Exercises: Beam and truss calculations

Explain what internal force diagrams of a beam are and what they are used for: what the shear force diagram represents and what the bending moment diagram represents, and why the cross-section sized is the one where the bending moment is maximum (the "critical section").

Distinguish between shear force and bending moment in a beam cross-section: what each tends to do (sliding one part vertically relative to the other versus bending the member) and what SI units each is measured in.

Explain the common sign convention for bending moments: positive when the beam bends with the bottom face in tension and negative when it bends the other way. Explain on which face of a simply supported reinforced concrete beam the main reinforcement bars are placed and why.

Compare a simply supported beam with a cantilever beam from the perspective of their internal forces: in which one a positive moment appears at mid-span and in which one a negative moment appears next to the support. Give a real example of each structural type (floor slab between joists, balcony) and indicate on which face the material works in tension in each case.

Explain what a uniformly distributed load is and how it is replaced by its resultant to calculate reactions (total force $q \cdot L$ applied at the centre of the loaded area). Give two real-life examples: snow on a flat roof and the self-weight of a beam.

Describe what the method of joints for solving a truss consists of: what equilibrium equations are set up at each joint ($\sum F_x = 0$ and $\sum F_y = 0$), why it is convenient to start with a joint that has only two unknown members, and how the calculation propagates to the rest of the structure.

Compare tension and compression in a truss member: in which direction each "pulls" or "pushes" on the joint, and what additional phenomenon threatens a slender compressed member (buckling) that does not affect a member in tension. What consequence does this have when sizing it?

Explain the concept of Cremona's graphical method for trusses: what is drawn joint by joint (force polygons to scale), what is measured on paper to obtain the internal forces, and why this method was so valuable before calculators. Comment on its accuracy compared to the analytical method.

You can do it on paper and send a photo with your answer.

The figure shows a simply supported beam of $5\ \text{m}$ with a point load of $2\,000\ \text{N}$ applied at $2\ \text{m}$ from support A. a) Calculate the reactions at A and B. b) Draw by hand the shear force diagram indicating its value in each section. c) State the value of the maximum bending moment and the section at which it occurs.

You can do it on paper and send a photo with your answer.

A simply supported beam of $6\ \text{m}$ carries a load of $3\,000\ \text{N}$ at its centre. a) Calculate the reactions. b) Draw the bending moment diagram (a symmetrical triangle) and calculate the maximum moment using $M_{max} = \frac{P \cdot L}{4}$. c) Explain why the shear force diagram changes sign right at the section where the moment is maximum.

You can do it on paper and send a photo with your answer.

A simply supported beam of $4\ \text{m}$ carries a uniformly distributed load of $500\ \text{N/m}$ along its entire length. a) Calculate the reactions. b) Calculate the maximum moment at the centre using $M_{max} = \frac{q \cdot L^2}{8}$. c) Draw the shape of the shear force diagram (an inclined line crossing zero) and the bending moment diagram (a parabola), marking their key values.

You can do it on paper and send a photo with your answer.

A cantilever shelf of $1{,}5\ \text{m}$, fixed into the wall, carries $800\ \text{N}$ at its free end. a) Calculate the vertical reaction and the fixing moment at the wall. b) Draw the shear force diagram (constant) and the bending moment diagram (triangle with its maximum at the wall). c) Explain why the cantilever bracket would break at the fixed support rather than at the free end.

You can do it on paper and send a photo with your answer.

A cantilever canopy of $2\ \text{m}$ supports a distributed load of $300\ \text{N/m}$ (self-weight plus snow). a) Replace the load with its resultant and calculate the moment at the fixed support. b) Check the result using the formula for maximum moment, $M_{max} = \frac{q \cdot L^2}{2}$. c) Draw the shape of the bending moment diagram (a parabola increasing towards the wall) and indicate its sign.

You can do it on paper and send a photo with your answer.

At a joint of a truss, a horizontal member and an inclined member sloping upwards to the left at $60^\circ$ to the horizontal meet, and a load of $1\,000\ \text{N}$ hangs from it. a) Set up $\sum F_y = 0$ at the joint and calculate the force in the inclined member, explaining whether it is in tension. b) Set up $\sum F_x = 0$ and calculate the force in the horizontal member, explaining its sign (you may use $\sen 60^\circ = 0{,}866$ and $\cos 60^\circ = 0{,}5$).

The shear force diagram for a simply supported beam of $4\ \text{m}$ consists of a rectangle of $+1\,500\ \text{N}$ between support A and the mid-span, and another of $-1\,500\ \text{N}$ from the mid-span to support B. a) Deduce what load the beam supports, where it is applied, and the values of the reactions. b) Draw the corresponding bending moment diagram and calculate the maximum moment.

You can do it on paper and send a photo with your answer.

Explain what zero-force members are in a truss and state the rules for identifying them by inspection (at an unloaded joint with three members, two of which are collinear, the third is a zero-force member; at an unloaded joint with only two non-collinear members, both are zero-force members). Draw an example of each case and explain why these members are kept even if they carry no force (stability, assembly, future loads).

You can do it on paper and send a photo with your answer.

Based on the beam and the internal force diagram schematics shown in the figure:

1. a) Determine the reactions at supports A and B using the fundamental equilibrium equations of statics and deduce the exact numerical values missing from the shear force diagram.
2. b) Analytically prove the value of the maximum bending moment in the section under the load without using memorized formulas, by deducing and checking the relationship between the area of the shear force diagram and the change in bending moment.
3. c) If the beam is made of steel with an allowable yield strength $\sigma_{\text{adm}} = 150\ \text{MPa}$, calculate the minimum section modulus ($W_z$) required to ensure safety against bending failure. Reasonably predict how the maximum bending moment would change if the point load of $2\,400\ \text{N}$ were moved to the central section of the beam ($x = 3\ \text{m}$), justifying whether this new position is more or less unfavorable for the component.

A simply supported beam of $6\ \text{m}$ supports a uniform continuous load $q = 200\ \text{N/m}$ due to its self-weight. To serve an industrial facility, a stationary machine of $1\,200\ \text{N}$ must be placed, and two alternatives are proposed: the central position A ($x = 3\ \text{m}$) or the lateral position B ($x = 1\ \text{m}$ from the left support).

1. a) Applying the principle of superposition, obtain and quantitatively justify the value of the maximum bending moment for position A ($x = 3\ \text{m}$). Determine the percentage increase in the maximum bending moment in the span compared to the case where the beam only supported its self-weight.
2. b) Compare the two alternatives with respect to the shear forces at the supports: predict and explain in which of the two situations the most heavily loaded support suffers more, and calculate the value of its reaction in alternative B ($x = 1\ \text{m}$).
3. c) Justify under what mechanical conditions of the material and deformations it is physically valid to apply the principle of superposition in structural analysis, and state a specific example where this principle ceases to be applicable.

A cantilever canopy with a length of $2{,}5\ \text{m}$ supports a uniform load of $400\ \text{N/m}$ (self-weight plus snow). The cantilever is anchored to the wall using a vertical base plate with a depth of $30\ \text{cm}$, secured with two steel through-bolts located at the top and a compression bearing against the wall at its bottom edge.

1. a) Derive the shear force and bending moment equations at the fixed support using static equilibrium equations. Sketch both diagrams and physically explain why the bending moment curve has a parabolic curvature and reaches its extreme value at the fixed support.
2. b) Interpret the stress state of the cross-section at the fixed support: explain why the top fiber is in tension and the bottom fiber in compression. If the vertical distance (internal lever arm) between the line of the top bolts and the bottom compression bearing is $25\ \text{cm}$, calculate the tensile force that each of the two bolts must withstand to maintain bending equilibrium.
3. c) If the total equivalent load ($1\,000\ \text{N}$) were accidentally concentrated as a point load at the free end of the cantilever instead of being uniformly distributed, predict quantitatively how it would affect the bending moment at the fixed support and calculate the factor by which the tensile force in the bolts would be multiplied.

In the symmetrical roof truss with a span of $6\ \text{m}$ and a height of $2\ \text{m}$ shown in the figure, a load of $5\,000\ \text{N}$ is suspended from the bottom joint D:

1. a) Set up the method of joints to calculate the axial forces in members AB, AD, and BD. Rigorously identify which are in tension and which are in compression.
2. b) The diagonal compression member AB has a length of $3{,}61\ \text{m}$. Explain why, in the structural design of pin-jointed trusses, members under compression are much more critical and limiting than those under tension, naming the responsible physical phenomenon and justifying which geometric characteristics of their cross-section should be maximized to prevent it.
3. c) Predict what would happen to the force distribution if the point load of $5\,000\ \text{N}$ were moved vertically from the bottom joint D to the top joint B, acting downward. Analytically prove what value the axial force in post BD would take and explain whether the forces in the perimeter members AB, BC, AD, and CD would increase, decrease, or remain identical.

Consider joint $A$ of a simply supported truss, whose roof carries downward vertical loads. Only the following meet at this joint: the support reaction, $R_A = 10$ kN, vertical and upward, the upper chord $AB$, inclined $30^\circ$ to the horizontal, and the lower chord $AD$, horizontal (see figure).

A student tries to solve joint $A$ graphically using the force polygon (Cremona / Maxwell method), but makes a mistake when orienting the vectors and concludes that the lower chord $AD$ works in compression and the upper chord $AB$ in tension.

1. a) Refute this claim: justify physically and geometrically, based on the equilibrium of joint $A$ and on the deformation expected in the truss when it receives a downward load, why it is impossible for the lower chord to be in compression.

1. b) Explain the rigorous procedure for constructing the closed vector equilibrium polygon of the joint ($\sum \vec{F} = \vec{0}$) with the data in the figure: the chosen scale (for example, $1$ cm $= 2$ kN), how the sequential path around the joint must be carried out, and what rule allows you to unambiguously deduce whether a member enters or leaves the joint (distinguishing compression from tension). Also state what is measured on the paper to obtain the forces.

1. c) Evaluate the use of graphical methods in modern engineering: what insurmountable limitations does the Cremona method have compared with computer-based matrix analysis when analyzing three-dimensional or statically indeterminate structures, and in what on-site situations does it still have practical use?

You can do it on paper and send a photo with your answer.

To span a total length of $10\ \text{m}$ with a beam supporting a uniformly distributed load of $100\ \text{N/m}$, it is planned to place two symmetrical supports A and B at a distance $a$ from each end, creating a central span with two lateral overhangs. In an initial proposal, $a = 2\ \text{m}$ is set.

1. a) Obtain the reactions and calculate, by applying static section equilibrium, the value of the bending moment at the supports and at the midpoint of the central span for $a = 2\ \text{m}$.
2. b) Compare the maximum bending moment in absolute value obtained in this configuration with the one the same beam would experience if it were supported only at its ends ($10\ \text{m}$ span without overhangs). Quantitatively justify the structural advantage provided by the overhangs.
3. c) Design the optimal position of the supports: predict and calculate analytically what value the overhang length $a$ must have so that the negative moment over the supports has exactly the same absolute magnitude as the positive moment at the center of the span ($|M_{\text{support}}| = M_{\text{center}}$), achieving a design with the minimum possible beam depth and material weight.

For three different spans ($4\ \text{m}$, $8\ \text{m}$, and $20\ \text{m}$) with similar loads, choose and justify the most reasonable structural type from solid timber beam, rolled steel beam, lattice girder (truss), and composite reinforced concrete beam. Support your answer using the increase of bending moment with span, available depth, self-weight, and ease of construction for each option.

A cantilever balcony has cracks on its top surface near the support fixity. Analyse the failure using the concepts from this topic: what sign the bending moment has there, which face of the slab is in tension, what rebar placement error might have occurred during construction, what other aggravating factors are possible (rebar corrosion, overloading from plant pots and awnings), and what corrective and preventive measures you would propose.

Unit challenge. Design and validate the support structure (truss) to install two solar panels on your terrace or roof. Develop the following phases with the help of LibreOffice Calc and write a brief technical report:

1. Data collection and loads: Measure the available space (span $L \approx 2,0\text{ to }2,5\text{ m}$). Consult the Spanish Technical Building Code (CTE) or AEMET for your town's wind and snow zone to estimate the live loads ($q$ in $\text{kN/m}^2$). Determine the dead weight of the panels and profiles.

2. Stability and overall equilibrium: Propose a symmetrical triangulated truss. Analytically obtain the reactions at the supports by applying the static equilibrium equations ($\sum F_x = 0$, $\sum F_y = 0$, $\sum M = 0$).

3. Bar calculation: Apply the method of joints to calculate the axial force ($N$) in at least three adjacent bars. Classify each bar according to whether it works in tension or compression.

4. Reasoned proposal: Select a commercial aluminum profile and justify that the most compressed bar withstands the load without risk of buckling ($N \le N_{cr}$).

Evaluation criteria:

• Rigor in obtaining and justifying actual loads according to regulations.

• Mathematical correctness in static equilibrium and joint resolution.

• Correct identification of the type of stress and buckling verification.

• Clarity of explanation and presentation of data in the final report.