Whitney Rectangular Stress Distribution(derivation of beam expressions) (2024)

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Whitney Rectangular Stress Distribution(derivation of beam expressions)

ByMohammad -May 05, 2019

experimental tests confirm that the strain varies in proportion to a distance from the neutral axis in compression and tension zone within and near the ultimate stress. compression stress will vary linearly as long the maximum stress is less than 0.5fc'. once the stress exceeds 0.5fc' the distribution of compression stress will change from linear distribution to the shape shown in figure 1.
compressive stress will vary from zero at the neutral axis to a maximum value at or near the extreme fiber. neutral axis location and stress variation will differ from a beam to another. this variation depends on several factors such as magnitude and history of post loading, shrinkage, creep and speed of loading.

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Figure 1

for simplification whiteny replaced the curved stress block with a rectangular stress block as shown in the figure 2. compressive stress assumed to be constant over the depth of block and equal to 0.85fc'. depth of whiteny block equala=β1c. the area of whiteny rectangular stress block is equal to actual area of curved stress distribution. also, the centroid of both block should coincide.β1 is equal to 0.85 for fc'=4000 psi or less and for fc' greater than 4000 psi it can be calculated using the following equation

for si units.β1 is equal to 0.85 for fc'=30 Mpa or less and it is reduced by 0.05 for each increase of 7 Mpa for a minimum value of 0.65. for concrete with a compressive strength greater than 30 Mpaβ1 can be calculated using the following equation

Figure 2

using force and moment equilibrium equation we can get expressions for a and Mn.

as shown in figure 3. c=0.85fc'ab, T=Asfy

C=T

0.85fc'ab=Asfy

a=Asfy/(0.85fc'b)

a=ρfyd/(0.85fc'),ρ=As/(bd)

Mn=0.85fc'ab(d-a/2)

Mn=Asfy(d-a/2)

Mn should be multiplied by a reduction factorϕ to account for material strength, dimensions and workmanship uncertainty

Mu=ϕMN=ϕ0.85fc'ab(d-a/2)=ϕAsfy(d-a/2)

if we substitute a withρfyd/(0.85fc') which derived earlier

Mu=ϕAsfy(d-ρfyd/(0.85fc'*2))

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Mohammad: My name is Mohammad Hamdan, and I am a Civil/bridge Engineer with extensive professional background and experience in the field of construction. I have 10+ years of experience in the construction field. During my professional career, my emphasis has been placed on structural projects, mainly bridges, infrastructure and roads. Reading and practicing the design of structures is my passion.i was lucky to be a part of a big team that executed major highway interchanges. I believe in spreading the knowledge, thus, i compose these courses to share my experience and knowledge. My course focuses on the practical side of construction field. On other word, i am sharing some of what new engineers need to know about construction field. Beside working on construction field, i love writing some articles on my website ( curious civil engineer) Moreover, i wrote some engineering papers, you can check them by searching my name on Google (Mohammad mamon Hamdan)

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Whitney Rectangular Stress Distribution(derivation of beam expressions) (2024)

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