Report: Testing of a Beam in Pure Bending

DESIGN OF RCC STRUCTURES (ENCE 352)

Report: Testing of a Beam in Pure Bending

1. Objective of Testing a Beam in Pure Bending

i. To observe the cracking load, yielding load, ultimate load, as well as crack propagation and crushing of concrete at the ultimate state when analyzing a Beam in Pure Bending.
ii. To understand the ductile response of reinforced concrete under monotonic loading.

2. Theory: Understanding a Beam in Pure Bending

When studying a Beam in Pure Bending, subjected to monotonically increasing loading to resist an imposed bending moment, it goes through three distinct stages before complete failure. This principle is fundamental in structural engineering.

Un-cracked stage: Concrete resists tension up to its cracking moment.
Cracked stage: Tension is transferred completely to the steel reinforcement.
Ultimate strength stage: Extreme compression fibers of concrete reach their ultimate strain, and the reinforcement yields.

Behavior Under Load

Bending causes tension in the portion lying below the neutral axis and compression above the neutral axis. Therefore, the beam is reinforced on the tension side with steel so that the tension force is taken up by the reinforcement.

Concrete and steel act simultaneously with increasing load. The tension steel undergoes large plastic deformation while the tension force remains constant. This results in increased strain in the extreme compression fiber, shifting the neutral axis upwards. The flexural strength of the section is reached when the extreme compression fiber of the concrete attains its ultimate strain.

Limit State Method Design

The depth of the Neutral Axis (x_u) is determined by equating internal compression (C) and tension (T) forces:

C = T \] \[ 0.36f_{ck}bx_{u} = 0.87f_{y}A_{st}

Moment of Resistance (MoR) Equations:

x_{u} = \frac{0.87f_{y}A_{st}}{0.36f_{ck}b}

i. Under-Reinforced Section (x_u < x_u,max): $MoR = 0.87f_{y}A_{st}(d – 0.42x_{u})$
ii. Over-Reinforced Section (x_u > x_u,max): $MoR_{limiting} = 0.36f_{ck}bx_{u,max}(d – 0.42x_{u,max})$

3. Materials and Apparatus for Beam in Pure Bending Test

3.1 Materials

i. Cement
ii. Sand
iii. Aggregates
iv. Water
v. Steel bars (2 numbers of 8 mm Fe500)

3.2 Apparatus

i. Beam and Cube Moulds
ii. Universal Testing Machine (UTM) and Compressive Testing Machine (CTM)
iii. Vibrator
iv. Shovels and Mixing Trays
v. Weighing Machine
vi. Hack-saw and Rebar Bending Tool
vii. Measuring Tape

4. Procedure for Testing a Beam in Pure Bending

The following steps outline the meticulous process required to prepare and test a Beam in Pure Bending.

The formwork was cleaned, prepared, and properly greased with oil.
Two 8 mm steel bars were cut to a straight length of 700 mm, and the edges were bent to 100 mm for side anchorage (total bar length 900 mm).
The concrete mix was prepared from the calculated amounts of cement, sand, aggregate, and water in the specified proportions.
The bottom of the formwork was filled with the concrete mix up to a 25 mm cover depth.
The steel bars were placed on top of the cover, and the remaining concrete mix was poured, vibrated, and allowed to settle for 24 hours.
Six concrete cubes (150 mm × 150 mm × 150 mm) were cast simultaneously with the beam.
After 24 hours, the formwork was removed. The beam and cubes were cured in water for 28 days before testing was conducted in the laboratory.

5. Observation and Calculation for Beam in Pure Bending

Accurate observations are crucial when evaluating the performance of a Beam in Pure Bending. The following data details the mix design and theoretical calculations.

