A Survey on Structural Plan Density.

By Nan Gao.

In the illustrated essay of 2005 CUREE calendar The Expression of Seismic Design, the author Robert Reitherman discussed the concept of Structural Plan Density. Basically, it is the ratio of the area of vertical structural components to the whole plan area of a building structure. It could be used as a parameter of structural efficiency and reliability. As the figures showed in The Expression of Seismic Design, the plan density of the Temple of Khons in Karnak is over 50%. The famous Taj Mahal in India also has a 50% structural plan density. The plan density of the Parthenon in Athens and the Pantheon in Rome was about 20%. In contrast, the 442 meters high Sears Tower has a plan density of only 2%. From stone to steel, from masonry to bundle-tubes, and from 50% to 2%, this is a phenomenal progress in the field of structural engineering. The overall structural plan density has decreased a lot due to the development of more efficient materials and systems.

Besides the materials and structural systems, the level of plan density is also influenced by the local seismic level, the height of the building, and the architectural design. Typically, structures in high-seismic regions have larger plan density than those in non-seismic regions. Low-rise buildings usually have lower plan density than high-rise buildings. Maybe unnoticed, but the architectural design has a huge impact on structural efficiency, especially in high-rise structures. An irregular architectural plan will cause larger seismic response and thus a higher plan density. Sometimes this impact could be very huge.

What is the overall structural plan density in current Chinese industry? Maybe I could conduct a survey on this topic. As a simple survey, I used the database of my company as my resource. For purpose of comparison, the scope is confined to high-rise concrete structures. All the samples are completed designs in the last 8 years and they are classified as one of these three structural types: moment frame (MF), shearwall (SW), and moment frame-shearwall (MF-SW) or moment frame-corewall (MF-CW).

Plans of moment frame-corewall or moment frame-shearwall structures

Table of moment frame-corewall or moment frame-shearwall structures

Plans of shearwall structures

Table of shearwall structures

Plans of moment frame structures

Table of moment frame structures

We can see the relationship between structural types and plan density. Shearwall structures have relatively high plan density. The range of their plan densities is 5% to 12%. The reason is that all the gravity load and lateral loads are resisted by concrete walls. The whole structure needs a high level of stiffness to maintain reliability.

Moment frame-shearwall structures have a lower plan density, from 7% to 2%. This type of structure is a dual system. Most lateral loads are resisted by the shearwalls. Moment frames resist a small portion of lateral loads. The gravity loads are resisted by the two systems together. The moment frame part could provide more flexible arrangement for architectural design. The efficiency of moment frame-corewall structures is better than the moment frame-shearwall structures, because the arrangement of shearwalls in moment frame-corewall structures is more efficient. The shearwalls in the middle act like a tube. Thus, they could perform better when resisting lateral loads than the dispersed shearwalls in the moment frame-shearwall structures.

Moment frame structures have the lowest plan density and the lowest structural height. They could be used when the height is under 35m or 40m. However, the reliability of this type of structure is not as good as these with shearwalls. Stiffness of moment frames is much smaller. Thus, it may cause larger displacement and more non-structural damage. Also, the beam-column joint is the Achilles’ heel of the moment frame structure. It needs very special treatment and reinforcement.

Structural plan density of different types of structures

Structural plan density in different seismic regions

Also, several samples corroborate the relation between structural efficiency and architectural design. For example, the No. 16 sample has a height of 84.6m and a plan density of 5.99%. As a contrast, the No. 19 sample has a height of 72.8m but a plan density of 11.5%. These two shearwall structures are in the same seismic region and their heights are similar. However, No. 16 sample has a much better structural efficiency since its plan density is about the half of the No. 19 sample. Its rectangle plan shape and its regular arrangement of shearwalls might be the main reasons.

The No.19 sample and the No. 21 sample are from the same project. The plan density of them is more than 10%, much higher than other shearwall structures. Why? A main factor is the irrational design of the architectural plan. This project is a copycat of another design project. The client wanted to build their own buildings exactly the same as that project. However, the original edition is in non-seismic region while their own site is in a high-seismic region. The results are nearly disasters. Thus, their structural efficiency is extremely low.

In conclusion, this simple survey supported the statement of the relationship between structural efficiency and architectural plan. The efficiency can be approximately measured by the parameter of structural plan density. In order to achieve higher structural efficiency and lower cost of investment and material, more rational and regular architectural designs should be recommended.

Strut-and-tie A to Z

By Nan Gao.

