Learning topic

Strength and Stiffness Design in Tension and Compression

Strength and stiffness checks for axially loaded bars, required cross-sectional area, and allowable axial load calculations.

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This topic systematizes the main engineering calculations for bars under centric tension and compression: strength verification, sizing of the required cross-sectional area, determination of allowable axial load, and stiffness verification. It uses |σmax| ≤ [σ], |Δl| ≤ [Δl], σ = N/A, and Δl = NL/(EA).

An axially loaded bar must not only carry the applied forces safely but also remain within acceptable deformation limits. Therefore, calculations distinguish between strength and stiffness requirements.

For centric tension or compression, the strength condition is:

$$|\sigma_{\max}|\le[\sigma].$$

For a uniform cross-section, $\sigma=N/A$, so the required area for a known axial force can be estimated as:

$$A_{\mathrm{req}}\ge\frac{|N|}{[\sigma]}.$$

The stiffness check limits deformation or displacement, for example:

$$|\Delta l|\le[\Delta l].$$

$\sigma_{\max}$ — maximum absolute normal stress;
$[\sigma]$ — allowable stress;
$A_{\mathrm{req}}$ — required cross-sectional area;
$\Delta l$ — calculated change in length;
$[\Delta l]$ — allowable change in length.

Verification calculation

If geometry and loading are known, determine the axial force $N$, calculate $\sigma=N/A$, and compare the largest absolute stress with the allowable value. If the strength condition is satisfied, the section meets the adopted allowable-stress criterion.

Design calculation

If the load and allowable stress are known but the cross-section must be selected, estimate the required area from $A_{\mathrm{req}}\ge |N|/[\sigma]$. Then choose an actual standard or constructively acceptable section with an area not smaller than the calculated requirement and verify it again.

Allowable load

For a given section, the allowable axial force under the adopted condition can be estimated from $|N|\le[\sigma]A$. If the member has several segments, the governing external load is determined by the most critical segment together with the relation between its internal force and the applied load.

Stiffness check

For a straight prismatic bar under centric tension or compression, when $N$, $E$, and $A$ are constant over the segment and the material behaves linearly elastically:

$$\Delta l=\frac{NL}{EA}.$$

$\Delta l$ — change in bar length;
$N$ — axial force;
$L$ — initial segment length;
$E$ — Young's modulus;
$A$ — cross-sectional area.

The axial strain is $\varepsilon=\Delta l/L$. For a stepped bar, the total change in length is the sum of segment deformations: $\Delta l=\sum_i N_iL_i/(E_iA_i)$.

Even when stresses are acceptable, excessive elongation or shortening may interfere with service. The calculated displacement is therefore compared with an allowable value specified by service requirements or the applicable design method.

Area-sizing example

Let a bar carry $N=60\ \text{kN}$ in tension and let the allowable stress be $[\sigma]=150\ \text{MPa}$. Using N and mm:

$$A_{\mathrm{req}}=\frac{60000}{150}=400\ \text{mm}^2.$$

An actual section with area not less than $400\ \text{mm}^2$ should be selected and then checked using its real properties and, where required, the stiffness condition.