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Article / Updated 03-20-2024

In mechanics of materials, Hooke's law is the relationship that connects stresses to strains. Although Hooke's original law was developed for uniaxial stresses, you can use a generalized version of Hooke's law to connect stress and strain in three-dimensional objects, as well. Eventually, Hooke's law helps you relate stresses (which are based on loads) to strains (which are based on deformations). For a three-dimensional state of stress, the normal strain in a given direction (such as x) is a function of the stresses in all three orthogonal directions (usually the Cartesian x-, y-, and z-directions), as shown by this equation: where E is the modulus of elasticity and ν is Poisson's ratio for the material. For a uniaxial stress, two of the stresses in the equation are zero. For a biaxial stress condition, one of the stresses in this equation is zero. The generalized relationship for Hooke's law for shear in the XY plane can be given as

Cheat Sheet / Updated 02-25-2022

Students and professional engineers in the mechanical sciences know that mechanics of materials deals extensively with stress on objects — from determining stress at a particular point to finding stresses in columns. Knowing how to apply some important laws and graphic representations can help you tackle stressful mechanics of materials problems with ease.

Cheat Sheet / Updated 02-02-2022

As with any branch of physics, solving statics problems requires you to remember all sorts of calculations, diagrams, and formulas. The key to statics success, then, is keeping your shear and moment diagrams straight from your free-body diagrams and knowing the differences among the calculations for moments, centroids, vectors, and pressures.

Article / Updated 08-11-2021

In the world of bird watching, a field trip is defined as going afield — that is, beyond your immediate home surroundings or backyard to observe birds. For many beginning bird watchers, their first organized field trip is with a bird club. This field trip can be an educational experience as you observe how other bird watchers act in the field, how they spot and identify birds, and where they go to find birds. In the more traditional sense, a bird-watching field trip is going to a particularly good birding area for several hours or several days. Some folks decided to go see some birds in Montana, which was a long drive from their Pennsylvania home. They liked it so much that they stayed. Now that's the field trip of a lifetime! Load the car with food, coats, boots, and other gear, more food, cold drinks of all types, the optics, a field guide or two, and yourselves, if you can fit. Then head off to see what you can see. Oftentimes you come back after a long day of bird watching exhausted, sunburned, and happy. Sometimes, your luck just runs out and you get to a birding spot that has no birds! This happens a lot in the heat of a midsummer's day, or in the dead of winter. Here are some suggestions for keeping yourself interested on a birdless day: If it's sunny and warm, look for butterflies. If it's the middle of winter, go find a warm, greasy-spoon diner and get a cup of hot chocolate. You can count the days until spring. If you have common sense, you can plan a field trip just fine. Here's a quick checklist to help you: Plan where you want to go. Plot your route and determine your schedule (early wake-up and departure to get there for prime dawn birding, and so on). Gather binoculars, spotting scope, field guide, and bird checklist (if you use one). Check the weather and plan the clothing and outerwear you'll need. Then take one extra layer. Make sure that you have the right footwear (boots, rubber boots, extra socks). Pack or wear a hat with a visor. Add food, glorious food. Take some even if you plan to eat in restaurants or at quickie-marts. Get stuff to drink. You'll be thirsty more often than not. Take money. You may need cash for an emergency (especially coins for a phone call). Include other gear for comfort or necessity. Depending on the weather, the gear can include sunscreen, lip balm, sunglasses, binocular rain guard, emergency survival kit, and other stuff. Tell someone where you're going. You can't be too safe. Get a magnetic key-holder, put an extra set of keys in it, and affix it under the front bumper of your vehicle. When traveling, always pocket the keys before you slam the doors; the aggravation you save may be your own.

