Shear, Torsion and Force Transfer

Learning purpose. Develop an explicit force, geometry, materials and evidence model before any final code compliance calculation. Numerical examples in this unit are educational and not an authorization to construct.

Guided investigation: shear mechanisms and the limits of a single formula

Diagonal cracking develops from combined bending and shear, but its physical behavior is more complicated than a single stress check. Under a shear force, principal tension may lead to inclined cracks. Concrete compression fields, aggregate interlock across rough crack faces, dowel action from longitudinal steel and the action of stirrups all contribute. Their balance depends on load history, section size, longitudinal strain, reinforcement detailing and axial force. This is why shear rules across code generations often change substantially even when introductory truss diagrams remain the same.

An elementary vertical-stirrup truss argument gives a reinforcement term proportional to Av fy d/s. Explain each variable on a drawing: Av is the relevant area of transverse bar legs, not the total length or cross-sectional area of every stirrup segment; fy is an appropriate reinforcement strength; d is an effective depth; and s is center-to-center spacing along the member. The units are N if MPa, mm and mm² are used. A simplification of concrete contribution such as 0.17√f′c bw d must be presented as an illustrative expression with limitations, not a universally valid current code requirement.

Now consider three geometries. A long slender beam with a uniform load has a recognizable one-dimensional shear-force diagram. A short transfer beam with a high concentrated load near its support may have a direct compression strut; ordinary beam shear modeling can be unrepresentative. An edge beam carrying a cantilever balcony develops torsion, meaning that one set of open stirrups would not provide the force closure expected from a closed spatial torsional truss. An apparently small drawing detail can therefore decide which analysis framework applies.

A useful laboratory exercise is to compare a model response for several spacings and to calculate the stirrup force contribution independently. But an engineer must subsequently assess maximum allowable spacing, minimum reinforcement, maximum nominal shear resistance, concrete strength, anchorage and appropriate critical sections. For torsion, identify whether it is required for equilibrium or arises from compatibility. Where a design includes a joint, a corbel or a bearing node, draw a strut-and-tie model and confirm its nodal forces and reinforcement development lengths.

Produce an inspection-oriented checklist: closed stirrup shape and hooks; number of effective legs; cage support and cover; congestion at beam-column joints; appropriate spacing transitions near critical zones; and reinforcement continuity through openings. Distinguish a calculation input from a site-verifiable detail. A structurally sound mathematical truss is not a constructed truss if one anchorage is missing.

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Lesson 3.1: Beam shear and diagonal cracking

Physical meaning and governing ideas

Shear is carried by several interacting mechanisms: the uncracked compression zone, aggregate interlock, dowel action, transverse steel and strut action. A truss analogy helps explain stirrups but does not fully capture size effect, axial stress or disturbed regions. The design shear at a critical section must be determined from the actual load pattern and support geometry.

Analysis and design reasoning

Establish demand from equilibrium, determine the appropriate critical section, and check whether minimum stirrups, spacing limits or other detailing applies even when a nominal calculation suggests little transverse reinforcement. A simple sum of concrete and stirrup contributions is a learning model—not a substitute for the selected code’s detailed resistance expressions.

Worked numerical or geometric illustration

A beam with bw=280 mm, d=500 mm, f′c=32 MPa, Av=160 mm², fy=420 MPa and s=175 mm gives a surrogate Vc=0.17√32×280×500/1000≈134.63 kN and Vs=Avfy d/(1000s)=192.00 kN. These mechanics-derived values must not be used directly as a permitted design strength.

Interpretation and failure checks

When a shear response looks unexpectedly high, verify area of stirrup legs, dimensions in mm, stirrup orientation, and code caps on concrete strut capacity.

Study task 3.1. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

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Lesson 3.2: Torsion and combined shear

Physical meaning and governing ideas

Members experience torsion when the line of action of loads does not pass through the member shear center or when edge beams restrain slab deformation. Equilibrium torsion must be resisted; compatibility torsion may redistribute as cracking occurs. In reinforced concrete, closed transverse reinforcement and longitudinal bars develop a spatial resisting truss.

Analysis and design reasoning

Distinguish a torsion demand needed for overall stability from a redundant compatibility moment. Closed stirrup geometry, anchorage, longitudinal corner bars and section dimensions are essential to the path. Numerical expressions for torsional resistance differ significantly across code systems and cannot be interchanged casually.

