ENGINEERING WITH INSIGHT · RC BEAMS

RC Beams: Beyond Strength

The theory behind the response

Hands-on Workshop · Thursday 10 September 2026

Dr. Naveed Anwar

Founder and CEO, CSi Bangkok Co. Ltd. · Adjunct Faculty, Structural Engineering, Asian Institute of Technology

Elastic behaviour, ductility and performance

The Strength Design and its Limitations

A simple structure to be Designed

350

x700

Fc=30 Mpa

Fy=415 Mpa

DL = 25 kN/m

LL = 15 kN/m

Designed and Detailed Beam

The code-based strength design model

Beyond this moment, tension steel alone is no longer enough.

The code requires ductile behaviour�steel strain at least 0.005 when the concrete reaches 0.003

This limits the neutral axis depth�c no deeper than 0.375 d for a tension-controlled section

Which limits the moment capacity�for 300 x 500: φMn,max = 356 kN·m at As = 2 554 mm²

For gravity strength design, tension control is the only ductility requirement. Rotation capacity and moment-curvature are covered in Workshop 2.

Beam Vs Full Interaction Computations – ACI-318

Moment Capacity = 366 kN-m

DCR = 0.8

Approx 369 kN-m

THE SUBJECT

The designed beam:

Analyse

Bring results from the ETABS model, or analyse in the app: patterns, envelopes, the full response

→

Design

Group and design the beams; required against provided reinforcement at every station

→

Detail

Bar selection, curtailment and stirrups resolved along the span

→

Report

The calculation report and design flow, every step visible

A simple structure – some more questions!

• What is stress principal stress at any location before cracking?
• What is Code based Demand Capacity Ratio?
• What is the actual maximum load this beam can carry?
• What will happen when it reaches its maximum capacity?
• When and where will it crack?
• How much will it deflect?
• Will it fail in shear or bending ?
• Can I control how it will fail?

Limitations of Strength Based Design

• Cross-sections are capable of resisting a certain value of actions based on assumed failure criterion

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• Actions are obtained often from linear elastic analysis, and are factored to provide certain factor of safety

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• Strength design itself provides no information or control on the level of deformation produced at that factored load level

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• No information about behavior of the member if loads or actions were to exceed the factored design load

Advance Concrete Structures: Class Project - 2019

THE SUBJECT

Questions beyond section and member strength

Workshop 1 focused on designing a “safe” beam. Loads were traced from the slab, moments and shears computed, the section designed and detailed, and every station passed. Strength design is complete.

1

Stress

What does the section actually feel at service load, and where does the elastic picture end?

2

Stiffness

What is the real EI of a cracked section, and how does it change with the load?

3

Deflection

How much does the beam move, now and after years of sustained load?

4

Rotation

How much can the beam rotate before failing, and how much will be asked of it?

Elastic Stresses

PART 1

The working section under service load, and where the elastic picture ends

PART 1 · ELASTIC STRESSES

Elastic Stresses on a Beam

Demo: Elastic Stresses app

PART 1 · ELASTIC STRESSES

Every point carries a combined state of stress – the Mohr’s Circle

Principal stresses reveal what bending and shear do together, not separately.

Computed axial and shear stresses from shear and moment

Combining these stresses using Mohr’s circle

Getting principal and maximum stress

The anatomy of stresses on a Beam

Deeper Insights to Beam Elastic Stresses

Moment-Curvature

PART 2

The section's true stiffness, read from its own response

We need to bring in Deformations

M

y

θ

Curvature

Curvature

The strain gradient through the section depth

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Curvature is the slope of the strain distribution through the depth. It equals the extreme concrete fibre strain divided by the neutral axis depth, and equals the reciprocal of the radius of curvature.

Curvature is a deformation quantity. No force appears in its definition.

θ

Φ = ϴ when dx = 1

One Model, Two Paths

The P-M interaction and the M-φ response come from the same computation

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The difference between the two is not in the model. It is how the iteration runs.

Axial force and moment on the section

with the curvature that produced them

The first expression sums stress over the concrete area and over the bars to give the axial force. The second takes the same stresses about the section centroid to give the moment.

