Q1. A hollow circular mild steel column of external diameter 300 mm and internal diameter of 250 mm carries an axial load of 1500 kN.

a) Determine the compressive stress in the column.

b) If the initial length of the column is 3.75 m, find the decrease in length of the column. Take E = 2 x 10⁵N/mm².

Q2. The diameter of an aluminum rod serving as a compression member is 30 mm. The yield stress for aluminum is σy = 270 N/mm² and the ultimate stress is σu = 310 N/mm2.

a) Find the allowable compressive force if the factors of safety with respect to the yield stress and the ultimate stress are 4 and 5 respectively.

Q3. Find the maximum bending stress produced in a round steel bar 50 mm in diameter and 9 m long due to its own weight when it is simply supported at the ends.

Note: Steel weighs 77000 N/m³. (Use the Section Modulus, Z = πd³/32 and bending stress f = M/Z). 

Q4. Determine the end support reactions and maximum moment for the beam loaded as shown in the figure below.

Q5. Using appropriate equations for deflection, slope, curvature and moment, explain how the solution of beam deflection problems can be carried out by direct integration of differential equations.

Q6. A cast iron pipe of external diameter 60 mm and 10 mm thickness and 5m long is supported at its ends. The pipe carries a point load of 100 N at its centre.

a) Calculate the maximum flexural stress induced in the pipe due to the point load. (Take Section Modulus Z = I/c = π(D4 - d4)/32D).

Q7. What is the equation for the fixed end moments (M_AB and M_BA) in the following case?

Q8. A simply supported beam has a concentrated downward force P at a distance of a from the left support, as shown in the figure below. The flexural rigidity EI is constant. Find the equation of the Elastic Curve by successive integration.

Q9. Determine the rotations at A and B due to an applied moment MB on the beam, as shown in the figure below. Use the Method of Virtual Work.

Q10. Find the strain energy stored per unit volume for the materials listed below when they are axially stressed to their respective proportional limits.

Q11. As shown in the figure below, find the downward deflection of the end C caused by the applied force of 2 kN in the structure. Neglect deflection caused by shear. Let E = 7 x 10⁷ kN/m².

Q12. For the loaded beam, as shown in the figure below, determine the magnitude of the counter weight Q for which the maximum absolute value of the bending moment is as small as possible. If this beam section is 150 mm x 200mm, determine the maximum bending stress. Neglect the weight of the beam.

Q13. A wooden beam with sectional dimensions of 150 mm x 300 mm, carries the loading as shown in the figure below. Determine the maximum shearing and bending stress for the beam.

Q14. For the box beam shown in the figure below, determine the maximum intensity w of the distributed loading that can be safely supported if the permissible stresses in bending and shear are 10 N/mm² and 0.75 N/mm² respectively.

Q15. A beam of rectangular section 450 mm wide and 750 mm deep has a span of 6 metres. The beam is subjected to a uniformly distributed load of 20 kN per metre run (including the self-weight of the beam) over the whole span. The beam is also subjected to a longitudinal axial compressive load of 1500 kN. Find the extreme fibre stresses at the middle section span.

Q16. A hollow alloy tube 5 metres long with external and internal diameters equal to 40 mm and 25 mm respectively, was found to extend by 6.4 mm under a tensile load of 60 kN. Find the buckling load for the tube when it is used as a column with both ends pinned. Also find the safe compressive load for the tube with a Factor of Safety of 4.

Q17. A cantilever beam of length l carrying a distributed load varies uniformly from zero at the free end to w per unit run at the fixed end. Find the slope and downward deflection of the free end B.

Q18. a) Define strain and provide its equation.

b) What are the different types of strains normally encountered in practice?

c) How are longitudinal strains different from bending strains?

Illustrate with a diagram the bending strains in a beam subjected to shear forces perpendicular to its principal axis.

Q19. a) Briefly discuss strain development in rectangular sections.

b) In relation to reinforced concrete sections, write short notes with illustrations on:

i. Over-reinforced sections.

ii. Under-reinforced sections.

iii. Balanced sections.  

Which one of these section is preferred in practice from a safety point of view?

Q20: a) Discuss various types of end conditions for beams with illustrations. 

b) How do these end conditions affect the equilibrium conditions of displacement and moment in beams?

Q21. Write a note with illustrations on the types of homogeneous boundary conditions in structural beams.

Q22. a) Discuss the importance of the study of deflection in beams with respect to structures. (Discuss this with examples).

Q23. a) Discuss the concept of the development of a bending moment in beams under the application of external loads.

b) What are the three equilibrium conditions for the static equilibrium of beams? 

c) Using the method of sections, draw a neat illustration showing the development of axial force, shear, and bending moment at a beam section under external loads.

Q24. a) Define elastic strain energy as applied to structural members under external loading. 

b) Identify the expressions for elastic strain energy and strain energy density stored in members due to axial force and pure shear.

Q25. Discuss the principle of Virtual Work and its applications for finding deflections in structural members through the Unit Load or Dummy Load Method.

Q26. a) With the help of a diagram, discuss the development of flexural stresses in a beam section under external loading.

b) Specify the flexure equation for calculating bending stresses and discuss its importance.

Q27. a) Identify the equations for the general expressions of normal and shearing stresses on any plane located by the angle θ and caused by a known system of stresses.

b) Specify the expression for locating the planes on which the largest normal and shearing stresses occur.

c) Detail the expression for the largest shearing stress in terms of the principal normal stresses σ₁ and σ₂.

Q28. a) Discuss the procedure for the design of a beam whilst ensuring adequate structural safety of the structure.

b) How are the flexure and shear stress equations helpful while designing beams for structural adequacy?

Q29. a) What is a column?  

b) Specify the buckling or crippling load for a compression member. 

c) Using Euler's theory of long columns, develop an expression for the crippling load of a column with both ends fixed.

Q30. Specify the expressions for crippling loads in the following cases: 

i. When both ends of the column are pinned (hinged). 

ii. When one end of the column is fixed and the other end is free. 

iii. When both ends of the column are fixed.  

When one end of the column is fixed, and the other end is pinned (hinged).

Q31. With the aid of an illustration and Euler's equations, briefly discuss the load carrying capacity of columns as determined by experimentation on real columns.

Q32. a) Why is there a need to adopt a Factor of Safety while designing columns?

b) Explain the criteria an engineer uses to determine a particular Factor of Safety for a design. 

Q33. 

a) Discuss the assumptions which are made while deriving Euler's theory of long columns. 

b) What the limitations of this theory are as applied to real columns? 

c) What is the limiting value of the slenderness ratio of columns beyond which Euler's theory will not be valid?

Q34. Choose the wrong statement from among the following choices. 

The Principal Stresses are: 

a) Planes on which maximum or minimum normal stresses occur, there are no shearing stresses.  

b) Planes on which maximum or minimum shearing stresses occur, there are no normal stresses.  

c) Shown to have an angle of 45^o with the planes of maximum or minimum shearing stress. 

Maximum shearing stress differs from the minimum shearing stress only in sign.