Article / Updated 03-26-2016

Birds may stop by a place where they see other birds (that's why ducks check out a pond with decoys), but if that spot doesn't supply their needs (food, water, shelter, a place to breed), they move on. What you're looking for is a place with a lot of different kinds of birds and an element of surprise: where odd birds sometimes show up, far from their normal haunts. Location, location, location Typically, what makes a place great for birding is either its location or its habitat. Geography turns some places into big concentration points for migrating birds. A peninsula on the coast or on the shore of a major lake can be a place where birds pause before setting out on their next flight, especially under certain weather conditions. A wooded park in the middle of a large city may have the same kind of concentrating effect during the migration season. One of the best migration hotspots in the Northeast is Central Park in New York City. It doesn't offer an expanse of habitat, but what it does have — green trees and thick underbrush, water, and shelter — is the only choice migrant birds have. The park is surrounded by miles of concrete, skyscrapers, bustling traffic, and people. During spring migration, incredible concentrations of birds can be found in this oasis in the center of Manhattan. Habitat More reliable for birding at other seasons is a place with really good habitat. Of course, no two kinds of birds will see "good habitat" in exactly the same way. A horned lark may be happy in a plowed field, while a wood thrush seeks out the forest shadows. Therefore, habitat variety is a key point. It helps to have lots of plant life: A woodland with a brushy understory is likely to hold more birds than a park with nothing but mowed lawns under the trees. Water is almost always an element of good bird habitat. A stream or small pond attracts certain types of birds while a large marsh or lake attracts many more. Odd spots Some kinds of hotspots for bird watching can seem bizarre to normal people. But if you pursue birds for long, you'll find yourself comparing notes with other birders about your favorites among the sewage ponds and garbage dumps you've visited. But if you're trying to get a friend interested in birds, it's probably best not to take them to such a place on the first date. Trial and error is one way to find hotspots. You can pick a random spot and hang around to see if any birds show up. But birds are practically everywhere, so it can take a while to figure out whether a spot is average or great. In most parts of North America, good birding spots are already known. Nearly all bird watchers are willing to share the good places. It's not like fishing — if too many people go fishing at one spot, they may use up the fish. You're not likely to use up all the birds. Go clubbing Unless you live in a really remote region, a bird club is probably located somewhere near you. Bird watchers with local experience can give you a big head start in finding the hotspots. They may be able to recommend a nearby park, lake, refuge, or other location where the birding is great. Read up You also may be able to find published information on where to go birding. Hundreds of bird-finding guides are available for North America — some cover large areas, such as several states or provinces; others detail the birding spots in a small area, such as a single county. If you don't know any bird watchers who can advise you about such guides, your state or provincial wildlife department may be able to steer you to these publications. Lewis-and-Clark it You can, of course, discover your own hotspots. If you develop an eye for habitat, you may recognize productive spots at first sight — you may predict, for example, that a marshy pond next to a forest is likely to be good. You may predict that an empty parking lot is likely to be a lousy bird habitat — unless it happens to be next to a fast-food restaurant, in which case the gulls are probably going to show up soon. Unusual usual places for birds Birds can commonly be found in spots you wouldn't think to look. Here are some of them: * Sewage treatment plants and ponds: Fertile water means lots of food for birds. * Landfills: Gulls are legendary garbage pickers. Sometimes the regular gull species attract their rarer fellow gulls. * Urban boating/shipping harbors: Lots of handouts for gulls, ducks, geese, and swans. * Urban/suburban lakes, ponds, and reservoirs: Calm water with little or no disturbance from watercraft. Birds like that. * Vacant lots or old industrial sites that are overgrown: Hawks, owls, killdeer, swallows, and sparrows — lots of species move in when humans move out. * Arboretums and tree farms, even in urban areas: Great habitat for attracting all kinds of birds. Can be especially good for owls in winter and warblers in spring. * Cemeteries: Quiet places with good habitat where birds can feed and rest in peace. * Roadside rest stops: Ornamental plantings and the presence of water attract birds. Convenient bathrooms attract bird watchers. * Powerline right-of-way cuts: These often pass right through dense woodland habitats. Poles and lines provide perches for birds. * Edge habitats along highways and roadways: Any edge habitat is good for birds. Notice how many red-tailed hawks you see along the highway. The grassy medians are perfect for mammal-hunting birds of prey.