Anatomy of the Curve

Three transitions on the way to failure

Cracking

Concrete on the tension side stops contributing. The first reduction in slope.

Yield

Tension steel reaches its yield stress. The slope falls sharply and the curve flattens.

Ultimate

Concrete reaches its crushing strain. The moment falls away.

The section changes stiffness three times before it fails.

Moment, M

Curvature, φ

The Moment-Curvature Response�

• The ratio M/φ gives the secant flexural stiffness—close to the gross value before cracking and much lower afterward

Steel

Concrete

Yield Capacity

Max Capacity

Ductility

+ve curvature

-ve curvature

Secant Stiffness

Moment

First Cracking

1

This curve holds the beam's stiffness, ductility and strength, in both curvature directions

What Moment Curvature Can Give

• Many response quantities can be derived from a single curve and a single moment diagram.

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What Moment Curvature Can Give

Moment

M-φ

Stiffness

Rotation

Deflection

Cracking

Strain

Beam Design to Section to Curvature

Deflection

PART 3

From hand formula to code procedure to nonlinear integration

PART 3 · DEFLECTION

The Deflection Paths

M-φ at a section

Curvature Variation

Deflection

Moment

Rebar variation

Stiffness Variation

Rotation

Linear Procedure

Nonlinear Procedure

PART 3 · DEFLECTION

The code's answer: an effective stiffness

Demo: Beam Deflection app

Cracked section

Transformed-section analysis gives Icr, the moment of inertia after cracking, a fraction of Ig.

→

Effective inertia Ie

ACI 318-19, 24.2.3.5: Ie interpolates between Ig and Icr from the ratio Mcr / Ma. One stiffness for the whole span.

→

Immediate deflection

δi computed with Ie for each service combination: dead, plus dead and live.

→

Long-term multiplier

λΔ = ξ / (1 + 50ρ′), with ξ from the sustained-load duration. Creep and shrinkage in one calibrated factor.

The code compresses cracking into one stiffness and time into one multiplier. Calibrated, practical, and honest about being approximate.

Two calibrated factors stand in for everything Part 2 computed explicitly.

“Real” Deflection, Not Gross-Section Deflection

• Once the section cracks, effective stiffness drops sharply.

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• Deflections corresponding to the cracked section exceed those estimated using the gross section, and the consideration of long-term effects further increases them. The value that governs serviceability corresponds to the cracked section with long-term effects considered, and it can fail the allowable limit checks, which the estimate with gross-section assumption may pass.

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The deflection that matters is the cracked, long-term one.

Instantaneous linear based on gross stiffness

Gross Stiffness

Effective Stiffness

Instantaneous using cracked stiffness

Longterm using Moment Curvature

PART 3 · DEFLECTION

Three methods, one beam, three numbers

Method

Midspan deflection

Basis

Verdict

Elastic hand formula

6.6 mm

Gross EI, uncracked, short-term

comfortable, and false

Instantaneous using Moment Curvature

13.5 mm

Ie and λΔ, one stiffness, one multiplier

Better estimate of immediate

Longterm Moment-curvature integration

32.7 mm

EIeff at every station, integrated

the near-true answer

Allowable: L/360 = 22.2 mm. Bracketed values are recorded from the demonstration models at the dry run.

The three numbers are not competing answers to one question; they are the same question asked with three different sets of assumptions.

The differences are not errors; they are assumptions made visible.

ACI Code Comparison

• Very Close to Moment Curvature computations for simply supported beams

PART 3 · DEFLECTION

Continuous beams: the codes may not work well

Demo: Beam Deflection app

Why the code procedure strains

• Cracking differs at supports and midspan; one Ie must represent all of it
• Pattern live loads move the cracked regions between combinations
• Averaging rules for Ie exist, but they are conventions, not mechanics

Why the integration does not

• Every station carries its own moment, its own cracking state, its own EIeff
• Support and midspan regions are simply different stations, not special cases
• The demonstration runs Beam C, the Workshop 1 beam, by exactly the same method as Beam S

Performance

PART 5

From curvature to rotation to acceptance

Link the Hazard to Structural Response Performance Levels

Nonlinearity Builds up

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Nonlinear Pushover Analysis

Nonlinear Time History Analysis

Realistic Site-specific Hazard Analysis

Realistic Performance Evaluation

Key Performance Indicators and Criteria

PART 5 · PERFORMANCE

The curve becomes a hinge

Demo: Beam Performance app

Curvature is local

M-φ describes one cross-section. The structure does not ask sections to curve; it asks member ends to rotate.

