Structural Analysis: How Forces Find Their Way to the Ground

On the 80th floor of a tall building on a windy day, a glass of water on a desk can ripple even though nothing is wrong. The building is swaying, slowly and by a few centimetres, the way a tree does. Nobody designed it to be still. Someone decided how far it would be allowed to move, and checked that the people on the 80th floor would not be seasick, that the lifts would still run, and that the cladding would not crack.

That decision comes from structural analysis: given a structure and the loads on it, find every force, every bending moment and every movement inside it. Statics (34/10) gave you the forces in a structure that is simple enough to solve by balance alone. Real structures are not that simple. They are continuous frames with more members than the minimum, loaded by wind, earthquakes and lorries as well as their own weight, and the answer has to say not just whether they stay up but how much they move.

Loads find a path to the ground, and they never arrive alone

Every force on a building has to reach the ground by a continuous route, called the load path. A floor slab spreads weight to its beams; the beams pass it to girders; the girders pass it to columns; the columns pass it to footings; the footings pass it to the soil (34/66). Break the chain anywhere and the load looks for another route or the structure falls. Engineers read a building by tracing these paths, and count how much floor each column is responsible for, its tributary area. An interior column in a 6 m by 6 m grid under ten floors of 5 kPa carries 10 × 36 × 5 = 1,800 kN at its base, about 180 tonnes.

The loads have different characters. Dead load is the structure itself, permanent and well known. Live load is people, furniture and stored goods, which come and go. Wind pushes sideways and its effect grows with height. Earthquake does not push at all: the ground moves, and the building's own mass resists the movement, so the force is mass times acceleration and a heavier building attracts a bigger force. That is why a heavy building is often governed by earthquake and a light, tall one by wind.

These loads never act alone and never all peak together. Codes give load combinations: a recipe of factors that multiplies each load by how uncertain it is and how likely it is to coincide with the others. Two examples of the shape: ASCE 7 in the United States uses 1.2 D + 1.6 L, and the Eurocodes use 1.35 G + 1.5 Q (G is permanent load, Q variable). Here is the surprise: another combination takes only 0.9 of the dead load and subtracts the wind. When wind lifts a light roof, the dead weight is helping, so you must assume you have less of it. The factors above are examples, not design values.

Redundancy: spare load paths forgive mistakes

A structure is statically determinate when the equilibrium equations alone give every force, with no member to spare: remove any one and it becomes a mechanism. It is indeterminate when it has more members or supports than that. For a flat pin-jointed truss, count members m, support reactions r and joints j. If m + r = 2j it is determinate. The truss in Statics has 7 members, 3 reactions and 5 joints: 7 + 3 = 10 = 2 × 5. Add one extra diagonal and the sum exceeds 10 by one: one degree of indeterminacy, one spare load path.

Spare paths are what you want when something goes wrong, and something does. On 15 December 1967 the Silver Bridge over the Ohio River, between Point Pleasant, West Virginia and Gallipolis, Ohio, collapsed in rush-hour traffic and 46 people died. A single eyebar in one suspension chain failed: a crack about 2.5 mm deep, grown by stress corrosion, broke it. Each link of the chain had only two eyebars, so when one went the other was overloaded. Investigators concluded that with three or more eyebars per link, one failure might not have been fatal; the crack was beyond what 1960s inspection could find. The collapse led to the first national bridge inspection programme in the Federal-Aid Highway Act of 1968.

The price of redundancy is that indeterminate structures are sensitive to things statics does not see: if one support settles, or one member heats up, the forces redistribute (the soil enters the problem again, 34/66).

Beams and frames: the moment diagram is the picture of the structure

The bending moment at each point of a beam is the quantity that decides how big the beam must be (34/11), so the central drawing of structural analysis is the moment diagram: the moment plotted along the member. For a simply supported beam under a uniform load w, the moment is a parabola with a peak of wL²÷8 at midspan. For a beam fixed at both ends, the ends hold it: the peak moment at each end is wL²÷12 and at midspan only wL²÷24. In numbers, for 15 kN/m on 6 m: 67.5 kN·m simply supported against 45 at the ends and 22.5 at the middle if fixed. Same beam, same load, one third less peak moment, and much less deflection, because indeterminate beams are stiffer.

Read it as: sag grows with the fourth power of the span, so doubling the span makes the sag sixteen times larger unless the beam is made stiffer. A frame is beams and columns joined by stiff joints; the joint passes moment between them, which is why a beam's end moments land in the columns and why the column sizes depend on the floors above. Codes limit deflection as a fraction of the span, with values such as span ÷ 360 under live load common for floors; the actual limit depends on the code and what the floor carries. The steel beam here sags 15.1 mm over 6 m, which is span ÷ 396.

Influence lines: ask what one load does, wherever it stands

A bridge carries a moving load, so no single picture of forces is enough. The trick is to turn the question round. Put a single unit load at every position along the span and plot what it does to one quantity, such as the moment at midspan. That plot is an influence line. For a simply supported span the midspan moment is a triangle: zero with the load at either support, largest with it in the middle. Then for any set of loads, the effect is the sum of each load times the height of the line under it.

Take a truck with three axles of 50, 150 and 150 kN spaced 4 m apart, crossing a 20 m span. The worst midspan moment is 1,350 kN·m, reached when the lead axle is 14 m along and a heavy axle sits at midspan. Parking all 350 kN at midspan would give 1,750 kN·m, so a truck is kinder than its total weight because its axles spread out. Real codes use design vehicles and lane loads of this kind (the AASHTO HL-93 loading in the United States or Load Model 1 in the Eurocodes, for example), and the maximum is found with influence lines. The worst place for the truck is not where you would guess.

The stiffness method: stiff things attract force

Indeterminate structures were once solved by hand with clever iterative tricks, such as Hardy Cross's moment distribution (1930). Computers need something more mechanical, and the answer is to treat every member as a spring. A bar of area A, modulus E and length L is a spring with stiffness k = EA÷L. A beam is a spring for bending, stiffness proportional to EI over length. Join the springs at nodes, write the force balance at each node in terms of unknown displacements, and you get one equation:

Read it as: the stiffness matrix times the displacements gives the applied loads. Solve for the displacements, then every member force follows from its own stretch times its own stiffness. A stiffer member moves the same distance but carries more force, so stiff things attract load. The finite element method, FEA, is exactly this with the structure cut into thousands or millions of small elements. Only the size of K changes.

Try three bars between two walls: stiffnesses 100, 50 and 100 kN/mm, with a 100 kN pull on the first node. Equilibrium alone cannot say how the load splits, but K = [150, −50; −50, 150] does. The nodes move 0.75 mm and 0.25 mm. Bar 1 is pulled with 75 kN, bars 2 and 3 are pushed with 25 kN each.

Tall buildings: a vertical cantilever fighting the wind

A tall building is a cantilever sticking out of the ground, loaded sideways by wind. For a uniform sideways load w, the top moves by wH⁴÷8EI: fourth power of the height again. So the whole art of tall-building structure is to raise I, and I grows with the square of how far material sits from the centre line. Material in the middle is nearly useless for sway; material at the edge is worth far more.

That gives the three systems. A core, a hollow tube of concrete or steel walls round the lifts and stairs, is the spine. A tube puts the stiff material on the outside wall: closely spaced columns joined by deep beams, as in the framed, braced and bundled tubes developed by Fazlur Khan in the 1960s and 70s (examples are the John Hancock Center, Chicago, and the Willis Tower). Outriggers are stiff arms or trusses that tie the core to the perimeter columns, usually at a few floors, so that the outer columns pull and push like the flanges of a giant I-beam when the core tries to bend. Where the structure alone cannot calm the sway, a tuned mass damper (a heavy pendulum, 660 tonnes in Taipei 101) takes the energy out.

What a graduate just sees

• A structure is a set of paths that carry load to the ground; tracing the path is how you read it.
• Loads come in families, and codes combine them with factors. The worst case is not always the heaviest one: wind uplift beats dead weight.
• Count members, reactions and joints. A determinate structure has none to spare; an indeterminate one has spare paths.
• The moment diagram is the beam's story. Fixing the ends lowers the peak moment and the sag.
• Sag grows with the fourth power of span and falls with EI.
• An influence line turns "where is the worst place for the load?" into reading a graph.
• Everything is a spring. Stiff members attract force, and the computer solves K u = P.
• To stop a tall building swaying, put the stiff material as far from the centre as you can.

Where it shows up next

• Structural Design: Concrete & Steel (34/65): this course gives the forces and moments; design sizes the section and the bars for them, using the load combinations shown here.
• Geotechnical Engineering (34/66): the supports in every model are soil, and a support that settles changes the forces in an indeterminate frame.
• Vibrations (34/17): earthquake and wind response of a building is its mass and stiffness as a spring-and-mass problem.
• Linear Algebra (34/04): K u = P with a million unknowns is a sparse linear system.
• In the world: the structural model behind every skyscraper, bridge rating, stadium roof and aircraft wing, and the FEA package an engineer runs before anything is built.

If you want to go deeper

• 📘 Standard textbook: R. C. Hibbeler, Structural Analysis (Pearson). Widely assigned, with many worked problems on beams, frames, influence lines and matrix methods.
• 📘 Standard textbook: Aslam Kassimali, Structural Analysis (Cengage). Another common course text, clear on the stiffness method.
• 📘 Gentler: Mario Salvadori, Why Buildings Stand Up: The Strength of Architecture (W. W. Norton, 1980). Load paths, arches, frames and skyscrapers for the general reader. And J. E. Gordon, Structures: Or Why Things Don't Fall Down.
• 🌐 Free course: MIT OpenCourseWare 1.050 Solid Mechanics, which covers determinate planar structures and statically indeterminate systems. For a graduate step, 1.571 Structural Analysis and Control.