**Definition**

Shear Force Diagramis the Graphical representation of the variation of Shear Force Over the cross-section along the length of the beam. With the Shear Force Diagram’s help, we can identify Critical sections Subjected to Shear and design amendments to be made to avoid failure.

Similarly,

Bending Moment Diagramis the Graphical representation of the Bending moment’s variation over the cross-section along the length of the beam. With the Bending moment Diagram’s help, we can identify Critical sections Subjected to bending and design amendments to be made to avoid failure. While constructing the Shear Force Diagram [S.F.D.], There is a sudden rise or sudden drop due to point load acting on the beam while constructing Bending moment Diagram [BMD]; there is a sudden rise or sudden drop due to couples acting on the beam.

**Q.1) What is the Formula for Bending Moment?**

The Algebraic sum of the moments over a particular cross-section of the beam due to clock or anticlockwise moments is called bending moment at that point.

Let W be a force vector acting at a point A in a body. The moment of this force about a reference point (O) is defined as

M = W x p

Where M = Moment vector, p = the position vector from the reference point (O) to the point of application of the force A. The symbol indicates the vector cross product. it is easy to compute the moment of the force about an axis that passes through the reference point O. If the unit vector along the axis is ”i”, the moment of the force about the axis is defined as

M = i . (W x p)

Where [.] represent Dot product of the vector.

**Q.2) What is Bending moment and Shear force?**

**Ans: **

Shear Forceis the Algebraic sum of forces Parallel to cross-section over a particular cross-section of the beam due to action and reaction forces. Shear Force tries to shear off the beam’s Cross section perpendicular to the beam’s axis, and due to this, the developed shear stress distribution is Parabolic from the neutral axis of the beam.

A Bending moment is a summation of the moments over a particular cross-section of the beam due to Clockwise and Counter Clockwise Moments. Bending moment tries to bend the beam in the plane of the member, and due to transmission of Bending moment over a Cross-section of the beam, the Developed Bending stress distribution is Linear from the neutral axis of the beam.

**Q.3) What is Shear Force Diagram S.F.D. and Bending Moment Diagram B.M.D?**

**Ans: Shear Force Diagram** [S.F.D.] Shear Force Diagram can be described as the Pictorial representation of the variation of Shear Force that is generated in the beam, Over the cross-section and along the length of the beam. With the Shear Force Diagram help, we can identify Critical sections Subjected to Shear and design amendments to be made to avoid failure.

Similarly, **Bending Moment Diagram** [BMD] is the Graphical representation of the Bending moment’s variation over the cross-section along the beam’s length. With the Bending moment Diagram’s help, we can identify Critical sections Subjected to bending and design amendments to be made to avoid failure. While constructing the Shear Force Diagram [S.F.D.] There is a sudden rise or sudden drop due to point load acting on the beam while constructing Bending moment Diagram [BMD]; there is a sudden rise or sudden drop due to couples acting on the beam.

**Q.4) What is the unit of Bending moment?**

**Ans: **Bending Moment has a unit similar to a couple as **Nm.**

**Q.5) Why is moment at hinge zero?**

**Ans: **In hinge Support, the movement is restricted in Vertical and Horizontal Direction. It offers no resistance for the rotational motion about the support. Thus, support offers a rection towards horizontal and vertical motion and no reaction to the moment. Thus, the Moment is Zero at the hinge.

**Q.6) What is the bending of beam?**

**Ans: ** If the moment applied to the beam tries to bend the beam in the plane of the member, then it is called a bending moment, and the phenomenon is called bending of beam.

