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Structural analysis principles such as shear force and bending moment calculations are essential in architectural design to ensure the safety and stability of building elements under load. In Kenya, architects must understand these principles to collaborate effectively with structural engineers and ensure compliance with local building codes. This chapter focuses on applying shear force and bending moment analysis to typical architectural structural elements like beams and frames, emphasizing practical calculations relevant to Kenyan construction contexts.
Structural elements in buildings are subjected to various types of loads and are supported in different ways that influence their behavior under stress. Understanding the types of loads and supports is critical for calculating shear forces and bending moments accurately.
Loads on structural elements can be classified based on their nature and application point. These loads create internal forces such as shear and bending moments that must be analyzed to design safe structures.
A point load is a concentrated force acting at a single point on a beam or frame member. In reality every load is spread over some small area, but when that area is very short compared with the span, the load is treated as acting at one point, which simplifies analysis. It causes abrupt changes in shear force and bending moment diagrams. Point loads are measured in kilonewtons (kN) and are drawn as a single arrow at the point of application. Example: a 2 000 kg steel water tank resting on a short stand over the middle of a roof beam at a primary school is treated as a point load on that beam.
Force = mass × gravitational acceleration
The tank and its water have a mass of 2 000 kg, with g = 9.81 m/s².
Force = 2 000 × 9.81
Force = 19 620 N
Force = 19.62 kN
A uniformly distributed load (UDL) is a load spread evenly along the length of a beam, so that every metre carries the same load. It is expressed as a load per unit length, in kilonewtons per metre (kN/m), and is drawn as a row of equal arrows or a rectangle along the loaded length. A UDL produces a linear variation in shear force and a quadratic variation in bending moment. For calculating reactions, a UDL is replaced by a single total load equal to the intensity multiplied by the loaded length, acting at the middle of that length. Example: a reinforced concrete beam supporting the floor slab of a hospital ward carries the weight of the slab, finishes and occupants as a UDL along its span.
UDL = width × depth × unit weight of concrete
A beam is 0.2 m wide and 0.4 m deep, and reinforced concrete weighs 24 kN/m³.
UDL = 0.2 × 0.4 × 24
UDL = 1.92 kN/m
Total load = UDL × length
A beam 8 m long carries a UDL of 5 kN/m.
Total load = 5 × 8
Total load = 40 kN, acting at 8 ÷ 2 = 4 m from either end
A varying load, also called a non-uniformly distributed load, is a distributed load whose intensity changes along the length of the member. The most common forms are the triangular load, which rises from zero at one end to a maximum at the other, and the trapezoidal load, which changes from one value to another. Load intensity varies over the length, requiring integration, or the use of its resultant, to find shear and moment values. For a triangular load, the total load equals half the maximum intensity multiplied by the length, and it acts at the centroid of the triangle, one third of the length from the maximum end. Example: the wall of a rainwater storage tank on a dairy farm carries water pressure that increases from zero at the surface to a maximum at the base, a triangular varying load.
Total load = ½ × maximum intensity × length
A beam 10 m long carries a load rising from zero at the left end to 6 kN/m at the right end.
Total load = 0.5 × 6 × 10
Total load = 30 kN
Position from the zero end = 2 × L ÷ 3
Position = 2 × 10 ÷ 3
Position = 6.67 m from the left end (3.33 m from the maximum end)
A moment load is a couple applied at a point creating bending without shear, important in analyzing frames and cantilever beams. It is measured in kilonewton metres (kNm) and is drawn as a curved arrow showing its direction of rotation.
Supports in structural systems restrain movement and rotation in different ways, influencing the internal forces generated in beams and frames. The type of support determines the boundary conditions for analysis.
A simply supported beam is a beam carried on two supports, one at each end, usually a pin (hinge) at one end and a roller at the other, so that the beam is free to rotate at both ends and free to expand along its length. This arrangement gives three unknown reactions (two at the pin and one at the roller), which can all be found from the three equations of static equilibrium, so the beam is statically determinate. Under downward loads a simply supported beam sags, with zero bending moment at the supports and maximum moment within the span. Example: the reinforced concrete lintels over windows in a SACCO office block are designed as simply supported beams bearing on the walls at each side.
Degree of indeterminacy = number of unknown reactions − number of equilibrium equations
A simply supported beam has 3 unknown reactions and 3 equilibrium equations.
Degree of indeterminacy = 3 − 3
Degree of indeterminacy = 0, so the beam is statically determinate.
A fixed support restrains translation and rotation, generating reaction forces and moments; it is common in rigid connections. Because it prevents horizontal movement, vertical movement and rotation, it provides three reactions: a vertical force, a horizontal force and a fixing moment. A beam built into a wall or cast monolithically with a column behaves as if fixed, and a cantilever is a member fixed at one end and free at the other. Example: a reinforced concrete balcony at a hotel is cast as a cantilever fixed into the floor slab and supporting beam, with the outer edge free.
A beam fixed at one end and resting on a roller at the other (a propped cantilever) has 3 reactions at the fixed end and 1 at the roller.
Number of unknown reactions = 3 + 1 = 4
Degree of indeterminacy = 4 − 3
Degree of indeterminacy = 1, so equilibrium equations alone cannot find the reactions.
A simple support, the most basic form of simply supported end, is one where a member merely rests on its support without being fixed or pinned to it, such as a beam bearing on a wall or a knife-edge. It provides only a vertical reaction pushing up on the member; it cannot hold the member down, so uplift or overturning must be prevented by other means. Horizontal movement is resisted only by friction, which is usually ignored in analysis. Example: the timber floor joists in the gallery of a church are laid directly on the masonry walls, so each end acts as a simple support.
A hinged support, also called a pinned support, allows rotation but prevents translation in both horizontal and vertical directions. It produces vertical and horizontal reaction forces but no moment, because the pin lets the member turn freely. Hinged supports are used where a member must be held in position but allowed to rotate as it deflects, as in roof trusses and pinned steel frames. Example: the steel roof trusses of a contractor's warehouse are bolted to the column tops with single-bolt connections that act as hinged supports.
A roller support allows horizontal movement but prevents vertical translation. It generates a vertical reaction only, perpendicular to the surface on which it rolls, and is used to accommodate expansion or contraction. By allowing the member to move freely along its length as temperature changes, it prevents thermal stresses that would otherwise crack the member or its supports. Example: a pedestrian footbridge built by a county government over a river has a pinned bearing at one end and a roller bearing at the other, so the deck can expand on hot days.
A free end has no restraint; the element is free to move and rotate, producing no reactions. It occurs at the unsupported tip of a cantilever, where both the shear force and the bending moment are zero unless a load acts exactly at the tip.
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Create a free accountThis chapter introduced the fundamental principles of shear force and bending moment analysis in structural engineering. It began by explaining different types of loads including point loads, moments, uniformly distributed loads, and varying loads, which are essential for assessing structural behavior. Various support types were detailed, such as simply supported, fixed, hinged, and roller supports, highlighting their influence on load transfer and structural response. Methods for calculating support reactions on beams and frames were then presented, providing the basis for further analysis. The chapter also covered the preparation of shear force and bending moment diagrams, which visually represent internal forces within structural elements. Finally, the concept of the point of contraflexure was introduced, indicating where bending moments change sign along a beam. Together, these topics equip learners with the skills to analyze and design safe and efficient structural systems.
A simply supported beam of length \(6\,m\) carries a point load of \(12\,kN\) at its mid-span. Calculate the reactions at the supports. (3 marks)
A beam \(8\,m\) long is simply supported and carries a uniformly distributed load of \(5\,kN/m\) over its entire length. Determine the support reactions. (3 marks)
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