Types of Beams in Civil Engineering
Simply Supported, Cantilever, Fixed, Propped, Continuous & Overhanging — Support Conditions, Reactions & Applications
Last Updated: March 2026
Key Takeaways 📌
- A beam is a structural member that carries transverse loads primarily through bending and shear — it is the most fundamental element in structural engineering.
- Beams are classified by their support conditions: the type and number of supports determine the reactions, the degree of indeterminacy, and the shape of the bending moment diagram.
- A simply supported beam has a pin at one end and a roller at the other — 2 unknown reactions, statically determinate.
- A cantilever beam is fixed at one end and free at the other — 3 unknown reactions, statically determinate.
- A fixed beam is built-in (encastré) at both ends — 6 unknown reactions, statically indeterminate to the 3rd degree.
- A continuous beam spans over more than two supports — highly indeterminate, analysed by the Moment Distribution or Slope-Deflection Method.
- Understanding support conditions and reactions is the essential first step before drawing any SFD or BMD.
1. What is a Beam?
A beam is a structural member whose length is significantly greater than its cross-sectional dimensions, and which is loaded primarily in a direction perpendicular (transverse) to its longitudinal axis. Under transverse loading, a beam develops internal bending moments and shear forces — these are what cause it to curve and, if overloaded, to fail.
Beams are everywhere in civil and structural engineering. Floor beams in buildings carry the weight of slabs and people down to columns. Bridge girders span rivers and valleys, carrying traffic loads to piers. Lintels over door and window openings are short beams that prevent the masonry above from collapsing. Roof purlins carry roofing loads to rafters. In each case, the beam’s job is to collect distributed or concentrated loads and transfer them to supports as shear forces.
The behaviour of a beam — how much it deflects, what bending moments develop, where it might fail — depends critically on two things: the loading (magnitude, type, and position of applied forces) and the support conditions (how the beam is held at its ends and intermediate points). Understanding support conditions is therefore the starting point for all beam analysis.
Internal Forces in a Beam
Shear Force (V or S): The algebraic sum of all transverse forces acting on one side of a section. Tries to slide one part of the beam relative to the other. Units: kN or N.
Bending Moment (M): The algebraic sum of all moments of forces acting on one side of a section about the centroid of that section. Causes the beam to curve (sag or hog). Units: kN·m or N·m.
Axial Force (P): Force along the longitudinal axis. Usually zero for transversely loaded beams unless there is horizontal loading or the beam is part of a frame.
2. Types of Supports
A support is a point of attachment between the beam and the ground (or the structure below). Every support restrains certain degrees of freedom of the beam — translation and rotation — and in return develops reaction forces and moments. There are three fundamental support types in 2D structural analysis:
| Support Type | Symbol | Reactions Provided | No. of Unknowns | Movements Allowed |
|---|---|---|---|---|
| Roller Support | Circle on flat surface | 1 reaction — vertical force only (perpendicular to rolling surface) | 1 | Horizontal translation + rotation (free to move horizontally and rotate) |
| Pin (Hinge) Support | Triangle on point | 2 reactions — vertical force + horizontal force | 2 | Rotation only (free to rotate, cannot translate in any direction) |
| Fixed Support | Wall with hatching | 3 reactions — vertical force + horizontal force + moment | 3 | None (no translation, no rotation) |
Important distinction between pin and roller: A pin support prevents movement in both horizontal and vertical directions but allows rotation. A roller support prevents movement in only one direction (perpendicular to the rolling surface) and allows both rotation and translation parallel to the rolling surface. This is why a simply supported beam (pin + roller) can expand or contract thermally without developing axial stresses — the roller allows horizontal movement.
Internal hinge: Sometimes a beam has an internal hinge (also called a condition of release) at a point within its span. An internal hinge allows relative rotation between the two parts of the beam at that point, which means the bending moment at an internal hinge is always zero. This is an additional equation that helps in solving otherwise indeterminate structures.
3. Simply Supported Beam
A simply supported beam rests on a pin support at one end and a roller support at the other end. It is the most commonly encountered beam type in structural analysis and the starting point for almost all beam problems.
Support reactions: The pin provides a vertical reaction (RA) and a horizontal reaction (HA). The roller provides only a vertical reaction (RB). For transverse loading only, HA = 0, leaving two unknown vertical reactions — RA and RB. These can be found directly from ΣFy = 0 and ΣM = 0.
Degree of static indeterminacy: 0 — fully determinate.
Bending moment at supports: Zero at both ends (the pin and roller cannot resist moment).
Deflection pattern: The beam sags (positive bending) between the supports. Maximum deflection occurs near midspan.