Article / Updated 03-26-2016

Shiny, clanky, commercially available bird feeders have a lot going for them. First and foremost, they keep the seed fresh and dry. It can be snowing like mad, covering up your bird table and ground feeding area in seconds, and a good hanging feeder will still be calmly dispensing seed to the hungry. Commercial feeders come in three basic types: hopper, tube, and satellite. Hoppers (not Dennis) Hopper feeders have a lot of different styles, but the old favorite looks like a little barn or covered bridge, minus the young lovers and graffiti. The sides are usually panels of clear acrylic that allow you to see how much seed remains, and access is usually through the top (see Figure 1). Figure 1: Hopper feeders are old favorites. A downy woodpecker (left) and a male rose-breasted grosbeak are at this one. A good hopper feeder can be disassembled for cleaning. Hoppers can be pole-mounted, often with a threaded sleeve that screws onto the threaded top of a plumber's galvanized pipe. They can be suspended, too. The two best features of hoppers are the following: They hold a lot of seed, so that you don't have to go out every day to refill them. They're big and bird-friendly. Shy birds, or big birds like doves and jays and woodpeckers, are able to land and feed from them comfortably. When unusual species such as rose-breasted grosbeaks show up at a feeder, they'll usually come to hopper feeders. Birds that are reluctant to perch on tube feeders or cling to satellite feeders will happily come to a hopper feeder. You can feed any kind of seed in a hopper feeder because the seed usually comes out of slots at the bottom of the acrylic panels. Sunflower seed is a favorite of many hopper feeder visitors, but seed mixes containing millet, corn, and peanut hearts can be fed in these feeders, too. Totally tubular Tube feeders — long cylinders with perches at the feeding ports — are the classic feeders for woodland birds like chickadees, titmice, woodpeckers, and nuthatches, as well as for finches like goldfinches, siskins, and house finches. All these birds are small, and they can hang out comfortably on the usually short metal perches most tube feeders have (see Figure 2). Figure 2: Small, seed-eating birds, such as finches and chickadees, like tube feeders. Tube feeders are great for screening out big birds like blue jays, grackles, blackbirds, and doves, if you're into avian discrimination. But they also keep grosbeaks and cardinals away, too, because these birds aren't so good at clinging, and they're just too big for the perches. They'll go to the hopper or platform feeders, or onto the ground. When you're looking at tube feeders, make sure the seed you're planning to put in them will fit through the holes at the feeding ports. Most have big holes that let sunflower seed through, but others are made especially for the tiny thistle or niger seed. Satellites (going global) Speaking of tiny, if you want to cater only to little birds like chickadees, titmice, nuthatches, goldfinches, and siskins, then pick a globe feeder. These look like flying saucers, or satellites, and they're suspended from a wire so that they spin when a bird lands on them (see Figure 3). Dizzying, but no problem for all these little clinging birds. Figure 3: Clinging birds can hang from a globe feeder. If you're sick of shoveling feed into the maws of flocks of house finches, try a satellite feeder. Birds must approach them from the bottom and cling upside-down to feed from them. But you fill them with seed from a small, capped opening in the top. Some satellite feeders have a domed baffle on top to keep squirrels out, too. House finches just can't do the cling thing; their legs aren't strong enough. So if you see your chickadees politely trying to fit in edgewise among the house finches at the tube feeder, hang a satellite feeder just for them. What to look for when buying Many feeder styles are on the market, but any feeder you buy should be easily filled, emptied, and cleaned. Beware of feeders that require you to use a funnel to fill them because you'll quickly tire of lugging a funnel out every time you have to replenish the seed. Wooden parts of hopper feeders should be made of weather-resistant cedar, or stained or painted to protect against moisture. Plastic feeders should be reinforced with metal around the feeding ports to ward off chewing squirrels. Perches should be metal or replaceable dowel, for the same squirrelly reasons. Because you may be looking at a feeder for a decade or more, it pays to buy the sturdiest and most easily maintained one you can. With tube feeders, look at the bottom port. Is there dead space beneath it where seed can collect because the birds can't reach it? This seed gets all icky and moldy — a waste of food, and dangerous for the birds. Can you take the feeder apart to scrub and clean it? If it looks like you'll need fancy bottle brushes or an act of Congress to get it clean, pick one of simpler design. Beware of super-cheap feeders. Not all of the $5 to $10 feeders available for sale are going to last you for more than one season of feeding. Remember, these things are going to be filled with seed and hung out in the weather. Look for durable construction if you want to get your money's worth. By now, a picture should be emerging of an ideal feeding station — one with a variety of different feeder styles, at different heights, where the greatest diversity of birds can find food. It should provide shelter from wind, rain, and predators in the form of shrubbery or the instant habitat of brushpiles and found materials. It should have a wide variety of foods, and some water should be available year-round.