**Q.7) What is the condition of deflection and bending moment in a simply supported beam?**

**Ans: The conditions of deflection and bending moment in a simply supported beam are:**

**The maximum Bending Moment that yields bending stress must be equal to or less than the Permissible strength bearing capacity of the material of the beam.****The maximum induced deflection should be less than acceptable level based on Durability for the given length, the period, and material of the beam.**

**Q.8) What is the difference between bending moment and bending stress?**

**Ans: Bending Moment** is the Algebraic sum of the moments over a particular cross-section of the beam due to Clockwise and Counter Clockwise Moments. Bending moment tries to bend the beam in the plane of the member, and due to transmission of Bending moment over a Cross-section of the beam, the Developed Bending stress distribution is Linear from the neutral axis of the beam. **Bending **stress can be defined as resistance induced due to Bending Moment or by two equal and opposite couples in the plane of the member.

**Q.9) How are the intensity of load shear force and bending moments related to mathematically?**

**Ans:** Relations: Let f = load intensity

Q = Shear Force

M = Bending Moment

The rate of change of shear force will give the intensity of the distributed load.

The rate of change of bending moment will give shear force at that point only.

**Q.10) What is the relation between loading shear force and bending moments?**

**Ans: **The rate of change of Bending moment will give Shear force at that particular point only.

**Q.11) What is the difference between a plastic moment and a bending moment?**

**Ans: **The plastic moment is defined as the maximum value of the moment when the complete cross-section has reached its yielding limit or permissible stress value. Theoretically, It is the maximum bending moment that the entire section can bear before yielding any load beyond this point will result in large plastic deformation. While Bending Moment is the Algebraic sum of the moments over a particular cross-section of the beam due to Clockwise and Counter Clockwise Moments. Bending moment tries to bend the beam in the plane of the member, and due to transmission of Bending moment over a Cross-section of the beam, the Developed Bending stress distribution is Linear from the neutral axis of the beam.

**Q.12) What is the difference between the moment of force, couple, torque, twisting moment, and bending moment? If any two are the same, what is the use of assigning different names?**

**Ans: **A moment, a torque, and a couple are all similar concepts which rests on a basic principle of the product of a force (or forces) and a distance. A Moment of force can be formulated as the product of force and the length of the line crossing over the point of support and is vertical to the acting force. Bending moment tries to bend the beam in the plane of the member and due to transmission of Bending moment over a Cross-section of the beam.

A couple is a moment that is generated of two forces having the same magnitude, acting in the opposite direction equidistant from the reaction point. Therefore, a couple is statically equivalent to a simple Bending. Torque is a moment when functional have a tendency to to twist a body around its axis of rotation. A Typical example of torque is a torsional moment applied on a shaft.

**Q.13) Why maximum bending moments is smallest when the numerical value is the same in positive and negative directions?**

**Ans: **The maximum bending moment and minimum bending moment are dependent on the condition and direction of application of stress rather than the magnitude of stress. A positive sign denotes tensile stress, and the negative sign denotes compression. The maximum magnitude of the bending moment is taken for designing, while the sign denotes whether the beam is designed for compressive loading or tensile loading conditions. Usually, beams are designed for tensile stress as a material is likely to yield under tension and ultimately rupture.

**Q.14) What is bending moment equation as a function of distance x calculated from the leftside for a simply supported beam of span L carrying U.D.L. w per unit length?**

**Ans: **

The resultant load acting on the Beam Due to U.D.L. can be given by

W = Area of a rectangle

W = L * w

W=wL

Equivalent Point Load **wL** will act at the center of the beam. i.e., at L/2

The value of the reaction at A and B can be calculated by applying Equilibrium condition

[latex]\sum F_y=0,\sum M_A=0[/latex]

For vertical Equilibrium,

[latex]R_A+R_B=wL[/latex]

Taking Moment about A, Clockwise moment positive, and Counter Clockwise moment is taken as negative

[latex]\frac{wL^2}{2}-R_B*L=0[/latex]

[latex]R_B=\frac{wL}{2}[/latex]

Putting the value of R_{B} in Vertical equilibrium equation we get,

[latex]R_A=wL-R_B[/latex]

[latex]R_A=wL-\frac{wL}{2}=\frac{wL}{2}[/latex]

Let X-X be the section of interest at a distance of x from end A

According to the Sign convention discussed earlier, if we start calculating Shear Force from the Left side or Left end of the beam, Upward acting force is taken as Positive, and Downward acting Force is taken as Negative. For Bending Moment Diagram, if we start calculating Bending Moment from the Left side or Left end of the beam, Clockwise Moment is taken as positive. Counter Clockwise Moment is taken as Negative.

