Hi friends,

In the last 14 episodes of this reinforced concrete series, we learned all fundamentals and the most important design verifications.

In the next couple of weeks, we design structural elements like beams, columns, frames, slabs, and much more using the verifications that we learned in the last weeks.

In this newsletter, we'll design and verify a reinforced concrete beam. We design the rc beam 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)

You can use the sheet with a 1-month free trial of Maple Flow (all details below).

I would like to thank the sponsor of this episode, Maplesoft for supporting me in producing this series. You can thank them by clicking on the info below.

​Maple Flow calculation software from Maplesoft helps structural engineers perform preliminary calculations with ease. Work with equations using natural notation, track units automatically, and present results in the “classic hand calculation” style for clarity and precision.

As a Structural Basics reader, you also get a special deal. You get a 1-month free trial, instead of 14 days, if you sign up with this link:

Download link for Maple Flow: For your extended-length trial, visit https://www.maplesoft.com/fulfillment/ and enter coupon code SBFLOWPRO (code valid up to JUN 1, 2025).

Special for students: If you are enrolled in a university and like the software, you can also get a Maple Flow standard license (lifetime – no expiry) for only 150 euro → here ←.

You can also check out my getting-started guide for Maple Flow→ here ←.

Alright let's get into it..

Process of reinforced concrete beam design

Before we dive into the nerdy calculations, it’s good to get an overview of the steps that need to be taken to design a reinforced concrete beam.

1. Calculate characteristic and design loads that act on the beam
2. Define properties of concrete and reinforcement
3. ULS bending verification
4. ULS shear verification
5. SLS crack verification
6. SLS deflection verification

You don't have to verify a rc beam in this order. This is just the order that I personally follow.

Design of a reinforced concrete beam in 6 steps

In this tutorial, we'll verify a simply supported reinforced concrete beam with the following cross-sectional properties.

Step #1: Calculation of the characteristic and design loads that act on the beam

First, we need to calculate the characteristic loads that act on the beam such as dead load, live load or snow load. We won't show how to calculate the loads in this tutorial, 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​

And as most of you know, I will publish my first book soon which is a step-by-step guide to calculate all loads you need to design a residential house. Unfortunately, I don't have any updates about the publishing date as I am still waiting for my tax number.

In this article, we'll define the design internal forces later in the article.

But if you don't know how to calculate design loads from characteristic loads, then check out our → detailed guide to load combinations. ←

Step #2: Define the properties of concrete and reinforcement

Here are the material properties we are going to use:

You can find many of these material parameters on eurocodeapplied.com.

Step #3: ULS bending verification

We first need to calculate the bending moment from the design line load. For the simply supported beam, we'll find the max. bending moment at midspan as:

In the bending verification, we design the longitudinal reinforcement of the beams.

Let’s assume the diameter of the longitudinal reinforcement is d_s = 25 mm and the stirrups d_v = 10 mm, then the lever arm is calculated as:

The lever arm is the distance from the most outer concrete fibre in compression to the longitudinal reinforcement taking up the tension forces.

In our example (for a positive bending moment), we’ll get compression in the top and tension in the bottom.

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

Required longitudinal reinforcement:

We picked a rebar diameter of 25 mm, which has a cross-sectional area of:

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

Now, we can calculate the total reinforcement area for 4 rebars of diameter 25 mm.

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

Verification:

Verification:

Verification:

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

Now, comparing A_s with A_s.min shows that the minimum reinforcement is less than the reinforcement we get from 4 x d=25mm rebars.

Now, this was only the verification for the bottom longitudinal reinforcement. Don’t forget to also verify the reinforcement in the top. If you don’t have negative moments, it should at least have the minimum reinforcement.

Step #4: ULS shear verification

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.

The shear force we’ll use is 450 kN.

There are different approaches to the steps, we do in the following. We are going to verify the diameter of the stirrups. But you can also choose to verify shear capacity. This is up to you.

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

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 450 kN and a shear resistance of 479 kN lead to a utilization of:

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:

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)):

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

→ Required cross-sectional area of the shear reinforcement of 1 stirrup:

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

And the required bar diameter:

We could reduce the stirrup diameter from 10mm to 8mm, or just leave it to have some extra capacity. In general, I always like to have some extra capacity. It often times saves you (and your design) when design changes come from the architect or if someone forgot to tell you that the live load was changed from 1.5 kN/m2 to 3.0 kN/m2.

Step #5: SLS crack verification

First, we’ll define the quasi-permanent bending moment for the quasi-permanent load as:

We’ve already written an in-depth article about the moment of inertia of a crack cross-section. Go check it out. Now, we’ll do a quick version of it.

Creep coefficient:

​

E-modulus of concrete long-term:

​

Long-term reinforcement-concrete ratio:

We’ll find the depth of the neutral axis by equilibrium of the first moment of areas of the tension (steel) and compression (triangle). But don’t get confused by the triangle here. When calculating the static moment of the compression zone, we only need the area of it and the lever arm is x/2.

Equilibrium of the 1st moment of areas:

Neutral axis depth:

Calculation of the tensile stress in the reinforcement due to the quasi-permanent moment:

Calculation of the strain. Factor dependent on the duration of the load EN 1992-1-1 (7.10) (long-term loading):

Tensile strength concrete:

Effective height (EN 1992-1-1 7.3.2 (3)):

EN 1992-1-1 (7.10):

EN 1992-1-1 (7.9):

Next, we'll calculate the max. crack spacing. Coefficient EN 1992-1-1 (7.11) for high bond bars:

Coefficient EN 1992-1-1 (7.11) for bending:

Max. crack spacing (EN 1992-1-1 (7.11)):

Verification (EN 1992-1-1 (7.11)):

Finally, we calculate and verify the crack width according to EN 1992-1-1 (7.8):

Utilization:

Cracking is now verified for our reinforced concrete beam!!

Step #6: SLS deflection verification

The calculation of the deflection of reinforced concrete elements is not as straightforward as for timber or steel structures. However, according to Eurocode EN 1992-1-1 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.

Verification

Final Words

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

Beams were the first structural elements that we designed and verified in our reinforced concrete series. Many more are comming in the next weeks.

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.