Article / Updated 03-26-2016

The best place to start watching birds is close to home. Birds are present in most yards and gardens, no matter how small or urban the space may be. But why settle for the random robin? If you provide birds with a few perks and pleasures, they'll hang around your yard because it's a nice place to be. Birds need four basic things to survive: Food, water, shelter, and a place to nest. You can attract birds with these four offerings, no matter where you live, even in the urban heart of a large city. The first three — food, water, and shelter — are fairly simple to offer. The last, a place to nest, requires a bit more effort, but offers far greater rewards to you and the birds. Eating like a bird Consider your bird feeding to be an ongoing experiment. Use trial and error to determine what foods your birds like. Also experiment with how you offer the foods to birds, because not all birds like to eat in the same way. Not all people like lasagna, or drumsticks, or drive-through windows, or fancy restaurants, and not all birds like all bird foods or all bird feeders. Where you live in North America has a bearing on what birds you can attract to your yard. The birds at the feeders in a rural Ohio farmyard are very unlike those in a central California yard. The latter gets California quail and yellow-billed magpies, the former cardinals and blue jays. Seeds. Bird feeding involves seeds. Black-oil sunflower seed is the most universally used seed for bird feeding because it's eaten by many feeder visitors, including chickadees, titmice, finches, grosbeaks, cardinals, jays, nuthatches, and woodpeckers, among others. Bird feeding's other popular seeds include: striped sunflower seeds, sunflower hearts, millet, safflower, cracked corn, peanuts and peanut bits, niger or thistle seed, and mixed seed containing milo, wheat, millet, and cracked corn. Feeders. Today, as many feeder types and styles are available as there are sunflower kernels in a 50-pound bag of seed. The ideal feeding station has: A large platform feeder a few feet off the ground with sunflower seed and mixed seed on it A tube feeder or two offering sunflower seed, sunflower hearts, or thistle seed A hopper feeder dispensing sunflower seed A satellite feeder with sunflower seed, hearts, or peanut bits An area of open ground for scattering mixed seed, with a nearby shelter or brushpile Start simply, and then expand the feeding operation at your own pace. After you've got the basics down, you and your birds can move on to other areas of bird feeding, including offering fruits, nuts, suet, and specialized foods and feeders to cater to (or discourage) certain feeder visitors. Getting water Few creatures on the planet can survive without water. Birds are no different. Birds need water for drinking, to help with digestion, and for bathing (see Figure 1). Figure 1: An American goldfinch and a brown thrasher enjoy a birdbath with a dripper. Fine tubing connects to an adapter at the hose spigot; a stopcock adjusts the rate of drip. There's nothing worse for a bird than a bunch of dirty, matted, haven't-been-washed-in-days feathers. Dirty feathers don't insulate well, they don't offer optimal flight, and, let's face it, dirty feathers don't look good. You can do your part for avian hygiene by offering water to the birds in your yard. If you don't have a naturally occurring stream or pond on or near your property, consider putting out a birdbath. Remember the following for an ideal birdbath set-up: The bath should have a large, shallow, cement stone, or composite bowl elevated slightly above ground. By shallow, say less than 3 inches deep, and the shallower the better. (Most birds don't like the deep end.) The surface of the bath shouldn't be so smooth that birds can't get a secure footing. If it feels slippery to you, consider roughing up the surface of the bath a bit. Or consider adding a few handfuls of small, coarse gravel to the bottom of the bath. This makes for better bird footing. Flat rocks should be placed in the water to provide shallow areas and secure footing. Not all birds like total-immersion bathing. If you can't put your birdbath in the shade of a tree, place it near some type of cover. Birds get nervous when they're all wet (wet birds can't fly well). Shelter makes them feel better and, if shelter is near, they're more likely to stop and bathe than to sip and fly, or not stop at all. Regularly clean your birdbath, at least once a week. Birds drink this water. If you had to drink out of your tub, wouldn't you rather it had just been cleaned? Gimme shelter Suppose you're a white-throated sparrow absent-mindedly kicking through the seeds scattered on the ground below a feeding station. Suddenly, one of those annoyingly perky black-capped chickadees sitting on the tube feeder above you gives an alarm call that means a hawk is nearby. Swooooooosh! Lucky for you, the kind human owner of this backyard has placed the feeders near several evergreen trees and a humongous brushpile (see Figure 2). Figure 2: This brushpile provides welcome respite for a tree sparrow and an eastern towhee. Good shelter. That's why you liked this yard and decided to stop here for a bite to eat. Being a denizen of the brushy field edges, you calmly hop into the center of the brushpile and wait for the sharpie to get bored (or lucky) and leave. A backyard can have all the perfect feeders and best bird foods, but the birds will ignore it if no decent shelter, or cover, is available nearby. They're not stupid. Like people, birds need shelter from bad weather. And birds and people want a cozy place to sleep, roost, and hide from predators. Shelter can come in many forms: weedy areas, shrubs and brush, trees, brushpiles, woods, and even buildings (barn owls and barn swallows got their names in this way). When you look at the setup of your feeding station, try thinking like a bird: Where is the nearest place you can go to hide from danger? What about in bad weather? Are the feeders exposed to direct wind, snow, or rain? If no shelter is convenient to the area of the feeders, create some shelter, such as an instant brushpile. Nest boxes: Making a house a home All birds that venture into your yard benefit from a varied habitat. Some species take advantage of human-made shelter — commonly called birdhouses or nest boxes. Birdhouses come in hundreds of shapes and sizes, designs, and colors. As you get to know the birds in your yard, you can target the housing you provide to maximize the birds' benefit. Among the species that use housing are bluebirds, chickadees, titmice, woodpeckers, nuthatches, swallows, some flycatchers, and even a few warblers, ducks, hawks, and owls. But remember, not all birds use birdhouses, just as not all birds visit bird feeders. Any experienced bird watcher will tell you that you find the most birds in the "edge habitat." Edge habitat is the area where two or more habitat types meet, such as where a meadow habitat meets the edge of a woodland habitat. Or where a thick, overgrown brushy area abuts a roadway. The greatest variety of habitat occurs where habitat types meet, so it's not just a coincidence that edge habitat is where the birds are. Birds preferring each individual type of habitat can potentially be in the place where these habitats meet. And with some creative thinking, that place could be in your backyard.

