Category: Writing

For the square two-dimensional body shown in Figure P13–21,
determine the temperature distribution. Let K
xx
= K
yy
=
25 Btu/(h-ft-F) and h = 10 Btu/(h-ft
2
– F). Convection occurs across
side 4–5. The free-stream temperature is Ty = 50 F. The temperatures at nodes 1
and 2 are 100 F. The dimensions of the body are shown in the figure. Take the
thickness of the body to be 1 ft.

1. For the one-dimensional rod shown in Figure P13–2
(insulated except at the ends), determine the temperatures at L/3, 2L/3, and L.
Let K
xx
= 3 Btu/(h.-in.-F), h = 1:0 Btu/(h-in
2
– F), and
Ty = 0 F. The temperature at the left end is 200 F.
2. A rod with uniform cross-sectional area of 2 in
2
and thermal conductivity of 3 Btu/ (h-in.-F) has heat flow in the x direction
only (Figure P13–3). The right end is insulated. The left end is maintained at
50 F, and the system has the linearly distributed heat flux shown.
Use a two-element model and estimate the temperature at the
node points and the heat flow at the left boundary.

Calculate the [k] and { f } matrices for the element shown
in Figure P13–20. The conductivities are K
xx
= K
yy
= 15
W/(m C) and the convection coefficient is h = 20 W/(m
2
C).
Convection occurs across the i-m surface. The free-stream temperature is Ty =
15 C. The coordinates are shown expressed in units of meters. Let the line
source be Q = 100 W/m as located in the figure. Take the thickness of the
element to be 1 m.

For the one-dimensional composite bar shown in Figure P13–1,
determine the interface temperatures. For element 1, let K
xx
= 200
W/(m C); for element 2, let K
xx
= 100 W/(m C); and for element 3, let
K
xx
= 50 W/(m C). Let A = 0:1 m
2
. The left end has a
constant temperature of 100 C, and the right end has a constant temperature of
300 C.

1. Express the stiffness matrix and the force matrix for
convection from the left end of a bar, as shown in Figure P13–18. Let the
cross-sectional area of the bar be A, the convection coefficient be h, and the
free stream temperature be T∞.
2. For the element shown in Figure P13–19, determine the [k]
and { f } matrices. The conductivities are K
xx
= K
yy
= 15
Btu/(h-ft-F) and the convection coefficient is h = 20 Btu/(h-ft
2
–
F). Convection occurs across the i-j surface. The free-stream temperature is Ty
= 70 F. The coordinates are expressed in units of feet. Let the line source be
Q = 150 Btu/(h-ft) as located in the figure. Take the thickness of the element
to be 1 ft.

A tractor bucket with dimensions is shown in Figure P12–16.
The load on the bucket is 600 lb spread uniformly over the inside surface.
Three split surfaces are fully constrained. The bucket material is A36
structural steel. Determine the von Mises stress throughout the bucket. (This
problem was done in Cosmos Works and created by John Mirth and Brian
Niggemann.)
Draw the typical bucket using your own dimensions or use the
scaled drawing with dimensions shown in Figure P12–16.

Use the direct method to derive the element equations for
the one-dimensional steady-state conduction heat-transfer problem shown in
Figure P13–17. The bar is insulated all around and has cross-sectional area A,
length L, and thermal conductivity K
xx
. Determine the relationship
between nodal temperatures t
1
and t
2
( F) and the thermal
inputs F
1
and F
2
(in Btu). Use Fourier’s law of heat
conduction for this case.

A hot surface is cooled by attaching fins (called pin fins)
to it, as shown in Figure P13–16. The surface of the plate (left end of the
pin) is 90 C. The fins are 6 cm long and have a cross-sectional area of 5
10
6
m
2
with a perimeter of 0.006 m. The fins are made of copper (k
= 400 W/ m-C). The temperature of the surrounding air is T
=
20 C with heat transfer coefficient on the surface (including the end surface)
of h = 10W/ m
2
– C. A model of the typical fin is also shown in
Figure P13–16. Use three elements in your finite element model to determine the
temperatures along the fin length.

The base plate of an iron is 0.6 cm thick. The plate is
subjected to 100 W of power (provided by resistance heaters inside the iron, as
shown in Figure P13–15), over a base plate cross-sectional area of 250 cm
2
,
resulting in a uniform flux generated on the inside surface. The thermal
conductivity of the metal base plate is k = 20 W/ m-C. The outside ambient air
temperature is 20 C with a heat transfer coefficient of 20W/ m
2
– C
at steady-state conditions. Assume one-dimensional heat transfer through the
plate thickness. Using three elements, model the plate to determine the
temperatures at the inner surface and interior one-third points.

A large plate of stainless steel with thickness of 5 cm and
thermal conductivity of k = 15 W/ m-C is subjected to an internal uniform heat
generation throughout the
plate at constant rate of Q = 10 10
6
W/ m
3
.
One side of the plate is maintained at 0 C by ice water, and the other side is
subjected to convection to an environment at T
x
= 35 C, with heat
transfer coefficient h = 40 W/ m
2
– C, as shown in Figure P13–14. Use
two elements in a finite element model to estimate the temperatures at each
surface and in the middle of the plate’s thickness. Assume a one-dimensional
heat transfer through the plate.