Double Corbel Design using CAST (Part 2)

 

 In the above figure, STM shape that is adopted for design the double corbel is shown. To illustrate in more details all components of this STM constructed by CAST, the following info is provided below.

• D-Region Boundary:
  • A closed, non-intersecting polygon that defines the geometry of the structure.
  • There are two types of D-Region Boundaries: Outer Boundary (Perimeter) and Inner Boundary (for openings).
  • You are allowed to create only one Outer Boundary, but you can create as many Inner Boundaries as you want.
  • Because there is only one Outer Boundary that should exist in your model and the region inside the Outer Boundary represents the concrete continuum, you are not allowed to copy, cut, or remove it from your model once you create it. However, you are surely allowed to change the positions of the Boundary Corners. You are also allowed to add and remove Boundary Corners as you wish.
• D-Region Boundary Corner:
  • Intersection of two Boundary Edges.
• D-Region Boundary Edge:
  • Line connecting two Boundary Corners.
  • Place for STM Nodes that possess Bearing Plates, Point Loads, or Point Supports.
• Strut-and-Tie Model:
  • Internal truss in the D-Region.
  • Consists of STM Elements interconnected at STM Nodes.
• STM Element:
  • Component of Strut-and-Tie Model.
  • STM Elements can be in the form of Struts (compressive STM Elements) or Ties (tensile STM Elements).
  • STM Elements have Effective Widths, representing the extent of idealized compressive stress fields (for Struts) or tensile stress fields (for Ties).
• STM Node:
  • Component of Strut-and-Tie Model.
  • Place where one or more STM Elements meet, also called a nodal zone.
  • There is only one STM Element allowed to frame into an STM Node having Bearing Plates, Point Loads, or Point Supports.
• Stabilizer:
  • STM Element whose member force is zero.
  • Stabilizers are not included in the nodal zone construction.
  • Stabilizers are required to avoid ill-conditioned structure stiffness matrix in truss analysis.
  • You must always create a stable Strut-and-Tie Model; Stabilizers are identified by CAST during truss analysis.
  
  
  After you are familiar with all components given above, it is time to show you steps by steps how to construct STM using CAST. Please follow the instruction in the next post. For the previous post about designing of double corbel using CAST (part1) please see here.
  
  

Double Corbel Design using CAST (Part 1)

Overview

In this document, the task of designing a double corbel is completed to illustrate how the CAST Design Tool (Version 0.9.10) can be used for the design of D- (Discontinuity) Regions. Following a brief introduction and description of the CAST graphical user interface, a step-by-step solution is presented. In order to describe many of the important features of this program, complete details are provided with associated images from CAST. The completion of this exercise and thus familiarization with CAST is estimated to take 60 minutes.

Introduction

Figure 1 describes the geometry and loadings for the double corbel structure being considered. The thickness of the corbel is 600 mm. The concrete strength is 35 MPa, and the yield strength of reinforcement is taken as 420 MPa. The corbel supports an ultimate vertical force of 1000 kN and an ultimate horizontal force of 100 kN at each end and two ultimate point loads of 3000 kN in the supporting column region. Bearing plates of 150 mm length x 600 mm width x 25 mm thick are provided at each end of the corbel. Figure 2 shows the selected strut-and-tie model for this structure. The design will be completed to meet ACI 318-02 Appendix A requirements. Figure 1   The Geometry and the Loadings of the Corbel under Consideration Figure 2   Strut-and-Tie Model Employed in this Corbel Design

STM Design Steps using CAST

The design using CAST can be summarized into five steps. They are illustrated in Figure 1 below.

Figure 1   Summary of Design Procedure Using CAST

Note that the steps are given here for guidance only; you do not have to strictly follow the steps in the order presented. For example, the first item in Step 1, i.e., Define Project Description, can be done in any step. However, some steps cannot be performed prior to the completion of related steps. For example, the Define Strut-and-Tie Model in Step 2 cannot be performed without the step Define Outer and Inner Boundaries being completed even though you still can modify the Outer and Inner Boundaries later. Another example is the step Define Strut, Tie, and Node Property Types in Step 4. This step cannot be completed without having information about concrete compressive and steel yield strengths defined in step Define D-Region Thickness and Material Strengths in Step 1. CAST implicitly notifies users of what the proper order of steps by three approaches:

• Enabling or disabling menus and buttons of toolbars associated with the current modeling process.
• Displaying dialog boxes that need to be completed before proceeding to the step you want to perform.
• Showing warning messages of improper order of steps in the status bar.