ACRONYM

STM is the acronym for “Strut and Tie Models” or “Strut and Tie Methods“. It is a design method of reinforced concrete structures by idealizing structural components as truss models which are composed of axially loaded members, including compression bearing members (strut) and tension bearing members (tie).

B REGION

The word “B region” is short for “Bernoulli region” or “Beam region“. According to Bernoulli hypothesis (Plane sections remain plane after bending…), the strains in concrete structures follow a linear distribution. This is the theoretical basis for flexural design of concrete components. Those regions which follow Bernoulli hypotheses belong to B regions. They could be designed by simple calculations.

CORBEL

Corbel is an example which does not follow the Bernoulli hypotheses. It is discontinued in geometry. Or we can say the stress and strain in corbel is disturbed.

D REGION

Examples of D regions

D region means “discontinued region” or “disturbed region“, such as corbel. These D regions do not follow Bernoulli hypothesis. Thus, they could not be designed or analyzed by simple calculations. The solution is either empirical approximation or very complicated computations such as FEA methods.

EXAMPLES

Besides corbels, there are many other types of D regions, such as deep beams, pile caps, beams with opening holes, and beam-column joints.

FLEXURE

Take the design of pile caps as an example. In many cases, pile caps are designed by beam theory. They are assumed to fail in flexure, which is a ductile break and has warning cracks. However, according to several research works, the failure modes of most of them are brittle shear failures.

GRAPHICAL PROCEDURE

2D and 3D graphic of STM

Like Maxwell method for analyzing trusses, STM is also a graphical procedure. It is based on graphics of structures, either two-dimensional or three-dimensional. The angles, dimensions, and areas of struts and ties could be obtained by graphical methods.

HISTORY

STM was presented by Schlaich et al. in the year 1987. Other research works include Collins and Mitchell (1991) and MacGregor (1992). It has been adopted by many codes or standards, such as AASHTO LRFD Specifications, ACI 318, CSA Standard, FIP Recommendations, and European Code.

INCLINED ANGLE

One of the definitions in these codes for STM is the geometric rules in creating a proper STM model. Feasible inclined angle between strut and tie members is an important factor. There are different provisions in different codes. Approximately, the inclined angle is limited between 25° and 60°. For example, in the ACI 318M-05, the provision is “The angle between the axes of any strut and any tie entering a single node shall not be taken as less than 25 degrees.“

JOINT

STM of beam-column joints

The common beam-column joints in moment frame structures are examples of D regions. They could be analyzed by STM models.

KIT

Nothing is elixir. STM is only a tool kit for structural engineers. It is not a cookbook procedure. It could perform well only when used correctly. In order to use STM properly, engineers should have a good understanding of structural behavior and an accurate judgment of design issues.

LOAD PATH

Good and poor load paths of the same deep beam

One characteristic of a proper STM model is that the load path is simple and direct. There are several possible load paths in reinforced concrete structures since there are several possible reinforcement arrangements. Nevertheless, we should be meticulous when choosing the load path. Load path should be as elegant as possible. Models which have unnecessarily complicated load paths are not good choices.

METHOD

The basic method for STM is showed in this flow chart (C. C. Fu, 2001).

Flow chart of the method of STM

NODE

Different types of Nodes

In STM truss models, nodes are the connections of members. Usually, there are three members converging into one node. Based on the internal force of members (Compression or Tension), common nodes could be classified as CCC, CCT, and CTT. They have different calculation factors when checking node strengths.

OPTIMAL MODEL

Since there are several possible models, we should use the most ideal model in our analysis. Generally, the common criterion is the amount of reinforcement. A model with the minimum amount of reinforcement is usually the optimal model.

POTENTIAL

STM could provide relatively easy and accurate analysis for D regions in reinforced concrete. It is also useful in the shear design of structural members. Thus, it has a great potential in the field of concrete structures.

QUANTITATIVE

Although STM is a graphical procedure, quantitative method is still required. The calculation of the amount of ties and the checking of nodes and struts all need quantitative procedures. We can see the detailed procedure in the following example.

REINFORCEMENT

STM of joint and the corresponding reinforcement

Detailed reinforcement could be arranged by following the location and distribution of idealized ties in the STM models. Pay attention to the anchorage. Otherwise, brittle anchorage failures might happen.

STRUT

Basically, there are three types of struts: prism, fan, and bottle. They have different factors in the calculation of compression strengths.

TIE

Ties are tension members in the STM models. They should have adequate anchorage in the nodes.

UNIFIED APPROACH

The tile of the paper of Schlaich et al. in the year 1987 is Toward a Consistent Design of Structural Concrete. Since STM considers all load effects simultaneously, it is a unified approach for concrete structures.

