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of elasticity allows the solution of structural elements of general geometry under general loading conditions, in principle. Analytical solution, however, is limited to relatively simple cases. The solution of elasticity problems also requires the solution of a system of partial differential equations, which is considerably more mathematically demanding than the solution of mechanics of materials problems, which require at most the solution of an ordinary differential equation. The finite element method is perhaps the most restrictive and most useful at the same time. This method itself relies upon other structural theories (such as the other two discussed here) for equations to solve. It does, however, make it generally possible to solve these equations, even with highly complex geometry and loading conditions, with the restriction that there is always some numerical error. Effective and reliable use of this method requires a solid understanding of its limitations.
3096:. The stiffness method is the most popular by far thanks to its ease of implementation as well as of formulation for advanced applications. The finite-element technology is now sophisticated enough to handle just about any system as long as sufficient computing power is available. Its applicability includes, but is not limited to, linear and non-linear analysis, solid and fluid interactions, materials that are isotropic, orthotropic, or anisotropic, and external effects that are static, dynamic, and environmental factors. This, however, does not imply that the computed solution will automatically be reliable because much depends on the model and the reliability of the data input.
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elements interconnected at finite number of nodes and the overall stiffness is the result of the addition of the stiffness of the various elements. The behaviour of individual elements is characterized by the element's stiffness (or flexibility) relation. The assemblage of the various stiffness's into a master stiffness matrix that represents the entire structure leads to the system's stiffness or flexibility relation. To establish the stiffness (or flexibility) of a particular element, we can use the
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girders, the floor slab, roofing, walls, windows, plumbing, electrical fixtures, and other miscellaneous attachments. The second type of loads are live loads which vary in their magnitude and location. There are many different types of live loads like building loads, highway bridge loads, railroad bridge loads, impact loads, wind loads, snow loads, earthquake loads, and other natural loads.
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only 3 members of the truss structure. This restriction is because this method uses the force balances in the x and y direction and the moment balance, which gives a maximum of 3 equations to find a maximum of 3 unknown truss element forces through which this cut is made. Find the forces FAB, FBD and FCD in the above example
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3044:. The equations of elasticity are a system of 15 partial differential equations. Due to the nature of the mathematics involved, analytical solutions may only be produced for relatively simple geometries. For complex geometries, a numerical solution method such as the finite element method is necessary.
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Since there is a pin joint at A, it will have 2 reaction forces. One in the x direction and the other in the y direction. At point B, there is a roller joint and hence only 1 reaction force in the y direction. Assuming these forces to be in their respective positive directions (if they are not in the
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There are 2 commonly used methods to find the truss element forces, namely the method of joints and the method of sections. Below is an example that is solved using both of these methods. The first diagram below is the presented problem for which the truss element forces have to be found. The second
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Each method has noteworthy limitations. The method of mechanics of materials is limited to very simple structural elements under relatively simple loading conditions. The structural elements and loading conditions allowed, however, are sufficient to solve many useful engineering problems. The theory
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The finite element method approximates a structure as an assembly of elements or components with various forms of connection between them and each element of which has an associated stiffness. Thus, a continuous system such as a plate or shell is modeled as a discrete system with a finite number of
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This method can be used when the truss element forces of only a few members are to be found. This method is used by introducing a single straight line cutting through the member whose force has to be calculated. However this method has a limit in that the cutting line can pass through a maximum of
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The solutions are based on linear isotropic infinitesimal elasticity and Euler–Bernoulli beam theory. In other words, they contain the assumptions (among others) that the materials in question are elastic, that stress is related linearly to strain, that the material (but not the structure) behaves
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include buildings, bridges, and towers; and in other branches of engineering, ship and aircraft frames, tanks, pressure vessels, mechanical systems, and electrical supporting structures are important. To design a structure, an engineer must account for its safety, aesthetics, and serviceability,
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approach. The first two make use of analytical formulations which apply mostly simple linear elastic models, leading to closed-form solutions, and can often be solved by hand. The finite element approach is actually a numerical method for solving differential equations generated by theories of
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There are two types of loads that structure engineering must encounter in the design. The first type of loads are dead loads that consist of the weights of the various structural members and the weights of any objects that are permanently attached to the structure. For example, columns, beams,
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Once the dimensional requirement for a structure have been defined, it becomes necessary to determine the loads the structure must support. Structural design, therefore begins with specifying loads that act on the structure. The design loading for a structure is often specified in
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for large rigid frames. Except for moment distribution, which came into use in the 1930s, these methods were developed in their current forms in the second half of the nineteenth century. They are still used for small structures and for preliminary design of large structures.
