cross-sectional
area, modulus of elasticity, and length, respectively.
b. Now let A
1
= A
2
= A
3
=
A; E
1
= E
2
= E
3
= E, and L
1
= L
2
= L
3
= L. If nodes 1 and 4 are fixed and a force P acts at node 3 in
the positive x direction, find expressions for the displacement of nodes 2 and
3 in terms of A; E; L, and P.
c. Now let A = 1 in
2
, E = 10 10
6
psi,
L = 10 in., and P = 1000 lb.
i. Determine the numerical values of the displacements of
nodes 2 and 3.
ii. Determine the numerical values of the reactions at nodes
1 and 4.
iii. Determine the stresses in elements 1–3.

1. For the bar assemblages shown in Figures P3–2 through P3–11, determine the nodal displacements, the forces in each element, and the reactions. Use the direct stiffness method for these problems.
2. Determine the stiffness matrix for the bar element with end nodes and midlength node shown in Figure P3–13. Let axial displacement u = a
1
a
2
x
a
3
x
2
. (This is a higher-order element in that strain now varies linearly through the element.)

1. Consider the following displacement function for the two-noded bar element:
u = a
bx
2
Is this a valid displacement function? Discuss why or why not.
2. For each of the bar elements shown in Figure P3–15, evaluate the global x-y stiffness matrix.

For the bar elements shown in Figure P3–16, the global
displacements have been determined to be u
1
= 0:5 in., v
1
= 0:0, u
2
= 0:25 in., and v
2
= 0:75 in. Determine the
local x displacements at each end of the bars. Let E = 12 106 psi, A = 0:5 in
2
,
and L = 60 in. for each element.

1. For the bar elements shown in Figure P3–17, the global
displacements have been determined to be u
1
= 0:0; v
1
=
2:5 mm, u
2
= 5:0 mm, and v
2
= 3:0 mm. Determine the local
x0 displacements at the ends of each bar. Let E = 210 GPa, A = 10 10
4
m
2
, and L = 3 m for each element.
2. Using the method of Section 3.5, determine the axial
stress in each of the bar elements shown in Figure P3–18.