updated publication and MEE342 hw1

Author: antantantant

_images/mechdesign/hw1_6.png

@@ -16,15 +16,17 @@ <h3>Research Interests</h3>
   <p>I'm interested in open questions related to game theory, optimization, and AI for science that are technically within my reach. 
     Current projects include:</p>
   <ul>
-    <li>2p0s differential games with incomplete information and temporal logic specifications (e.g., for football strategic planning)</li>
+    <li>2p0s differential games with imperfect information and temporal logic specifications (e.g., football strategic planning)</li>
     <li>Learning PDEs from experimental observations (e.g., for modeling materials structure-property mapping)</li>
     <li>GenAI robust watermarking and attribution</li>
   </ul>
 </section>
 
 <section id="papers">
   <h3>Selected Publications</h3>
-  
+  <ul>
+    <li>Ghimire, M., Zhang, L., Xu, Z., & Ren, Y. (2026). <i>Solving Football by Exploiting Equilibrium Structure of 2p0s Differential Games with One-Sided Information</i> ICLR 2026. <a href="https://arxiv.org/abs/2502.00560">arXiv:2502.00560</a></li>
+  </ul>  
   <ul>
     <li>Ghimire, M., Zhang, L., Xu, Z., & Ren, Y. (2025). <i>A Scalable Solver for 2p0s Differential Games with One-Sided Payoff Information and Continuous Actions, States, and Time.</i> <a href="https://arxiv.org/abs/2502.00560">arXiv:2502.00560</a> (Best paper award at AAAI Multiagent AI in Real World Workshop 2025) </li>
   </ul>

_images/mechdesign/hw1_7.png

@@ -85,6 +85,50 @@ $I_2$ be for the beam to have minimal maximum deflection? (Optional)
 <img src="/_images/mechdesign/hw1_5.png" alt="Drawing" style="height: 200px;"/> 
 
 
+### Problem 5 (From bending moment to deflection)
+
+Consider beam analysis where the beam is positioned along the $x$-axis and deflection along $y$-axis. Let $M(x)$, $E(x)$, $I(x)$, $\rho(x)$ be the bending moment, Young's modulus, moment of inertia along $y$-axis, and the curvature radius, respectively. Show step-by-step the following relationship:
+
+$$
+\frac{1}{\rho(x)} = \frac{M(x)}{E(x)I(x)}.
+$$
+
+Then explain that for small beam defection, we have
+
+$$
+\frac{1}{\rho(x)} \approx \frac{d^2y}{dx^2}.
+$$
+
+### Problem 6 (Stress due to bending)
+
+Consider beam analysis where the beam is positioned along the $x$-axis and deflection along $y$-axis. Focus on a cross-section at some location $x$ along the beam. Let $M$, $V$ and $I$ be the bending moment, shear force, and moment of inertia along $y$-axis, respectively, at this cross-section. Let $\sigma(y)$, $\tau(y)$, and $b(y)$ be the normal stress, shear stress, and cross-section width, respectively, along $y$-axis, *considering the neutral axis of the cross-section as the origin*. Let $c$ be the distance from the neutral axis of the cross-section to its top (we assume that the cross-section is symmetric along its neutral axis). Show step-by-step the following relationship:
+
+$$
+\sigma(y) = -\frac{M}{I}y,
+$$
+
+and
+
+$$
+\tau(y) = \frac{V}{Ib(y)}\int_y^c y'b(y')dy'.
+$$
+
+### Problem 6 (Shigley's 11th Exercise 4-11 Superposition)
+For the wire form of diameter d shown in figure below, determine the deflection of
+point B in the direction of the applied force F (neglect the effect of transverse shear).
+
+<img src="/_images/mechdesign/hw1_6.png" alt="Drawing" style="height: 400px;"/> 
+
+### Problem 7 (Shigley's 11th Exercise 4-122)
+The steel beam ABCD shown is simply supported at C as shown and supported at B
+and D by shoulder steel bolts, each having a diameter of 8 mm. The lengths of BE
+and DF are 50 mm and 65 mm, respectively. The beam has a second area moment of
+21 $\times 10^3$ $mm^4$. Prior to loading, the members are stress-free. A force of 2 kN is then
+applied at point A. Determine the stresses in the bolts and the deflections of points A, B, and D.
+
+<img src="/_images/mechdesign/hw1_7.png" alt="Drawing" style="height: 400px;"/> 
+
+
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