Simply Supported Beam — Key Results
Central point load W:
RA = RB = W/2 | Mmax = WL/4 (at midspan) | δmax = WL³/48EI
UDL of intensity w per unit length over full span L:
RA = RB = wL/2 | Mmax = wL²/8 (at midspan) | δmax = 5wL⁴/384EI
Point load W at distance a from A (b = L − a from B):
RA = Wb/L | RB = Wa/L | M under load = Wab/L
Real-world examples: Simply supported floor beams in steel-framed buildings, precast concrete bridge beams spanning between abutments, and timber joists in residential construction.
4. Cantilever Beam
A cantilever beam is fixed at one end (built into a wall or column) and completely free at the other. The fixed support provides all three reactions — vertical force, horizontal force, and a fixing moment — making the structure statically determinate despite having only one support.
Degree of static indeterminacy: 0 — fully determinate (3 unknowns, 3 equilibrium equations).
Bending moment pattern: Maximum bending moment occurs at the fixed support, not at midspan. The bending moment at the free end is always zero. The entire beam hogsw (negative bending — concave downward when loaded from above).
Critical feature: The maximum bending moment in a cantilever (at the fixed end) is much larger than in an equivalent simply supported beam for the same load and span. A cantilever of span L with UDL w develops a maximum moment of wL²/2 at the fixed end — four times the maximum moment of wL²/8 in a simply supported beam of the same span and loading.
Cantilever Beam — Key Results
Point load W at free end:
RA = W (upward) | MA = WL (hogging, at fixed end) | δmax = WL³/3EI (at free end)
UDL of intensity w per unit length over full span L:
RA = wL (upward) | MA = wL²/2 (hogging) | δmax = wL⁴/8EI (at free end)
Point load W at distance a from fixed end:
MA = Wa | δ at free end = Wa²(3L − a)/6EI
Real-world examples: Balconies projecting from building facades, roof canopies over entrance porticos, bridge approach slabs, retaining wall stems, and aircraft wings (the classic cantilever in aerospace).
5. Fixed Beam (Built-in or Encastré Beam)
A fixed beam (also called a built-in beam or encastré beam) is rigidly restrained at both ends — neither end can rotate or translate. Each fixed support provides three reactions (vertical, horizontal, moment), giving a total of 6 unknowns for 3 equilibrium equations. The beam is therefore statically indeterminate to the 3rd degree.
The key characteristic of a fixed beam is the development of hogging moments at the supports in addition to the sagging moment at midspan. These support moments partially counteract the midspan moment, resulting in significantly smaller deflections and smaller maximum bending moments compared to an equivalent simply supported beam. This is why fixed beams are structurally more efficient — but they also transfer moments into the supporting columns or walls, which must be designed for this.
Fixed Beam — Key Results (Fixed End Moments)
Central point load W:
FEM at each end = WL/8 (hogging) | Mmidspan = WL/8 (sagging) | δmax = WL³/192EI
UDL of intensity w per unit length:
FEM at each end = wL²/12 (hogging) | Mmidspan = wL²/24 (sagging) | δmax = wL⁴/384EI
Point load W at distance a from A, b from B:
FEMA = Wab²/L² (hogging) | FEMB = Wa²b/L² (hogging)
Comparison with simply supported beam (UDL):
Fixed beam Mmax = wL²/12 vs SS beam Mmax = wL²/8 — fixed beam has 33% lower maximum moment.
Fixed beam δmax = wL⁴/384EI vs SS beam δmax = 5wL⁴/384EI — fixed beam deflects 5× less.
Real-world examples: Beams monolithically cast with columns in RCC frames, prestressed concrete bridge beams with moment-resisting connections, and steel beams with fully welded end-plate connections.
6. Propped Cantilever
A propped cantilever is fixed at one end and supported by a roller (prop) at the other end. The fixed end provides 3 reactions and the roller provides 1 reaction — a total of 4 unknowns against 3 equilibrium equations. It is therefore statically indeterminate to the 1st degree (one redundant reaction).
To solve a propped cantilever, the prop reaction is treated as the redundant. The compatibility condition is that the deflection at the propped end is zero — the prop prevents any vertical movement there.
Propped Cantilever — UDL w over full span L
Prop reaction (at roller end B): RB = 3wL/8
Reaction at fixed end A: RA = 5wL/8 (upward)
Fixing moment at A: MA = wL²/8 (hogging)
Maximum sagging moment: Mmax = 9wL²/128 (at x = 5L/8 from A)
Point of contraflexure: The BM changes sign once between A and B — at x = 3L/4 from the fixed end.
Real-world examples: Retaining walls with a base slab that acts as a prop, beams in building frames where one end is rigidly connected to a column and the other rests on a wall, and basement floor slabs fixed to the foundation wall.