VERTICAL REINFORCEMENT

STM for shear design of beams

One of the usages of STM is the shear design of concrete components. The vertical reinforcement (stirrups) in beams could be decided by STM analysis.

WALL

Like deep beams, load bearing walls could also be analyzed by STM.

X-RAY

“X-ray” of pier cap

Sometimes, STM models of concrete structures could be regarded as the x-ray pictures of the structure members, such as this STM model of a pier cap.

YIELD

In order to guarantee the safety of structural members, reinforcement should yield before the brittle crush of concrete.

ZUM BEISPIEL (FOR EXAMPLE)

STM and FEA analysis of a deep beam

Prof. Wright provided an example of STM in his work published in the year 2003. I converted it into metric units and tried another version of STM model. Also, I compared it with the calculation based on beam theory in Chinese code GB 50010 and the results of FEA analysis. Please click this web link to see the detailed graphics and calculations of this example: http://studystructural.wordpress.com/2012/12/01/stm-deep-beam/

References:

• ACI Committee 318, Building code requirements for structural concrete and commentary (ACI 318M-05), 2005
• C. C. Fu, Keynote “The Strut-and-tie model of concrete structures“, 2001
• de Souza RA, Kuchma DA, Park JW, et al. “Nonlinear finite element analysis of four-pile caps supporting columns subjected to generic loading“, Computers and concrete, Vol. 4,  Iss. 5, 2007, pp. 363-376
• Schlaich, J.; Schäfer, K.; and Jennewein, M., “Toward a Consistent Design of  structural Concrete”, PCI Journal, V. 32, No. 3, 1987, pp. 74-150.
• Wight, J.K., and Parra-Montesinos, G. “Strut-and-tie model for deep beam design“, ACI Concrete International, Vol. 25, No. 5, 2003, pp. 63-70
• Wight, J.K. Keynote “Development of the Strut-and-Tie Method for Appendix A of the Building Code (ACI 318-08)“, 2008

Strut-and-Tie Model of a deep beam

By Nan Gao.

STM is a useful method in the analysis of the D regions in concrete structures. Today’s case study is using STM to design a deep beam. This case is from Prof. James K. Wight’s paper “Strut-and-tie model for deep beam design“. First of all, I repeated the STM model of Prof. Wight, but converted all the English units into metric units. It is a practice for me. Secondly, I choose a different STM model and try to figure about whether this simpler triangle STM model could work. Thirdly, I used traditional beam theory to design this deep beam and compared the result with the STM models. The calculation is based on Chinese code GB 50010-2010. Finally, I used SAP2000 to analyze the same deep beam and made a comparison with results of STM.

1. REPEAT PROF. WIGHT’S STM MODEL IN METRIC UNITS

2. TRY ANOTHER VERSION OF STM MODEL

Compared with former STM model, the longitudinal reinforcement is the same, but the vertical reinforcement in this model is much small. However, there are some issues. The inclined angle is very small and is very near to the lower limit. Considering the decrease in the amount of reinforcement and the smaller inclined angle, maybe a better solution is to make the height of the beam a little higher. Thus, the angle could be a bit bigger and the amount of reinforcement will not increase significantly.

3. DESIGN THE SAME DEEP BEAM FOLLOWING CHINESE GB 50010-2010

Compared with STM models, the longitudinal reinforcement is similar. However, the required amount of vertical reinforcement is much larger. The traditional beam theory is much more conservative in the shear design of STM models.

4. ANALYZE THE SAME DEEP BEAM WITH SAP2000

The required amount of longitudinal reinforcement in the left part is larger than the right part. Only the first STM model reflects this characteristic. The second STM model and the calculation based on GB 50010-2010 has the same required amount of reinforcement in the whole span. Thus, Prof. Wight’s STM model is more rational.

In this image, we can see the two zones which required larger amount of vertical reinforcement. They are the locations of Struts in STM models. Also, the Tie 3-4 in Prof. Wight’s model is reflected in this image.

We can see the two bottle-shaped concrete struts very clearly. The stress in the left strut is larger than the right strut. This corresponds with the analysis of the STM model.

References:

• Wight, J.K., and Parra-Montesinos, G. (2003). “Strut-and-tie model for deep beam design“, ACI Concrete International, Vol. 25, No. 5, pp. 63-70
• Wight, J.K.(2008). Keynote “Development of the Strut-and-Tie Method for Appendix A of the Building Code (ACI 318-08)”

More documents about Structures.