Shear Force and Bending Moment as Structural Basics

Basic structural learning begins with an analyzing of a simply supported beam. A beam is a structural member (horizontal) that is design to support the applied load (vertical). It resists the applied loading by a combination of internal transverse shear force and bending moment. An accurate analysis required in order to make sure the beam is construct without any excessive loads which affect its strength.

Types Of Load and Support

The Men-at-Work in construction...

Two types of typical loadings:

• Concentrated load is one which can be considered to act at a point although of course in practice it must be distributed over a small area (normally vertical or incline loads). (Unit in kN)
• Distributed load is one which is spread in some manner over the length or a significant length of the beam. It is usually quoted at a weight per unit length of beam and it may either be uniform or varying loading from point to point. (Unit in kN/m)

Three types of support:

• Namely as Pinned support, Roller support and Fixed or Built-in support.
• Covered in previous post together the reactions explained in diagrams.

The Sign Conventions

The general sign convention of a beam.

The sign convention depends on the direction of the stress resultant with respect to the material against which it acts. It is used for both shear force and bending moments in analyzing the directions. Positive (+ve) shear forces always deform right hand face downward with respect to the left hand face and negative (-ve) would be the other way round. Positive (+ve) bending moments always elongate the lower section of the beam and negative (-ve) would elongate the mid-section upward of the beam.

Shear Force and Bending Moment in Simply Supported Beam
For a simply supported beam, the reactions are generally simple forces. When the beam is built-in, the free body diagram will show the relevant support point as a reaction force and a reaction moment. It is normal practice to produce a free body diagram with the shear diagram and bending moment diagram position.

A simply supported beam with brief details.

The unknown forces (generally the support reactions) are then determined using the Equation of Equilibrium:

• Horizontal reaction respect to x-axis; ΣF_x = 0
• Vertical reaction respect to y-axis; ΣF_y = 0
• Bending reaction (clockwise or anti-clockwise), ΣM = 0

Shear Force and Bending Moment Diagrams
The shear force diagram indicates the shear force withstood by the beam section along the length of the beam. The bending moment diagram indicates the bending moment withstood by the beam section along the length of the beam.

Shear Force and Bending Moment Diagrams in simply supported beam with two different loads.

Simple calculation of determine the reactions:
* Take example the 1st one with point load, W…
ΣM_c = 0 (taking clockwise as +ve at point C)
(R_a x L) – (W x L/2) = 0
So, R_a = W/2 [in kN]

ΣF_y = 0 (taking vertical ↑ as +ve)
R_a + R_c – W = 0
W/2 + R_c – W = 0
So, R_c = W/2 [in kN]

The shear force diagram (SFD) is simply constructed by moving a section along the beam from the left origin and summing the forces to the left of the section. The equilibrium condition states that the forces on either side of a section balance and therefore the resisting shear force of the section is obtained by this simple operation.

On the other hand, bending moment diagram (BMD) is obtained in the same way except that the moment is the sum of the product of each force and its distance of x from the section either left or right. Distributed loads are calculated buy summing the product of the total force (to the left of the section) and the x distance of the distributed load centroid.

Shear Force and Bending Moment Diagrams in cantilever beam with two different loads.

The basic procedure for determining the shear and moment is to determine the values of V and M at various sections along the beam and plotting the results from point to point. By doing so, we will be able to determine critical sections within the beam where a critical or maximum stress might occurs:

• Section of Maximum Shear – Since the shear, V, at any transverse section of the beam is the algebraic sum of the transverse forces to the left of the section, the shear, in most cases can be estimated at a glance.
• Section of Maximum Moment – It can be obtained mathematically, that when the shear force is zero or changes sign; the bending moment, M will be either a maximum or relative maximum.

In conclusion, we could obtained the reactions value at the supports by using the equation of equilibrium. The shear force and bending moment diagram used for determine both of force, V and moment, M maximum values.