Article / Updated 03-26-2016

Solving statics problems can be complicated; each problem requires a list of items to account for and equations to create and solve. Solve statics problems with ease by using this checklist: Draw a free-body diagram of the entire system. In addition to dimensions and angles, you must include four major categories of items on a properly constructed free-body diagram: Applied external loads Revealed internal loads Support reactions Self weight Write equilibrium equations to compute as many unknown support reactions as possible. To solve for internal forces, identify the type of structure and write your equilibrium equations. After you identify the type of structure, you then know which technique to use to help you write your equilibrium equations: Trusses/axial members: Members are loaded with internal axial forces only. To solve, you can use the method of joints or the method of sections. Beams (bending members): Members are loaded with internal axial forces, shear forces, and moments. To solve, cut the member at the desired location, draw a new free-body diagram of the cut section, and write equilibrium equations. Frames/machines: Members are loaded with internal axial forces, shear forces, and moments. To solve, use the blow-it-all-apart approach to break the structure into smaller pieces. Look for internal hinges as common places to separate your structure and draw free-body diagrams to solve for the connecting pin forces. Cable structures: Members are constructed from axially loaded cables. Identify the type of cable loading (concentrated, parabolic/uniform, or catenary). Compute the cable tension at the location of known maximum sag (or vice versa). Submerged surfaces: Members are subjected to fluid pressure. To solve, draw a free-body diagram of the hydrostatic pressure diagram which is zero at the fluid surface and linearly increases with depth. Include the weight of the fluid on objects with non-vertical faces.

Article / Updated 03-26-2016

When working submerged surface problems in statics, remember that all submerged surfaces have a fluid acting upon them, causing pressure. You must compute two pressures: the hydrostatic pressure resultant and the fluid self weight. Hydrostatic pressure resultant: The hydrostatic pressure resultant acts horizontally at 0.333z (z is the depth of the fluid) from the bottom of the pressure distribution. Fluid self weight: The fluid self weight is acting downward at the centroid (geometric center) of the area of fluid.

Article / Updated 03-26-2016

In many statics problems, you must be able to quickly and efficiently create vectors in the Cartesian plane. Luckily, you can accomplish your Cartesian vector creations easily with the handy vector formulas in this list: Force vectors and distance vectors are the most basic vectors that you deal with.