7. Continuous Beam
A continuous beam spans over more than two supports without any internal hinges, forming a single structural member that is continuous over intermediate supports. Each intermediate support is typically a roller (1 redundant reaction per intermediate support), making a two-span continuous beam indeterminate to the 1st degree, a three-span beam indeterminate to the 2nd degree, and so on.
Continuous beams are structurally very efficient — the intermediate supports reduce the bending moments in each span compared to a series of simply supported beams. However, settlement of any intermediate support causes significant redistribution of moments throughout the beam, which is a critical design consideration.
Analysis methods: Theorem of Three Moments (Clapeyron’s equation) for hand calculation, and Moment Distribution or Slope-Deflection methods for complex cases. For GATE CE, the Moment Distribution Method is the standard tool for continuous beam problems.
Theorem of Three Moments (Clapeyron’s Equation)
For any three consecutive supports A, B, C of a continuous beam with spans L₁ (A to B) and L₂ (B to C):
MAL₁ + 2MB(L₁ + L₂) + MCL₂ = −6A₁ā₁/L₁ − 6A₂b̄₂/L₂
Where A₁, A₂ = areas of free BMDs in spans 1 and 2 respectively, ā₁ = distance of centroid of A₁ from A, b̄₂ = distance of centroid of A₂ from C.
For uniform EI and UDL w on each span: 6Aā/L = wL³/4
Real-world examples: Multi-span bridge girders, floor beams continuous over several column supports in a building frame, and railway bridge stringers continuous over cross-girders.
8. Overhanging Beam
An overhanging beam is a simply supported beam with one or both ends extending beyond the supports. The overhanging portion behaves like a cantilever, developing hogging (negative) bending moment. At the support nearest the overhang, the bending moment may change sign — there is a point of contraflexure between the support and the point of maximum sagging moment.
Degree of static indeterminacy: 0 — fully determinate, solved by equilibrium alone.
An interesting feature of overhanging beams is that a load on the overhanging portion can actually reduce the sagging moment in the main span. This principle is used in balanced cantilever bridge construction and in the design of crane beams — the back span acts as a counterbalance to the overhanging load.
Overhanging Beam — Point of Contraflexure
The point of contraflexure is where the bending moment is zero and changes sign (from sagging to hogging or vice versa). For an overhanging beam, there is always at least one point of contraflexure between the outer support and the location of maximum sagging moment.
To find it: write the bending moment expression M(x) and set M(x) = 0. Solve for x.
In the overhanging portion beyond the outer support: BM varies linearly from zero at the free end to the hogging value at the support, just like a cantilever.
Real-world examples: Eaves beams in industrial sheds that overhang to support cladding, crane rails that project slightly beyond their last support, and bridge approach slabs.
9. Comparison Table — All Beam Types
| Beam Type | Supports | Unknowns | DSI | M at Supports | Mmax (UDL w, span L) | δmax (UDL) |
|---|---|---|---|---|---|---|
| Simply Supported | Pin + Roller | 2 | 0 | Zero | wL²/8 (midspan) | 5wL⁴/384EI |
| Cantilever | Fixed + Free | 3 | 0 | wL²/2 (hogging at fixed end) | wL²/2 (at fixed end) | wL⁴/8EI |
| Fixed Beam | Fixed + Fixed | 6 | 3 | wL²/12 (hogging at each end) | wL²/12 (at supports) / wL²/24 (midspan) | wL⁴/384EI |
| Propped Cantilever | Fixed + Roller | 4 | 1 | wL²/8 (hogging at fixed end) | 9wL²/128 (at 5L/8 from fixed) | wL⁴/185EI (approx) |
| Continuous Beam | Multiple supports | n+2 | n−1 (n = intermediate supports) | Hogging at intermediate supports | Less than wL²/8 per span | Less than SS beam |
| Overhanging Beam | Pin + Roller (extended) | 2 | 0 | Hogging at outer support (overhang) | Depends on overhang load | Depends on geometry |
10. Calculating Support Reactions — Worked Examples
Example 1: Simply Supported Beam with UDL and Point Load
Problem: A simply supported beam AB of span 8 m carries a UDL of 10 kN/m over the entire span and a point load of 40 kN at 3 m from A. Find the support reactions RA and RB.
Solution
Total UDL load: 10 × 8 = 80 kN, acting at midspan (4 m from A)
Take moments about A (ΣMA = 0):
RB × 8 = (80 × 4) + (40 × 3) = 320 + 120 = 440
RB = 440/8 = 55 kN
ΣFy = 0:
RA + RB = 80 + 40 = 120
RA = 120 − 55 = 65 kN
Check (moments about B): 65 × 8 = 520 = (80 × 4) + (40 × 5) = 320 + 200 = 520 ✓
Example 2: Cantilever Beam with Combined Loading
Problem: A cantilever beam AC is fixed at A and free at C. Span = 5 m. It carries a point load of 20 kN at the free end C, and a UDL of 6 kN/m over the entire span. Find the fixing moment and vertical reaction at A.
