SS - Determinacy

In statics, when the static equilibrium equations are sufficient to determine the internal forces and reactions of the structure, that structure is said to be statically determinate.

In solving for determinacy, three summations of forces (which are obtained by Newton's laws of motion) are used to determine whether a structure is determinate. If for three equations, three variables can be solved, then that structure is called statically determinate.

The three equations for equilibrium state the following:

Σ N = 0: the sum of the horizontal components of the forces equals zero;

Σ V = 0: the sum of the vertical components of forces equals zero;

Σ M = 0: the sum of the horizontal components of the forces equals zero;

Where N is considered the normal force, V the shear force and M the moment force. The units for N and V are either pounds (lb), kips (thousands of pounds) (k) or Newtons (N) / kiloNewtons (kN).

The Free Body Diagram

In beams, columns or other composite structural members, it is best to break the key components of bar forces into separate reactions acting against or along those members. This is called a free body diagram, and it is the fundamental building block of all statics in determining force vectors.

The classical scenario typically used for physics is the block on a slope. The block has some weight, w, which acts downward toward the surface of the earth. Against the slope, however, a partial fraction of that weight, called the normal force (-N) acts perpendicularly against the slope. An equal, but opposite reaction (N) is the result of the sum of vertical forces equaling zero.

In structural beams, N is a horizontal force, acting linearly throughout the member. A change of connotation in how one determines force is all structural analysis suggests. Instead of negative forces acting downward and positive forces acting upward, structural analysis is more interested in what members are in tension, and which are in compression. If a member is in tension, that force is outward and its value is positive. Conversely, if it is in compression, the force is inward and its value is negative.

Or put it more simply, when your muscles are in tension, you build bigger biceps, and that typically is a positive outcome. Structurally, you're better with bigger, stronger muscles. Alternatively, when your muscles are compressed, there is a force acting against you, a negative force (which may include a bench press, dumbbells or some other voluntary load). You must prove elastic against that force in order to withstand it (which is the definition of compression).

A standard free body diagram may look something like this:

Where there may be a point load (or concentrated load), distributed load (in kN/m), reaction forces (at A in this diagram), distances for the loading forces and reaction forces, and subsequent shear vectors (V) and moment vectors (M) at cut cross-sections in the member. As a result of where the cut is taken, V and M with vary either linearly or parabolically. Or not at all. It all depends on the degree of loading (whether it is one point, continuously linear or curvilinear). The way it affects V and M, however, is another blog, however, and will not be included here.

All in all, though, there are two main considerations when analyzing a structure. One is stability, which is also another post. And the other one (which tells you something about stability) is determinacy. Without determinacy, it is very hard to predict what happens for a structure. Therefore, it is the preeminent consideration for all of structural analysis.

Now take a look at the top diagram. Why are the members and trusses either determinate or indeterminate (not solvable by equilibrium equations)? Recall that roller reactions have only one force they exhibit (vertical to loading), pins have two (vertical and horizontal) and fixed connections have three (vertical, horizontal and rotational to loading). Consider this a bit of a quiz. But feel free to take your time and look closely.

SS - Cables

SS - Shear, Moment, Deflection Diagrams Review

SS - Fatigue

Structural design takes into consideration two different modes for failure - fatigue and fracture. Whereas it is commonplace (and correct) to believe fracture is the more dangerous form for failure, the fatigue of a member is more critical to design for. If a material is about to fail due to cyclic loading, this means the maximum stress values are less than the ultimate tensile stress limit, and thus the material will eventually fail to a smaller subjected load.

Continual loads will form gradual cracks that accumulate at a microscopic level in a material. These cracks eventually find their ways to the face of a structural member, and the compilation of the cracks will eventually result in fracture. If a structural member is formed into a certain shape, such as rectangular holes or sharp edges, like the reentrant corners in a building undergoing seismic surface waves, the material will be subjected to elevated local stresses, which will result in the increased rate of fatigue in the member.

Fatigue Life

The fatigue life, N(f), is the number of cyclic loadings a material sustains before it fails. It is NOT measured in seconds or time. Only in number of loadings.