Worked numerical or geometric illustration

A floor edge load of 10 kN/m acting at an eccentricity of 0.4 m creates a distributed torsional action of 4 kN·m per metre along the supporting member. A 6 m member cannot be checked by simply multiplying this by six without considering restraint and actual torsional equilibrium diagrams.

Interpretation and failure checks

Sketch load placement in three dimensions. If a cantilever balcony is fixed to an edge beam, explicitly show the torsional transfer into the frame.

Study task 3.2. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

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Lesson 3.3: Deep beams, corbels and strut-and-tie models

Physical meaning and governing ideas

In regions near concentrated loads, supports and abrupt changes in geometry, plane-section behavior may be inadequate. A strut-and-tie model replaces the disturbed stress field with a balanced network of compressive concrete struts and tensile steel ties. Nodal zones, bearing stresses and anchorage determine whether the assumed load path is physically realizable.

Analysis and design reasoning

Choose a plausible truss geometry from the expected compressive and tensile fields. Resolve nodal equilibrium, check strut and tie forces with governing design strengths, then detail continuous reinforcement through the nodes. Two different equilibrium trusses may both be statically admissible but impose different local detailing and cracking behavior.

Worked numerical or geometric illustration

A 600 kN vertical load transferred through two equal diagonal struts at 45° to horizontal requires each strut’s vertical component of 300 kN, so each axial compressive force is about 424.3 kN. The horizontal components are resisted by a tie or by an appropriate boundary system; the geometry must close all forces.

Interpretation and failure checks

A well-balanced truss diagram is not enough: anchorage length and nodal bearing areas commonly control a realizable detail.

Study task 3.3. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

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Lesson 3.4: Interfaces, joints and bearing regions

Physical meaning and governing ideas

Construction joints, precast interfaces and bearing pads create interfaces where shear transfer cannot be taken for granted. Surface roughness, clamping stress, reinforcement crossing the interface, shrinkage and local crushing affect behavior. Beam–column joints, transfer girders and brackets require a three-dimensional understanding of force flow.

Analysis and design reasoning

Identify which reinforcement actually crosses a potential sliding plane and where its tensile force develops. For a bearing zone, compare load dispersion with plausible concrete strut geometries and contact pressure distribution. Document joint preparation and inspection because a perfectly rough interface assumed on paper may be smooth or contaminated on site.

Worked numerical or geometric illustration

If 400 kN is transferred through a nominal 300 mm×400 mm contact area under uniform idealization, the average contact pressure is 400000/(300×400)=3.33 MPa. This single number cannot verify nodal crushing, punching, local confinement or other connection checks.

Interpretation and failure checks

When designing a retrofit or staged pour, investigate whether the existing member can carry the temporary load path before the joint has gained strength.

Study task 3.4. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

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Numerical practice — solved and interpreted

The eight problems below vary geometry and actions within a consistent instructional model. They develop reproducible calculation habits, but do not substitute for independent code checks.

Practice scenario 3.1

Problem. A beam shear teaching model uses f′c=25 MPa, web b=228 mm, d=375 mm, Av=112 mm², fy=420 MPa and stirrup spacing s=135 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=72.67 kN and Vs=Avfy d/s=130.67 kN. Their illustrative sum is 203.34 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 3.2

Problem. A beam shear teaching model uses f′c=26 MPa, web b=236 mm, d=390 mm, Av=124 mm², fy=420 MPa and stirrup spacing s=145 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=79.78 kN and Vs=Avfy d/s=140.08 kN. Their illustrative sum is 219.86 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 3.3

Problem. A beam shear teaching model uses f′c=27 MPa, web b=244 mm, d=405 mm, Av=136 mm², fy=420 MPa and stirrup spacing s=155 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=87.29 kN and Vs=Avfy d/s=149.25 kN. Their illustrative sum is 236.54 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 3.4

Problem. A beam shear teaching model uses f′c=28 MPa, web b=252 mm, d=420 mm, Av=148 mm², fy=420 MPa and stirrup spacing s=165 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=95.21 kN and Vs=Avfy d/s=158.23 kN. Their illustrative sum is 253.43 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 3.5

Problem. A beam shear teaching model uses f′c=29 MPa, web b=260 mm, d=435 mm, Av=160 mm², fy=420 MPa and stirrup spacing s=175 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=103.54 kN and Vs=Avfy d/s=167.04 kN. Their illustrative sum is 270.58 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 3.6