5.1 Mix Design & Weight Calculations

Given Parameters:

Mix Proportion: 1: 1.5: 3 (Grade M20)
Sum of Ratio: 1 + 1.5 + 3 = 5.5
Theoretical W/C Ratio: 0.45

Material Densities:

Cement: 1440 kg/m³
Sand: 1600 kg/m³
Aggregate: 1700 kg/m³

Volume Computations:

i. Volume of 2 Beams:
2 × 0.75 m × 0.15 m × 0.15 m = 0.03375 m³
ii. Volume of 6 Cubes:
6 × 0.15 m × 0.15 m × 0.15 m = 0.02025 m³
iii. Total Wet Volume:
0.03375 + 0.02025 = 0.054 m³
iv. Total Dry Volume (+55%):
0.054 m³ × 1.55 = 0.0837 m³

Theoretical Material Weights (w/c = 0.45):

i. Cement: (1/5.5 × 0.0837) × 1440 = 21.91 kg
ii. Sand: (1.5/5.5 × 0.0837) × 1600 = 36.52 kg
iii. Aggregate: (3/5.5 × 0.0837) × 1700 = 77.61 kg
iv. Water: 0.45 × 21.91 kg = 9.86 kg

Actual Batch Weights Taken:

i. Cement: 26.0 kg
ii. Sand: 39.0 kg
iii. Aggregate: 78.0 kg
iv. Water: 11.7 kg

5.2 Beam Data & Theoretical Calculations

Section Properties:

Provided length (l): 750 mm
Effective span (L): 600 mm = 0.60 m
Breadth (b): 150 mm
Overall depth (D): 150 mm
Clear cover: 25 mm
Effective depth (d): 150 – 25 – (8/2) = 121 mm

Material Properties:

Area of steel (A_st): 2 × (π/4) × 8² = 100.53 mm²
Grade of concrete (f_ck): 20 MPa
Grade of steel (f_y): 500 MPa

Neutral Axis & Moment of Resistance

x_{u} = \frac{0.87 \times 500 \times 100.53}{0.36 \times 20 \times 150} = 40.49\text{ mm} \] \[ x_{u,max} = 0.46 \times 121 = 55.66\text{ mm} \] Since xu < xu,max, the section is under-reinforced. \[ MoR = 0.87 \times 500 \times 100.53 \times (121 – 0.42 \times 40.49) = 4.55\text{ kNm}

5.3 Experimental Observations

Concrete Cube Test for Beam in Pure Bending analysis — Fig 1: Concrete Cube Compression Test

Flexure Test Setup of a Beam in Pure Bending — Fig 2: Beam Flexure Test Setup

Concrete Cubes Test Data (150 mm × 150 mm)

Test Period	Cube No.	Weight (kg)	Failure Load (kN)	Compressive Strength (MPa)
7 Days	1	8.415	430.0	19.11
	2	8.590	385.0	17.11
	3	8.520	377.0	16.75
	Mean	–	397.3	17.65
28 Days	1	8.375	650.0	28.89
	2	8.580	750.0	33.33
	3	8.145	810.0	36.00
	Mean	–	736.7	32.54

Beam Flexure Test Data

Beam No.	Crack Width	First Crack Load (kN)	Failure Load P (kN)
1	6 mm	35.9	83.3
2	6 mm	29.7	85.0
Mean	6 mm	32.8	84.15

Experimental Moment Calculation

For two-point loading configuration M = (P · L) / 6:

M_{exp} = \frac{84.15\text{ kN} \times 0.60\text{ m}}{6} = 8.42\text{ kNm}

6. Results of the Beam in Pure Bending Test

The final outcomes summarize the structural capabilities observed during the testing of the Beam in Pure Bending.

7 Days Compressive Strength

17.65 MPa

28 Days Compressive Strength

32.54 MPa

Initial Crack Load

32.8 kN

Moment at Mid Span

8.42 kNm

Avg. Ultimate Load of Beam

84.15 kN

7. Discussion and Conclusion on Beam in Pure Bending

The flexure test demonstrated a classic ductile failure for an under-reinforced section, characteristic of a Beam in Pure Bending, marked by progressive flexural cracking, steel yielding, and ultimate concrete crushing. The experimental moment capacity (8.42 kNm) exceeded the theoretical design (4.55 kNm). This was primarily due to the actual 28-day concrete strength (32.54 MPa) testing significantly higher than the nominal M20 grade. Ultimately, the experiment successfully verified the theoretical equations and the safe, ductile warning behavior of under-reinforced concrete when subjected to pure bending.