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composing that structure. The structural elements guiding the systemic forces through the materials are not only such as a connecting rod, a truss, a beam, or a column, but also a cable, an arch, a cavity or channel, and even an angle, a surface structure, or a frame.
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It is common practice to use approximate solutions of differential equations as the basis for structural analysis. This is usually done using numerical approximation techniques. The most commonly used numerical approximation in structural analysis is the
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3241:, H. C. Martin, and L. J. Topp's paper on the "Stiffness and Deflection of Complex Structures" introduces the name "finite-element method" and is widely recognized as the first comprehensive treatment of the method as it is known today
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3088:, plates, shells, and three-dimensional solids. Commercial computer software for structural analysis typically uses matrix finite-element analysis, which can be further classified into two main approaches: the displacement or
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mechanics such as elasticity theory and strength of materials. However, the finite-element method depends heavily on the processing power of computers and is more applicable to structures of arbitrary size and complexity.
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Since the system is in static equilibrium, the sum of forces in any direction is zero and the sum of moments about any point is zero. Therefore, the magnitude and direction of the reaction forces can be calculated.
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Elasticity methods are available generally for an elastic solid of any shape. Individual members such as beams, columns, shafts, plates and shells may be modeled. The solutions are derived from the equations of
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is the combination of structural elements and their materials. It is important for a structural engineer to be able to classify a structure by either its form or its function, by recognizing the various
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are small, and that beams are long relative to their depth. As with any simplifying assumption in engineering, the more the model strays from reality, the less useful (and more dangerous) the result.
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1378:{\displaystyle \sum F_{y}=0=-10-F_{AD}\sin(60)-F_{BD}\sin(60)=-10-\left(-{\frac {10}{\sqrt {3}}}\right){\frac {\sqrt {3}}{2}}-F_{BD}{\frac {\sqrt {3}}{2}}\Rightarrow F_{BD}=-{\frac {10}{\sqrt {3}}}}
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The simplest of the three methods here discussed, the mechanics of materials method is available for simple structural members subject to specific loadings such as axially loaded bars, prismatic
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393:. There are two types of codes: general building codes and design codes, engineers must satisfy all of the code's requirements in order for the structure to remain reliable.
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291:. In contrast to theory of elasticity, the models used in structure analysis are often differential equations in one spatial variable. Structures subject to this type of
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Early applications of matrix methods were applied to articulated frameworks with truss, beam and column elements; later and more advanced matrix methods, referred to as "
1587:{\displaystyle \sum F_{x}=0=-F_{AD}\cos(60)+F_{BD}\cos(60)+F_{CD}=-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+F_{CD}\Rightarrow F_{CD}=0}
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3021:{\displaystyle \sum F_{x}=0=-F_{AB}-F_{BD}\cos(60)-F_{CD}=-F_{AB}-\left(-{\frac {10}{\sqrt {3}}}\right){\frac {1}{2}}-0\Rightarrow F_{AB}={\frac {5}{\sqrt {3}}}}
3084:", model an entire structure with one-, two-, and three-dimensional elements and can be used for articulated systems together with continuous systems such as a
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3216:' publication of the moment distribution method which was later recognized as a form of the relaxation method applicable to the problem of flow in pipe-network
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which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of
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Although the forces in each of the truss elements are found, it is a good practice to verify the results by completing the remaining force balances.
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2576:{\displaystyle \sum F_{x}=0=F_{AB}+F_{BD}\cos(60)+F_{CD}={\frac {5}{\sqrt {3}}}-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+F_{CD}\Rightarrow F_{CD}=0}
1154:{\displaystyle \sum F_{x}=0=R_{Ax}+F_{AD}\cos(60)+F_{AB}=0-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+F_{AB}\Rightarrow F_{AB}={\frac {5}{\sqrt {3}}}}
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The truss elements forces in the remaining members can be found by using the above method with a section passing through the remaining members.
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to analyze a member undergoing combined loading. Solutions for special cases exist for common structures such as thin-walled pressure vessels.