Problem. A beam shear teaching model uses f′c=30 MPa, web b=268 mm, d=450 mm, Av=172 mm², fy=420 MPa and stirrup spacing s=185 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=112.29 kN and Vs=Avfy d/s=175.72 kN. Their illustrative sum is 288.01 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 3.7

Problem. A beam shear teaching model uses f′c=31 MPa, web b=276 mm, d=465 mm, Av=184 mm², fy=420 MPa and stirrup spacing s=195 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=121.48 kN and Vs=Avfy d/s=184.28 kN. Their illustrative sum is 305.76 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 3.8

Problem. A beam shear teaching model uses f′c=32 MPa, web b=284 mm, d=480 mm, Av=196 mm², fy=420 MPa and stirrup spacing s=205 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=131.09 kN and Vs=Avfy d/s=192.75 kN. Their illustrative sum is 323.84 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Knowledge check with explanatory answers

Question 3.1

Why is a stirrup shear contribution inversely related to spacing in an elementary truss model?

Discussion answer. A diagonal crack crosses more stirrup sets per unit beam length when longitudinal spacing is reduced. With a constant area Av and yield stress fy, the transferred vertical force per length grows approximately as Avfy/s. Multiplying by an effective internal lever arm yields a contribution proportional to Avfy d/s. This analogy omits multiple concrete mechanisms and cannot independently establish code shear strength.

Question 3.2

How does torsion differ from simple transverse shear?

Discussion answer. Torsion causes shear flow around a cross-section and usually requires a spatially closed force-resisting mechanism. Closed transverse reinforcement and longitudinal bars are critical, and members need a path for torsional equilibrium into adjoining elements. A member with adequate vertical web stirrups for transverse shear may still lack proper torsion reinforcement. Whether torsion is necessary for equilibrium or is compatibility-driven affects analysis decisions.

Question 3.3

What makes a deep transfer region unsuitable for ordinary beam assumptions?

Discussion answer. Strain distributions near concentrated supports and abrupt geometry changes are not well described by plane-section bending theory. Compression struts can carry load directly into supports while steel ties close equilibrium. A strut-and-tie model or appropriate nonlinear analysis can represent this disturbed region, but concrete nodal capacity and bar anchorage must also be checked. A statically balanced sketch does not alone guarantee a safe detail.

Question 3.4

What is missing from the average bearing pressure P/A at a beam seat?

Discussion answer. The load may be concentrated, uneven, eccentric or transmitted through a contact region with local confinement and tensile splitting. A uniform average contact pressure does not establish nodal stress state, bearing enhancement, anchor force, shear transfer or construction-joint performance. Proper design requires geometry, material and code-specific local checks.

Question 3.5

Why does a small numerical concrete shear contribution not remove every need for stirrups?

Discussion answer. Codes often require minimum transverse reinforcement to maintain ductile resistance and control diagonal cracking even when a simplified calculation predicts modest shear demand. Spacing limits, support regions and seismic detailing can also govern independently. The engineer must apply the actual chosen code rather than comparing one modeled force to a guessed material limit.

Unit project and experiment

A 7 m beam supports a secondary beam reaction at a point 2 m from the left support and a uniform load along the entire span. Build accurate reaction, shear and bending diagrams with dimensions. Select two candidate support conditions and examine how the reaction distribution changes. Establish a preliminary shear-reinforcement schedule from an illustrative truss contribution, then clearly separate it from required code expressions and minimum reinforcement. Sketch a bearing node and consider whether a concentrated load near support creates a disturbed region requiring a strut-and-tie model. Show the intended concrete placement sequence and the accessibility of cage corners. Identify which tests or analyses would be needed before detailed design.

Deliverable: A transparent engineering memo with assumptions, diagrams, units, hand-checks, source references and a list of issues that need a governing-code and professional review.

Related interactive simulations

• [Stirrup spacing and shear-force components](../labs/lab-shear.html) — adjust inputs, inspect curves, export data and explain limits of the model.

Unit 3: Shear, torsion and force transfer

Learning purpose. Develop an explicit force, geometry, materials and evidence model before any final code compliance calculation. Numerical examples in this unit are educational and not an authorization to construct.

