Unit II Shear and Bending in Beams

Transcription

1 Beams and Bending Unit II Shear and Bending in Beams 2 Marks questions and answers 1. Mention the different types of supports. i. Roller support ii. Hinged support iii. Fixed support 2. Differentiate between uniformly distributed load and uniformly varying load. Uniformly distributed load (UDL) Uniformly varying load (UVL) The loads are uniformly applied over the entire length of the beam. Triangular or trapezoidal loads fall under this category. The variation in intensities of such loads is constant. It can be shown as follows: It can be shown as follows: 3. Distinguish between cantilever beam and simply supported beam. Cantilever beam Simply supported beam A cantilever beam which is fixed at one end and free at the other end. A simply supported beam rests freely on hinged support at one end and roller support at the other end. 4. What is the advantage of roller support in simply supported beam? Roller support allows free displacements in the beam due to temperature changes. 5. Differentiate between overhanging beam and continuous beam Overhanging beam Continuous beam If a beam extends beyond its supports it is called an overhanging beam. Over hanging portion could be either any one of the sides or both the sides. A beam which rests on more than two supports is called a continuous beam. 6. Define shear force Shear force at a section of a loaded beam may be defined as the algebraic sum of all vertical forces acting on any one side of the section. 7. Define bending moment Bending moment at a section of a loaded beam may be defined as the algebraic sum of all moments of forces acting on any one side of the section.

2 8. Explain point of contraflexure Point of contraflexure is the point where bending moment changes sign from positive to negative or vice- versa. 9. Distinguish between sagging bending moment and hogging bending moment. Sagging moment Moments which bend the beam downwards and cause compression in the top fibre and tension in the bottom fibre are taken as positive. Hogging moment Moments which bend the beam upwards and cause compression in the bottom and tension in the top fibre are taken as negative. 10. If a beam is to be supported at one end only, mention the type of support. Fixed support. Example: Cantilever beams are supported at one end only with fixed support. 11. A cantilever beam of length L carries a concentrated load W at the free end. What is the maximum bending moment? The maximum bending moment is hogging and it is, M max = WL 12. A cantilever beam of span 3m carries point load of 10 kn at the free end. What is the value of support moment? The support moment is, M = 10 3 = 30 kn. m -ve sign indicates hogging moment. 13. A cantilever beam of length 4m carries a concentrated load of 10 kn at 2m from the free end. Sketch the bending moment diagram. 14. A cantilever beam of length L carries a u.d.l. of w per unit length throughout its length L, write the value of maximum bending moment? The maximum bending moment is hogging and it is, M max = WL2 2

3 15. A simply supported beam of span L carries a central concentrated load W. What is the maximum bending moment? The maximum bending moment is, M max = WL A simply supported beam of span 6 m carries a concentrated load of 12 kn at 2 m from the left support. What is the maximum shear force? Maximum shear force is = 8 kn. 17. simply supported beam of span 8 m carries a concentrated load of 10 kn at 2 m from the left support. What is the maximum bending moment? Maximum bending moment, M max = Wab L = = 15 kn.m 18. A simply supported beam of span L carries a U.D.L throughout its length. What is the maximum shear force? The shear force diagram is shown in figure. Maximum shear force = wl A simply supported beam of span L carries a U.D.L of w unit length throughout its length. Write the value of maximum bending moment? Maximum bending moment, M max = WL In a simply supported beam, where the maximum shear force will occur? At the supports.

4 Theory of simple bending 1. State assumptions in the theory of simple bending. Assumptions in the theory of simple bending: i). Modulus of elasticity is the same for the beam material in tension as well as in compression. ii). Transverse section of the beam is plane before bending remains plane after bending. iii). The beam material is homogeneous and isotropic. iv). The elastic limit is not exceeded. v). On a transverse section of a beam the resultant pull or thrust is zero. 2. Write the theory of simple bending formula. Theory of simple bending formula is, M I = f y = E R Where, M= bending moment at any section of a beam, I = Moment of inertia of a beam cross section about neutral axis, f= bending stress in the beam cross section at a distance y from neutral axis, E= modulus of elasticity and R= radius of curvature. 3. How section modulus is related to the flexural strength of a section. As per theory of simple bending, M = f, and hence the flexural strength M and section I y modulus Z is related by, M = fz, where section modulus Z= I y 4. Define modulus of section. Modulus of section is defined as the moment of inertia divided by the distance of the extreme fibre of the section from the axis through the centroid of the section i.e., Z= I y. 5. Define neutral axis of a cross section. The intersection of the neutral surface with any cross section of the beam perpendicular to its longitudinal axis is called neutral axis. 6. Define the term bending stress. The stress produced at the section to resist the bending moment is known as the bending stress. 7. Sketch the bending stress distribution in a cantilever beam indicating the nature of stresses. Tensio At any section of a cantilever beam the nature of bending stress is shown in figure with top fibres N are in tension and bottom fibres are in compression from neutral axis of a beam. Compression 8. What is the maximum value of shear stress in a rectangular cross section? Maximum shear stress in a rectangular section is, = 1.5 q average = 1.5 S A = 1.5 S bd, where S= shear force, b= breadth and d= depth of rectangular section. 9. A rectangular section 30 mm x 40 mm is subjected to a shear force of 12 kn. What is the maximum shear stress setup in the beam? = 1.5 S = 1.5 = 15 N/mm2 bd A simply supported beam of span 10 m carries a central concentrated load of 20 kn. If the cross section of the beam is a rectangle 40 mm x50 mm, what is the maximum shear stress setup in the beam?

5 = 1.5 S = 1.5 = 7.5 bd N/mm A rectangular section is subjected to a transverse shear force. Sketch the shape of shear stress distribution. A 12 kn +ve 20 kn 10 m B - ve 12 kn 12. The maximum shear stress in a circular section is times the average stress What is the maximum value of shear stress in a circular cross section? = 4 3 q average = 4 3 = 4 3 S πr S A 2, where S= shear force and R= Radius of a circle. 14. A circular section is subjected to a transverse shear force. Sketch the shape of shear stress distribution. 15. A T-section is subjected to a transverse shear force. Sketch the shape of shear stress distribution. 16. An I-section is subjected to a transverse shear force. Sketch the shape of shear stress distribution. 17. Define flitched beam. A beam of composite section consisting of a wooden beam strengthened by mild steel plates is known as flitched beam. 18. State the condition to avoid slip between wood and steel of a flitched beam. In order to avoid slip between wood and steel, at any distance y from neutral axis strains in steel and wood must be equal.