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for more complex two- and three-dimensional elements. The analytical and computational development are best effected throughout by means of
1948:{\displaystyle \sum F_{y}=R_{B}+F_{BD}\sin(60)+F_{BC}=5+\left(-{\frac {10}{\sqrt {3}}}\right){\frac {\sqrt {3}}{2}}+0=0\Rightarrow verified}
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for computing displacement as partial derivative of the strain energy. This theorem includes the method of 'least work' as a special case
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2102:{\displaystyle \sum F_{x}=-F_{AB}-F_{BD}\cos(60)={\frac {5}{\sqrt {3}}}-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}=0\Rightarrow verified}
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465:. The solutions are approximate when any of these relations are only approximately satisfied, or only an approximation of reality.
2398:{\displaystyle \sum F_{y}=0=R_{Ay}-F_{BD}\sin(60)-10=5-F_{BD}{\frac {\sqrt {3}}{2}}-10\Rightarrow F_{BD}=-{\frac {10}{\sqrt {3}}}}
421:. This information is then compared to criteria that indicate the conditions of failure. Advanced structural analysis may examine
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976:{\displaystyle \sum F_{y}=0=R_{Ay}+F_{AD}\sin(60)=5+F_{AD}{\frac {\sqrt {3}}{2}}\Rightarrow F_{AD}=-{\frac {10}{\sqrt {3}}}}
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include all that must withstand loads, such as buildings, bridges, aircraft and ships. Structural analysis uses ideas from
2824:{\displaystyle \sum F_{y}=0=F_{BD}\sin(60)+R_{B}=F_{BD}{\frac {\sqrt {3}}{2}}+5\Rightarrow F_{BD}=-{\frac {10}{\sqrt {3}}}}
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This type of method uses the force balance in the x and y directions at each of the joints in the truss structure.
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For the analysis of entire systems, this approach can be used in conjunction with statics, giving rise to the
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refers to a body or system of connected parts used to support a load. Important examples related to
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2234:{\displaystyle \sum M_{D}=0=-5*1+{\sqrt {3}}*F_{AB}\Rightarrow F_{AB}={\frac {5}{\sqrt {3}}}}
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Regardless of approach, the formulation is based on the same three fundamental relations:
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To perform an accurate analysis a structural engineer must determine information such as
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This article is about structural studies in engineering. For social-science usage, see
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diagram is the loading diagram and contains the reaction forces from the joints.
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on the discretization of plane elasticity problems using a lattice framework
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while considering economic and environmental constraints. Other branches of
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Geometrically and materially nonlinear analysis with imperfections included
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approach (which is actually a special case of the more general field of
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2675:{\displaystyle \sum M_{B}=0={\sqrt {3}}*F_{CD}\Rightarrow F_{CD}=0}
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identically regardless of direction of the applied load, that all
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749:{\displaystyle \sum F_{y}=0=R_{Ay}+R_{B}-10\Rightarrow R_{Ay}=5}
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655:{\displaystyle \sum M_{A}=0=-10*1+2*R_{B}\Rightarrow R_{B}=5}
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published a treatise on the elastic behaviors of structures
1767:{\displaystyle \sum F_{x}=-F_{CD}=-0=0\Rightarrow verified}
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approach for simple one-dimensional bar elements, and the
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behavior. There are three approaches to the analysis: the
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3120:" in which he examined the failure of simple structures
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approach (also known as strength of materials), the
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555:positive directions, the value will be negative).
319:, support reactions, velocity, accelerations, and
478:Strength of materials methods (classical methods)
327:. Structural analysis is thus a key part of the
548:A simple triangular truss with loads imposed .