Guided investigation: shear mechanisms and the limits of a single formula

Diagonal cracking develops from combined bending and shear, but its physical behavior is more complicated than a single stress check. Under a shear force, principal tension may lead to inclined cracks. Concrete compression fields, aggregate interlock across rough crack faces, dowel action from longitudinal steel and the action of stirrups all contribute. Their balance depends on load history, section size, longitudinal strain, reinforcement detailing and axial force. This is why shear rules across code generations often change substantially even when introductory truss diagrams remain the same.

An elementary vertical-stirrup truss argument gives a reinforcement term proportional to Av fy d/s. Explain each variable on a drawing: Av is the relevant area of transverse bar legs, not the total length or cross-sectional area of every stirrup segment; fy is an appropriate reinforcement strength; d is an effective depth; and s is center-to-center spacing along the member. The units are N if MPa, mm and mm² are used. A simplification of concrete contribution such as 0.17√f′c bw d must be presented as an illustrative expression with limitations, not a universally valid current code requirement.

Now consider three geometries. A long slender beam with a uniform load has a recognizable one-dimensional shear-force diagram. A short transfer beam with a high concentrated load near its support may have a direct compression strut; ordinary beam shear modeling can be unrepresentative. An edge beam carrying a cantilever balcony develops torsion, meaning that one set of open stirrups would not provide the force closure expected from a closed spatial torsional truss. An apparently small drawing detail can therefore decide which analysis framework applies.

A useful laboratory exercise is to compare a model response for several spacings and to calculate the stirrup force contribution independently. But an engineer must subsequently assess maximum allowable spacing, minimum reinforcement, maximum nominal shear resistance, concrete strength, anchorage and appropriate critical sections. For torsion, identify whether it is required for equilibrium or arises from compatibility. Where a design includes a joint, a corbel or a bearing node, draw a strut-and-tie model and confirm its nodal forces and reinforcement development lengths.

Produce an inspection-oriented checklist: closed stirrup shape and hooks; number of effective legs; cage support and cover; congestion at beam-column joints; appropriate spacing transitions near critical zones; and reinforcement continuity through openings. Distinguish a calculation input from a site-verifiable detail. A structurally sound mathematical truss is not a constructed truss if one anchorage is missing.

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Lesson 3.1: Beam shear and diagonal cracking

Physical meaning and governing ideas

Shear is carried by several interacting mechanisms: the uncracked compression zone, aggregate interlock, dowel action, transverse steel and strut action. A truss analogy helps explain stirrups but does not fully capture size effect, axial stress or disturbed regions. The design shear at a critical section must be determined from the actual load pattern and support geometry.

Analysis and design reasoning

Establish demand from equilibrium, determine the appropriate critical section, and check whether minimum stirrups, spacing limits or other detailing applies even when a nominal calculation suggests little transverse reinforcement. A simple sum of concrete and stirrup contributions is a learning model—not a substitute for the selected code’s detailed resistance expressions.

Worked numerical or geometric illustration

A beam with bw=280 mm, d=500 mm, f′c=32 MPa, Av=160 mm², fy=420 MPa and s=175 mm gives a surrogate Vc=0.17√32×280×500/1000≈134.63 kN and Vs=Avfy d/(1000s)=192.00 kN. These mechanics-derived values must not be used directly as a permitted design strength.

Interpretation and failure checks

When a shear response looks unexpectedly high, verify area of stirrup legs, dimensions in mm, stirrup orientation, and code caps on concrete strut capacity.

Study task 3.1. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

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Lesson 3.2: Torsion and combined shear

Physical meaning and governing ideas

Members experience torsion when the line of action of loads does not pass through the member shear center or when edge beams restrain slab deformation. Equilibrium torsion must be resisted; compatibility torsion may redistribute as cracking occurs. In reinforced concrete, closed transverse reinforcement and longitudinal bars develop a spatial resisting truss.

Analysis and design reasoning

Distinguish a torsion demand needed for overall stability from a redundant compatibility moment. Closed stirrup geometry, anchorage, longitudinal corner bars and section dimensions are essential to the path. Numerical expressions for torsional resistance differ significantly across code systems and cannot be interchanged casually.

Worked numerical or geometric illustration

A floor edge load of 10 kN/m acting at an eccentricity of 0.4 m creates a distributed torsional action of 4 kN·m per metre along the supporting member. A 6 m member cannot be checked by simply multiplying this by six without considering restraint and actual torsional equilibrium diagrams.

