Grid Strut-Tie Model – Strength Analysis of Torsional Beam

The evaluation of the 3-D grid approach was carried out using the ultimate strength of 60 reinforced concrete beams subjected to pure torsion to failure by Hsu (1968), Rasmussen & Baker (1995), and Koutchoukali & Belarbi (2001). For purposes of comparison with current codes of practice, the ultimate strength of the beams was also evaluated using the AASHTO-LRFD (2014) and ACI 318-14 (2014) design provisions, Hsu & Mo (1983)’s method, and method of Rahal & Collins (1996).
The ultimate strength of each of the 51 torsional beams evaluated with the existing and proposed methods are compared in Table. It must be noted that all the methods overestimated the experimental capacities. On the other hand, the ultimate strengths were evaluated satisfactorily by the proposed 3-D grid strut-tie model approach. Thus, the proposed approach can be conservatively used for members under torsion.
ReferenceHsu, T. T. C. (1968) Torsion of Structural Concrete, ACI Journal, Vol. 18, pp. 261-306.Rasmussen, L. J. and Baker, G. (1993) Torsion in Reinforced Normal and High-Strength Concrete Beams – Part 1: Experimental Test Series, ACI Structural Journal, Vol. 92, No. 1, pp. 56-62.Koutchoukali, N. and Belarbi, A. (2001) Torsion of High-strength Reinforced Concrete Beams and Minimum Reinforcement Requirement, ACI Structural Journal, Vol. 98, No. 4, pp. 462-469.American Concrete Institute (2014) Building Code Requirements for Structural Concrete (ACI 318M-14) and Commentary(ACI 318RM-14), Farmington Hills, Michigan, USA.American Association of State Highway and Transportation Officials (2014) AASHTO LRFD Bridge Design Specifications, 6th Edition, Washington D.C., USA.Hsu, T. T. C. and Mo, Y. L. (1983) Softening of Concrete in Torsional Members, Research Report No. ST-TH-001-83, Department of Civil Engineering, University of Houston, Texas, USA.Rahal, K. N. and Collins, M. P. (1995) Analysis of Sections Subjected to Combined Shear and Torsion – A Theoretical Model, ACI Structural Journal, Vol. 92, No. 4, pp. 459-469.

Grid Strut-Tie Model – Design of Pier Coping

To examine the capability and effectiveness of the 3-D grid strut-tie model approach in the design of a concrete member with multiple load combinations, they selected a bridge pier cap subjected to three load cases with 3-D loads. The shear span-to-effective depth ratio of the cap is 1.25.In load case 1 with the dead and live loads acting vertically, most of the design loads at loading plates 2, 3, and 4 were transferred to the pier directly by the concrete struts connecting the loading plates and column. On the other hand, most of the design loads at loading plates 1 and 5 were transferred to the pier by the curved-shaped paths described by multiple concrete struts. The pier cap subjected to load cases 2 and 3 with dead and live loads acting in three directions were designed with the same 3-D grid strut-tie model. This illustrates the versatility of the proposed approach to handle multiple loading, including 3-D loads, by using a single type of 3-D grid strut-tie model. The conditions of the strut-tie model’s geometrical compatibility under all three load cases were satisfied, and the strengths of all nodal zones were enough to transfer the strut and tie forces. Thus, the maximum cross-sectional area of a steel tie at a certain position under the three load cases was taken as the required area of the reinforcing bar that should be distributed with the effective area of the steel tie. The required areas of reinforcing bars corresponding to the other steel ties were obtained and distributed in the same way.
To compare the design results of the proposed approach, the designs of the pier cap were carried out using the ACI 318-14 flexural design method, ACI 318-14 bracket provisions, and ACI 318-99 deep beam provisions. The pier cap was also designed based on the strut-tie model approaches of current design codes, with the 2-D strut-tie models representing arch and truss load transfer mechanisms, respectively. As the 2-D strut-tie models are valid for 2-D load cases, only load case 1 was considered. The vertical positions of both nodes are the same as those of corresponding nodes of the 3-D grid strut-tie model. The horizontal positions of both nodes can be moved toward the boundary of the pier by considering the required cross-sectional areas of the concrete struts located in the pier.From the design results, two features can be observed. First, more flexural reinforcing steel is required by the strut-tie model approach than by the other methods. This is mainly due to the positions of nodes F and G and the corresponding nodes. If these nodes are moved toward points F’ and G’ by considering the critical section defined by the ACI 318-14 flexural design and bracket provisions, similar flexural design results will be obtained. Second, more vertical shear reinforcing bars are required by the strut-tie model of truss mechanism than by the 3-D grid strut-tie model. This is because the design loads at loading plates 1 and 5 are transferred to the column mainly by truss action in the strut-tie model of the truss mechanism. On the other hand, the design loads are transferred to the pier by the combined arch and truss mechanisms in the 3-D grid strut-tie model. In other words, a portion of the design loads are transferred directly to the pier through arch-type concrete struts in the 3-D grid strut-tie model.
ReferenceKim B. H. (2004) Grid Mesh Strut-Tie Model Approach for Design of Reinforced Concrete Members, Ph. D Dissertation, Kyungpook National University, Daegu, South Korea (in Korean).American Concrete Institute (2014) Building Code Requirements for Structural Concrete (ACI 318M-14) and Commentary(ACI 318RM-14), Farmington Hills, Michigan, USA.American Concrete Institute (1999) Building Code Requirements for Structural Concrete (ACI 318-99) and Commentary (ACI 318R-99), Farmington Hills, Michigan, USA.

Grid Strut-Tie Model – Max Widths of 3D

he maximum width calculation method for evaluating the geometrical compatibility of grid strut-tie model.  This concept employed here originates from a rule stating that the overlaps of areas (or widths) of neighboring components are not allowed.
In this approach, the method employed in the 2-D grid strut-tie model approach is extended. Namely, in addition to the strut-tie model approaches of current design codes, the maximum cross-sectional areas of struts and ties are defined as the maximum areas that struts and ties can take without overlaps in a grid element.
The maximum cross-sectional area of a component of a grid element is determined by multiplying two maximum widths of the component evaluated on the component’s two orthogonal planes. For example, the maximum cross-sectional area of the x-directional component of length lx1 is determined by multiplying the z-directional maximum width WHE and the y-directional maximum width WVE. Here, the maximum width WHE is taken as 0.2lz1+0.2lz2, where lz1 and lz2 are the lengths of the z-directional components connected to the ends of the x-directional component. In the same way, the maximum width WVE is defined as 0.2ly1+0.2ly2, where ly1 and ly2 are lengths of the y-directional components connected to the ends of the x-directional component.
The maximum cross-sectional area of an inclined plane component of a grid element is determined in the same manner. For example, the maximum cross-sectional area of the inclined plane component of x-directional length lx1 in the xy-plane is determined by multiplying the maximum width WIPE of the xy-plane and the maximum width WHEZ of the plane orthogonal to the xy-plane. The maximum width WIPE of the xz-plane are taken as the length of the perpendicular lines connecting the points (a, b, c, d), which are determined by considering the maximum widths of the neighboring horizontal and vertical elements. Following the methods, the maximum widths WIPE and WHEZ of the xy-plane and its orthogonal planes, respectively, are expressed as
The cross-sectional area of an inclined space component that exists only in a 3-D grid element is defined as the area excluding all the cross-sectional areas of horizontal components, vertical components, and inclined plane components in a grid element. For example, the maximum cross-sectional area of the space component is determined by multiplying the maximum width WIPE of the xz-plane and the maximum width WISE of the yz-plane rotated about the y-axis. The maximum widths WIPE and WISE are expressed as


In the grid strut-tie model approach, a component of a grid element is assumed to fail if its maximum cross-sectional area is less than that required for carrying the cross-sectional force. To reflect more precisely the characteristics of the load transfer mechanism in a grid element, the modified maximum cross-sectional area of a component that is determined by considering the required cross-sectional areas of neighboring components (instead of the maximum cross-sectional area defined above) is used for examining the geometrical compatibility. Here, the modified maximum cross-sectional area of a component is determined by multiplying two modified maximum widths of the component in two orthogonal planes.
ReferenceKim, B. H. and Yun, Y. M. (2014) Strut-Tie Model Approach Associated with 3-Dimensional Grid Elements for Design of Structural Concrete – (I) Proposal of Approach, Journal of the Korean Society of Civil Engineers, Vol. 34, No. 2, pp. 425-436.

Grid Strut-Tie Model – Max Widths of 2D

The maximum width calculation method for evaluating the geometrical compatibility of grid strut-tie model.
The maximum widths of the horizontal and vertical elements are determined by taking 20% of the length of the neighboring vertical and horizontal elements respectively. The maximum widths of the inclined elements are taken as the lengths (Wd) of the perpendicular lines connecting the points (a, b, c, d), which are determined by considering the maximum widths of the neighboring horizontal and vertical elements (V1, V3, H1, H3). The reason for taking 20% of the lengths of the neighboring elements is based on a rule whereby the maximum widths of all elements in a square grid should be the same.

In a grid strut-tie model approach, a concrete strut is assumed to fail if its maximum width is less than the required width needed for carrying its cross-sectional force. Here, to reflect the actual characteristics of the load transfer mechanism within a grid, the modified maximum width of a concrete strut is used for examining the geometrical compatibility of a strut-tie model. The maximum width of the element increases as long as the required widths of the neighboring element B~E are less than their maximum load-carrying capacities (maximum widths). When the required widths of elements B~E is equal to zero, the modified maximum width of element A that can be maximized. This originated from the rule whereby the overlaps of widths of the neighboring elements are not allowed.
Appendix : Mathematical proof for taking 20% of the length of neighboring horizontal (vertical) components in the determination of the maximum width of vertical (horizontal) components
By using the equation of perpendicular, the half of the maximum width of the inclined plane component of the square grid element shown above is described as follows:
where the coefficients a, b, and c are a=y1-y2=-1.0, b=x2-x1=1.0, and c=x1y2-y1x2=0, respectively. Substituting the coefficients into above equation, we haveAs the value 0.5 is the maximum half width that the vertical component can take, i.e., as wa=0.5, abs(-wa+0.5) becomes -wa+0.5. By using the condition that the halves of the maximum widths of the vertical and inclined components are the same, i.e., by equating wa to wb, we have
Therefore, the maximum width of the inclined plane component equals to those of neighboring vertical (horizontal) components if we take 20% of the length of neighboring horizontal (vertical) components.
ReferenceYun, Y. M. and Kim B. H. (2008) Two-Dimensional Grid Strut-Tie Model Approach for Structural Concrete, Journal of Structural Engineering, ASCE, Vol. 134, No. 7, pp. 1199-1214.

Grid Strut-Tie Model – Shear Wall with Openings

A shearwall with two openings, which was used for the illustration of the ACI 318-02’s strut-tie model approach (Barnes, 2002), was selected in order to verify the effectiveness of the grid strut-tie model. The whole region of the shearwall belongs to a disturbed region because of openings, concentrated loads, and support conditions. The five types of load combinations are as follows:– Load Combination 1: Two 2000kN vertical loads acting on the top of the shearwall– Load Combination 2: Two 760kN horizontal loads acting on the right side of the shearwall– Load Combination 3: Load combinations 1 and 2– Load Combination 4: Two 760kN horizontal loads acting on the left side of the shearwall– Load Combination 5: Load combinations 1 and 4ReferenceYun Y. M. and Kim B. H. (2008) A 2-Dimensional Grid Strut-Tie Model Approach for Structural Concrete, Journal of Structural Engineering, ASCE, Vol. 134, No. 7, pp. 1199-1214.Barnes, R.W. (2002) High Wall with Two Openings – Examples for the Design of Structural Concrete with Strut-and-Tie Models, Reineck, K.H. Ed., American Concrete Institute, Michigan, pp. 195-211.

What is GSTM?

A grid strut-tie model, composed of grids which are used to allow load transfers in all directions, is selected regardless of the stress trajectories and a designer’s subjectivity on stress flows.The approach enables a structural design deliberating all kinds of load combinations by using a single type of grid strut-tie model.A structural design is possible by using an indeterminate strut-tie model with appropriate element stiffness derived by applying a simple iterative technique to a grid strut-tie model.
Construction of Grid Strut-Tie Model1. 2-D Grid Strut-Tie ModelA grid strut-tie model, composed of a number of grids, is selected by considering the loading and geometrical conditions of the structural concrete. A grid consists of four horizontal elements, four vertical elements, and four inclined elements, as shown in above figure. The ratio of vertical length to horizontal length of a grid is limited within the range of 0.75~1.33. If the ratio is not satisfied, the maximum width of an inclined element is reduced up to 90% of the width of the horizontal or vertical element, thus causing an improper load transfer capacity of the inclined element compared to the other elements.2. 3-D Grid Strut-Tie ModelA 3-D grid strut-tie model is composed of a number of basic grid elements, and is constructed by considering the loading and geometrical conditions of the structural concrete. A basic grid element with twelve horizontal/vertical components, twelve inclined plane components, and four inclined space components is shown in above figure. The basic grid elements allow for load transfers in all directions at a node. Application Procedure of the Grid Strut-Tie Model ApproachThe analysis or design of 3-D structural concrete with D-regions using the proposed approach incorporating basic grid elements to construct a 3-D strut-tie model is conducted using the procedure shown in below figure.A considerable amount of time and effort for determining the effective strengths and cross-sectional areas of struts and ties, and for examining the conditions of geometrical compatibility and nodal zone strength, is required in the proposed approach. In particular, the time and effort needed for numerous numerical calculations and graphical decisions are significantly increased when multiple load combinations are required. Therefore, we developed a computer graphics program to automate the design procedure, such as AStrutTie3D, by handling numerical calculations and graphical decisions effectively.ReferenceKim B. H. (2004) Grid Mesh Strut-Tie Model Approach for Design of Reinforced Concrete Members, Ph. D Dissertation, Kyungpook National University, Daegu, South Korea (in Korean).Yun Y. M. and Kim B. H. (2008) A 2-Dimensional Grid Strut-Tie Model Approach for Structural Concrete, Journal of Structural Engineering, ASCE, Vol. 134, No. 7, pp. 1199-1214.