Newton's Law of Cooling

Newton's Law of Cooling states that the temperature of a body changes at a rate proportional to the difference in temperature between its own temperature and the temperature of its surroundings.

We can therefore write

$ \dfrac{dT}{dt} = -k(T - T_s) $

T = temperature of the body at any time, t
Ts = temperature of the surroundings (also called ambient temperature)
To = initial temperature of the body
k = constant of proportionality

Problem 1010 and Problem 1011 | Investigation of timber reinforced by two steel channels

Problem 1010
A pair of C250 × 30 steel channels are securely bolted to wood beam 200 mm by 254 mm, as shown in Fig. P-1010. From Table B-2 in Appendix B, the depth of the channel is also 254 mm.) If bending occurs about the axis 1-1, determine the safe resisting moment if the allowable stresses σs = 120 MPa and σw = 8 MPa. Assume n = 20.



Problem 1011
In Problem 1010, determine the safe resisting moment if bending occurs about axis 2-2.

Problem 1009 | Width of aluminum plate reinforcement for the wood section to resist 14 kN-m moment

Problem 1009
A timber beam 150 mm wide by 200 mm deep is to be reinforced at the top and bottom by aluminum plates 6 mm thick. Determine the width of the aluminum plates if the beam is to resist a moment of 14 kN·m. Assume n = 5 and take the allowable stresses as 10 MPa and 80 MPa in the wood and aluminum, respectively.

Problem 1005 | Maximum concentrated load at the midspan that the reinforced timber beam can carry

Problem 1005
A timber beam 6 in. by 10 in. is reinforced only at the bottom by a steel plate as shown in Fig. P-1005. Determine the concentrated load that can be applied at the center of a simply supported span 18 ft long if n = 20, fs ≤ 18 ksi and fw ≤ 1200 psi. Show that the neutral axis is 7.1 in. below the top and that INA = 1160 in.4.