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3140:Philosophiae Naturalis Principia Mathematica
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339:In the context to structural analysis, a
260:Learn how and when to remove this message
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3234:divided a domain into finite subregions
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3282:Probabilistic Assessment of Structures
3306:"Science Direct: Structural Analysis"
3048:Methods using numerical approximation
7:
198:adding citations to reliable sources
802:{\displaystyle \sum F_{x}=0=R_{Ax}}
25:
3267:Structural integrity and failure
3208:Timoshenko–Ehrenfest beam theory
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2127:
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329:engineering design of structures
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2122:Method 1: Ignore the right side
185:needs additional citations for
27:Calculation of structural loads
3223:submitted his D.Sc. thesis in
3075:partial differential equations
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3262:Structural engineering theory
3151:Euler–Bernoulli beam equation
3187:presented his dissertation "
3159:introduced the principle of
513:for small rigid frames, and
360:Classification of structures
3189:Intorno ai sistemi elastici
107:Metal-induced embrittlement
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511:moment distribution method
381:
352:work on a wide variety of
97:Liquid metal embrittlement
36:Structure (disambiguation)
29:
3277:von Mises yield criterion
3185:Carlo Alberto Castigliano
307:to compute a structure's
112:Stress corrosion cracking
3169:developed the theory of
44:Mechanical failure modes
3145:Newton's laws of motion
3109:made many contributions
3082:finite element analysis
492:superposition principle
354:non-building structures
117:Sulfide stress cracking
3272:Stress–strain analysis
3063:mechanics of materials
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34:. For other uses, see
3143:" which contains the
3055:Finite Element Method
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3331:Structural analysis
3178:Claude-Louis Navier
3067:elasticity approach
443:continuum mechanics
305:applied mathematics
273:Structural analysis
102:Mechanical overload
18:Structural research
3311:2021-05-16 at the
3287:Structural testing
3257:Limit state design
3200:Stephen Timoshenko
3191:", which contains
3094:flexibility method
3035:Elasticity methods
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2113:Method of sections
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401:Analytical methods
3237:1956: J. Turner,
3204:Applied mechanics
3202:father of modern
3107:Leonardo da Vinci
3092:and the force or
3042:linear elasticity
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16:(Redirected from
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3118:Two New Sciences
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632:
631:
592:
591:
503:method of joints
423:dynamic response
407:structural loads
283:on the physical
265:
258:
254:
251:
245:
243:
202:
178:
170:
160:
153:
146:
41:
21:
3346:
3345:
3341:
3340:
3339:
3337:
3336:
3335:
3321:
3320:
3319:
3318:
3313:Wayback Machine
3304:
3300:
3295:
3248:
3114:Galileo Galilei
3102:
3086:pressure vessel
3050:
3037:
2990:
2950:
2946:
2930:
2911:
2880:
2864:
2842:
2834:
2833:
2790:
2756:
2743:
2712:
2693:
2685:
2684:
2653:
2637:
2608:
2600:
2599:
2589:
2554:
2538:
2482:
2451:
2435:
2416:
2408:
2407:
2364:
2330:
2287:
2271:
2252:
2244:
2243:
2203:
2187:
2143:
2135:
2134:
2124:
2115:
1998:
1982:
1966:
1958:
1957:
1873:
1869:
1847:
1816:
1803:
1790:
1782:
1781:
1709:
1693:
1685:
1684:
1648:
1632:
1610:
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1601:
1565:
1549:
1480:
1449:
1418:
1396:
1388:
1387:
1344:
1316:
1280:
1276:
1236:
1205:
1177:
1169:
1168:
1123:
1107:
1060:
1029:
1013:
994:
986:
985:
942:
914:
877:
861:
842:
834:
833:
815:
786:
767:
759:
758:
727:
708:
692:
673:
665:
664:
636:
623:
583:
575:
574:
536:
480:
471:
403:
386:
384:Structural load
380:
362:
337:
277:solid mechanics
275:is a branch of
266:
255:
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246:
203:
201:
191:
179:
164:
39:
28:
23:
22:
15:
12:
11:
5:
3344:
3342:
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3333:
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3317:
3316:
3297:
3296:
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3291:
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3289:
3284:
3279:
3274:
3269:
3264:
3259:
3254:
3247:
3244:
3243:
3242:
3235:
3228:
3217:
3210:
3206:including the
3196:
3181:
3174:
3167:Leonhard Euler
3163:
3153:
3147:
3131:
3121:
3110:
3101:
3098:
3071:matrix algebra
3049:
3046:
3036:
3033:
3029:
3028:
3014:
3010:
3005:
3000:
2997:
2993:
2989:
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2863:
2860:
2857:
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2800:
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2774:
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2746:
2742:
2739:
2736:
2733:
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2711:
2708:
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2640:
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2597:
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2532:
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2517:
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2255:
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2202:
2197:
2194:
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2176:
2173:
2170:
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2158:
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2146:
2142:
2132:
2123:
2120:
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2111:
2110:
2109:
2098:
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2080:
2077:
2074:
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2068:
2063:
2060:
2052:
2048:
2043:
2037:
2033:
2028:
2025:
2022:
2019:
2016:
2013:
2008:
2005:
2001:
1997:
1992:
1989:
1985:
1981:
1978:
1973:
1969:
1965:
1955:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1903:
1899:
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1885:
1881:
1876:
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1857:
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1371:
1367:
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1359:
1354:
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1215:
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1189:
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1176:
1162:
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1147:
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1126:
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1039:
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1023:
1020:
1016:
1012:
1009:
1006:
1001:
997:
993:
983:
969:
965:
960:
957:
952:
949:
945:
941:
936:
932:
924:
921:
917:
913:
910:
907:
904:
901:
898:
895:
892:
887:
884:
880:
876:
871:
868:
864:
860:
857:
854:
849:
845:
841:
827:
826:
814:
811:
810:
809:
796:
793:
789:
785:
782:
779:
774:
770:
766:
756:
745:
742:
737:
734:
730:
726:
723:
720:
715:
711:
707:
702:
699:
695:
691:
688:
685:
680:
676:
672:
662:
651:
648:
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639:
635:
630:
626:
622:
619:
616:
613:
610:
607:
604:
601:
598:
595:
590:
586:
582:
567:
566:
552:
551:
535:
532:
486:in a state of