Interpretation and failure checks

Sketch load placement in three dimensions. If a cantilever balcony is fixed to an edge beam, explicitly show the torsional transfer into the frame.

Study task 3.2. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

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Lesson 3.3: Deep beams, corbels and strut-and-tie models

Physical meaning and governing ideas

In regions near concentrated loads, supports and abrupt changes in geometry, plane-section behavior may be inadequate. A strut-and-tie model replaces the disturbed stress field with a balanced network of compressive concrete struts and tensile steel ties. Nodal zones, bearing stresses and anchorage determine whether the assumed load path is physically realizable.

Analysis and design reasoning

Choose a plausible truss geometry from the expected compressive and tensile fields. Resolve nodal equilibrium, check strut and tie forces with governing design strengths, then detail continuous reinforcement through the nodes. Two different equilibrium trusses may both be statically admissible but impose different local detailing and cracking behavior.

Worked numerical or geometric illustration

A 600 kN vertical load transferred through two equal diagonal struts at 45° to horizontal requires each strut’s vertical component of 300 kN, so each axial compressive force is about 424.3 kN. The horizontal components are resisted by a tie or by an appropriate boundary system; the geometry must close all forces.

Interpretation and failure checks

A well-balanced truss diagram is not enough: anchorage length and nodal bearing areas commonly control a realizable detail.

Study task 3.3. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

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Lesson 3.4: Interfaces, joints and bearing regions

Physical meaning and governing ideas

Construction joints, precast interfaces and bearing pads create interfaces where shear transfer cannot be taken for granted. Surface roughness, clamping stress, reinforcement crossing the interface, shrinkage and local crushing affect behavior. Beam–column joints, transfer girders and brackets require a three-dimensional understanding of force flow.

Analysis and design reasoning

Identify which reinforcement actually crosses a potential sliding plane and where its tensile force develops. For a bearing zone, compare load dispersion with plausible concrete strut geometries and contact pressure distribution. Document joint preparation and inspection because a perfectly rough interface assumed on paper may be smooth or contaminated on site.

Worked numerical or geometric illustration

If 400 kN is transferred through a nominal 300 mm×400 mm contact area under uniform idealization, the average contact pressure is 400000/(300×400)=3.33 MPa. This single number cannot verify nodal crushing, punching, local confinement or other connection checks.

Interpretation and failure checks

When designing a retrofit or staged pour, investigate whether the existing member can carry the temporary load path before the joint has gained strength.

Study task 3.4. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

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Numerical practice — solved and interpreted

The eight problems below vary geometry and actions within a consistent instructional model. They develop reproducible calculation habits, but do not substitute for independent code checks.

Practice scenario 3.1

Problem. A beam shear teaching model uses f′c=25 MPa, web b=228 mm, d=375 mm, Av=112 mm², fy=420 MPa and stirrup spacing s=135 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=72.67 kN and Vs=Avfy d/s=130.67 kN. Their illustrative sum is 203.34 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 3.2

Problem. A beam shear teaching model uses f′c=26 MPa, web b=236 mm, d=390 mm, Av=124 mm², fy=420 MPa and stirrup spacing s=145 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=79.78 kN and Vs=Avfy d/s=140.08 kN. Their illustrative sum is 219.86 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 3.3

Problem. A beam shear teaching model uses f′c=27 MPa, web b=244 mm, d=405 mm, Av=136 mm², fy=420 MPa and stirrup spacing s=155 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=87.29 kN and Vs=Avfy d/s=149.25 kN. Their illustrative sum is 236.54 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 3.4

Problem. A beam shear teaching model uses f′c=28 MPa, web b=252 mm, d=420 mm, Av=148 mm², fy=420 MPa and stirrup spacing s=165 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=95.21 kN and Vs=Avfy d/s=158.23 kN. Their illustrative sum is 253.43 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 3.5

Problem. A beam shear teaching model uses f′c=29 MPa, web b=260 mm, d=435 mm, Av=160 mm², fy=420 MPa and stirrup spacing s=175 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=103.54 kN and Vs=Avfy d/s=167.04 kN. Their illustrative sum is 270.58 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 3.6

