Hi friends,

Reinforced concrete frames are used to resist both vertical and horizontal loads.

In this newsletter, I'll show you how to design and verify a reinforced concrete frame. We design the rc frame for bending, shear, cracks and deflection.

I did the design calculations in Maple Flow and you can download the sheet here. ↓

→ Download Maple Flow sheet ← (with this link you can also download the PDF file of the Maple Flow sheet)

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Alright let's get into it..

The 7 Steps to Design Reinforced Concrete Frames

Step #1: Geometry of the Frame

First, we define the geometry of the frame.

• Width of the pile cap: w_c = 45 cm
• Height of the pile cap: h_b = 45 cm
• Depth of the column and beam d_f = 35 cm

Step #2: Concrete and Reinforcement Properties

Here are the material properties we are going to use:

Step #3: Calculation of the Loads / Internal Forces

First, we need to calculate the characteristic horizontal and vertical loads that act on the frame. We won't show how to calculate the loads in this newsletter, as each calculation of the individual load is an article for itself. I've written detailed articles and published video tutorials about loads, which you can follow to understand how to calculate these loads:

• ​Dead load​
• ​Live load​
• ​Snow load​

The most detailed resource is, however, my e-book → Loads on Residential Buildings ←. There is only so much you can cover in a video or a blog post. But in the book I covered all loads of a whole residential building (all loads I include in my structural designs at work).

If you are not ready yet to purchase the book, I made a free email course introducing the different loads. It's not as detailed as the book, but it's a good resource to learn what loads you need to consider and a starting point to load calculation.

→ Click here to sign-up. ←

In this article, we'll design the frame for the following internal forces:

• Bending moment M_d = 55 kNm
• Shear force V_d = 60 kN

Step #4-1: Bending Verification of the Cross-Section

The bending moment needs to be checked for the different cross-section. In our case, the cross-section of the beam and the columns is the same. Therefore, one verification with the biggest bending moment is enough. Another thing that needs to be checked is the bending moment of the corners because the moments in a momentstiff frame corner needs to travel from the column to the beam and the other way around. I show you how after the cross-section verification.

Let’s assume the diameter of the longitudinal reinforcement is d_s = 20 mm and the stirrups d_v = 8 mm, then the lever arm is calculated as:

d = h_b – c – d_v – d_s/2 = 402 mm

Next, we'll calculate the required longitudinal reinforcement to take up the tension force with the following formulas:

μ = M_d/(d_f ⋅ d² ⋅ f_cd) = 0.042

ω = 1 - √(1 - 2μ) = 0.043

Required longitudinal reinforcement:

A_s = ω ⋅ d_f ⋅ f_cd/f_yd = 3.22 cm²

We picked a rebar diameter of 20 mm, which has a cross-sectional area of:

A_s.1 = π ⋅ (d_s/2)² = 3.14 cm²

Next, we calculate the amount of rebars we need as a minimum to fulfill the required reinforcement demand:

n = ceil(A_s/A_s.1) = 2

Now, we can calculate the total reinforcement area for 2 rebars of diameter 20 mm.

A_s = n ⋅ A_s.1 = 6.28 cm²

As a next step, we basically verify that the cross-section is not over and underreinforced.

ω_min = max(0.26 ⋅ f_ctm/f_yk ⋅ f_yd/f_cd ; 0.0013 ⋅ f_yd/f_cd) = 0.031

Verification:

ω > ω_min = 1 → OK!

Verification:

ω < ω_bal = 1 → OK!

Verification:

ω < ω_max = 1 → OK!

Finally, we verify the minimum reinforcement. The minimum reinforcement is calculated with EN 1992-1-1 9.2.1.1 (9.1N):

A_s.min = max(0.26 ⋅ f_ctm/f_yk ⋅ d_f ⋅ d; 0.0013 ⋅ d_f ⋅ d) = 2.33 cm²

Now, comparing A_s with A_s.min shows that the minimum reinforcement is less than the reinforcement we get from 2 x d=20mm rebars.

A_s.min = 2.33 cm² < A_s = 6.28 cm²

Step #4-2: Bending Verification of the Frame Corner

Another thing we need to check is the moment stiff frame corner. We need to check that the corners (concrete and reinforcement) can resist the bending moments.

To verify the corners, we use the strut & tie method. Here are the 2 strut & tie models for positive and negative bending moment.

We won't go into detail any more how the strut & tie method works, because we covered this in detail in a newsletter in January.

Now, with the strut & tie method, we need to check that the concrete can resist the compression forces C1 of the nodes.

Let's do that..

Distance between the strut and tie (beam):

a = h_b – (c + d_v + d_s/2) = 40.2 cm

Distance between the strut and tie (column):

b = w_c – (c + d_v + d_s/2) = 40.2 cm

Angle:

α = arctan(b/a) = 45 deg

Max. design bending moment:

M_d = 55 kNm

Compression / tension force in the struts / ties:

T = M_d/a = 136.8 kN

The bending radius of the reinforcement is used to transfer the tension force from the rebars to the concrete in compression.

Bending diameter of the rebars:

d_bend = 4 ⋅ d_s = 80 mm

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Compression length (length of bending radius):

Compression area:

A_t = d₁ ⋅ d_s = 1131 mm²

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For local crushing, the distribution of the load can be increased to A_c0 if the conditions of EN 1992-1-1 6.7 (3) are fullfilled.

