where b = 2r is the diameter (width) of the cross section, Ic = πr4/4 is the centroidal moment of inertia, and A = πr2 is the area of the cross section.

When the beam is cut at the section, either side of the beam may be considered when solving for the internal reactions. The side that is selected does not affect the results, so choose whichever side is easiest. In the figure above, the side of the beam to the right of the section cut was selected. The selected side is shown as the blue section of beam, and section shown in grey is ignored. The internal reactions at the section cut are shown with blue arrows. The reactions are calculated such that the section of beam being considered is in static equilibrium.

Understanding how a camera lens works is fundamental for anyone interested in photography, whether as a hobby or a profession. The camera lens is a crucial component that determines the quality and characteristics of the images captured. This article will delve into the mechanics of camera lenses, their types, and how they influence the final photograph. By the end of this article, you will have a comprehensive understanding of how camera lenses work and how to choose the right one for your needs.

The signs of the shear and moment are important. The sign is determined after a section cut is taken and the reactions are solved for the portion of the beam to one side of the cut. The shear force at the section cut is considered positive if it causes clockwise rotation of the selected beam section, and it is considered negative if it causes counter-clockwise rotation. The bending moment at the section cut is considered positive if it compresses the top of the beam and elongates the bottom of the beam (i.e., if it makes the beam "smile").

Based on the above discussion, we can see that a fixed boundary condition can develop axial and transverse reaction forces as well as a moment. Likewise, we see that a pinned boundary condition can develop axial and transverse reaction forces, but it cannot develop a reaction moment.

Every lens has an aperture range where it performs best in terms of sharpness and minimal aberrations. This is often referred to as the "sweet spot." Typically, it is found two to three stops down from the maximum aperture. Experiment with different apertures to find your lens's sweet spot.

Prime lenses have a fixed focal length, meaning they do not zoom in or out. They are known for their sharpness, wide apertures, and compact size. Prime lenses are ideal for portrait photography, low-light conditions, and achieving a shallow depth of field.

Wide-angle lenses have short focal lengths, typically ranging from 10mm to 35mm. They capture a wide field of view, making them perfect for landscape photography, architecture, and interior shots.

where Ic = b·h3/12 is the centroidal moment of inertia of the cross section. The maximum shear stress occurs at the neutral axis of the beam and is calculated by:

The equations for shear stress in a beam were derived using the assumption that the shear stress along the width of the beam is constant. This assumption is valid over the web of an I-Beam, but it is invalid for the flanges (specifically where the web intersects the flanges). However, the web of an I-Beam takes the vast majority of the shear force (approximately 90% - 98%, according to Gere), and so it can be conservatively assumed that the web carries all of the shear force.

This section covers shear force and bending moment in beams, shear and moment diagrams, stresses in beams, and a table of common beam deflection formulas.

Understanding how a camera lens works is essential for capturing high-quality images. The lens plays a pivotal role in focusing light, determining the field of view, and influencing image characteristics. By familiarizing yourself with different types of lenses and their features, you can make informed decisions and elevate your photography skills. Whether you're a beginner or an experienced photographer, mastering the use of camera lenses will open up new creative possibilities and help you achieve your desired photographic outcomes.

The shear stress is zero at the free surfaces (the top and bottom of the beam), and it is maximum at the centroid. The equation for shear stress at any point located a distance y1 from the centroid of the cross section is given by:

The shear force, V, along the length of the beam can be determined from the shear diagram. The shear force at any location along the beam can then be used to calculate the shear stress over the beam's cross section at that location. The average shear stress over the cross section is given by:

Selecting the right lens depends on your photography style and the subjects you intend to capture. Here are some considerations to keep in mind:

Dust, fingerprints, and smudges can degrade image quality. Regularly clean your lens using a microfiber cloth and lens cleaning solution. Avoid touching the glass elements with your fingers.

The figure below shows the standard sign convention for shear force and bending moment. The forces and moments on the left are positive, and those on the right are negative.

Consider the size and weight of the lens, especially if you plan to carry it around for extended periods. Prime lenses are generally more compact and lightweight compared to zoom lenses.

where M is the bending moment at the location of interest along the beam's length, Ic is the centroidal moment of inertia of the beam's cross section, and y is the distance from the beam's neutral axis to the point of interest along the height of the cross section. The negative sign indicates that a positive moment will result in a compressive stress above the neutral axis.

