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- Calculation of Bending Moments & Shear Forces of Simple Beams
- Calculation of Bending Moments & Shear Forces of Simple Beams
- Calculation of Bending Moments & Shear Forces of Simple Beams
- Relationship Between Load, Shear, and Moment

*Given below are solved examples for calculation of shear force and bending moment and plotting of the diagrams for different load conditions of simply supported beam, cantilever and overhanging beam.*

In solid mechanics , a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The diagram shows a beam which is simply supported free to rotate and therefore lacking bending moments at both ends; the ends can only react to the shear loads. Other beams can have both ends fixed; therefore each end support has both bending moments and shear reaction loads.

Beams can also have one end fixed and one end simply supported. The simplest type of beam is the cantilever , which is fixed at one end and is free at the other end neither simple or fixed. In reality, beam supports are usually neither absolutely fixed nor absolutely rotating freely. The internal reaction loads in a cross-section of the structural element can be resolved into a resultant force and a resultant couple.

For equilibrium, the moment created by external forces and external moments must be balanced by the couple induced by the internal loads. The resultant internal couple is called the bending moment while the resultant internal force is called the shear force if it is transverse to the plane of element or the normal force if it is along the plane of the element.

The bending moment at a section through a structural element may be defined as the sum of the moments about that section of all external forces acting to one side of that section. The forces and moments on either side of the section must be equal in order to counteract each other and maintain a state of equilibrium so the same bending moment will result from summing the moments, regardless of which side of the section is selected.

If clockwise bending moments are taken as negative, then a negative bending moment within an element will cause " hogging ", and a positive moment will cause " sagging ". It is therefore clear that a point of zero bending moment within a beam is a point of contraflexure —that is, the point of transition from hogging to sagging or vice versa.

The concept of bending moment is very important in engineering particularly in civil and mechanical engineering and physics. Tensile and compressive stresses increase proportionally with bending moment, but are also dependent on the second moment of area of the cross-section of a beam that is, the shape of the cross-section, such as a circle, square or I-beam being common structural shapes.

In structural analysis, this bending failure is called a plastic hinge, since the full load carrying ability of the structural element is not reached until the full cross-section is past the yield stress. It is possible that failure of a structural element in shear may occur before failure in bending, however the mechanics of failure in shear and in bending are different.

Moments are calculated by multiplying the external vector forces loads or reactions by the vector distance at which they are applied. When analysing an entire element, it is sensible to calculate moments at both ends of the element, at the beginning, centre and end of any uniformly distributed loads, and directly underneath any point loads. Of course any "pin-joints" within a structure allow free rotation, and so zero moment occurs at these points as there is no way of transmitting turning forces from one side to the other.

It is more common to use the convention that a clockwise bending moment to the left of the point under consideration is taken as positive. This then corresponds to the second derivative of a function which, when positive, indicates a curvature that is 'lower at the centre' i. When defining moments and curvatures in this way calculus can be more readily used to find slopes and deflections.

Critical values within the beam are most commonly annotated using a bending moment diagram , where negative moments are plotted to scale above a horizontal line and positive below. Bending moment varies linearly over unloaded sections, and parabolically over uniformly loaded sections. Engineering descriptions of the computation of bending moments can be confusing because of unexplained sign conventions and implicit assumptions.

The descriptions below use vector mechanics to compute moments of force and bending moments in an attempt to explain, from first principles, why particular sign conventions are chosen. An important part of determining bending moments in practical problems is the computation of moments of force. The moment of this force about a reference point O is defined as [2]. For many problems, it is more convenient to compute the moment of force about an axis that passes through the reference point O.

The negative value suggests that a moment that tends to rotate a body clockwise around an axis should have a negative sign. For this new choice of axes, a positive moment tends to rotate body clockwise around an axis. In a rigid body or in an unconstrained deformable body, the application of a moment of force causes a pure rotation. But if a deformable body is constrained, it develops internal forces in response to the external force so that equilibrium is maintained.

An example is shown in the figure below. These internal forces will cause local deformations in the body. For equilibrium, the sum of the internal force vectors is equal to the negative of the sum of the applied external forces, and the sum of the moment vectors created by the internal forces is equal to the negative of the moment of the external force.

The internal moment vector is called the bending moment. Though bending moments have been used to determine the stress states in arbitrary shaped structures, the physical interpretation of the computed stresses is problematic. However, physical interpretations of bending moments in beams and plates have a straightforward interpretation as the stress resultants in a cross-section of the structural element.

For example, in a beam in the figure, the bending moment vector due to stresses in the cross-section A perpendicular to the x -axis is given by. For this situation, the only non-zero component of the bending moment is. The minus sign is included to satisfy the sign convention. To obtain each reaction a second equation is required. Balancing about the point O is simplest but let's balance about point A just to illustrate the point, i.

We can check this answer by looking at the free body diagram and the part of the beam to the left of point X , and the total moment due to these external forces is. Thanks to the equilibrium, the internal bending moment due to external forces to the left of X must be exactly balanced by the internal turning force obtained by considering the part of the beam to the right of X. In the above discussion, it is implicitly assumed that the bending moment is positive when the top of the beam is compressed.

That can be seen if we consider a linear distribution of stress in the beam and find the resulting bending moment. The bending moment due to these stresses is. Therefore, the bending moment is positive when the top of the beam is in compression. In that case, positive bending moments imply that the top of the beam is in tension. Of course, the definition of top depends on the coordinate system being used.

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Thus, the rate of change of the bending moment with respect to x is equal to the shearing force, or the slope of the moment diagram at the given point is the shear at that point. Thus, the rate of change of the shearing force with respect to x is equal to the load or the slope of the shear diagram at a given point equals the load at that point. Properties of Shear and Moment Diagrams The following are some important properties of shear and moment diagrams:. Sign Convention The customary sign conventions for shearing force and bending moment are represented by the figures below. A force that tends to bend the beam downward is said to produce a positive bending moment. A force that tends to shear the left portion of the beam upward with respect to the right portion is said to produce a positive shearing force. An easier way of determining the sign of the bending moment at any section is that upward forces always cause positive bending moments regardless of whether they act to the left or to the right of the exploratory section.

After the support reactions are calculated, the shear force and bending moment diagrams can be drawn. Shear force is the force in the beam acting perpendicular to its longitudinal x axis. For design purposes, the beam's ability to resist shear force is more important than its ability to resist an axial force. Axial force is the force in the beam acting parallel to the longitudinal axis. The following is a drawing of a simply-supported beam of length L under a uniform load, q:. This beam has the following support reactions:.

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Problem Bending Moment and Shear force. Problem Find the reactions at the supports for a simple beam as shown in the diagram. Weight Find the reactions for the partially loaded beam with a uniformly varying load shown in. Figure.

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In solid mechanics , a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend.