Table of contents

Lecture 1 - Introduction and Problem Setting
            Some general practicalities
            Reading resources
      Introduction to Structural Engineering
            Design of a roof - a motivating example
      Quiz

Lecture 1 - Introduction and Problem Setting

The learning goal of today is:

• getting started
• become aware about the practicalities of the course
• gaining principle understanding of the structural design process

Reading / Homework:

• Per Kr. Larsen - Konstruksjonsteknikk: Chapters 1 + 2
• Answer questionair on BB.

Some general practicalities

Lectures and exercises will be held by:

Jochen Köhler:

Materialteknisk 3-209/ jochen.kohler@ntnu.no, and,

Jorge Mendoza Espinosa:

Materialteknisk 3-201/ jorge.m.espinosa@ntnu.no

Lectures and exercises will take place Tuesdays 10-12 and Fridays 8-10 both in S8.

The following course related material can be found on Blackboard:

• Lecture slides
• Exercises
• Exercise Solutions
• Supplemental material

Students have to engage in 3 compulsory exercises and 1 project work. The project work is intended to be a group work.

Preliminary lecture plan:

Preliminary plan of assignments:

Reading resources

• Per Kristian Larsen— Konstruksjonsteknikk, Laster og bæresystemer
• We will have special emphasis to chapters 1 -5
• Can be ordered via Fagbokforlaget

• Jörg Schneider— Introduction to Safety and Reliability of Structures
• We have special emphasis to chapter 4.3
• Is available online via NTNU Library

• The EUROCODES, in particular:
• NS EN1990 (Basis of Design)
• NS EN1991 (Loads)
• Available online via NTNU Library

• EUROCODE— background document
• Interesting background information and examples
• Available online via NTNU Library

Introduction to Structural Engineering

Civil Engineers play and will play an important role in our society. Many current and future challenges, as related to the efficient management of (limited) financial and natural resources or the appropriate mitigation to the possible effects of climate change, are closely related to the build environment and require good civil engineering. As the challenges will become bigger, the demand for new and innovative solutions, i.e. well beyond traditional engineering practice, will increase. The role of future civil engineers is very nicely defined by the American Society of Civil Engineers (ASCE) 1 as a professional who is:

Entrusted by society to create a sustainable world and enhance the global quality of life, civil engineers serve competently, collaboratively, and ethically as master:

• planners, designers, constructors, and operators of society’s economic and social engine, the built environment;
• stewards of the natural environment and its resources;
• innovators and integrators of ideas and technology across the public, private, and academic sectors;
• managers of risk and uncertainty caused by natural events, accidents, and other threats; and
• leaders in discussions and decisions shaping public environmental and infrastructure policy.

1: ASCE is the permanent organisation representing the civil engineering profession in the United States.

The planning, design, maintenance and reconstruction of structures plays a key role in the general civil engineering area and the engineering discipline dedicate to structures is called Structural Engineering.

Structures are the "bones and muscles" of the build environment. They are intended to bear loads and enable all kinds of societal utility by creating sheltered space (e.g. buildings), supporting infrastructure (e.g. bridges) or facilitate industrial or energy production (e.g. wind energy converter), see figure 1. Thereby, structures have played and will play a key role for successful societal development.

At the same time structures are responsible for the consumption of large amounts of natural and economic resources. The construction industry is responsible for an overall share of around 40% of total energy consumption, 90% of global raw materials usage (in tons). Moreover, in most developed countries the construction industry contributes with 10% or more to the GDP and constitutes a key prerequisite for all critical infrastructures, including transport, communication, energy, production, food and housing.

A solid basic understanding of structural engineering principles is paramount for all cilvil engineers that work in connection of the build environment. The transmission of the most basic principles is the objective of this part of the course. The topics adressed in the course therefore are:

• Principle load bearing systems
• Attaining structural requirements, structural safety
• Representation of loads on structures

Design of a roof - a motivating example

In order to get started let's consider typical structural design situation as an example:

A flat roof exposed to snow load has to be designed.

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Figure 1: A flat roof is often exposed to considerable snow loads.

Every educated structural engineer would follow a rather standardised procedure to do the job. However, let's assume we do this the first time and just use our already developed engineering understanding to identify a systematic approach for solving this problem.

So, how could we start, what would be the first step?