If the beam is asymmetric about the neutral axis such that the distances from the neutral axis to the top and to the bottom of the beam are not equal, the maximum stress will occur at the farthest location from the neutral axis. In the figure below, the tensile stress at the top of the beam is larger than the compressive stress at the bottom.

Identify the primary purpose of the lens. Are you shooting landscapes, portraits, sports, or macro photography? Each genre has specific lens requirements.

Zoom lenses have a variable focal length, allowing you to zoom in and out without changing the lens. They offer versatility and convenience, making them suitable for various photography genres, including landscapes, sports, and wildlife.

Ensure that the lens is compatible with your camera body. Different camera brands have their own lens mounts, so double-check compatibility before making a purchase.

Telephoto lenses have long focal lengths, usually starting from 70mm and going up to 600mm or more. They are designed to magnify distant subjects, making them ideal for wildlife, sports, and portrait photography.

If you have a zoom lens, experiment with different focal lengths to see how they affect composition and perspective. Wide-angle lenses can exaggerate the sense of space, while telephoto lenses compress the scene and bring distant subjects closer.

Notice the Free boundary condition in the table above. This boundary condition indicates that the beam is free to move in every direction at that point (i.e., it is not fixed or constrained in any direction). Therefore, a constraint does not exist at this point. This highlights the subtle difference between a constraint and a boundary condition. A boundary condition indicates the fixed/free condition in each direction at a specific point, and a constraint is a boundary condition in which at least one direction is fixed.

Common boundary conditions are shown in the table below. For each boundary condition, the table indicates whether the beam is fixed or free in each direction at the point where the boundary condition is defined.

The equations for shear stress in a beam were derived using the assumption that the shear stress along the width of the beam is constant. This assumption is valid at the centroid of a circular cross section, although it is not valid anywhere else. Therefore, while the distribution of shear stress along the height of the cross section cannot be readily determined, the maximum shear stress in the section (occurring at the centroid) can still be calculated. The maximum value of first moment, Q, occurring at the centroid, is given by:

If the boundary condition indicates that the beam is fixed in a specific direction, then an external reaction in that direction can exist at the location of the boundary condition. For example, if a beam is fixed in the y-direction at a specific point, then a transverse (y) external reaction force may develop at that point. Likewise, if the beam is fixed against rotation at a specific point, then an external reaction moment may develop at that point.

The bending stress is zero at the beam's neutral axis, which is coincident with the centroid of the beam's cross section. The bending stress increases linearly away from the neutral axis until the maximum values at the extreme fibers at the top and bottom of the beam.

For a beam to remain in static equilibrium when external loads are applied to it, the beam must be constrained. Constraints are defined at single points along the beam, and the boundary condition at that point determines the nature of the constraint. The boundary condition indicates whether the beam is fixed (restrained from motion) or free to move in each direction. For a 2-dimensional beam, the directions of interest are the x-direction (axial direction), y-direction (transverse direction), and rotation. For a constraint to exist at a point, the boundary condition must indicate that at least one direction is fixed at that point.

The aperture is the opening in the lens through which light enters the camera. It is measured in f-stops (e.g., f/2.8, f/4, f/5.6). A larger aperture (smaller f-stop number) allows more light to enter, which is beneficial in low-light conditions and for achieving a shallow depth of field. Conversely, a smaller aperture (larger f-stop number) allows less light and provides a greater depth of field.

The focal length of a lens is the distance between the lens and the image sensor when the subject is in focus. It is usually measured in millimeters (mm). The focal length determines the field of view and magnification of the image. Lenses with shorter focal lengths (wide-angle lenses) capture a broader scene, while lenses with longer focal lengths (telephoto lenses) magnify distant subjects.

Lenses can vary significantly in price. Determine your budget and look for lenses that offer the best value for your money. Remember that investing in high-quality lenses can make a noticeable difference in your photography.

The maximum value of shear stress occurs at the neutral axis ( y1 = 0 ), and the minimum value of shear stress in the web occurs at the outer fibers of the web where it intersects the flanges y1 = ±hw/2 ):

Macro lenses are specialized lenses designed for close-up photography. They have a high magnification ratio, allowing you to capture intricate details of small subjects like insects, flowers, and textures.

