Design of a Continuous Span Beam
Step 1: Design Moments
Find out the maximum moments at supports and at mid spans from the moment diagrams for
various load combinations.
At mid span, beam may be a rectangular beam or T beam,
At supports, beam must be a rectangular beam.
Step 2: Assuming Beam section (including Slab Depth)
Beam Size, 12″ ×18″ (including Slab Depth)
Step 3: Moment Capacity of the singly reinforced beam
Maximum Reinforcement ratio allowed by the ACI code
𝑓′
0.003
𝜌𝑚𝑎𝑥 = 0.85 × 𝛽1 × 𝑓𝑐 × 0.003+0.004 ; 𝛽1 = 0.85 − 0.05 ×
𝑦
𝑓𝑐′ −4000
1000
Layer of bars
1 layer: X = 2.5in
2 layers: X = 3.5~4 in
3 layers: X = 4.5~5 in
x
Effective depth, d = Depth of beam – X
𝐴𝑠 = 𝜌𝑚𝑎𝑥 × 𝑏𝑑
𝑎=
𝐴𝑠 𝑓𝑦
0.85𝑓𝑐′ 𝑏
𝑎
𝑀𝑢(𝑐𝑜𝑛𝑐𝑟𝑒𝑡𝑒) = ∅ × 𝐴𝑠 × 𝑓𝑦 × (𝑑 − )
2
𝑀𝑢(𝑐𝑜𝑛𝑐𝑟𝑒𝑡𝑒) is the Moment Capacity of the singly reinforced beam
; 0.65 ≤ 𝛽1 ≤ 0.85
Step 4: Calculation of Steel area
If the moment capacity of the singly reinforced beam is less than the required capacity, then
compression steel is need. Otherwise, we have to change the depth to make it singly reinforced.
Suppose, required capacity = −𝑀𝑢 k-ft (at support)
Assuming that, 𝑓𝑠′ = 𝑓𝑦
∅𝑀𝑛 = 𝑀𝑢 − 𝑀𝑢(𝑐𝑜𝑛𝑐𝑟𝑒𝑡𝑒)
∅𝑀𝑛 = ∅𝐴′𝑠 𝑓𝑦 (𝑑 − 𝑑′)
𝐴′𝑠 = ∅𝑓
𝐴′𝑠
∅𝑀𝑛
𝑦 (𝑑−𝑑′)
Compressive steel area = 𝐴′𝑠
Tensile steel area = 𝐴𝑠 + 𝐴′𝑠
Suppose, required capacity = 𝑀𝑢 k-ft (at mid span)
Effective width:
1. 𝑏 =
𝑆𝑝𝑎𝑛
4
2. 𝑏 = 16ℎ𝑓 + 𝑏𝑤
3. 𝑏 = 𝑐𝑒𝑛𝑡𝑒𝑟 𝑡𝑜 𝑐𝑒𝑛𝑡𝑒𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑏𝑒𝑎𝑚𝑠
Assume, 𝑎 = ℎ𝑓
Calculate, 𝐴𝑠 =
𝑀𝑢
∅𝑓𝑦 (𝑑−
ℎ𝑓
2
𝐴𝑠 + 𝐴′𝑠
)
𝐴𝑠 𝑓𝑦
Check, 𝑎 = 0.85𝑓′ 𝑏
𝑐
If 𝑎 > ℎ𝑓 ; T-beam analysis is required
If 𝑎 < ℎ𝑓 ; Rectangular beam analysis is required
Step 5: Design for shear (Stirrups design)
Vcr at a distance d from face of support = 𝑉𝑢
Shear resisted by concrete, ∅𝑉𝑐 = 2∅√𝑓𝑐′ 𝑏𝑤 𝑑
𝑉𝑠 = 𝑆ℎ𝑒𝑎𝑟 𝑡𝑜 𝑏𝑒 𝑡𝑎𝑘𝑒𝑛 𝑏𝑦 𝑠𝑡𝑒𝑒𝑙
Conditions:
1
1. If 𝑉𝑢 < 2 ∅𝑉𝑐 , theoretical no stirrup required
2. If 𝑉𝑢 < ∅𝑉𝑐 , Minimum stirrup required
3. If 𝑉𝑢 > ∅𝑉𝑐 , Stirrup required. Spacing are specified by design.
4. If 𝑉𝑠 > 4∅√𝑓𝑐′ 𝑏𝑤 𝑑, all spacing above are halved.
5. If 𝑉𝑠 > 8∅√𝑓𝑐′ 𝑏𝑤 𝑑, Section must be changed.
Spacing are specified by design
∅𝐴𝑣 𝑓𝑦 𝑑
𝑆=𝑉
𝑢 −∅𝑉𝑐
; ∅ = 0.75
Minimum stirrup
𝑑
Spacing = 2
= 24 inch
𝐴𝑣 𝑓𝑦
= 50𝑏
𝑤