Most structural elements in practice are exposed to loads that produce a combination of different internal stresses due to tension, compression shear, flexure, torsion, fatigue, etc.
Flexural stresses, also known as bending stresses, are commonly encountered in beams and slabs, normally in combination with shear stresses. Compression members, such as columns, are usually exposed to axial, shear, and bending stresses. This article is an overview of the bending moment concept and its application in practical structural design. We will use ASDIP CONCRETE and ASDIP STEEL to support our discussion. You can click here to start a free 15-day trial of ASDIP software.
What is a bending moment?
In short, the definition of bending moment is a measure of the effect produced by a load which tends to bend a structural member. The higher the bending moment at a given beam section, the higher the probability that the beam will bend at that location. A simply supported beam loaded with a uniform load will be more likely to bend at the midspan than at the supports, so it's said that the maximum bending moment occurs at the midspan.
Engineers quantify the bending moments in terms of k-ft or KN-m, in other words, in force-distance units. Therefore a moment can be represented by a couple of parallel and opposite forces P acting at a distance x between them, or M = P x. When the curvature of the beam produces tension at the bottom, the bending moment is positive. When the curvature of the beam produces tension at the top, the bending moment is negative, as shown below.
The philosophy in the current design codes is to identify all possible limit states, or failure modes, and then quantify the corresponding internal force, either axial, shear, moment, etc. This value will then be compared versus the corresponding structural strength for compliance.
Bending moment equations
Loads, shears, and bending moments are closely related by the well known equations below, where q is the load, V is the shear, and M is the bending moment:
The first formula above is the load-shear relationship, and it indicates that the shear difference between two points equals the corresponding area of the load diagram. Likewise, the second formula is the shear-moment relationship, and it indicates that the moment difference between two points equals the corresponding area of the shear diagram. If we remove the integrals, V = dM/dx indicates that the shear value at any point represents the slope of the moment diagram at that point. This is useful to identify the location of the maximum moment, which corresponds to zero slope, or V = 0.
Depending on the applied loads, both the shear forces and bending moments vary along the beam to comply with the static equilibrium of forces. Formulas have been developed to calculate the shear forces, bending moments, and deflections for single-span beams with different loading and support conditions. These equations are useful to quickly calculate the maximum internal forces in the beam.
Below are shown some examples of bending moment equations. More cases can be found in multiple publications. Note that the typical provided information includes formulas for reactions, shear, bending moment, and deflection, as well the location of maximum values.
Shear and bending moment diagrams
Since the shears and bending moments vary along the beam, the best way to represent their values at any point is by the use of shear and bending moment diagrams, as shown in the examples above.
For continuous beams, the generation of the shear and moment diagrams may become complex and time-consuming if done by hand, particularly because the negative moments and reactions at the supports are unknown. Many analytical and numerical methods have been developed to calculate the shear and bending moments along a continuous beam. At the end, the shear and bending moment diagrams will reflect the static equilibrium and the compatibility of deformations in the system.
Some simple cases of continuous beams can be found in some publications, as the case shown below, but for unequal span lengths or more complex types of loads it's necessary to develop the shear and bending moment diagrams from scratch.
Calculating shear and moment diagrams in ASDIP
The shear and bending moment diagrams can be calculated using ASDIP software for continuous beams with multiple types of loads and support conditions. ASDIP shows the loads, reactions and diagrams per load combination. The example below shows a 3-span continuous beam with a cantilever, under the action of multiple uniform and concentrated loads. The calculated support reactions and diagrams correspond to the load combination shown.
The image updates with any change of the input data, so the effect of the change can be evaluated immediately. It can be sorted by load combination, and also it can show the entire beam, as shown below, or just a specific span. For concrete beams, ASDIP CONCRETE has the option to show the shear and bending strengths as a background of the corresponding diagrams, for visual comparison purposes. This way any problem can be quickly identified.
The calculation of the shear and bending moment diagrams is very important for the design of flexural structural elements, such as beams and slabs. Using ASDIP software to calculate these diagrams is quick, simple, and efficient.
For software usage, please read the blog post Continuous Beam Design Using ASDIP CONCRETE. For a beam design example, please see the blog post Continuous Beam Design Example Using ASDIP CONCRETE. For our collection of blog posts about concrete design please visit Structural Concrete Design.
Javier Encinas, PE
ASDIP Structural Software