The benefit of the section modulus is that it characterizes the bending resistance of a cross section in a single term. The section modulus can be substituted into the flexure formula to calculate the maximum bending stress in a cross section:

The shear force and bending moment throughout a beam are commonly expressed with diagrams. A shear diagram shows the shear force along the length of the beam, and a moment diagram shows the bending moment along the length of the beam. These diagrams are typically shown stacked on top of one another, and the combination of these two diagrams is a shear-moment diagram. Shear-moment diagrams for some common end conditions and loading configurations are shown within the beam deflection tables at the end of this page. An example of a shear-moment diagram is shown in the following figure:

The external reactions should balance the applied loads such that the beam is in static equilibrium. After the external reactions have been solved for, take section cuts along the length of the beam and solve for the internal reactions at each section cut. (The reaction forces and moments at the section cuts are called internal reactions because they are internal to the beam.) An example section cut is shown in the figure below:

There are several types of camera lenses, each designed for specific purposes. Understanding these types will help you choose the right lens for your photography needs.

A camera lens is essentially a curved piece of glass or other transparent material that focuses light onto the camera's sensor or film. The primary function of the lens is to gather light rays from the scene and bend them to form a sharp image on the sensor. The lens achieves this through a combination of refraction and focusing mechanisms.

Modern lenses consist of multiple glass elements arranged in groups. These elements are designed to correct various optical aberrations, such as chromatic aberration, distortion, and vignetting. Additionally, lens coatings are applied to reduce reflections, flare, and ghosting, enhancing contrast and color accuracy.

The bending moment, M, along the length of the beam can be determined from the moment diagram. The bending moment at any location along the beam can then be used to calculate the bending stress over the beam's cross section at that location. The bending moment varies over the height of the cross section according to the flexure formula below:

where V is the shear force acting at the location of the cross section, Ic is the centroidal moment of inertia of the cross section, and b is the width of the cross section. These terms are all constants. The Q term is the first moment of the area bounded by the point of interest and the extreme fiber of the cross section:

Depth of field refers to the range of distance within a photo that appears acceptably sharp. A shallow depth of field (achieved with a wide aperture) isolates the subject from the background, creating a pleasing bokeh effect. A deep depth of field (achieved with a narrow aperture) keeps more of the scene in focus, which is ideal for landscapes.

The tables below give equations for the deflection, slope, shear, and moment along straight beams for different end conditions and loadings. You can find comprehensive tables in references such as Gere, Lindeburg, and Shigley. However, the tables below cover most of the common cases.

Some lenses come with built-in image stabilization (IS) technology, which compensates for camera shake. This feature is particularly useful when shooting handheld or in low-light conditions, as it helps produce sharper images.

General rules for drawing shear-moment diagrams are given in the table below. All of the rules in this table are demonstrated in the figure above.

PDH Classroom offers a continuing education course based on this beam analysis reference page. This course can be used to fulfill PDH credit requirements for maintaining your PE license.

To find the shear force and bending moment over the length of a beam, first solve for the external reactions at each constraint. For example, the cantilever beam below has an applied force shown as a red arrow, and the reactions are shown as blue arrows at the fixed boundary condition.

where b = 2 (ro − ri) is the effective width of the cross section, Ic = π (ro4 − ri4) / 4 is the centroidal moment of inertia, and A = π (ro2 − ri2) is the area of the cross section.

Refraction is the bending of light as it passes through different mediums. In the case of a camera lens, light enters through the front element and is bent or refracted by the lens elements inside. These elements are carefully shaped and positioned to direct the light rays to converge at a single point, known as the focal point.

Many structures can be approximated as a straight beam or as a collection of straight beams. For this reason, the analysis of stresses and deflections in a beam is an important and useful topic.

Based on this sign convention, the shear force at the section cut for the example cantilever beam in the figure above is positive since it causes clockwise rotation of the selected section. The moment is negative since it compresses the bottom of the beam and elongates the top (i.e., it makes the beam "frown").

We can see from the previous equation that the maximum shear stress in the cross section is 50% higher than the average stress V/A.