Ultimate strength analysis and design for bending
3.9 Continuous beams
The analysis and design procedures presented in this chapter are applicable to sections in statically determinate beams, and to sections in continuous beams and rigid frame
structures. A simplified method for determining the design bending moment (M∗) and shear (V∗) in continuous beams is given in Section 8.2.1. Alternatively, linear elastic methods of analysis may be used, in which case the Standard allows redistribution of moment at interior supports. Details may be found in Clause 6.2.7 of the Standard.
The design of continuous beams for serviceability requirements is discussed in Sections 4.3.5 and 4.5. The formulas given in Chapters 5, 6 and 7, respectively, for shear, torsion and stress development, are equally applicable to continuous structures.
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3.10 Problems
1. Repeat the example in Section 3.3.6 with the equivalent stress block, assuming an intensity of 0.85 fcwhere fc= 65 MPa. Draw the Mversus ptcurves for the two
values of equivalent stress block intensities on the same diagram (similar to Figure 3.3(4)) and discuss the significance of usingα2in place of 0.85 in the new Standard.
2. A singly reinforced rectangular section having a cross section b= 300 mm and d = 600 mm is reinforced by 10 N32 bars. Assuming fc= 32 MPa, compute the
reliable moment capacity of the section (i.e.φMu).
3. Figure 3.10(1) details a square beam vertically loaded symmetrically in the diagonal direction. Given fc= 32 MPa, compute Mu.
500 50 0 50 0 4N28 0 50 d = 500 353.6
Figure 3.10(1) Cross-sectional details of a square beam Note: all dimensions are in mm
4. A symmetrically loaded triangular beam is shown in Figure 3.10(2) with
fc= 32 MPa. Compute Mu.
5. Details of a one-way slab are illustrated in Figure 3.10(3). Based on the load combination formula: ultimate load= 1.2g + 1.5q, compute the uniformly distributed live load (q) that may be carried by the slab. Take fc= 25 MPa and
Chapter 3 Ultimate strength analysis and design for bending 67 10 00 100 0 6N36 d =600 1000
Figure 3.10(2) Cross-sectional details of a triangular beam Note: all dimensions are in mm
a a Simple support 550 0 N16@ 200 mm 12 5 16 0 200 Section a-a
Figure 3.10(3) Details of a slab Note: all dimensions are in mm
ρ = 24 kN/m3. (Hint: take a typical strip 1000 mm wide and analyse as a simply supported beam.)
6. A simply supported beam with a span of 8 m is to carry, in addition to its own weight, a superimposed dead load of 18 kN/m and a live load of 30 kN/m, both over the entire span.
The beam has a rectangular section, which is to be singly reinforced. Given
pt= 1.1%, design and detail the steel reinforcement for the section where the
Take R= d/b ≈ 1.5, ρ = 24 kN/m3, f
c= 20 MPa and use N36 bars only.
Exposure classification A1 applies; use R10 bars only for closed ties; maximum aggregate size= 10 mm.
7. Given a beam section b× D = 450 mm × 950 mm, M∗= 1500 kNm, fc= 32
MPa, fsy= 500 MPa and the maximum aggregate size = 20 mm. Design and
detail the section. Use N28 bars for the main reinforcement and R10 bars for stirrups. Exposure classification A2 applies.
8. A beam section having b× D = 400 mm × 800 mm is required to develop an
effective ultimate moment (φMu) of 1800 kNm. Design the reinforcement using compression steel if necessary.
Assume fc= 32 MPa and fsy= 500 MPa. Sketch the cross-section showing the
reinforcement details. Use N36 bars only with R10 ties. Exposure classification A2 applies.
9. Evaluate Mu for the section shown in Figure 3.10(4). Assume fc= 20 MPa.
200
500
2N24
4N24 100
Figure 3.10(4) Cross-sectional details of the example beam section Note: all dimensions are in mm
10. Design the reinforcement for the section shown in Figure 3.10(5) so as to resist a design ultimate moment (M∗) of 900 kNm. If multiple bar layers are required, they are to be placed 75 mm centre-to-centre. Use N28 bars only and assume
fc= 25 MPa. 300 60 650 C of Asc C of bottom layer of Ast
Figure 3.10(5) Cross-sectional details of the example design section Note: all dimensions are in mm
