Keywords: concrete deep beams, strut-and-tie model, web openings, support conditions, topology optimisation, stress and displacement constraints.

Reinforced concrete deep beams have useful applications in tall buildings and foundations. Over the past two decades, numerous design models for deep beams were suggested. However even the latest design manuals still offer little insight into the design of deep beams in particular when complexities exist in the beams like openings. Numerous investigations have been conducted into various attributes of deep beams [1,2,3,4,5,6,7,8,9]. However only limited research has been conducted dealing with simply supported deep beams with openings [10,11,12,13,14]. Having different failure mechanism to simply supported deep beams, continuous deep beams with web openings present further complications in the analysis and design. Fewer publications in relation to experimental investigations of continuous deep beams with openings have been discovered [15].

A method commonly suggested for the design of deep beams with openings is the strut-and-tie model which is primarily used to represent the actual load transfer mechanism in a structural concrete member under ultimate load. In this study, the development of the strut-and-tie model is transformed to the topology optimisation problem of continuum structures. The optimal strut-and-tie model is generated by gradually removing inefficient material from an over-designed area [16]. During the optimisation process, both the stress and displacement constraints are satisfied and the performance of progressive topologies is evaluated [17].

In the present investigation, the influence on the strut-and-tie model in relation to different support conditions of deep beams is examined. In total, five deep beams with web openings [15,18] are optimised and compared in three groups. The comparison is made based on different support conditions, as well as different locations and sizes of web openings. The optimal strut-and-tie models achieved are also compared with published experimental crack patterns [15,18]. The resulted optimal strut-and-tie models have shown to efficiently represent the load transfer mechanism in concrete deep beams under ultimate load. Numerical results also confirm the experimental observations in that the modes of failure in continuous deep beams depend on the location of web openings regardless of their sizes.

1

D.M. Rogowsky, J.G. MacGregor, S.Y. Ong, "Test of Reinforced Concrete Deep Beams", ACI Structural Journal, 83(4), 614-623, 1986.

2

F.K. Kong, "Reinforced Concrete Deep Beams", Blackie and Sons Ltd, Glasgow, 1990.

3

K.H. Tan, F.K. Kong, S. Teng, L. Guan, "High-Strength Concrete Deep Beams With Effective Span and Shear Span Variations", ACI Structural Journal, 92(S37), 395-405, 1995.

4

K.H. Tan, F.K. Kong, L.W. Weng, "High Strength Concrete Deep Beams Subject to Combined Top-and Bottom-Loading", The Structural Engineer, 75(11), 191-197, 1997.

5

K.H. Tan, F.K. Kong, S. Teng, L.W. Weng, "Effect of Web Reinforcement on High Strength Concrete Deep Beams", ACI Structural J., 94(5), 572-582, 1997.

6

K.H. Tan, S. Teng, F.K. Kong, H.Y. Lu, "Main Tension Steel in High Strength Concrete Deep and Short Beams", ACI Structural J., 94(6), 752-768, 1997.

7

K.H. Tan, C.Y. Tang, K. Tong, "A Direct Method for Deep Beams with Web Reinforcement", Magazine of Concrete Research, 55(1), 53-63, 2003. doi:10.1680/macr.55.1.53.37556

8

A.F. Ashour, "Tests of Reinforced Concrete Deep Beams", ACI Structural Journal, 94(1), 3-12, 1997.

9

S. Foster, R. Gilbert, "Experimental Studies on High Strength Concrete Deep Beams", ACI Structural Journal, 95(4), 382-390, 1998.

10

F.K. Kong, G.R. Sharp, "Structural Idealisation for Deep Beams with Web Openings", Magazine of Concrete Research, 29(99), 81-91, 1977.

11

F.K. Kong, G.R. Sharp, "Structural Idealisation for Deep Beams with Web Openings: Further Evidence", Magazine of Concrete Research, 30(103), 89-95, 1978.

12

M.A. Mansur, W.A.M. Alwis, "Reinforced Fibre Concrete Deep Beams with Web Openings", International Journal of Cement Composites and Lightweight Concrete", 6(4), 263-271, 1984.

13

B.S. Maxwell, J.E. Breen, "Experimental Evaluation of Strut-And-Tie Model Applied to Deep Beam with Opening", ACI Structural Journal, 97(1), 142-148, 2000.

14

K.H. Tan, K. Tong, C.Y. Tang, "Consistent Strut-And-Tie Modelling of Deep Beams with Web Openings", Magazine of Concrete Research, 55(1), 65-75, 2003. doi:10.1680/macr.55.1.65.37555

15

A.F. Ashour, G. Rishi, "Tests of Reinforced Concrete Continuous Deep Beams with Web Openings", ACI Structural Journal, 97(3), 418-426, 2000.

16

Y.M. Xie, G.P. Steven, "Evolutionary Structural Optimisation", Springer-Verlag, London, 1997.

17

H. Guan, Y.J. Chen, Y.C. Loo, "Topology Optimisation of Bridge Type Structures with Stress and Displacement Constraints", International Journal Computational Engineering Science, 2(2), 199-221, 2001. doi:10.1142/S1465876301000295

18

S. Foster, R. Gilbert, "Tests on High Strength Concrete Beams", UNICIV Report, R-354, 1996.

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