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🔥 Content Introduction
In the fields of construction, machinery, and bridges, beam frames are the most common load-bearing structures (such as concrete frame beams in buildings and support frame beams in machinery). The safety of beam frames fundamentally depends on the stress state of each beam — if the stress of a single beam exceeds the allowable stress of the material, it may lead to cracking, deformation, or even structural collapse. This article will focus on the “stress calculation of a single beam in a beam frame,” covering the overall force transfer in the frame, internal force calculation of a single beam, stress formula derivation, and engineering examples, providing a clear guide for structural design and verification.
1. Basics of Beam Frame Loading: Load Transfer and Characteristics of Single Beam Stress
Before calculating the stress of a single beam, it is essential to clarify the load transfer path of the beam frame and the stress form of the single beam — this is a prerequisite for subsequent calculations to avoid isolating the analysis of a single beam from the overall structure.
1. Types of Loads and Transfer Paths in Beam Frames
The loads borne by beam frames can be divided into “vertical loads,” “horizontal loads,” and “local loads,” with their transfer paths determining the stress magnitude of each beam:
- Vertical Loads: Such as self-weight, live loads from floors, and equipment weight, transferred through the floor to secondary beams, then to main beams, and finally to columns, ultimately reaching the foundation;
- Horizontal Loads: Such as wind loads and seismic loads, transferred through walls or lateral force components to the frame columns, and then from the columns to the main beams (causing horizontal shear and bending moments in the main beams);
- Local Loads: Such as concentrated equipment loads on beams, directly acting on the corresponding single beam, requiring separate verification of the local stress of that beam.
Taking a “two-story, two-span reinforced concrete frame” as an example, the middle main beam not only bears the vertical concentrated force from the secondary beams but also the horizontal shear force from the columns, thus the stress of a single beam is a combination of “bending, shear, and torsion” (in most cases dominated by “bending and shear,” with torsion considered only when loads are eccentric).
2. Stress Forms and Internal Force Components of a Single Beam
In a beam frame, the core stress forms of a single beam (whether main or secondary) are plane bending and shear, with corresponding internal force components including:
- Bending Moment (M): The internal force that causes the beam to deform in bending (for example, when a simply supported beam is subjected to a downward load at mid-span, a positive bending moment is generated at mid-span, with tension at the bottom and compression at the top of the beam);
- Shear Force (V): The internal force that causes the beam to deform in shear (for example, at the supports of the beam, the shear force is maximum, which may lead to shear failure of the beam);
- Axial Force (N): Generated only when the frame is subjected to horizontal loads or when there is eccentricity at the ends of the beam (for example, lateral displacement of the frame columns causes tension or compression in the main beam), usually negligible, requiring only verification of bending and shear combined stress;
- Torque (T): Generated when the line of action of the load deviates from the centroidal axis of the beam (for example, when a secondary beam is eccentrically placed on a main beam), requiring separate verification of torsional shear stress.
The core logic of stress calculation is: first calculate the internal forces of the single beam (M, V, T), then calculate the corresponding stresses based on the internal forces, and finally determine whether the stresses are less than the allowable stresses of the materials.
⛳️ Results
📣 Sample Code
%—————————————————————
E=200e6;% evaluate 10^6 (this is GPa) to leave the dynamics in kN
%I=200;
%———-shape Functions—————-
N1=1/L^3*(2*x^3-3*x^2*L+L^3);
N2=1/L^3*(L*x^3-2*x^2*L^2+x*L^3);
N3=1/L^3*(-2*x^3+3*x^2*L);
N4=1/L^3*(L*x^3-x^2*L^2);
N=[N1 N2 N3 N4];
%——————————————
syms d_1x d_1y phi_1 d_2x d_2y phi_2;
d_beam=[d_1y; phi_1; d_2y; phi_2;];%—4 degrees of freedom for each beam—!!!!!!—–axial freedom is missing, remains in line 35
u=N*d_beam;%—the transformation distribution
%==========Output relations (for the beam matrix)===================
f1y=subs(diff(u,x,3),x,0);
m1=-subs(diff(u,x,2),x,0);
f2y=-subs(diff(u,x,3),x,L);
m2=subs(diff(u,x,2),x,L);
F_loc=[f1y; m1; f2y; m2];
🔗 References
[1] Wang Lan, Zheng Yipeng. Calculation of Bending Load Capacity of Reinforced Concrete Beams Based on MATLAB [J]. Journal of Huaihai Institute of Technology: Natural Science Edition, 2017, 26(4):5. DOI:10.3969/j.issn.1672-6685.2017.04.011.
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