Chapter 3 Ultimate strength analysis and design for bending 69
11. For the section shown in Figure 3.10(6), compute Mu. Take fc= 20 MPa.
1200 800 65 4N24 4N28 C of Ast 300 40
Figure 3.10(6) Cross-sectional details of a T-section Note: all dimensions are in mm
12. Design and detail the reinforcement for the T-section shown in Figure 3.10(7) for
M∗= 3700 kNm. Use N32 bars only; centre-to-centre spacing of steel layers is set
at 75 mm. Assume fc= 20 MPa and an A1 exposure classification. A final check
must be made on your design for adequacy. 1650
1100 to centre-line of bottom layer 100
450
Figure 3.10(7) Cross-sectional details of a T-section Note: all dimensions are in mm
13. The cross-section of a footbridge structure shown in Figure 3.10(8)a may be idealised as the flanged beam illustrated in Figure 3.10(8)b. For the given loading plus self-weight, design and detail the longitudinal steel reinforcement.
Take fc= 32 MPa and use only N32 bars; exposure classification B1 applies.
Note that the full widths of the top and bottom flanges may be used to accommodate the steel reinforcing bars.
14. The beam-and-slab floor system detailed in Figure 3.10(9) is of reinforced concrete design. What is the effective moment capacity of a typical T-beam unit, assuming that do= 800 mm? Assume fc= 25 MPa.
900 1200 150 150 3000 150 1000 240 100 20 m x x g =75 kN/m q =35 kN/m 100 mm
(a) Box-beam bridge and loading
(b) Section x-x
Figure 3.10(8) Loading configuration and sectional details of a footbridge structure Note: all dimensions are in mm
As As As = 8 N28 As
Figure 3.10(9) Details of a beam-and-slab floor system Note: all dimensions are in mm
15. A doubly reinforced beam section is detailed in Figure 3.10(10). (a) Compute M, assuming fc= 40 MPa.
(b) If the beam is simply supported over a span of 12 m, what is the maximum superimposed (uniformly distributed) working load permissible?
Chapter 3 Ultimate strength analysis and design for bending 71
Figure 3.10(10) Cross-sectional details of a doubly reinforced section Note: all dimensions are in mm
16. A reinforced concrete beam section for a precast concrete structural system is as shown in Figure 3.10(11). Assuming fc= 32 MPa
(a) show that the section is under-reinforced (b) compute the ultimate moment Mu.
Hint: make use of the compatibility and equilibrium conditions.
125 75 125 10 0 48 0 70 3N28
Figure 3.10(11) Cross-sectional details of a precast reinforced concrete beam section Note: all dimensions are in mm
17. For the doubly reinforced section detailed in Figure 3.10(12), compute the ultimate moment. Take fc= 32 MPa.
6 N28
6 N28
Figure 3.10(12) Cross-sectional details of a doubly reinforced section Note: all dimensions are in mm
18. The L-beam illustrated in Figure 3.10(13) is to carry, in addition to its own weight, a dead load g= 25 kN/m and a live load q = 100 kN/m over a
simply-supported span of 8 m. Design and detail the tension reinforcement. Use N32 bars and take fc= 50 MPa; maximum aggregate size is 20 mm, ρ = 24
kN/m3and exposure classification B1 applies.
Figure 3.10(13) Details of an L-beam in a beam and slab system Note: all dimensions are in mm
Chapter 3 Ultimate strength analysis and design for bending 73
19. The cantilever beam detailed in Figure 3.10(14) forms part of a loading platform. In addition to self-weight, the beam carries a uniform dead load of 10 kN/m and a concentrated live load (q), positioned as shown in Figure 3.10(14)a. Assuming fc
= 32 MPa, what is the maximum allowable q?
10 N28
5 N28
q =?
(a) (b)
Figure 3.10(14) Details of a cantilever beam Note: all dimensions are in mm
20. The overhang of a continuous T-beam, illustrated in Figure 3.10(15)a, can be treated as a cantilever beam. In addition to its own weight, the cantilever is to carry a concentrated live load of 75 kN at the tip.
q =
(a) (b)
q = 75 kN
Figure 3.10(15) Details of a continuous T-beam with overhang Note: all dimensions are in mm
Figure 3.10(15)b shows the dimensions of the T-section. Design and detail the reinforcement for the support (root) section. Use compression steel if necessary. Take fc= 25 MPa; the exposure classification is A2 and the maximum aggregate
size= 20 mm. Use only N28 bars, noting that reinforcing bars may be spread over the width of the flange.