1. Context and requirements: Upfront to any analysis we have to clarify some principle questions in regard to the requirements on the structure. The dimensions of the space that has to be sheltered is generally specified / required by the client. E.g. the client might specify that the required area is 20x8 meters. Furthermore the client has to specify what he expects from the roof in regard to its load bearing behavior. Generally a client expects a (roof-) structure to be:

• safe - it should carry the load and not fail,
• sufficiently stiff - i.e. should not deflect extensively (as this would jeopardise the functionality of the roof).
• cost efficient.

2. Conceptual design: The fist logical step is the development of a principal load bearing concept of a flat roof. With the dimensions specified above a possible structural concept would be an assembly of parallel arranged timber beams as illustrated in the Figure.

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Figure 2: Structural concept of a flat roof assembled from timber beams.

The load bearing behaviour of this concept is that the load on the roof is transfered to a beam action as illustrated in the next figure. The load bearing capacity of the system of parallel assembled beams is now triggered by:

• The material strength of the timber
• The distance between the beams (this is decisive for how much load each beam receives), and,
• The cross section dimension of the beam, i.e. the height \( h \) and the width \( b \) of the rectangular beam.

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Figure 3: The flat roof as a system of beams.

3. Simplifications and assumptions: Compared to other structures, the flat roof structure is relative simple, however, the realistic 1:1 mechanical representation of this system would be cumbersome already as we would have to represent:

• How is the snow spatially distributed on the roof?
• How is the load distributed among the beams? (it obviously depends on the stiffness of the different beams and the behaviour of the transversal beams at the supports, the load bearing behaviour of the columns, the board that rest on the beams etc.)
• etc.

The realistic representation of structural systems is in general rather difficult and an important step in a structural engineering project is the identification of reasonable assumptions and simplifications. In this example, it seems reasonable to assume the following:

• The (snow-) load is uniformly distributed on the roof surface,
• The beams share the load in equal proportions,
• The transversal support beams and the columns are considered stiff.

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Figure 4: The snow load is uniformly distributed, the system of beams can be decomposed to similar simple supported beams.

4. Mechanical Assessment: The load bearing behaviour has to be assessed. In this example we concentrate on the safety of the roof (it shall not fail) and therefore we have to assess the ultimate load bearing capacity and compare it with the load the structure is exposed to.

Bending capacity of an elastic rectangular cross section:

In order to evaluate the maximum moment load bearing behaviour of a rectangular cross section we evaluate the maximum inner moment that is dependent on the maximum stress \( \sigma_{max,timber} \) of the timber material:

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Figure 5: Maximum moment capacity, with \( b \) and \( h \) being the width and height of the rectangular cross section.

$$ \begin{equation} M_{max,out}=M_{max, in}=\frac{bh^2}{6}\sigma_{max,timber} \label{eq:inM} \end{equation} $$

Moment effect of the load: The system is a simple supported beam with uniform distributed load.

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Figure 6: The moment distribution alongside the beam can be expressed, i.e. as a function of \( x \).

$$ \begin{equation} M_{load}(x)=\frac{q_s l}{2}x-q_s \frac{x^2}{2} \label{eq:loaM} \end{equation} $$

The maximum moment is found at \( x=l/2 \): $$ \begin{equation} M_{load,max}=M_{load}\left(x=\frac{l}{2}\right)=\frac{q_s l^2}{8} \label{eq:loaMmax} \end{equation} $$

Accordingly, the beam would not fail if the resistance is larger than the effect of the load, i.e. the comparison of the moments reads: $$ \begin{equation} \frac{q_s l^2}{8}=M_{load,max} < M_{resistance}=\frac{bh^2}{6}\sigma_{max,timber} \label{eq:deq1} \end{equation} $$

Equation \eqref{eq:deq1} is generally referred to as design equation. It contains design variables that are the variables that can be chosen / controlled during design. In the present example it would be the width \( b \) and the height \( h \) of the cross section that might be chosen freely and this choice is called structural design. The span of the beam \( l \) is determined by the requirements for space (the client required a roof of specified dimensions) and the load \( q_s=a*s \) and the material strength \( \sigma_{max,timber} \) are variables that have to be quantified.

For given \( b= \) 300 mm the required \( h \) is plotted as a function of the snow load \( s \) and the material strength \( \sigma_{max,timber} \):

But how do we quantify the snow load and the timber strength?