Solution
ΣFy = 0 (upward positive):
RA = 20 + (6 × 5) = 20 + 30 = 50 kN (upward)
ΣMA = 0 (anticlockwise positive):
MA = (20 × 5) + (6 × 5 × 2.5) = 100 + 75 = 175 kN·m (hogging/clockwise)
The fixed support must apply a 175 kN·m anticlockwise moment to maintain equilibrium.
Example 3: Overhanging Beam
Problem: Beam ABCD is supported at A (pin) and C (roller). AB = 2 m, BC = 4 m, CD = 2 m (overhang). A point load of 30 kN acts at B and a point load of 20 kN acts at D. Find RA and RC.
Solution
Take moments about A (ΣMA = 0):
RC × 6 = (30 × 2) + (20 × 8) = 60 + 160 = 220
RC = 220/6 = 36.67 kN (upward)
ΣFy = 0:
RA = 30 + 20 − 36.67 = 13.33 kN (upward)
Note: The bending moment at C = 20 × 2 = 40 kN·m (hogging) — the overhang creates a hogging moment at the outer support.
11. Common Mistakes Students Make
- Confusing pin and roller reactions: A pin provides both horizontal and vertical reactions (2 unknowns). A roller provides only one reaction perpendicular to the rolling surface (1 unknown). Many students assign two reactions to a roller — this is wrong and leads to incorrect DSI calculations and redundant equilibrium equations.
- Forgetting the fixing moment in a cantilever: A fixed support provides three reactions — vertical force, horizontal force, AND a moment. Students often account for the vertical force but forget the fixing moment when taking moments for equilibrium. Always draw all three reaction components on the free body diagram before writing equilibrium equations.
- Assuming maximum BM always occurs at midspan: This is only true for symmetrically loaded simply supported beams. For cantilevers, the maximum moment is at the fixed end. For propped cantilevers, the maximum sagging moment is at 5L/8 from the fixed end under UDL — not at midspan.
- Misidentifying the degree of indeterminacy of a fixed beam: A fixed beam (fixed at both ends) has 6 unknown reactions (3 at each fixed end) and only 3 equilibrium equations, giving DSI = 3. Many students say DSI = 1 or 2 — this is incorrect. The correct answer is 3.
- Not checking the moment equilibrium after finding reactions: Always verify by taking moments about the second support — if the reactions are correct, the moments balance. This simple check catches arithmetic errors before they propagate into the SFD and BMD.
12. Frequently Asked Questions
What is the difference between a fixed beam and a propped cantilever?
Both are statically indeterminate, but they differ in their support conditions and degree of indeterminacy. A fixed beam is rigidly restrained (no rotation, no translation) at both ends — it is indeterminate to the 3rd degree. A propped cantilever is fixed at one end (full restraint) and supported by a roller at the other end — the roller allows rotation but prevents vertical movement, making it indeterminate to only the 1st degree. The propped cantilever therefore requires only one compatibility equation to solve, while a fixed beam requires three (or, in practice, is solved using fixed end moment formulae combined with equilibrium).
Why is the bending moment always zero at a roller or pin support?
A roller or pin support cannot resist rotation — it allows the beam end to rotate freely. Because the support exerts no moment on the beam, and equilibrium requires that the internal moment at any section equals the sum of external moments on one side of that section, the internal bending moment at a pin or roller support must be zero. Conceptually: if the support applied a moment to the beam end, the beam end would not be free to rotate, which contradicts the definition of a pin or roller.
What is a point of contraflexure and why does it matter?
A point of contraflexure is a location along a beam where the bending moment is zero and changes sign — from sagging (positive) to hogging (negative) or vice versa. It is important in RCC design because the tension side of the beam changes at this point. Longitudinal reinforcement in an RCC beam is placed on the tension face, so at a point of contraflexure, the reinforcement must be continued past this point and properly anchored. Points of contraflexure also indicate where a beam could theoretically be spliced (though in practice, a safety margin is applied).
Which beam type gives the lowest deflection for the same span and load?
The fixed beam gives the lowest deflection, followed by the propped cantilever, then the continuous beam, and finally the simply supported beam. For a UDL over span L, the maximum deflection of a fixed beam (wL⁴/384EI) is exactly one-fifth of that of a simply supported beam (5wL⁴/384EI). This dramatic reduction in deflection — achieved by providing end fixity — is the primary structural reason for designing beams as part of moment-resisting frames rather than as simple spans.