engineering

Hardy Cross method

Hardy Cross method is a engineering topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hardy Cross method rather than just read about it. In short: The Hardy Cross method is an iterative method for determining the flow in pipe network systems where the inputs and outputs are known, but the flow inside the network is unknown. The method was first published in November 1936 by its namesake, Hardy Cross, a structural engineering professor at the University of Illinois at Urbana–Champaign.

Key takeaways

Hardy Cross method belongs to engineering; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hardy Cross method to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hardy Cross method from memory before moving on to harder problems.

Reference excerpt

The Hardy Cross method is an iterative method for determining the flow in pipe network systems where the inputs and outputs are known, but the flow inside the network is unknown. The method was first published in November 1936 by its namesake, Hardy Cross, a structural engineering professor at the University of Illinois at Urbana–Champaign. The Hardy Cross method is an adaptation of the Moment distribution method, which was also developed by Hardy Cross as a way to determine the forces in statically indeterminate structures. The introduction of the Hardy Cross method for analyzing pipe flow networks revolutionized municipal water supply design. Before the method was introduced, solving complex pipe systems for distribution was extremely difficult due to the nonlinear relationship between head loss and flow. The method was later made obsolete by computer solving algorithms employing the Newton–Raphson method or other numerical methods that eliminate the need to solve nonlinear systems of equations by hand.

History In 1930, Hardy Cross published a paper called "Analysis of Continuous Frames by Distributing Fixed-End Moments" in which he described the moment distribution method, which would change the way engineers in the field performed structural analysis. The moment distribution method was used to determine the forces in statically indeterminate structures and allowed for engineers to safely design structures from the 1930s through the 1960s, until the development of computer oriented methods. In November 1936, Cross applied the same geometric method to solving pipe network flow distribution problems, and published a paper called "Analysis of flow in networks of conduits or conductors."

Derivation The Hardy Cross method is an application of continuity of flow and continuity of potential to iteratively solve for flows in a pipe network. In the case of pipe flow, conservation of flow means that the flow in is equal to the flow out at each junction in the pipe. Conservation of potential means that the total directional head loss along any loop in the system is zero (assuming that a head loss counted against the flow is actually a head gain). Hardy Cross developed two methods for solving flow networks. Each method starts by maintaining either continuity of flow or potential, and then iteratively solves for the other.

Assumptions The Hardy Cross method assumes that the flow going in and out of the system is known and that the pipe length, diameter, roughness and other key characteristics are also known or can be assumed. The method also assumes that the relation between flow rate and head loss is known, but the method does not require any particular relation to be used. In the case of water flow through pipes, a number of methods have been developed to determine the relationship between head loss and flow. The Hardy Cross method allows for any of these relationships to be used. The general relationship between head loss and flow is:

h f = k ⋅ Q n {\displaystyle h_{f}=k\cdot Q^{n}}

where k is the head loss per unit flow and n is the flow exponent. In most design situations the values that make up k, such as pipe length, diameter, and roughness, are taken to be known or assumed and thus the value of k can be determined for each pipe in the network. The values that make up k and the value of n change depending on the relation used to determine head loss. However, all relations are compatible with the Hardy Cross method.

It is also worth noting that the Hardy Cross method can be used to solve simple circuits and other flow like situations. In the case of simple circuits,

V = K ⋅ I {\displaystyle V=K\cdot I}

is equivalent to

h f = k ⋅ Q n {\displaystyle h_{f}=k\cdot Q^{n}} . By setting the coefficient k to K, the flow rate Q to I and the exponent n to 1, the Hardy Cross method can be used to solve a simple circuit. However, because the relation between the voltage drop and current is linear, the Hardy Cross method is not necessary and the circuit can be solved using non-iterative methods.

Method of balancing heads The method of balancing heads uses an initial guess that satisfies continuity of flow at each junction and then balances the flows until continuity of potential is also achieved over each loop in the system.

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Hardy Cross method

Start with the simplest possible case. Write down what Hardy Cross method claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In engineering, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hardy Cross method before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hardy Cross method ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hardy Cross method

In research: Hardy Cross method appears in engineering research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hardy Cross method in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hardy Cross method is common in secondary-school and first-year university syllabi. It links to neighbouring topics Hydraulic engineering, so understanding it makes those chapters shorter.
In everyday life: Look for Hardy Cross method outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Hardy Cross method in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hardy Cross method means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hardy Cross method out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hardy Cross method in simple terms?

Why does Hardy Cross method matter?

Because it connects several engineering ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hardy Cross method?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hardy Cross method.