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