Slope At Left End Of Simply Supported Beam Carrying Couple At Right End Calculator

Slope of Beam Formula:

\[ \theta = \frac{M_c \cdot l}{6 \cdot E \cdot I} \]

Moment of Couple (Mc):

N·m

Length of Beam (l):

Elasticity Modulus of Concrete (E):

Area Moment of Inertia (I):

m⁴

Slope of Beam (θ):

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1. What is the Slope at Left End of Simply Supported Beam?

The slope at the left end of a simply supported beam carrying a couple at the right end represents the angular displacement at that support. It is a crucial parameter in structural analysis for determining beam deflection behavior.

2. How Does the Calculator Work?

The calculator uses the slope formula:

\[ \theta = \frac{M_c \cdot l}{6 \cdot E \cdot I} \]

Where:

\( M_c \) — Moment of couple (N·m)
\( l \) — Length of beam (m)
\( E \) — Elasticity modulus of concrete (Pa)
\( I \) — Area moment of inertia (m⁴)

Explanation: This formula calculates the slope at the left end of a simply supported beam when a couple moment is applied at the right end, considering the beam's material properties and geometry.

3. Importance of Slope Calculation

Details: Calculating beam slope is essential for structural design, ensuring that deflections and rotations remain within acceptable limits for safety and serviceability requirements.

4. Using the Calculator

Tips: Enter moment of couple in N·m, length in meters, elasticity modulus in Pa, and area moment of inertia in m⁴. All values must be positive and non-zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a simply supported beam?
A: A simply supported beam is a structural element that rests on two supports at its ends, allowing rotation but preventing vertical displacement at the supports.

Q2: What is moment of couple?
A: Moment of couple is a system of two parallel forces equal in magnitude but opposite in direction, separated by a perpendicular distance.

Q3: Why is elasticity modulus important?
A: Elasticity modulus represents the material's stiffness and its ability to deform elastically under load, directly affecting beam deflection.

Q4: What is area moment of inertia?
A: Area moment of inertia is a geometric property that measures a cross-section's resistance to bending, depending on its shape and orientation.

Q5: Are there limitations to this formula?
A: This formula applies specifically to simply supported beams with a couple moment at the right end and assumes linear elastic material behavior and small deflections.