Fatigue Observations from Cracks

The process of fatigue is a gradual one, taking an extend period of time, so the slow build-up of stresses typically results in a darker, more layered cross section when a structural member finally breaks. The brighter more silvery portion of the cross is where a sudden fracture occurs in a member.

Also, fatigue is a stochastic process, meaning that it is random. The path for fatigue cannot be determined, although it might look similar from a quick observation.

Some obvious factors influencing fatigue would be temperature, surface finish, grain size, material type, texture, direction of loading, geometry, atomic structure, internal stresses and oxidation (if present). Fatigue strength can be measured to a maximum of 10^3 to 10^8 cycles (this is typically for steel members, which provide the greatest overall fatigue strength for an extended period of cycles).

Designing for Fatigue

1. Design to keep a structural member below it's fatigue limit

2. Design for a fixed life after which a user should replace the member with a new one (called a "lifed part" and this process is called "safe-life" design practice)

3. Inspect the member periodically for cracks. Replace when cracks exceed a critical length. Employ nondestructive techniques to determine crack length.

Where Fatigue Occurs:

In the graph above, fatigue occurs between the ultimate tensile strength and fracture stress. Although cracks form before the ultimate tensile strength, it is more important to design for this later region at an early stage. That way, one will prevent a critical build-up of stresses at any portion in the cross-section of structural member.

MIES - Glenn Murcutt on His Life

Glenn Murcutt has led an extraordinary life, from working for his father - a skilled craftsman in his own right - to moving to England and becoming an architect. Murcutt's work for environmental architecture put Australia in the forefront of design at the turn of the century, which helped him earn the Pritzker Prize and Aalto Medal (to name a few of the accolades he's earned). Glenn is widely perceived of as a shy person, but after this interview, you'll see just how determined he is. His belief on the results following a compromise highlight what's at the core of his modus operandi. The work has to be right, the opportunities have to make sense. He's a very sensible person from this interview, and when I met him in 2007, was every bit as fascinating as his buildings. I could go on for hours how I adore this man, but I will instead just post the clips and hope you find them as revealing as I do.

SS - Buckling

One of the simplest modes of elastic/plastic deformation and failure is buckling. Buckling is similar to bending in that the key consideration is the cross section of the member. If the cross sectional moment of inertia is increased (by the optimization of the material toward compressive and tensile loadings comparatively) then the new column will act stiffer and in turn will prove more resistant to buckling. Folding a sheet of paper, or rolling it into a tube can be seen as examples of making a simple 8.5 x 11 sheet of paper more resistant to bending under compressive loading. This is a simple exercise in modeling improved stiffness for a structural member as it relates to much heavier construction elements such as steel and engineered wood products.

The load at which a column begins to break is called the Critical Buckling Load or Euler Load. Leonard Euler (1707-1783) is most responsible for determining what this value was, and in 1744 he derived a formula that could accurately identify at what load a column would buckle. The equation is as follows, and applies to columns of any size:

Where F is the maximum load on a column, E is the modulus of elasticity, I the area moment of inertia, L the unsupported length and K (perhaps the most important consideration) the column effective length factor. K depends primarily on the end condition supports, which have much to do with the effective stability of a column. If the ends are fixed, a column or beam may be more resistant to moving than they would if pinned or hinged.

Some values of K are as follows (and as shown above in the photo):

For both ends pinned (hinged, free to rotate): K=1.0

For both ends fixed: K = 0.5

For one end fixed and the other end pinned: K=0.7

For one end fixed and the other end free to move laterally: K=2.0

One of the key considerations here is that columns buckling result in elastic instability, which by definition, is a mode of failure.

The Slenderness Ratio

Represented by the Greek letter lambda, λ , the slenderness ratio is one of the oldest forms used for classifying types of columns. Mathematically, it is the fraction of column length divided by a column's radius of gyration. The slenderness ratio is typically used for the following cases:

Steel

1. Short steel columns (lambda less than or equal to 50)

2. Average length steel columns (greater than 50, less than or equal to 200)

3. Long steel columns (greater than 200)

Concrete

1. Short concrete column (lambda less than or equal to 10)

2. Long concrete column (greater than 10)

Wood

1. Short wood columns (lambda less than 10)

2. Average length wood columns and long columns (greater than 10)

Self-buckling

Yes, buckling is a result of ALL weights, not just loads applied on a column. Therefore, internal stresses, if large enough, can cause a column to buckle. For a free-standing column with a particular density, Young's modulus of E, and radius r, the critical height for a column to buckle under it's own weight is the following:

Where g is gravity's acceleration, I is inertia and B is the Bessel function (which is usually 1.87).