Snow load

The relevant snow load for design is the maximum snow load in the intended service period of the structure. A typical time duration for buildings is 50 year, i.e. we are interested in the extrem snow load event in a period of 50 years. As the snow load will vary over time and future snow loads cannot be predicted with certainty, the snow load - as other types of loading - has to be represented with random variables that are specified based on measurement data.

The snow load is also very much dependent on the geographical location and on the altitude of the construction site. In Figure (7) the geographical distribution of the snow load with an average return period of 50 years is illustrated. This value is also referred to as the characteristic value of the snow load. This map is part of the national annex of Eurocode EN1991 for snow loads.

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Figure 7: The characteristic snow load on the ground in Norway.

From the map and corresponding tables a value of \( s_k=3.25 kN/m^2 \) for Trondheim could be identified.

Timber strength

The properties of solid timber are characterised as the load bearing performance properties of solid timber components exposed to different loading modes, see Figure (8). Accordingly, the strength and stiffness related properties of structural timber are communicated as the properties measured in standardised experiments i.e. with standardised dimensions, loading modes and duration, and surrounding climate.

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Figure 8: Different loading modes on structural timber that are generally referred to as material properties.

Timber material for structural purpose is generally associated to a certain grade or strength class. In Europe timber strength classes are defined as sub-populations with expected values for the 5th-percentile of the bending strength and timber density, and expected values for the mean value of the bending stiffness. Properties corresponding to other loading modes per timber strength class are also given, see Figure (9). (The mentioned material properties are defined as properties of standard test specimen according to corresponding test standards).

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Figure 9: Different loading modes on structural timber that are generally referred to as material properties.

For the very typical strength class C24 the bending strength of $\sigma_{max,timber}=f_{m,k}=$24 MPa can be specified.

Probability Distribution?

We can guess it already: The material strength and the snow load are uncertain - i.e. they cannot be specified unambiguously. What we can specify is with which probability a specific value is exceeded or not. The function that is representing this probability is called Probability Distribution Function. The Probability Distribution Function is defined over the possible domain of the variable \( X \) (this is a random variable and this is indicated by a capital letter) and attains values (or realisations - indicated by a small letter) in the closed interval \( x \in[0,1] \). Figure (10) shows a probability distribution function \( F_X(x) \) and so-called fractile values. The 5%-fractile value (or the 5% percentile) is the value with a non-exceedence probability of 0.05 or 5%, i.e. \( F_X(x)=Pr(X < x) \). The characteristic value of the snow load is defined as the 98%-fractile of distribution of the yearly extreme values.

It can be seen that the characteristic values of the material strength and the load are chosen as a lower and upper fractile correspondingly. But is this a sufficiently safe assumption?

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Figure 10: Probability distribution function and fractile values.

We will follow up on this question during this course - but first we will start with the introduction of principle structural systems and concepts in the next lecture.

Quiz

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Question: How is the snow load communicated?

 Choice A: With a load that is typically observed in average

 Choice B: With the load that is observed at January 1

 Choice C: As the maximum snow load with an average return period of 50 years

 Choice D: With the load observed at Lillehammer OL in 1994

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Question: How do we quantify the snow load at the spot where we want to design the roof?

 Choice A: We ask some locals

 Choice B: We look up a snow load map and adjust with the altitude

 Choice C: We wait for the next winter and measure it

 Choice D: We choose a value that cannot be overrun (the absolute maximum)

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Question: What is the typical range for the snow load in Norway?

 Choice A: 2 - 4 \( kN/m^2 \)

 Choice B: 10 - 15 \( kN/m^2 \)

 Choice C: 1 -2 \( kN/m^2 \)

 Choice D: 20 - 30 \( kN/m^2 \)

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Question: How do I quantify the material strength \( \sigma_{max,timber} \)?

 Choice A: I ask an old carpenter

 Choice B: I do one experiment and measure it

 Choice C: I choose a value that cannot be under-run (the absolute minimum)

 Choice D: I find out how the timber is graded and look up the characteristic value in a table

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Question: If i get some estimates for the snow load and the material strength, can I choose the height \( h \) and the width \( b \) such that the roof is absolute safe?

 Choice A: Absolute safety does not exist

 Choice B: I can calculate it if I use the equations above and set in the values - then it is absolute save

 Choice C: I better increase \( b \) and \( h \) by 10% in order to be really sure

 Choice D: I will listen to the lectures in the next weeks and then i learn how to be at least safe enough

Remarks. This script will be continuously updated during the course. In the longrun also a printable version will be available.