Frames, Cables and Three-Hinged Arches

Unit purpose

Frames and arches introduce thrust, bending and geometric coupling in ways that a straight beam diagram cannot capture. Hinges may make a system determinate, yet they also create meaningful kinematic behavior. Cables share some visual similarity with arches but carry a fundamentally different internal resultant.

Field scenario

Consider a short pedestrian crossing with two possible concepts: a three-hinged arch and a suspended cable. Compare geometry, support horizontal action, span-clearance implications and whether the member can resist an arbitrary load reversal.

Topics and worked reasoning

Topic 4.1: Frame equilibrium and internal force

A plane rigid-jointed frame can carry axial force, transverse shear and bending moment in the same member. A three-hinged frame or arch gains an additional internal hinge condition that can permit a global equilibrium solution without compatibility analysis. When isolating part of a frame at an internal hinge, the hinge transmits force but no resisting couple; the two parts have equal and opposite interface forces. A conventional beam-only model can miss large horizontal thrust or member axial forces when a structure contains inclined elements or rigid knees.

Engineering practice: Use clearly labelled joint and member free-body diagrams. For any inclined member, transform the global force components into local axial and transverse axes before sketching internal diagrams. Check the moment at a true pin: it should be zero in the simple idealization. A moment at a member center cannot be inferred from the center geometry alone; it follows from the external actions and constraints.

Fully evaluated study example

A three-hinged parabolic arch has horizontal span 8.0 m, crown rise f=2.5 m, and vertical point load 24 kN at 2.40 m from the left. Its vertical left reaction is 16.800 kN. The equivalent beam moment at crown is 28.800 kN·m; horizontal thrust is 11.755 kN. The crown hinge carries zero moment by construction. The large thrust sensitivity to small f can govern foundation assumptions even when the arch is statically determinate.

Reasoning and interpretation

Which member action becomes dominant at a rigid knee? Start by writing the physical reason, then use an equation or laboratory parameter sweep to support the claim. Compare the result against an extreme case: zero applied load, negligible rise, very long span, perfectly fixed restraint, or load at a support, whichever is physically appropriate. In a valid limiting case, the response should approach a predictable behavior. When a calculation fails an extreme-case check, investigate modeling assumptions before blaming numerical precision.

Practice tasks

1. Rework the given example with a 15% increase in the characteristic load and predict whether force, moment or displacement scales linearly. State all inputs kept fixed.
2. Investigate this potential failure of the model: Which member action becomes dominant at a rigid knee? Document the assumption at risk and describe a more complete model.
3. Draw the free-body diagram or curvature/influence diagram appropriate to frame equilibrium and internal force. Label axes, internal forces, signs, physical units and boundary conditions.
4. Use a unit or equilibrium residual to check your solution. Describe an error that would pass the numerical residual check but still lead to an incorrect engineering interpretation.

Applied investigation

Choose a structural example relevant to frame equilibrium and internal force. List the information required from a drawing, material schedule or site measurement. Separate what can be concluded directly from equilibrium from what depends on stiffness, geometry, or an unverified boundary condition. Run two different reasonable parameter settings and write a short explanation of why the predicted response changes. Avoid treating either result as a certified design action.

Topic 4.2: Three-hinged arch thrust

For a three-hinged arch with horizontal springing supports, span L, crown rise f and a crown hinge at midspan, the vertical end reactions match those of the corresponding simply supported beam under identical vertical loads, provided no applied horizontal actions. The crown hinge enforces zero bending moment there, so horizontal thrust H equals the simply supported beam moment at midspan divided by f. For a point load P at x=a, the beam crown moment is R_A L/2-P max(0,L/2-a). The arch section bending moment is M_arch(x)=M_beam(x)-H y(x), where y(x) is the arch ordinate above springing level.

Engineering practice: Test a symmetric point load and an off-center point load. Show that the arch moment at the crown is exactly zero while nonzero bending can remain elsewhere. If the rise is reduced at fixed span and load, required horizontal thrust increases. Do not interpret this idealized thrust as a safe bearing or foundation design without checking support capacity and movement.

Fully evaluated study example

A three-hinged parabolic arch has horizontal span 10.0 m, crown rise f=2.9 m, and vertical point load 45 kN at 3.00 m from the left. Its vertical left reaction is 31.500 kN. The equivalent beam moment at crown is 67.500 kN·m; horizontal thrust is 23.276 kN. The crown hinge carries zero moment by construction. The large thrust sensitivity to small f can govern foundation assumptions even when the arch is statically determinate.