Credits:http://dankuchma.com/stm/CAST/procedure.htm

Current Features of CAST

Graphical User Interface
	Define D-Region boundaries graphically
	Define strut-and-tie models graphically and numerically
	Guidelines, grid points, and snap tools for accurate dimensioning
	Toolbars and dialog boxes for defining common elements, such as the imposed loadings, material properties, supports, and bearing plates
	Templates for common truss models
	Standard drawing and editing features, such as cut, copy, paste, delete, undo, redo, move, find, glue, and unglue
	Easy access to member information by mouse right button click
	Display of structural model, including reinforcement positions, labels, and areas, in multiple windows with various zooming and panning options
	Context sensitive online help
Analysis Types
	Design calculations (evaluation of truss member forces and truss member dimensions)
	Detailed analysis of nodal zones
	Simple truss capacity prediction
	Load-deformation analysis
Truss Solution
	Truss analysis and stress check at command
	Statically indeterminate trusses
	Truss stability checked with warning messages
	Illustration of member forces and identification of member characteristics for dimensioning
	Pushover analysis per user-defined stress-strain relationships of struts and ties with options to use load-controlled or displacement-controlled

Multiple Load Cases and/or Strut-and-tie Models
	Organize multiple load cases and/or strut-and-tie models in one database
	Present multiple load cases and/or strut-and-tie models in multiple windows
Truss Member Dimensioning
	Manual or automatic selection of effective widths of struts
	Single or multiple layers of tie reinforcement
	Checks of bearing plate stresses
	Quick check of the adequacy of strut-and-tie model components through color identification
	Support explicitly ACI Code requirements
Output Features
	Graphical and tabular output
	Create customized text input echo and/or output files
	Create design summaries including graphics in Microsoft Word Document format
	Create input echo and/or output tables in Microsoft Excel format
	Create graphics printout with print preview
	Export graphics into AutoCAD DXF and Bitmap files
	Create response quantity plots obtained from load-deformation analysis in Microsoft Excel format
	Display response history obtained from load-deformation analysis
Miscellaneous
	Support SI, US customary, and old metric units interchangeable during the design
	Support user-defined and ASTM Standard Reinforcing Bars (ASTM A615/A615M)
	Unit converter for helping convert physical quantities from one unit to another
	Built-in calculator for editable text boxes
	Customizable windows and toolbars

CAST: Computer Aided Strut-and-Tie

    The CAST program is being developed to provide both students and practitioners with a graphical design tool that makes the design process more efficient and transparent. The principal investigator of this research has had the opportunity to teach the STM to students at the University of Illinois since Fall 1997. It has been his experience that students readily take to STM due to their familiarity and confidence in the simple mechanics of a truss. They have also been able to grasp proposed design provisions for the strength of struts, ties, and nodal zones. However, once a specific design assignment is given, they can quickly become bogged down in the details of calculating truss member forces and ensuring that the dimensions of the struts and nodal zones are sufficient to support the imposed loadings.

The CAST program is a graphically interactive design tool that has been under development since Fall 1998 and has been used by students taking the second course of Reinforced Concrete Design at the University of Illinois since Fall 1999. The NSF CAREER is making possible the continued development of CAST. This will enable the growth into a program that allows designers to quickly optimize their design, handle multiple load cases, and generate final drawings. This program also serves as an instructional device, familiarizing students and practitioners with both the program and the strut-and-tie design philosophy.
CAST utilizes a single interface for creation or modification of strut-and-tie models, truss analysis, selection of reinforcing steel, and capacity checks of the struts and nodes. CAST was developed within Windows 32-bit environments. This means that it will only work on Windows 95/98/2000/ME/XP or NT 4.0 machines.