479:
476:
470:
467:
447:finite element
402:
399:
391:building codes
382:Main article:
379:
376:
361:
358:
336:
333:
325:physical tests
268:
267:
182:
180:
173:
166:
165:
163:
162:
155:
148:
140:
137:
136:
135:
134:
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119:
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109:
104:
99:
94:
89:
84:
79:
74:
69:
64:
59:
54:
46:
45:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3343:
3332:
3329:
3328:
3326:
3314:
3310:
3307:
3302:
3299:
3292:
3288:
3285:
3283:
3280:
3278:
3275:
3273:
3270:
3268:
3265:
3263:
3260:
3258:
3255:
3253:
3250:
3249:
3245:
3240:
3236:
3233:
3229:
3226:
3222:
3218:
3215:
3211:
3209:
3205:
3201:
3197:
3194:
3190:
3186:
3182:
3179:
3175:
3172:
3168:
3164:
3162:
3158:
3154:
3152:
3148:
3146:
3142:
3141:
3136:
3132:
3130:
3126:
3122:
3119:
3115:
3111:
3108:
3104:
3103:
3099:
3097:
3095:
3091:
3087:
3083:
3078:
3076:
3072:
3068:
3064:
3058:
3056:
3047:
3045:
3043:
3034:
3032:
3012:
3008:
3003:
2998:
2995:
2991:
2984:
2981:
2976:
2973:
2967:
2960:
2956:
2951:
2947:
2943:
2938:
2935:
2931:
2927:
2924:
2919:
2916:
2912:
2908:
2902:
2896:
2893:
2888:
2885:
2881:
2877:
2872:
2869:
2865:
2861:
2858:
2855:
2852:
2847:
2843:
2839:
2832:
2815:
2811:
2806:
2803:
2798:
2795:
2791:
2784:
2781:
2776:
2772:
2764:
2761:
2757:
2753:
2748:
2744:
2740:
2734:
2728:
2725:
2720:
2717:
2713:
2709:
2706:
2703:
2698:
2694:
2690:
2683:
2669:
2666:
2661:
2658:
2654:
2645:
2642:
2638:
2634:
2629:
2624:
2621:
2618:
2613:
2609:
2605:
2598:
2595:
2591:
2590:
2586:
2570:
2567:
2562:
2559:
2555:
2546:
2543:
2539:
2535:
2530:
2527:
2519:
2515:
2510:
2504:
2500:
2495:
2490:
2487:
2483:
2479:
2473:
2467:
2464:
2459:
2456:
2452:
2448:
2443:
2440:
2436:
2432:
2429:
2426:
2421:
2417:
2413:
2406:
2389:
2385:
2380:
2377:
2372:
2369:
2365:
2358:
2355:
2350:
2346:
2338:
2335:
2331:
2327:
2324:
2321:
2318:
2315:
2309:
2303:
2300:
2295:
2292:
2288:
2284:
2279:
2276:
2272:
2268:
2265:
2262:
2257:
2253:
2249:
2242:
2225:
2221:
2216:
2211:
2208:
2204:
2195:
2192:
2188:
2184:
2179:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2148:
2144:
2140:
2133:
2130:
2126:
2125:
2121:
2119:
2112:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2069:
2066:
2061:
2058:
2050:
2046:
2041:
2035:
2031:
2026:
2020:
2014:
2011:
2006:
2003:
1999:
1995:
1990:
1987:
1983:
1979:
1976:
1971:
1967:
1963:
1956:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1915:
1912:
1909:
1906:
1901:
1897:
1890:
1883:
1879:
1874:
1870:
1866:
1863:
1860:
1855:
1852:
1848:
1844:
1838:
1832:
1829:
1824:
1821:
1817:
1813:
1808:
1804:
1800:
1795:
1791:
1787:
1780:
1779:
1778:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1734:
1731:
1728:
1725:
1722:
1717:
1714:
1710:
1706:
1703:
1698:
1694:
1690:
1683:
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1681:
1664:
1661:
1656:
1653:
1649:
1640:
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1633:
1629:
1626:
1623:
1620:
1615:
1611:
1607:
1600:
1599:
1598:
1581:
1578:
1573:
1570:
1566:
1557:
1554:
1550:
1546:
1541:
1538:
1530:
1526:
1521:
1516:
1513:
1505:
1501:
1496:
1493:
1488:
1485:
1481:
1477:
1471:
1465:
1462:
1457:
1454:
1450:
1446:
1440:
1434:
1431:
1426:
1423:
1419:
1415:
1412:
1409:
1406:
1401:
1397:
1393:
1386:
1369:
1365:
1360:
1357:
1352:
1349:
1345:
1336:
1332:
1324:
1321:
1317:
1313:
1308:
1304:
1297:
1290:
1286:
1281:
1277:
1273:
1270:
1267:
1264:
1258:
1252:
1249:
1244:
1241:
1237:
1233:
1227:
1221:
1218:
1213:
1210:
1206:
1202:
1199:
1196:
1193:
1190:
1187:
1182:
1178:
1174:
1167:
1166:
1165:
1145:
1141:
1136:
1131:
1128:
1124:
1115:
1112:
1108:
1104:
1099:
1096:
1088:
1084:
1079:
1076:
1073:
1068:
1065:
1061:
1057:
1051:
1045:
1042:
1037:
1034:
1030:
1026:
1021:
1018:
1014:
1010:
1007:
1004:
999:
995:
991:
984:
967:
963:
958:
955:
950:
947:
943:
934:
930:
922:
919:
915:
911:
908:
905:
899:
893:
890:
885:
882:
878:
874:
869:
866:
862:
858:
855:
852:
847:
843:
839:
832:
831:
830:
824:
820:
819:
818:
812:
794:
791:
787:
783:
780:
777:
772:
768:
764:
757:
743:
740:
735:
732:
728:
721:
718:
713:
709:
705:
700:
697:
693:
689:
686:
683:
678:
674:
670:
663:
649:
646:
641:
637:
628:
624:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
588:
584:
580:
573:
572:
571:
562:
558:
557:
556:
546:
542:
541:
540:
533:
531:
529:
523:
520:
516:
512:
508:
504:
500:
495:
493:
489:
485:
477:
475:
468:
466:
464:
463:compatibility
460:
456:
451:
448:
444:
440:
436:
432:
428:
424:
420:
419:displacements
416:
412:
408:
400:
398:
394:
392:
385:
377:
375:
372:
367:
359:
357:
355:
351:
346:
342:
334:
332:
330:
326:
322:
318:
314:
310:
306:
302:
298:
294:
290:
286:
282:
278:
274:
264:
261:
253:
250:December 2018
242:
239:
235:
232:
228:
225:
221:
218:
214:
211: –
210:
206:
205:Find sources:
199:
195:
189:
188:
183:This article
181:
177:
172:
171:
161:
156:
154:
149:
147:
142:
141:
139:
138:
133:
130:
128:
125:
123:
122:Thermal shock
120:
118:
115:
113:
110:
108:
105:
103:
100:
98:
95:
93:
90:
88:
85:
83:
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68:
65:
63:
60:
58:
55:
53:
50:
49:
48:
47:
42:
37:
33:
32:Structuralism
19:
3301:
3239:R. W. Clough
3188:
3161:virtual work
3138:
3135:Isaac Newton
3129:Robert Hooke
3079:
3066:
3062:
3059:
3051:
3038:
3030:
2116:
1776:
1679:
1596:
1163:
828:
816:
568:
553:
537:
528:deformations
524:
518:
515:portal frame
514:
502:
498:
496:
488:pure bending
481:
472:
459:constitutive
452:
404:
395:
387:
365:
363:
338:
309:deformations
272:
271:
256:
247:
237:
230:
223:
216:
204:
192:Please help
187:verification
184:
3214:Hardy Cross
3193:his theorem
3165:1707–1783:
3155:1700–1782:
3137:published "
3125:Hooke's law
469:Limitations
455:equilibrium
445:), and the
350:engineering
311:, internal
3293:References
3232:R. Courant
3198:1878-1972
3173:of columns
3105:1452–1519
3073:, solving
509:analysis,
431:non-linear
289:components
287:and their
285:structures
220:newspapers
2988:⇒
2982:−
2952:−
2944:−
2928:−
2909:−
2897:
2878:−
2862:−
2840:∑
2807:−
2788:⇒
2729:
2691:∑
2651:⇒
2635:∗
2606:∑
2552:⇒
2511:−
2468:
2414:∑
2381:−
2362:⇒
2356:−
2328:−
2316:−
2304:
2285:−
2250:∑
2201:⇒
2185:∗
2169:∗
2163:−
2141:∑
2073:⇒
2042:−
2015:
1996:−
1980:−
1964:∑
1919:⇒
1875:−
1833:
1788:∑
1738:⇒
1726:−
1707:−
1691:∑
1646:⇒
1630:−
1608:∑
1563:⇒
1497:−
1466:
1435:
1416:−
1394:∑
1361:−
1342:⇒
1314:−
1282:−
1274:−
1268:−
1253:
1234:−
1222:
1203:−
1197:−
1175:∑
1121:⇒
1080:−
1046:
992:∑
959:−
940:⇒
894:
840:∑
765:∑
725:⇒
719:−
671:∑
634:⇒
621:∗
609:∗
603:−
581:∑
427:stability
341:structure
321:stability
57:Corrosion
3325:Category
3309:Archived
3246:See also
3171:buckling
3100:Timeline
415:stresses
411:geometry
371:elements
317:stresses
293:analysis
132:Yielding
82:Fracture
52:Buckling
534:Example
234:scholar
77:Fouling
72:Fatigue
3230:1942:
3219:1941:
3212:1936:
3183:1873:
3176:1826:
3149:1750:
3133:1687:
3123:1660:
3112:1638:
1777:At B,
1597:At C,
1164:At D,
829:At A,
461:, and
313:forces
236:
229:
222:
215:
207:
92:Impact
507:truss
484:beams
378:Loads
281:loads
241:JSTOR
227:books
67:Creep
517:and
505:for
501:and
429:and
417:and
303:and
213:news
127:Wear
3225:MIT
3127:by
2894:cos
2726:sin
2465:cos
2301:sin
2012:cos
1830:sin
1463:cos
1432:cos
1250:sin
1219:sin
1043:cos
891:sin
196:by
3327::
3077:.
3057:.
2957:10
2903:60
2812:10
2735:60
2516:10
2474:60
2386:10
2359:10
2319:10
2310:60
2047:10
2021:60
1880:10
1839:60
1527:10
1502:10
1472:60
1441:60
1366:10
1287:10
1271:10
1259:60
1228:60
1200:10
1085:10
1052:60
964:10
900:60
722:10
606:10
457:,
425:,
409:,
364:A
356:.
331:.