Width of the reinforcement:

b₁ = d_s = 20 mm

Max. width of the compression area according to EN 1992-1-1 Figure 6.29:

b₂ = 3 ⋅ b₁ = 60 mm

Max. length of the compression area (there isn't much more concrete in the top, but we can use some more concrete area at the bottom; therefore, we multiply d₁ by 1.5).

d₂ = 1.5 ⋅ d₁ = 84.9 mm

Height/depth of the compression area (divided by 2 to take into account compression from both directions):

Verification:

h₁ > max(b₂ – b₁, d₂ – d₁) = 0.04 m < 0.284 m → OK!

Compression force in the diagonal strut:

C₁ = T/sin(α) = 193.5 kN

Max. compression area with a shape according to A_c0/A_t (EN 1992-1-1 (6.63)):

A_c1 = b₂ ⋅ d₂ = 50.9 cm²

Number of rebars (we have to increase the number of the longitudinal rebars; 2 are not enough):

n_r = 4

Compression stress:

Verification:

η = σ_t.1/f_cd = 0.86 → OK!

New reinforcement area:

A_s = n_r ⋅ π ⋅ (d_s/2)² = 12.6 cm²

Step #5: Shear Verification

If you are more of a visual learner, then check out my YouTube video where I go through the shear verification of rc beams step-by-step.

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Eurocode also covers formulas for shear verification without shear reinforcement, but for beams you usually always use stirrups (=shear reinforcement). In slabs, you often try to avoid shear reinforcement as these elements are not very tall, and therefore it's difficult to fit in vertical rebars.

First, we calculate the cross-sectional area of 1 stirrup with a diameter of 10mm with 2 vertical rebars as:

A_sw = 2 ⋅ π ⋅ (d_v/2)² = 1.0 cm²

Calculation of the design value of the shear resistance with shear reinforcement according to EN 1992-1-1 6.2.2

The design value of the shear resistance for members requiring shear reinforcement is calculated according to EN 1992-1-1 (6.9):

With,

• ν₁ = 0.6 according to EN 1992-1-1 (6.9) as a strength reduction factor for concrete cracked in shear
• α_cw = 1.0 as a coefficient taking into account the state of stress in the compression chord
• z = 0.9 ⋅ d = 448 mm (EN 1992-1-1 Figure 6.5)
• cot(θ) = 2.5 → EN 1992-1-1 (6.7N)
• tan(θ) = 0.4 → EN 1992-1-1 (6.7N)

Now, we verify that the shear force V_Ed < the design shear resistance with shear reinforcement V_Rd.max:

A design shear force of 60 kN and a shear resistance of 611 kN lead to a utilization of:

η = V_d/V_Rd.max = 0.1 → OK!

As a last step, we’ll calculate the spacing of the stirrups.

First, according to EN 1992-1-1 (6.8), we’ll reduce the design yield strength of the reinforcement:

f_ywd = 0.8 ⋅ f_y.k = 400 MPa

The required cross-sectional area of the shear links is calculated with EN 1992-1-1 (6.8):

The shear reinforcement (=stirrups) is vertical. Therefore, α is set to:

The maximum spacing is calculated with EN 1992-1-1 (9.6N)):

s_l.max = 0.75 ⋅ d ⋅ (1 + cot(α)) = 30.2 cm

We therefore set the spacing of the spacing of the stirrups to:

The required cross-sectional area of 1 stirrup is therefore (1 stirrup has 2 vertical “rebars”):

A_sw.req = A_sw ⋅ s/2 = 20.7 mm²

And the required bar diameter:

We could reduce the stirrup diameter from 8mm to 6mm, or just leave it as 8mm is already quite thin.

Step #6: Crack Verification

Next up crack verification.

We won't do it in this tutorial as we have written two really detailed articles where we have included the crack verification process.

• ​Design of reinforced concrete beams​
• ​Crack verification of reinforced concrete elements​

You can check these guides out or download the free Maple Flow calculation sheet in which I also included the crack verification.

→ Download Maple Flow sheet ← (with this link you can also download the PDF file of the Maple Flow sheet)

Step #7: Deflection Verification

The calculation of the deflection of reinforced concrete elements is not as straightforward as for timber or steel structures.

And for statically indeterminate structures with horizontal and vertical loads it's super hard to calculate the deflections accurately by hand and without using a FE program.

You could, however use the deflection requirement for simply supported beams according to Eurocode EN 1992-1-1 Table 7.4N. The deflection requirement is likely to be satisfied if the span – effective depth ratio is less than the values given in EN 1992-1-1 Table 7.4N.

For a simply supported beam where the concrete is highly stressed, the limit is 14.

Let's say the span l is 5m.

Verification

l/h_b = 11.1 < 14

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I will show you how to use and how to calculate deflections with FE programs in a future episode.

Final Words

Alright, this is how we design reinforced concrete frames according to Eurocode.

Thank you, Maplesoft, for sponsoring this episode of the Structural Basics newsletter.

I hope you enjoy the rest of the week and your weekend.

I’ll see you next Wednesday for the next newsletter.

Let’s design better structures together,

Laurin.

P.S.: In case you missed the Maple Flow trial link, here’s another chance to claim your 1-month free trial with coupon code SBFLOWPRO and get started with digital hand calculations today.

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