The plastic hinge

Inelastic curvature concentrates over a hinge length Lp at the member end. Integrating curvature over Lp turns the section curve into a member property.

M-θ

θp = (φu − φy) · Lp. The moment-rotation relation at each end is what the frame actually negotiates with during an earthquake.

Curvature is a section property; rotation is what the structure actually asks for.

The Simplified Backbone Curve

ASCE 41 defines component behavior through the generalized force-deformation relationship:

�

A → B: Elastic range

B → C: Strain hardening

C → D: Strength degradation

D → E: Residual strength

KEY INSIGHT

Parameters a, b, and c define the shape. They come from testing or ASCE 41 tables—not from material properties alone.

ACCEPTANCE CRITERIA

IO — Immediate Occupancy

Minor damage, full function

LS — Life Safety

Significant damage, safe egress

CP — Collapse Prevention

Severe damage, no collapse

Reference: ASCE 41-17 Tables 10-7, 10-8, 10-19 (RC beams, columns, walls)

PART 5 · PERFORMANCE

Where a, b and c come from: the beam table

Demo: Beam Performance app

Three parameters select the row

Reinforcement balance (ρ − ρ′) / ρbal�how far the section sits from the balanced condition; low values mean tension-controlled, ductile behaviour��Transverse reinforcement C or NC�conforming hoops confine the core and hold the compression zone together after cover loss��Shear level V / (bw d √f′c)�high shear on the hinge region erodes rotation capacity��Acceptance limit (total rotation) = θy + θp,limit

(ρ − ρ′)/ρbal

Transverse

Shear level

a (rad)

b (rad)

c

IO

LS

CP

≤ 0.0

C

low

0.025

0.050

0.2

0.010

0.025

0.050

≤ 0.0

C

high

0.020

0.040

0.2

0.005

0.020

0.040

≤ 0.0

NC

low

0.020

0.030

0.2

0.005

0.020

0.030

≥ 0.5

NC

high

0.005

0.010

0.2

0.0015

0.005

0.010

Excerpt, beams controlled by flexure: ASCE 41-17 Table 10-7 (i). Radians; interpolation permitted. Low and high shear levels are 3 and 6 in psi units, about 0.25 and 0.5 √f′c in MPa.

Ductile detailing is rewarded in this table, row by row; the app interpolates it for the actual section.

Putting it all Together for a Beam

Moment

Moment Curvature

Moment Rotation

Performance for given Moment

Section

Hinge Length

PART 4 · WHEN LATERAL LOAD ARRIVES

One demand, four ways in

1

Applied directly

End moments typed straight onto the beam in the analysis app: the fastest way to explore

2

From combinations

Lateral cases added to the load combinations, from Excel or in the app

3

From Frame 2D

The plane frame analysed in the platform, its beam moments extracted

4

From ETABS

The full building model, seismic combinations included, imported as a first-class source

The demonstration shows the first and the last: moments applied directly on Beam S, and the seismic ETABS model imported and carried through design.

The workflow meets the engineer wherever the model lives.

THE SESSION

The full picture

Demand

ETABS

SAP2000

Excel

Frame 2D · Beam analysis

→

Design

Multiple RC Beams

RC Beam

RC Beam Section

→

Behaviour

Elastic Stresses

Moment-Curvature

Beam Deflection

→

Performance

Beam Performance

M-θ backbone

IO · LS · CP

Demand may come from anywhere the model lives. The same visible path then runs from demand to design to behaviour to performance.

One visible path from demand to performance: the Workshop 1 promise completed.

THE SESSION

The demonstration scope

Understanding the beam response beyond strength .

NEXT : hands-on demonstration

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Engr. Muhammad Hammad