Problem. A beam shear teaching model uses f′c=30 MPa, web b=268 mm, d=450 mm, Av=172 mm², fy=420 MPa and stirrup spacing s=185 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=112.29 kN and Vs=Avfy d/s=175.72 kN. Their illustrative sum is 288.01 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 3.7

Problem. A beam shear teaching model uses f′c=31 MPa, web b=276 mm, d=465 mm, Av=184 mm², fy=420 MPa and stirrup spacing s=195 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=121.48 kN and Vs=Avfy d/s=184.28 kN. Their illustrative sum is 305.76 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 3.8

Problem. A beam shear teaching model uses f′c=32 MPa, web b=284 mm, d=480 mm, Av=196 mm², fy=420 MPa and stirrup spacing s=205 mm. Compute two basic shear-force contributions and discuss minimum reinforcement and upper limits.

Worked resolution. Vc=0.17√f′c bwd=131.09 kN and Vs=Avfy d/s=192.75 kN. Their illustrative sum is 323.84 kN before any strength factor, size limitation or nonlinear shear behavior. When s is reduced by 20%, Vs rises by 25% as long as the truss-model assumptions remain meaningful.

Engineering interpretation. Contrast the force model with a real code calculation, which must include the appropriate nominal strength expressions, maximum stirrup spacing, concrete strut checks and special seismic provisions where applicable.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Knowledge check with explanatory answers

Question 3.1

Why is a stirrup shear contribution inversely related to spacing in an elementary truss model?

Discussion answer. A diagonal crack crosses more stirrup sets per unit beam length when longitudinal spacing is reduced. With a constant area Av and yield stress fy, the transferred vertical force per length grows approximately as Avfy/s. Multiplying by an effective internal lever arm yields a contribution proportional to Avfy d/s. This analogy omits multiple concrete mechanisms and cannot independently establish code shear strength.

Question 3.2

How does torsion differ from simple transverse shear?

Discussion answer. Torsion causes shear flow around a cross-section and usually requires a spatially closed force-resisting mechanism. Closed transverse reinforcement and longitudinal bars are critical, and members need a path for torsional equilibrium into adjoining elements. A member with adequate vertical web stirrups for transverse shear may still lack proper torsion reinforcement. Whether torsion is necessary for equilibrium or is compatibility-driven affects analysis decisions.

Question 3.3

What makes a deep transfer region unsuitable for ordinary beam assumptions?

Discussion answer. Strain distributions near concentrated supports and abrupt geometry changes are not well described by plane-section bending theory. Compression struts can carry load directly into supports while steel ties close equilibrium. A strut-and-tie model or appropriate nonlinear analysis can represent this disturbed region, but concrete nodal capacity and bar anchorage must also be checked. A statically balanced sketch does not alone guarantee a safe detail.

Question 3.4

What is missing from the average bearing pressure P/A at a beam seat?

Discussion answer. The load may be concentrated, uneven, eccentric or transmitted through a contact region with local confinement and tensile splitting. A uniform average contact pressure does not establish nodal stress state, bearing enhancement, anchor force, shear transfer or construction-joint performance. Proper design requires geometry, material and code-specific local checks.

Question 3.5

Why does a small numerical concrete shear contribution not remove every need for stirrups?

Discussion answer. Codes often require minimum transverse reinforcement to maintain ductile resistance and control diagonal cracking even when a simplified calculation predicts modest shear demand. Spacing limits, support regions and seismic detailing can also govern independently. The engineer must apply the actual chosen code rather than comparing one modeled force to a guessed material limit.

Unit project and experiment

A 7 m beam supports a secondary beam reaction at a point 2 m from the left support and a uniform load along the entire span. Build accurate reaction, shear and bending diagrams with dimensions. Select two candidate support conditions and examine how the reaction distribution changes. Establish a preliminary shear-reinforcement schedule from an illustrative truss contribution, then clearly separate it from required code expressions and minimum reinforcement. Sketch a bearing node and consider whether a concentrated load near support creates a disturbed region requiring a strut-and-tie model. Show the intended concrete placement sequence and the accessibility of cage corners. Identify which tests or analyses would be needed before detailed design.

Deliverable: A transparent engineering memo with assumptions, diagrams, units, hand-checks, source references and a list of issues that need a governing-code and professional review.

Related interactive simulations

• [Stirrup spacing and shear-force components](labs/lab-shear.html) — adjust inputs, inspect curves, export data and explain limits of the model.