315:,
299:,
3013:3
3009:5
3004:=
2999:B
2996:A
2992:F
2985:0
2977:2
2974:1
2968:)
2961:3
2948:(
2939:B
2936:A
2932:F
2925:=
2920:D
2917:C
2913:F
2906:)
2900:(
2889:D
2886:B
2882:F
2873:B
2870:A
2866:F
2859:=
2856:0
2853:=
2848:x
2844:F
2816:3
2804:=
2799:D
2796:B
2792:F
2785:5
2782:+
2777:2
2773:3
2765:D
2762:B
2758:F
2754:=
2749:B
2745:R
2741:+
2738:)
2732:(
2721:D
2718:B
2714:F
2710:=
2707:0
2704:=
2699:y
2695:F
2670:0
2667:=
2662:D
2659:C
2655:F
2646:D
2643:C
2639:F
2630:3
2625:=
2622:0
2619:=
2614:B
2610:M
2571:0
2568:=
2563:D
2560:C
2556:F
2547:D
2544:C
2540:F
2536:+
2531:2
2528:1
2520:3
2505:3
2501:5
2496:=
2491:D
2488:C
2484:F
2480:+
2477:)
2471:(
2460:D
2457:B
2453:F
2449:+
2444:B
2441:A
2437:F
2433:=
2430:0
2427:=
2422:x
2418:F
2390:3
2378:=
2373:D
2370:B
2366:F
2351:2
2347:3
2339:D
2336:B
2332:F
2325:5
2322:=
2313:)
2307:(
2296:D
2293:B
2289:F
2280:y
2277:A
2273:R
2269:=
2266:0
2263:=
2258:y
2254:F
2226:3
2222:5
2217:=
2212:B
2209:A
2205:F
2196:B
2193:A
2189:F
2180:3
2175:+
2172:1
2166:5
2160:=
2157:0
2154:=
2149:D
2145:M
2097:d
2094:e
2091:i
2088:f
2085:i
2082:r
2079:e
2076:v
2070:0
2067:=
2062:2
2059:1
2051:3
2036:3
2032:5
2027:=
2024:)
2018:(
2007:D
2004:B
2000:F
1991:B
1988:A
1984:F
1977:=
1972:x
1968:F
1943:d
1940:e
1937:i
1934:f
1931:i
1928:r
1925:e
1922:v
1916:0
1913:=
1910:0
1907:+
1902:2
1898:3
1891:)
1884:3
1871:(
1867:+
1864:5
1861:=
1856:C
1853:B
1849:F
1845:+
1842:)
1836:(
1825:D
1822:B
1818:F
1814:+
1809:B
1805:R
1801:=
1796:y
1792:F
1762:d
1759:e
1756:i
1753:f
1750:i
1747:r
1744:e
1741:v
1735:0
1732:=
1729:0
1723:=
1718:D
1715:C
1711:F
1704:=
1699:x
1695:F
1665:0
1662:=
1657:C
1654:B
1650:F
1641:C
1638:B
1634:F
1627:=
1624:0
1621:=
1616:y
1612:F
1582:0
1579:=
1574:D
1571:C
1567:F
1558:D
1555:C
1551:F
1547:+
1542:2
1539:1
1531:3
1522:+
1517:2
1514:1
1506:3
1494:=
1489:D
1486:C
1482:F
1478:+
1475:)
1469:(
1458:D
1455:B
1451:F
1447:+
1444:)
1438:(
1427:D
1424:A
1420:F
1413:=
1410:0
1407:=
1402:x
1398:F
1370:3
1358:=
1353:D
1350:B
1346:F
1337:2
1333:3
1325:D
1322:B
1318:F
1309:2
1305:3
1298:)
1291:3
1278:(
1265:=
1262:)
1256:(
1245:D
1242:B
1238:F
1231:)
1225:(
1214:D
1211:A
1207:F
1194:=
1191:0
1188:=
1183:y
1179:F
1146:3
1142:5
1137:=
1132:B
1129:A
1125:F
1116:B
1113:A
1109:F
1105:+
1100:2
1097:1
1089:3
1077:0
1074:=
1069:B
1066:A
1062:F
1058:+
1055:)
1049:(
1038:D
1035:A
1031:F
1027:+
1022:x
1019:A
1015:R
1011:=
1008:0
1005:=
1000:x
996:F
968:3
956:=
951:D
948:A
944:F
935:2
931:3
923:D
920:A
916:F
912:+
909:5
906:=
903:)
897:(
886:D
883:A
879:F
875:+
870:y
867:A
863:R
859:=
856:0
853:=
848:y
844:F
795:x
792:A
788:R
784:=
781:0
778:=
773:x
769:F
744:5
741:=
736:y
733:A
729:R
714:B
710:R
706:+
701:y
698:A
694:R
690:=
687:0
684:=
679:y
675:F
650:5
647:=
642:B
638:R
629:B
625:R
618:2
615:+
612:1
600:=
597:0
594:=
589:A
585:M
263:)
257:(
252:)
248:(
238:·
231:·
224:·
217:·
190:.
159:e
152:t
145:v
38:.
20:)
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