Shear Force at A

[latex]S.F_A=R_A=\frac{wL}{2}[/latex]

Shear force at region X-X is

[latex] S.F_x=R_A-wx[/latex]

[latex] S.F_x=\frac{wL}{2}-wx[/latex]

[latex]S.F_x=w\frac{L-2x}{2}[/latex]

Shear Force at B

[latex]S.F_B=R_B=\frac{-wL}{2}[/latex]

Bending Moment at A = 0

Bending Moment at X

[latex]B.M_x=M_A-\frac{wx^2}{2}[/latex]

[latex]B.M_x=0-\frac{wx^2}{2}[/latex]

[latex]B.M_x=-\frac{wx^2}{2}[/latex]

Bending Moment at B = 0

**Q.15) Why does the cantilever beam have a maximum bending moment on its support? Why doesn’t it have a bending moment on its free end?**

**Ans: **For a cantilever beam with point loading, the beam has fixed support at one end, and another end is free. Whenever a load is applied on the beam, only the support resists the motion. At the free end, there is no restriction of motion. So, the moment will be maximum at support and minimum or zero at the free end.

**Q.16) What is the bending moment in a beam?**

**Ans: **Bending Moment tries to bend the beam in the plane of the member and due to transmission of Bending moment over a Cross-section of the beam.

**Q.17) Where do tension and compression act in bending of simply supported as well as in cantilever beams?**

**Ans: **For a simply supported beam with uniform Loading acting downward, the location of induced maximum bending tensile stress is acted on the bottom fiber of cross-section at the midpoint of the beam, while the maximum compression bending stress is acted on the top fiber of the cross-section at the midpoint of span. For a cantilever beam of a given span, the maximum bending stress will be at the beam’s Fixed end. For downward net load, maximum tensile bending stress is acted on top of cross-section, and max compressive stress is acted on the bottom fiber of the beam.

**Q.18) Why are we taking the bending moment left side of beam to the point that the shear force is zero?**

**Ans: **The bending moment can be taken on any side of the beam. It is generally preferred that If we start calculating Bending Moment from the Left side or Left end of the beam, Clockwise Moment is taken as Positive and Counter Clockwise Moment is taken as Negative. If we start calculating Shear Force from the Left side or Left end of the beam, Upward acting Force is taken as Positive and Downward acting Force is taken as Negative according to Sign Convention.

**Q.19) How do we use the sign convention in bending moments and shearing forces?**

**Ans:** If we start calculating Bending Moment from the **right side** or right end of **the beam**, **Clockwise Moment** is taken as **negative**, and **Counter-wise Moment** is taken as **Positive.** If we start calculating Bending Moment from the **Left side** or Left end of the beam, **Clockwise Moment** is taken as **Positive, **and **Counter Clockwise Moment** is taken as **Negative.**

**Q.20) How do I strengthen a simply supported steel I beam against Shear and bending?**

**Ans: **The strength of the I-Beam, which is simply supported, can be increased against Shear and bending conditions by increasing the Area Moment of Inertia of the beam, adding stiffeners to the web of I-Beam, changing the material of the beam to a higher strength material having greater yield strength. Changing the type of loading also affects the strength of the beam.

**Q.21) What is Point of Contraflexure?**

**Ans: **The point of Contraflexure can be defined as the point in the Bending Moment diagram where the bending moment is becomes ‘0 ’. This occasionally termed a Point of inflexion. At the point of contraflexure, the Bending moment curve of the beam will change sign. It is generally seen in a Simply supported beam subjected to moment at the mid-span of the beam and combined loading conditions of U.D.L. and point loads.

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