Credits: http://dankuchma.com/stm/CAST/

STM Design Steps

The design process using STM involves five major steps described below. These steps are illustrated in Figure 1 using the design example of a dapped-ended beam.

1.	Define the boundaries of the D-Region and determine the boundary forces (the ultimate design forces) from the imposed local and sectional forces.
2.	Sketch the truss, determine the equivalent boundary forces, and solve for the truss member forces.
3.	Select reinforcing or prestressing steel to provide the necessary tie capacity and ensure that this reinforcement is properly anchored in the nodes.
4.	Evaluate the dimensions of the struts and nodes such that the capacity of all struts and nodes is sufficient to carry the truss member forces.
5.	Provide distributed reinforcement to ensure ductile behavior of the D-Region.

Since equilibrium of the truss with the boundary forces must be satisfied (step 2) and stresses everywhere must be below the limits (step 3 and 4), one can see that the STM is a lower-bound (static or equilibrium) method of limit analysis.

                                     Figure 1   The Major Steps in STM Design Process
Credits: http://dankuchma.com/stm/STM/

Strut-and-Tie Model (STM)

The STM is based on the lower-bound theory of limit analysis. In the STM, the complex flow of internal forces in the D-Region under consideration is idealized as a truss carrying the imposed loading through the region to its supports. This truss is called strut-and-tie model and is a statically admissible stress field  in lower-bound (static) solutions. Like a real truss, a strut-and-tie model consists of struts and ties interconnected at nodes (also referred to as nodal zones or nodal regions). A selection of strut-and-tie models for a few typical 2-D D-Regions is illustrated in Figure 1. As shown in the figure, struts are usually symbolized using broken lines, and ties are usually denoted using solid lines.

Figure 1   Examples of Strut-and-Tie Models for Common Structural Concrete Members

Strut-and-Tie Model Components
Struts are the compression members of a strut-and-tie model and represent concrete stress fields whose principal compressive stresses are predominantly along the centerline of the strut. The idealized shape of concrete stress field surrounding a strut in a plane (2-D) member, however, can be prismatic (Figure 2(a)), bottle-shaped (Figure 2(b)), or fan-shaped (Figure 2(c)) [1]. Struts can be strengthened by steel reinforcement, and if so, they are termed reinforced struts.

Figure 2   Basic Type of Struts in a 2-D Member: (a) Prismatic (b) Bottle-Shaped (c) Fan-Shaped

Ties are the tension members of a strut-and-tie model. Ties mostly represent reinforcing steel, but they can occasionally represent prestressing steel or concrete stress fields with principal tension predominant in the tie direction.
Nodes are analogous to joints in a truss and are where forces are transferred between struts and ties. As a result, these regions are subject to a multidirectional state of stress. Nodes are classified by the types of forces being connected. Figure 3 shows basic types of nodes in a 2-D member; in the figure, C is used to denote compression and T is used to denote tension.

Figure 3   Basic Type of Nodes: (a) CCC  (b) CCT  (c) CTT  (d) TTT

Uniqueness of Strut-and-Tie Models
As a statically admissible stress field, a strut-and-tie model has to be in equilibrium externally with the applied loading and reactions (the boundary forces) and internally at each Node. In addition, reinforcing or prestressing steel is selected to serve as the ties, the effective width of each strut is selected, and the shape of each nodal zone is constructed such that the strength is sufficient. Therefore, only equilibrium and yield criterion need to be fulfilled for an admissible strut-and-tie model. The third requirement in solid mechanics framework, namely the strain compatibility, is not considered.
As a result of these relaxed requirements, there is no unique strut-and-tie model for a given problem. In other words, more than one admissible strut-and-tie model may be developed for each load case as long as the selected truss is in equilibrium with the boundary forces and the stresses in the struts, ties, and nodes are within the acceptable limits. The lower-bound theorem guarantees that the capacity obtained from all statically admissible stress fields is lower than or equal to the actual collapse load. However, as a result of limited ductility in the structural concrete, there are only a small number of viable solutions for each design region. Figure 4 illustrates an example in which one solution is preferable to another. Due to the point load at the tip of the cantilever portion, the upper part of the beam is likely to develop horizontal tensile stresses along the beam. Therefore, the model with the upper horizontal tie (Figure 4(a)) is preferable to that shown in Figure 4(b). The latter only effectively resists the tension in the upper region near the middle support.