One would typically see the Bessel function in electromagnetic wave equations or any calculation involving propagation or oscillation. Therefore, since columns tend to behave in a similar fashion, this second-order differential function would naturally make sense as part of the formula.

MIES - Architectural Credibility

Do we, as architects, have no strings to hold us down?

If you haven't listened to the AIA podcasts, I enjoin you to do so. Occasionally. Being an architect is a lot about what you ask out of yourself, and what you ask out of the profession as well. Architecture is a very damaged profession in that it is widely perceived as a luxury. This podcast goes into great detail as to how one can overcome this perception. Good stuff.

http://www.idimultimedia.net/clients/aia_podcast/10152008/credibility.mp3

SS - Deformation

Deformations can be the trickiest calculations for an architect/engineer to calculate. In nature, they are clear to the human eye. Deformations include any change in shape or size of an object, and generally are the result of an applied force (although chemical forces also can cause deformation on more microscopic scales). It is typically measured as strain, which is a unitless quantity, helpful in calculating the rate of elongation or contraction a member might undergo.

Internally, inter-molecular forces arise that act in an equal but opposite direction of the force applied onto the member. If a force isn't large enough, these internal stresses are enough to withstand permanent deformation (or plastic failure), and allow the member to completely resist the force. However, if the force is large enough, a new equilibrium stage will be reached via plastic failure, which allows the member to bend into a new geometric shape in order to compensate for a deficiency in internal strength. If an excessively large force is imposed on the structural shape, no matter what the material may be, structural failure will occur.

Stress-strain curves are the best way to plot this relationship. Although strain is actually the dependent variable we're talking about here, for engineering purposes, it is typically conveyed along the abscissa for relative ease. The linear portion of the curve is considered the elastic region, where Hooke's Law pertains most fiercely (and members retain their original form after being subjected to loads). It is followed by the plastic region (the curvilinear portion) and ultimately the finite fracture limit where the member fails to retain any structural integrity.

The above chart shows the particular strengths of utmost importance. Such quantities like Young's Modulus, the yield strength, strain hardening, ultimate strength, and necking vary for different materials. Concrete and steel, for instance, vary widely in every one of these quantities. And yet they both remain two of the most commonly used materials in construction. This is one fascinating example depicting the variety and versatility in modern day construction. Engineering properties can be worlds apart for our materials and manufactured products today, and yet still both be of great usage.

Four kinds of deformation: 1) Elastic, 2) Plastic and 3) Fatigue and 4) Failure

1. Elastic Deformation - The following is Hooke's Law. It pertains to elastic deformation:

Where σ is the applied stress, E is a material constant called Young's modulus, and ε is the resulting strain. This deformation is reversible and depends primarily on elastomers and shape memory in the construction materials used. Because of these stretching properties, engineers primarily calculate this region via the use of tensile tests (where a yield strength can quickly and methodically be obtained). At the yield strength, the element will follow the trajectory of the stress-strain curve and proceed into plastic deformation.

2. Plastic Deformation - Not reversible. But the element under stress will return to part of it's shape. Ductile metals, such as copper, silver and gold, will bend for an extended amount of time in the plastic range before failure, as will steel. But other elements, such as cast iron, will not. It all depends on the carbon content of a material and whether it the metallic bonds are enough to give way to internal stresses or are enough to hold firm. Carbon bonds stabilize a metallic element best with their versatility of valences.

Also of importance is the fact that the plastic region of a stress-strain curve has two portions: one, a strain hardening phase, where the material becomes stronger by a new movement of atoms to a stronger equilibrium state, and two, a necking region which holds off, but leads to the eventual failure of a member in structural loading. Necking usually results in a smaller cross-section in the member, which in turn results in the member stresses overcoming the internal axial stability of the member.

