Structural Analysis Fundamentals / Internal forces and structural diagrams
Sign conventions, sagging and hogging
Getting signs right, and knowing which face of the beam is in tension.
Lesson review: expert verification remains open
This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.
Checks needed for project use
- An independent structural reviewer must check remaining derivations, diagrams, sign conventions and the interactive models across their full input ranges.
- Model applicability, diagrams and implementation across all input cases remain subject to independent engineering review.
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesWhy this matters
State a sign convention before calculating. Use it consistently and explain what the resulting signs mean for the member.
By the end of this lesson you should be able to
- State a sign convention and stick to it
- Read sagging and hogging from the sign of the moment
- Say which face of the member is in tension
This course uses one convention throughout. Shear force at a section is the sum of the upward forces on the piece to the left of the cut. Bending moment is positive when the beam sags — bending into a smiling curve, with the bottom face stretched.
Sagging means the top of the beam shortens and the bottom stretches, so the bottom is in tension. Hogging is the reverse: the top stretches and is in tension. This matters enormously in reinforced concrete, where the steel must go on the tension face — put it in the wrong face and the member cracks and fails.
Predict first
A horizontal cantilever carries a downward point load at its free end. Which face is in tension from bending?
Practice
A horizontal cantilever 3.0 m long is fixed at the wall and carries a downward point load of 12 kN at its free end. What is the magnitude of the bending moment at the wall?
Practice
A horizontal cantilever 2.5 m long carries a downward UDL of 8 kN/m over its whole length. What is the magnitude of the bending moment at the fixed support?
Practice
A simply supported beam of 6.0 m carries a central point load of 24 kN. What is the bending moment 2.0 m from the left support?
Summary
- State your convention, then never change it
- Sagging positive: bottom in tension; hogging: top in tension
- Being able to say which face is in tension matters more than the sign itself