Reasoning and interpretation

Why is crown moment zero while thrust remains finite? Start by writing the physical reason, then use an equation or laboratory parameter sweep to support the claim. Compare the result against an extreme case: zero applied load, negligible rise, very long span, perfectly fixed restraint, or load at a support, whichever is physically appropriate. In a valid limiting case, the response should approach a predictable behavior. When a calculation fails an extreme-case check, investigate modeling assumptions before blaming numerical precision.

Practice tasks

1. Rework the given example with a 15% increase in the characteristic load and predict whether force, moment or displacement scales linearly. State all inputs kept fixed.
2. Investigate this potential failure of the model: Why is crown moment zero while thrust remains finite? Document the assumption at risk and describe a more complete model.
3. Draw the free-body diagram or curvature/influence diagram appropriate to three-hinged arch thrust. Label axes, internal forces, signs, physical units and boundary conditions.
4. Use a unit or equilibrium residual to check your solution. Describe an error that would pass the numerical residual check but still lead to an incorrect engineering interpretation.

Applied investigation

Choose a structural example relevant to three-hinged arch thrust. List the information required from a drawing, material schedule or site measurement. Separate what can be concluded directly from equilibrium from what depends on stiffness, geometry, or an unverified boundary condition. Run two different reasonable parameter settings and write a short explanation of why the predicted response changes. Avoid treating either result as a certified design action.

Topic 4.3: Cable mechanics and sag

A perfectly flexible cable carries tension but no bending moment. Under a vertical loading uniformly distributed along the horizontal projection, the ideal cable shape is parabolic. With horizontal span L, sag f and load intensity w per horizontal meter, horizontal tension component H=wL²/(8f), while the support vertical reaction is wL/2 for symmetric loading; resultant support tension is sqrt(H²+(wL/2)²). These formulas do not apply unchanged to a cable whose dominant uniform load is its own weight measured along cable length, which leads to a catenary.

Engineering practice: Explore the tension penalty of small sag. Doubling sag reduces H by half if load intensity and span remain fixed, but greater sag may conflict with clearance requirements. Compare parabolic arc-length estimates, total vertical loading, and support forces. Explain why a cable cannot resist an upward normal point load in the same way as a flexural beam.

Fully evaluated study example

A three-hinged parabolic arch has horizontal span 7.5 m, crown rise f=3.4 m, and vertical point load 38 kN at 2.25 m from the left. Its vertical left reaction is 26.600 kN. The equivalent beam moment at crown is 42.750 kN·m; horizontal thrust is 12.761 kN. The crown hinge carries zero moment by construction. The large thrust sensitivity to small f can govern foundation assumptions even when the arch is statically determinate.

Reasoning and interpretation

Does the specified load refer to cable length or horizontal projection? Start by writing the physical reason, then use an equation or laboratory parameter sweep to support the claim. Compare the result against an extreme case: zero applied load, negligible rise, very long span, perfectly fixed restraint, or load at a support, whichever is physically appropriate. In a valid limiting case, the response should approach a predictable behavior. When a calculation fails an extreme-case check, investigate modeling assumptions before blaming numerical precision.

Practice tasks

1. Rework the given example with a 15% increase in the characteristic load and predict whether force, moment or displacement scales linearly. State all inputs kept fixed.
2. Investigate this potential failure of the model: Does the specified load refer to cable length or horizontal projection? Document the assumption at risk and describe a more complete model.
3. Draw the free-body diagram or curvature/influence diagram appropriate to cable mechanics and sag. Label axes, internal forces, signs, physical units and boundary conditions.
4. Use a unit or equilibrium residual to check your solution. Describe an error that would pass the numerical residual check but still lead to an incorrect engineering interpretation.

Applied investigation

Choose a structural example relevant to cable mechanics and sag. List the information required from a drawing, material schedule or site measurement. Separate what can be concluded directly from equilibrium from what depends on stiffness, geometry, or an unverified boundary condition. Run two different reasonable parameter settings and write a short explanation of why the predicted response changes. Avoid treating either result as a certified design action.

Topic 4.4: Curved members and equilibrium

Curved members may experience strongly varying local axial and shear components even when global force resultants are simple. A three-hinged arch is determinate because the crown hinge supplies an additional moment condition. In a two-hinged arch the horizontal thrust generally depends on deformation and stiffness, requiring compatibility. Locally, a small element of curved structure transfers normal force, shear and bending according to the selected tangent orientation, with sign conventions that must be consistent along the curve. Centerline geometry is therefore an essential part of the analysis and should be recorded explicitly.

Engineering practice: For a parabolic arch y=4 f x(L-x)/L squared, calculate the slope at a selected section and transform global horizontal and vertical force resultants into tangent and normal components. Compare two arches having the same span and loading but different rise. State which changes are directly controlled by statics and which require stiffness assumptions.