Figure 4   Two statically admissible strut-and-tie models for a cantilevered deep beam under vertical loading: (a) Workable truss (b) Less favorable truss due to excessive ductility demands

Reference:
1) Schlaich, J. and Schäfer, K., "Design and Detailing of Structural Concrete Using Strut-and-Tie Models," The Structural Engineer, Vol. 69, No. 6, March 1991, pp. 113-125.

Credits:http://dankuchma.com/stm/STM

Historical Background of Strut-and-Tie Model (STM)

The idea of the strut-and-tie method came from the truss analogy method introduced independently by Ritter and Mörch in the early 1900s for shear design of B-Regions. This method employs the so-called truss model as its design basis. The model was used to idealize the flow of force in a cracked concrete beam. In parallel with the increasing availability of experimental results and the development of limit analysis in plasticity theory, the truss analogy method has been validated and improved considerably in the form of full member or sectional design procedures. The truss model has also been used as the design basis for torsion. Excellent summary of the development of truss model for shear design of B-Regions can be found in Ref. 1. The strut-and-tie method came after Marti [2] and Schlaich et al. [3,4] promoted the use of truss model in D-Regions.

 Reference:
1) ASCE-ACI Committee 445 on Shear and Torsion, "Recent Approaches to Shear Design of Structural Concrete," Journal of Structural Engineering, ASCE, Vol. 124, No. 12, December 1998, pp. 1375-1417.
2) Marti, P., "Basic Tools of Reinforced Concrete Beam Design," ACI Journal, Proceedings, Vol. 82, No. 1, January-February 1985, pp. 45-56.
3) Schlaich, J., Schäfer, K., and Jennewein, M., "Toward a Consistent Design of Structural Concrete," Journal of the Prestressed Concrete Institute, Vol. 32, No. 3, May-June 1987, pp. 74-150.
4) Schlaich, J. and Schäfer, K., "Design and Detailing of Structural Concrete Using Strut-and-Tie Models," The Structural Engineer, Vol. 69, No. 6, March 1991, pp. 113-125.

Credit:http://dankuchma.com/stm/STM/

B-Regions and D-Regions

   In selecting the appropriate design approach for structural concrete, it is useful to classify portions of the structure as either B- (Beam or Bernoulli) Regions or D- (Disturbed or Discontinuity) Regions. B-Regions are parts of a structure in which Bernoulli's hypothesis of straight-line strain profiles applies. D-Regions, on the other hand, are parts of a structure with a complex variation in strain. D-Regions include portions near abrupt changes in geometry (geometrical discontinuities) or concentrated forces (statical discontinuities). Based on St. Venant's principle, the extent of a D-Region spans about one section depth of the region on either side of the discontinuity.

Figure 1 and Figure 2 show examples of the division between B-Regions and D-Regions in building and bridge structures, respectively. In the figures, the unshaded area with a notation B indicates B-Region, and the shaded area with a notation D is used to indicate D-Region. The notations h₁, h₂, h₃, ... are used to denote the depth of structural members. The notations b₁ and b₂ denote the flange width of structural members.

Figure 1   Example of D-Regions in a Common Building Structure

Figure 2   Example of D-Regions in a Common Bridge Structure

Most design practices for B-Regions are based on a model for behavior. As examples, design for flexure is based on conventional beam theory while the design for shear is based on the well-known parallel chord truss analogy. By contrast, the most familiar types of D-Regions, such as deep beams, corbels, beam-column joints, and pile caps, are currently still designed by empirical approaches or by using common detailing practices. For most other types of D-Regions, code provisions provide little guidance to designers. The Strut-and-Tie Method (STM) is emerging as a code-worthy methodology for the design of all types of D-Regions in structural concrete.

It is worth noting that although the STM is equally applicable to both B- and D-Region problems, it is not practical to apply the method to B-Region problems. The conventional beam theory for flexure and parallel chord truss analogy for shear are recommended for those designs.

 Credits: http://dankuchma.com/stm/STM/