3. Fatigue - Happens most in ductile materials. Fatigue is the process of microscopic faults and molecular cracks compiling up over a period of time which eventually lead to the elimination of plastic deformation and results in fracture. An approximation of the number of deformations needed to result in fracture is somewhere between a thousand and a trillion depending on the structural members used. When compared to the short periods of impact loading in an earthquake, this concatenation of events truly conveys the power of the natural elements upon those built by man.

4. Fracture - Essentially, the breaking point in a structural member. All forces accumulate and overcome the internal forces of the beam, column, truss chord, etc. Fracture is, best put, the resultant death of a structure. Members should be either recycled or cast away after this point is attained as they are of no structural use.

MIES - Alleys of Seattle

http://alleysofseattle.com/

I recommend the above link to my friend's web blog devoted entirely to the renovation of Seattle city life. His focus of redefining what alleys are, to make them more than open space, has earned him some recognition from the likes of the AIA. If you are devoted to all thing built environment, then consider this an arrow in the right direction. Enjoy.

SS - Structural Analysis

(From Wiki)

Structural analysis helps engineers and architects alike predict the behavior of structures. The subjects of structural analysis are engineering artifacts whose integrity is judged largely based upon their ability to withstand loads; they commonly include buildings, bridges, aircraft, ships and cars. Structural analysis incorporates the fields of mechanics and dynamics as well as the many failure theories. The primary goal of structural analysis is the computation of deformations, internal forces, and stresses. In practice, structural analysis can be viewed more abstractly as a method to drive the engineering design process or prove the soundness of a design without a dependence on directly testing it.

Analysis

To perform an accurate analysis a structural engineer must determine such information as loads, geometry, support conditions, and materials properties. The results of such an analysis typically include:

1. Support Reactions

2. Internal Stresses

3. Displacements

This information is then compared to criteria that indicate the conditions of failure. Advanced structural analysis may examine dynamic response, non-linear behavior, and stability.

There are three approaches to the analysis:

1. Mechanics of Materials Approach

2. Elasticity Theory Approach

3. Finite Element Approach (Numerical Method for Larger Structural Systems)

Regardless of approach, the formulation is based on the same three fundamental relations: equilibrium, constitutive, and compatibility. The solutions are approximate when any of these relations are only approximately satisfied, or only an approximation of reality.

Limitations

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 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.

Mechanics of Materials Method

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 beams in a state of pure bending, and circular shafts subject to torsion. The solutions can under certain conditions be superimposed using the superposition principle to analyze a member undergoing combined loading. Solutions for special cases exist for common structures such as thin-walled pressure vessels.

For the analysis of entire systems, this approach can be used in conjunction with statics, giving rise to the method of sections and method of joints for truss analysis, moment distribution method for small rigid frames, and portal frame and cantilever method 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.

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 identically regardless of direction of the applied load, that all deformations 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.

Elasticity Methods

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 linear elasticity. 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.

Numerical Methods (Finite Analysis)

It is common practice to use approximations the solution of differential equations as the basis for structural analysis. This is usually done using numerical approximiation techniques. The most commonly used numerical approximation in structural analysis is the Finite Element Method.

The finite element method approximates a structure as an assembly of elements or components with various forms of connection between them. Thus, a continuous system such as a plate or shell is modeled as a discrete system with a finite number of elements interconnected at finite number of nodes. The behaviour of individual elements is characterised by the element's stiffness or flexibility relation, which altogether leads to the system's stiffness or flexibility relation. To establish the element's stiffness or flexibility relation, we can use the mechanics of materials approach for simple one-dimensional bar elements, and the elasticity approach for more complex two- and three-dimensional elements. The analytical and computational development are best effected throughout by means of matrix algebra.

Early applications of matrix methods were for articulated frameworks with truss, beam and column elements; later and more advanced matrix methods, referred to as "finite element analysis," 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 pressure vessel, 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 stiffness method and the force or flexibility method. 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.

SS - Load, Shear and Moment Diagrams

SS - Compound Truss Problem

SS - Method of Sections Problem

SS - Method of Joints, Method of Sections Problems

SS - Free Body Diagram and Determinacy Problems