Selected Problem Answers

1. What are the assumptions of the Bernoulli-Euler beam theorem?
2. Determine the slope $\theta(x)$ and deflection $\Delta(x)$ equations for the beam below using the direct integration method. Assume $EI$ is constant.
   
3. Determine the slope $\theta(x)$ and deflection $\Delta(x)$ equations for the beam below using the direct integration method. Assume $EI$ is constant.
   
4. For the beam shown below, determine the slope at midspan ($\theta_{mid}$), slope at point B ($\theta_B$), deflection at midspan ($\Delta_{mid}$) and deflection at point B ($\Delta_B$) using the moment area theorems.
   
5. For the beam shown below, determine the slope to the left of the hinge at point B ($\theta_{B,left}$), slope to the right of the hinge at point B ($\theta_{B,right}$), deflection at point B ($\Delta_{B}$) and deflection at point D ($\Delta_D$) using the moment area theorems.
   
6. For the beam shown below, determine the location and value of the maximum deflection using the moment area theorems. You must find and compare all potential local maxima and minima.
   
7. For the beam shown below, determine the location and value of the maximum deflection using the moment area theorems. You must find and compare all potential local maxima and minima.
   
8. Repeat problem 5 using the conjugate beam method.
9. For the beam shown below, determine the slope and deflection at Point B ($\theta_B$, $\Delta_B$) and at point D ($\theta_D$, $\Delta_D$) using the conjugate beam method.
   
10. For the beam shown below, determine the slope and deflection at Point B ($\theta_B$, $\Delta_B$) and at point C ($\theta_C$, $\Delta_C$) using the conjugate beam method.
    
11. For the beam shown below, determine the minimum moment of inertia ($I$) such that its deflection does not exceed $\frac{L}{360}$.
    
12. For the beam shown below, determine the minimum moment of inertia ($I$) such that its deflection does not exceed $\frac{L}{360}$. Use the length of each span (AB and BC) to calculate the maximum deflection for each.
    
13. For the beam shown below, determine the deflection at point C ($\Delta_C$) using the virtual work method.
    
14. For the beam shown below, determine the rotation at point A ($\theta_A$) and the deflection at point D ($\Delta_D$) using the virtual work method.
    
15. For the beam shown below, use virtual work to determine the minimum moment of inertia ($I$) such that its deflection does not exceed $\frac{L}{360}$.
    
16. For the beam shown below, use virtual work to determine the minimum moment of inertia ($I$) such that its deflection does not exceed $\frac{L}{360}$.
    
17. For the frame shown below, determine the vertical deflection at point C ($\Delta_C$) using the virtual work method.
    
18. For the frame shown below, determine the rotation and vertical deflection at point A ($\theta_A$, $\Delta_A$) using the virtual work method.
    
19. For the frame shown below, determine the rotation at point D ($\theta_D$) and the horizontal deflection at point E ($\Delta_E$) using the virtual work method.
    
20. For the truss shown below, determine the vertical deflection at point B ($\Delta_B$) and the horizontal deflection at point H ($\Delta_H$) using the virtual work method.
    
21. For the truss shown below, determine the vertical deflection at point C ($\Delta_C$) and the vertical deflection at point F ($\Delta_F$) using the virtual work method.
    
22. For the truss shown below, use virtual work to determine the minimum cross-sectional area ($A$) such that the horizontal deflection at point H ($\Delta_H$) does not exceed $20\mathrm{\,mm}$.
    
23. For the truss shown below, use virtual work to determine the minimum cross-sectional area ($A$) such that the vertical deflection at point C ($\Delta_C$) does not exceed $35\mathrm{\,mm}$.
    
24. For the truss from problem 20 but with no external loads, determine the horizontal deflection at point F ($\Delta_F$) if members AE, AB, BC and CG experience a temperature increase of $50\mathrm{\,^\circ C}$ and member BF experiences a temperature decrease of $20\mathrm{\,^\circ C}$ ($\Delta T = -20\mathrm{\,^\circ C}$). Assume that $\alpha=1.2\times 10^{-5}\mathrm{\,/ ^\circ C}$.
25. For the truss from problem 21 but with no external loads, determine the vertical deflection at point B ($\Delta_B$) if members BD, DF and AD were each fabricated $10\mathrm{\,mm}$ too short and member DE was fabricated $5\mathrm{\,mm}$ too long.