This document contains 3 math problems involving beam calculations. Problem 36 involves calculating beam deflection with no temperature change. Problem 79 calculates deflection with a 10 degree temperature change. Problem 80 involves calculating beam deflection and substituting the result into an integral equation. The problems utilize concepts like beam curvature, compatibility, and macro-equilibrium to set up and solve equations for determining beam deflection.

1. Prob # 36
E= 200 GPa = 200* 10^9 Pa D= 15 mm = 0.015 m
∆T = 0
Solution
0 x< 0.3
0.3< x < 1
From Bcs
--------------------------------
--------------------------------
So
From Compatibility
-------------------------------------- 1
From macro-equilibrium
Rl Rr
-------------------------2

2. From 1 &2
Prob # 79
∆T = 10
0< x < 0.3
0.3< x <0.6
From Bcs
--------------------------------
--------------------------------
From Compatibility
------------------------------------ 1

3. At joining point
------------------------------- 2
From 1 & 2
Prob # 80
– ------------ I
0.053 < x < 0.293
---------------------------------
0 < x < 0.3
---------------------------------

4. Subst in I
--------------------------- 1
From macro-equilibrium
---------------------------------- 2
From 1 & 2