Category: Writing

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.

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

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.

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.

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.

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.

1. For the spring assemblage in Figure P2–5, apply a
concentrated force of 2 kips at node 2 in the positive x direction and
determine the displacements at nodes 2 and 4.
2. Instead of assuming a tension element as in Figure P2–3,
now assume a compression element. That is, apply compressive forces to the
spring element and derive the stiffness matrix.

1. Solve Problem 2.3 with P ¼ 0 (no force applied at node 3) and with node 5 given a fixed, known displacement of d as shown in Figure P2–4.
2. For the spring assemblage shown in Figure P2–5, obtain the global stiffness matrix by the direct stiffness method. Let k
(1)
= 1 kip=in:; k
(2)
= 2 kip=in:; k
(3)
= 3 kip=in:; k
(4)
= 4 kip/in., and k
(5)
= 5 kip/in.

1. For the spring assemblage shown in Figure P2–2, determine the displacement at node 2 and the forces in each spring element. Also determine the force F3. Given: Node 3 displaces an amount d = 1 in. in the positive x direction because of the force F
3
and k
1
= k
2
= 500 lb/in.
2. a. For the spring assemblage shown in Figure P2–3, obtain the global stiffness matrix by direct superposition.
b. If nodes 1 and 5 are fixed and a force P is applied at node 3, determine the nodal displacements.
c. Determine the reactions at the fixed nodes 1 and 5.

1. For the truss shown in Figure P3–23, solve for the horizontal and vertical components of displacement at node 1. Also determine the stress in element 1. Let A = 1 in
2
, E = 10:0 106 psi, and L = -100 in.
2. Determine the nodal displacements and the element forces for the truss shown in Figure P3–24. Assume all elements have the same AE.