Fully evaluated study example

A three-hinged parabolic arch has horizontal span 5.5 m, crown rise f=3.8 m, and vertical point load 19 kN at 1.65 m from the left. Its vertical left reaction is 13.300 kN. The equivalent beam moment at crown is 15.675 kN·m; horizontal thrust is 4.125 kN. The crown hinge carries zero moment by construction. The large thrust sensitivity to small f can govern foundation assumptions even when the arch is statically determinate.

Reasoning and interpretation

How do local axes rotate along the arch centerline? Start by writing the physical reason, then use an equation or laboratory parameter sweep to support the claim. Compare the result against an extreme case: zero applied load, negligible rise, very long span, perfectly fixed restraint, or load at a support, whichever is physically appropriate. In a valid limiting case, the response should approach a predictable behavior. When a calculation fails an extreme-case check, investigate modeling assumptions before blaming numerical precision.

Practice tasks

1. Rework the given example with a 15% increase in the characteristic load and predict whether force, moment or displacement scales linearly. State all inputs kept fixed.
2. Investigate this potential failure of the model: How do local axes rotate along the arch centerline? Document the assumption at risk and describe a more complete model.
3. Draw the free-body diagram or curvature/influence diagram appropriate to curved members and equilibrium. Label axes, internal forces, signs, physical units and boundary conditions.
4. Use a unit or equilibrium residual to check your solution. Describe an error that would pass the numerical residual check but still lead to an incorrect engineering interpretation.

Applied investigation

Choose a structural example relevant to curved members and equilibrium. List the information required from a drawing, material schedule or site measurement. Separate what can be concluded directly from equilibrium from what depends on stiffness, geometry, or an unverified boundary condition. Run two different reasonable parameter settings and write a short explanation of why the predicted response changes. Avoid treating either result as a certified design action.

Deep dive and misconception repair

Curved systems clinic: an arch is not an inverted cable in every respect

A three-hinged arch and a hanging cable can share a parabolic appearance but resist loads by fundamentally different mechanisms. The arch is a compression-dominated structural element that can also carry shear and bending. A perfectly flexible cable carries tension only and must adopt a geometry compatible with the applied loading. A cable of fixed geometry under changed loading can redistribute shape substantially, whereas the first-order arch analysis generally holds its centerline geometry fixed.

The three-hinged arch's crown condition provides a particularly transparent force solution. First calculate reactions for an equivalent simply supported beam under the same vertical loading. Next evaluate beam bending moment at the crown. Because the arch has an internal hinge with zero transmitted moment, the horizontal thrust multiplied by rise must cancel the equivalent beam moment at that location. This gives H=Mbeam(crown)/f. Finally calculate any other section's arch moment by subtracting H times the arch ordinate from the equivalent beam moment.

This relation illustrates a geometric trade-off. A shallow arch has small rise f. If the equivalent crown moment is fixed, horizontal thrust must be large. More rise can reduce thrust but may increase overall height, change clearance and alter member lengths. Neither choice can be judged from the thrust formula alone. Soil capacity, footing restraint, construction sequence, environmental actions and material stability all matter in design practice.

Load mismatch and arch bending

For a parabolic arch under a full-span uniform load per horizontal length, an ideal funicular compression line can align well with the arch geometry, producing low bending when supports and loading match the assumptions. A concentrated off-center load destroys that exact match, leaving nonzero bending even though crown moment is constrained to zero. This distinction explains why a geometry that seems efficient for one load case may still experience flexure under real varying traffic or asymmetric snow.

The cable equation H=wL²/(8f) requires that w be uniform per horizontal projected distance. If w is instead a constant self-weight per cable arc length, the exact equilibrium shape is a catenary. In a shallow cable the parabolic approximation may be close, but the difference becomes relevant for large sag. One must not use the parabolic formula as a general exact solution for any cable material or loading.

Support horizontal action has an engineering consequence: foundations must supply the balancing thrust or an explicit tension tie must carry it. A three-hinged arch with a tie is not mechanically the same as an untied arch thrusting into the abutments. The tie changes the global force path. When presenting a computed H, label it 'horizontal thrust in this idealized arch' rather than 'safe foundation load'.

Multi-case calculations and validation

Case 4.1: Reaction shift

A three-hinged parabolic arch has horizontal span 6.0 m, crown rise f=1.8 m, and vertical point load 28 kN at 1.80 m from the left. Its vertical left reaction is 19.600 kN. The equivalent beam moment at crown is 25.200 kN·m; horizontal thrust is 14.000 kN. The crown hinge carries zero moment by construction. The large thrust sensitivity to small f can govern foundation assumptions even when the arch is statically determinate.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For frames, cables and three-hinged arches, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Case 4.2: Longer girder

A three-hinged parabolic arch has horizontal span 9.0 m, crown rise f=2.2 m, and vertical point load 34 kN at 2.70 m from the left. Its vertical left reaction is 23.800 kN. The equivalent beam moment at crown is 45.900 kN·m; horizontal thrust is 20.864 kN. The crown hinge carries zero moment by construction. The large thrust sensitivity to small f can govern foundation assumptions even when the arch is statically determinate.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For frames, cables and three-hinged arches, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Case 4.3: Footbridge loading

A three-hinged parabolic arch has horizontal span 12.0 m, crown rise f=2.6 m, and vertical point load 41 kN at 3.60 m from the left. Its vertical left reaction is 28.700 kN. The equivalent beam moment at crown is 73.800 kN·m; horizontal thrust is 28.385 kN. The crown hinge carries zero moment by construction. The large thrust sensitivity to small f can govern foundation assumptions even when the arch is statically determinate.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For frames, cables and three-hinged arches, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Case 4.4: Short lintel

A three-hinged parabolic arch has horizontal span 4.5 m, crown rise f=3.0 m, and vertical point load 15 kN at 1.35 m from the left. Its vertical left reaction is 10.500 kN. The equivalent beam moment at crown is 10.125 kN·m; horizontal thrust is 3.375 kN. The crown hinge carries zero moment by construction. The large thrust sensitivity to small f can govern foundation assumptions even when the arch is statically determinate.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For frames, cables and three-hinged arches, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Case 4.5: Beam with equipment

A three-hinged parabolic arch has horizontal span 8.0 m, crown rise f=3.4 m, and vertical point load 24 kN at 2.40 m from the left. Its vertical left reaction is 16.800 kN. The equivalent beam moment at crown is 28.800 kN·m; horizontal thrust is 8.471 kN. The crown hinge carries zero moment by construction. The large thrust sensitivity to small f can govern foundation assumptions even when the arch is statically determinate.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For frames, cables and three-hinged arches, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Case 4.6: Symmetric midspan

A three-hinged parabolic arch has horizontal span 10.0 m, crown rise f=3.8 m, and vertical point load 45 kN at 3.00 m from the left. Its vertical left reaction is 31.500 kN. The equivalent beam moment at crown is 67.500 kN·m; horizontal thrust is 17.763 kN. The crown hinge carries zero moment by construction. The large thrust sensitivity to small f can govern foundation assumptions even when the arch is statically determinate.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For frames, cables and three-hinged arches, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Case 4.7: Warehouse girder

A three-hinged parabolic arch has horizontal span 7.5 m, crown rise f=4.2 m, and vertical point load 38 kN at 2.25 m from the left. Its vertical left reaction is 26.600 kN. The equivalent beam moment at crown is 42.750 kN·m; horizontal thrust is 10.179 kN. The crown hinge carries zero moment by construction. The large thrust sensitivity to small f can govern foundation assumptions even when the arch is statically determinate.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For frames, cables and three-hinged arches, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Case 4.8: Roof beam

A three-hinged parabolic arch has horizontal span 5.5 m, crown rise f=4.6 m, and vertical point load 19 kN at 1.65 m from the left. Its vertical left reaction is 13.300 kN. The equivalent beam moment at crown is 15.675 kN·m; horizontal thrust is 3.408 kN. The crown hinge carries zero moment by construction. The large thrust sensitivity to small f can govern foundation assumptions even when the arch is statically determinate.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For frames, cables and three-hinged arches, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Interactive experiments

Use these standalone HTML laboratories. They load without a web connection and export response curves in CSV format. Each lab shows a calculation formula and limitations.

• [Simulation 06: Three-hinged arch bending](../labs/06-three-hinged-arch.html)
• [Simulation 07: Parabolic cable sag and thrust](../labs/07-cable-sag.html)

Unit assessment

1. Explain the key idealization governing frames, cables and three-hinged arches, and describe one condition that would invalidate it.
2. Explain what mathematical or physical equilibrium, compatibility, or limiting-case checks are appropriate for this unit.
3. Compare the results of two parameter settings from a linked lab. Name the inputs, calculate one numerical difference and interpret its sign.
4. Provide one real-world detail that the model does not represent and specify how an engineer should investigate it.

Project brief

Prepare a two-page calculation note using the methods of frames, cables and three-hinged arches. The note must include a problem definition, a labeled free-body diagram, a parameter and unit table, equations, at least two worked numerical results, an independently evaluated check, a figure or CSV trace from a linked simulation, and a discussion of three limitations. The note is not a structural design certification.